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References and Notes References and notes 1 Introduction 1. See especially K. Menninger (trans. by P. Broneer) Number Words and Number Symbols (MIT Press , 1969) and G. F1egg, Numbers: their History and Meaning (Deutsch and Schocken, 1983; Pelican Books , 1986). 2. A. Seidenberg, 'The Diffusion of Counting Practices', University of California Publica­ tions in Mathematics 3,4 (1960). 2 Counting systems 1. H Vedder, Die Bergdama (Hamburg, 1923) p. 165. 2. A. Seidenberg, 'The Diffusion of Counting Practices', University of California Publica­ tions in Mathematics 3, 4 (1960), 216. 3. S.S. Dorman, Pygmies and Bushmen ofthe Kalahari (London, 1925) p. 61. 4. K. von den Steinen , Unter den Naturvolkern Zentral Brasiliens (Berlin , 1894) pp. 406-7, (translated by Margot Mcintosh). 5. Seidenberg, op. cit., p. 218. 6. Ibid., pp. 218-9. 7. From Theodor Kluge, Zahlenbegriffe der Volker Amerikas, Nordeurasiens, der Munda , und der Palaioafrikaner (Berlin, 1939) p. 223. 8. K. Menninger, Zahlwort und Ziffer (1958) (trans. P. Broneer) Number Words and Number Symbols (MIT Press, 1969) p. 61. 9. Ibid., p. 69. to. O. Neugebauer, The Exact Sciences in Antiquity (Brown University Press, 1957). II. Menninger, op. cit., p. 49. 12. Seidenberg,op. cit., p. 225. 13. Menninger, op. cit., p. 36. 14. Seidenberg, op. cit., p. 263. 15. Menn inger , op. cit., p. 36. 16. Seidenberg, op. cit., p. 270. 17. Menninger ,op. cit. , p. 223. 18. Taken from science-history journal Isis, xxvii, 462-3. 19. From C. Zaslavsky, Africa Counts (Prindle, Weber , and Schmidt, 1973) p. 18. 20. Menninger, op. cit., p. 224. 21. Ibid. , p. 223. 22. See p. 8. 23. Menninger, op . cit., p. 39. 24. From Menninger, op. cit., p. 226. 25. G. Trevelyan, History of England , 2nd edn (Longman , Green, 1942) p. 125. 26. From which the expression 'bank stock'. 27. Menninger, op. cit.; p. 237. 28. From C. R. Jossett, Money in Britain (Warne , 1962) p. 17. 29. Ibid. , pp. 91-2. 30. From Menn inger , op. cit., p. 244. 31. Ibid. , pp . 253- 4. 32. Ibid ., pp. 254. 33. Zasvlasky, op. cit., pp. 93-5. References and notes 217 34. From the Latin sextarius, a 'sixth' (of the Roman measure of volume, the congius). 35. Menninger, op. cit., p. 256. 36. A. Seidenberg, 'The Ritual Origin of Counting', Archivefor History ofExact Sciences 2, 1-40. 3 Number words 1. See C. B. Boyer, A History of Mathematics (Wiley, 1968) p.683. 2. Raetia is now the canton of Grisons in Switzerland and part of the Tyrol. 3. That is, their endings are changed according to case or gender. 4. A word which is the subject of a verb is said to be in the nominative case, e.g. I give. The accusative denotes the direct object of a verb, e.g. I give a coat. The genitive denotes possession, e.g. I give my father's coat . The dative denotes the indirective (more remote) object, e.g. I give my father's coat to him. 5. K. Menninger, Zahlwort und Ziffer (1958) (trans. by P. Broneer) Number Words and Number Symbols (MIT Press, 1969) p.22. 6. Ibid ., p. 106. 7. Pronounced as in German acht or Scottish loch. 8. Gothic h = ch as in acht or loch. 9. Gothic ai was probably pronounced rather like a short e. 10. From Datta and Singh, History of Hindu Mathematics, vol. 1 (Asia Publishing House 1936) pp. 10-11. 11. Ibid., p. 12. 12. Ibid., p. 13. 13. See, for example , Menninger, op. cit., pp. 153-4. 4 Written numbers 1. All but a few fragments of the Rhind Papyrus are in the British Museum . It was published with commentary by T. E. Peet, London, 1923, and again by A. B. Chase , Ohio, 1929. 2. The numerals of Mesopotamia are discussed later. 3. From C. B. Boyer, 'Fundamental Steps in the Development of Numeration', Isis 35, 157-8. 4. Babylonian fractions and methods of calculating are discussed further in Chapter 5. 5. The sun's yearly path . 6. Though, of course, using Greek numerals and not Babylonian wedges . 7. J. Needham, Science and Civilization in China, vol. 3 (Cambridge University Press, 1970) p. 10. 8. Ibid ., p. 12. 9. Boyer, op . cit. , pp. 761-8. 10. O. Neugebauer, 'Babylonian Mathematics', Scripta Mathematica n, 312-5. 11. See Chapter 2, p. 17. 12. Quotations and opinions criticised by Boyer are found in the following works : Cajori, F. A History of Mathematical Notations (Open Court, 1928-9) Cantor, M. Varlesungen uber Geschicte der Mathematik (Leipzig, 1894) Crowther, J. G. The Social Relations of Science (New York , 1941) Delambre, J. B. J. Histoire de l'astronomie ancienne (Paris, 1917) Halsted, G. B. On the Foundation and Technic of Arithmetic (Chicago, 1912) Heath, T. L. A History of Greek Mathematics (Oxford, 1921) Hogben, L. Mathematics for the Million (New York, 1940) Karpinski, L. C. History ofArithmetic (Chicago and New York, 1925) Menninger, K. Zahlwort und Ziffer (1958), trans. P. Broneer, Number Words and Number Symbols (MIT Press, 1969) Tannery, P. Memoires , J. L. Heiberg and H. G. Zeuthen (eds) (Toulouse and Paris, 1912-37). 13. See Note 12. 218 Numbers Through the Ages 14. See Note 12. 15. Fractions are discussed in Chapter 5. 16. Decimal fractions are discussed in Chapter 5. 17. See Note 12. 18. See Note 12. 19. See pp. to2-30. 20. See p. to6 . 21. For a fuller discussion of alphabetical numerals see: G. Flegg, Numbers: their History and Meaning (Deutsch and Schocken, 1983; Pelican Books , 1986) Chapter 3. 22. K. Menninger, Zahlwort und ZifJer (1958) (trans. P. Broneer) Number Words and Number Symbols (MIT Press, 1969) pp.404-5. 23. The subject may be pursued further in B. Datta and A. N. Singh, History of Hindu Mathematics (Asia Publishing House, 1935-8). 24. For further details see O. Neugebauer and D. Pingree, The Paiicasiddhantlkii of Variihamihira (Copenhagen, 1970). 25. This can be followed up in W. E. Clark, The Aryabhatiya ofAryabhata (Chicago , 1930). 26. Datta and Singh, op. cit., p. 69. 27. Described in Datta and Singh, op. cit., pp. 69-72. 28. In a right-angled triangle with hypotenuse, the sine of an angle is equal to the length of the side opposite to it. 29. Datta and Singh (op. cit. , p. 58, footnote 5) describe Pargiter as 'probably the greatest Puranic scholar of modern times' . 30. It is translated with an excellent commmentary by Burgess and Whitney in Journal ofthe American Oriental Society 6 (1960) 141-498 . 31. See Datta and Singh, op. cit. , p. 40. 32. Extensive material on Arabic forms of numerals , with many examples from manuscripts, may be found in Rida A . K. Irani, 'Arabic Numeral Forms', Centaurus 4 (1955) 1-12. 33. Planudes (1255?-13to) was a Greek monk , Ambassador to Venice of the Emperor Andronicus II. He wrote a work on Hindu numerals and also a commentary on Diophantus' Arithmetic. 34. Bibliotheque Nationale, Paris, lat. 8663, f . 49. • 35. The orientations of the numerals are discussed in detail in G. Beaujouan 'Etude paleographique sur la rotat ion des chiffres et I'emploi des apices du X" au Xii" siecle', Revue d'histoire des sciences 1, 301-13. 36. Neither Pythagoras nor Boethius is a suitable historical choice. 5 Fractions and calculation 1. It was published with an excellent commentary by T. E. Peet (Liverpool and London, 1923, reprinted by Kraus Reprint, Nendeln, Lichtenstein, 1970). 2. See pp. 78-80 for a comment on the relative importance of hieroglyphic and hieratic script. 3. This hypothesis was put forward by O. Neugebauer in his pioneer paper in Quellen und Studien zur Geschichte der Math., Astron. und Physik 2 (1930) 301. 4. Published by Schack-Schackenburg in Aegypt. Zeitsch 38 and 40 (1900). 5. Yale Babylonian Collection (YBC) to529. The text is reproduced in O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts (Newhaven, Connecticut, 1945) p. 16. 6. See B. L. van der Waerden, Science Awakening I (Noordhoff) pp.97-1oo. 7. Ibid. , p. 66 for fuller discussion. 8. Ibid., pp. 70-1. 9. This explanation is a 'possible ' one only. Some scholars (e.g. Boyer and Neugebauer) specifically reject it. to . In a right-angled triangle with one further angle given, the ratio of the sides is determined. The ratio of the side opposite to a given angle to the hypotenuse is called the sine of the angle. The sine of 30° is t hence, in this example ~ = t and if e = I then g = 2. References and notes 219 11. K. Vogel, Griechische Logistik, Sitzungsber. Bayer Akad. Munchen (Math .-Nat .) p.357. 12. Manchu dynasty , 1644-1911. 13. For a discussion of the Euclidean algorithm see, for example , C. B. Boyer, A History of Mathematics (Wiley, 1968) pp. 126-7 . 14. From Datta and Singh, History of Hindu Mathematics, Part I (Asia Publishing House , 1962) p. 189. 15. Ibid., p. 190. 16. Ibid., p. 196. 17. Ibid., p. 196. 18. Ibid., p. 198. 19. Ibid. , p. 188. 20. Ibid., p. 170. 21. For example, ibid., pp. 155-60 . 22. Ibid., p. 160. 23. Ibid., p. 161. 24. There is an English translation with extensive commentary by Martin Luther d'Ooge (New York, 1926, reprinted, 1972). 25. From Datta and Singh, op. cit., pp. 291-20. 26. See P. Luckey , Der Lehrbrief uber Kreisumfrang von Gamsid b. Mas'ud al-Kashi (Berlin, 1953). 27. See A . P. Juschkowitsch, Mathematik im Mittelalter (Leipzig, 1964) p. 241. 28. Norton's English translation was reprinted, together with a facsimile of the original Dutch edition, in volume 2 of the Principal Works ofSimon Stevin, D. J. Struik (ed .) (Amsterdam, 1958). 6 Aids to Calculation 1.
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