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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 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 and methods of calculating are discussed further in Chapter 5. 5. The sun's yearly path . 6. Though, of course, using 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 (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. 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 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 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. K. Menninger, Zahlwort und Ziffer (1958) (trans. by P. Broneer) Number Words and Number Symbols (MIT Press, 1969) pp. 301-2. 2. Artes here means 'practical skills'. 3. See J. Needham, Science and Civilization in China, Vol. 3 (Cambridge University Press, 1970) p.8. 4. Ibid., p. 70. Bibliography

The following list includes histories of mathematics and source books which are commended for those wishing to read more widely in the subject.

Boyer, C. B. A History of Mathematics (Wiley, 1968). Dedron, P. and hard,J. (trans. Field, J.) Mathematics and Mathematicians (2 vols.) (Transworld/Open University Press, 1974). Eves, H. An Introduction to the History of Mathematics (Holt, 1964). Fauvel, J. and Gray , J. (eds) The History of Mathematics: a Reader (Macmillan , 1987). Flegg, G. (1983) Numbers: their History and Meaning (Deutsch and Schocken/Pelican, 1983). Flegg, G. Boolean Algebra (Macdonald/Transworld, 1971/2). Hollingdale, S. H. and Toothill, C. Electronic (Penguin, 1975). Karpinski, L. C. The History of Arithmetic (Russell and Russell, 1965). Kline, M. Mathematical Thought from Ancient to Modern Times (Oxford, 1972). Kline, M. Mathematics in Western Culture (Penguin, 1972). Lavington, S. Early British Computers (Manchester University Press, 1980). Martin, J. and Normand, A. R. D. The Computerised Society (Penguin , 1973). Menninger,K. (trans. Broneer, P.) Number Words and Number Symbols (MIT Press, 1970). Morrison , P. and E. (eds) Charles Babbage and his Calculating Engines (Dover, 1961). Midonick , H. The Treasury of Mathematics (2 vols) (Penguin, 1968). Neugebauer, O. The Exact Sciences in Antiquity (Dover, 1969). Newman, J. The World of Mathematics (4 vols) (Allen and Unwin , 1960). Popp , W. trans. Bruckheimer, M. History of Mathematics: Topics for Schools (Transworld/ Open University Press, 1975). Smith, D. E. A Source Book in Mathematics (2 vols) (Dover, 1959). Struik, D.J. A Concise History of Mathematics (Bell, 1954). Struik, D. J. A Source Book in Mathematics (Harvard University Press, 1969). Van der Waerden, B. L. Science Awakening I (Oxford, 1961). Wilder, R. L. Evolution of Mathematical Concepts (Transworld/Open University Press, 1974). Index

Aachen 61 Analytical Engine 201, 203-4, Auxiliary numbers 138, 140-3, Abacist(s) 118, 123, 128 209-10 170 Abacaus 4, 119-22, 163, 170, Analytical Society (Cambridge) Avestan 63, 65-7 176-84, 190-6, 214 200 Aztec(s) 2, 18, 20, 29 Abax/abakion 179 Anatolia(n) 63-4 Abstract place-value notation/ Anglo-Saxon 62-3, 66, 70 Babbage, Charles 200-6 system 101 Anuyogadvtira-stura 72 Economy of Manufacturers and Abu-l-Wafa 115 Apex/apices 117, 119, 122-3, Machinery 201 Addere 179 182-3, 185, 196 Babylon(ians) 24, 26, 93-5, 102 Adelhard of Bath 123 Apol1onius 94 Babylonian algebra/equations Adjectives 57-9 On Pythmenes (lost work) 94 148-56, 169-70 Afghan 63 Approximations 147, 149-50, Babylonian arithmetic 169 Afudu 15 157,168,170 Babylonian astronomy 167 Agni-Purana 109 A-rt:l 144-5, 148 Babylonian cubic equations 156 Aha-calculations 13~-40 , 142 Arabic letters 113 Babylonian division 146-7 Ahmes Papyrus (see Rhind 50, 94, 118, 218 Babylonian division of the zodiac Papyrus) Arabs 96, 110, 157 106 Aids to calculation 176-215 Aral Sea 113 Babylonian fractions 94, 144-7 Aiken, Howard 206 Aramaeans 96 Babylonian linear equations Ainu 22-4 Aramaic(s) 104 152-5 Akmin Papyrus 161 Arch of Pythagoras 182 Babylonian multiplication 91, Albanian 63, 65, 75 Archimedes 88-90, 94, 158-62 146-7 Albedo Monastery 116 Measurement of the Circle 90, Babylonian number system/ Albrecht, Johann 187 158, 160 notation/numerals 13, 27, Rechenbuchlein auf! der linien Psammites 94 56,76,80-4,90,95-6,129 187 The Sand-Reckoner 88 Babylonian quadratic equations Alcuin 182 Arithmetic unit 202,210-11 153-6, 169 Alexandria 109 Arithmoi paramekepipedoi 149 Babylonian simultaneous Algorist(s) 124, 126, 128 Arjuna 72 equations 153-6 Algorithm 123 Armenia 65 Babylonian tables 147-8 Algorithmists (see Algorists) Armenian 63 Bacon, Francis 127 AI-Karagi 115 A~abhata 73, 96, 106-9, 165-9 On Ambition 127 AI-Kashi , Jemshid 170-1 Aryabhatiya 106-7, 109, 166 Baghdad 112-13, 171 The Key to Arithmetic 171 Aryans 65 Bakairi 7-8, 11, 34, 40 Treatise on the Circumference Asia Minor 65 Baktun 100-1 171 Asoka era 102-4 Balkan Peninsular 65 Al-Khowarizrni 112-15, 123-4 Asoka inscription 102 Baltic 63-4, 70 Algebra 114 As-sifr 126 Bank notes 47 Arithmetic 114-15, 123 Assyrians 26 Banka Island (inscription) 87, 111 Al1emanni 41 Astronomy/astronomers 156-8, Bankire 187 Alphabetic(al) notation/numerals 166-7, 198 Bankrupt 128 88-92,96, 112, 115,218 Attic numerals 88, 91 Base 18,82 Alpine number billets 48-50 Automatic calculating machine Basel 43 Alpscheit 41, 44 200 Ba-si 148-9 Alsace-Lorraine 74 Automatic Sequence Control1ed Basic numerals (see Standard Alternating system 24 Calculator (ASCC) 206 numerals) Alveoli 181 Auxiliary equation 153 Basque(s) 29, 32-3, 64 222 Index

Bavarian reckoning cloths 186 Byron , Lord and Lady 205 Code civil of Napoleon 44 Bede, the Venerable 182 Byzantine sources 171 Codex Emilianus 117 Behistun inscription 12 Codes Vigilanus lt6-18 Beile 41 Calculating machines 4, 196-8, Colonia Aurelia Ovilana 180 Belhari inscription 111 200 Columna rostrata 99 Benrath line 61 Calculation (methods) 3,131-75 Commercial numerals 84-5 Bergdama 7 Calculation on the counting board Compound interest 169, 172 Berlin Museum 148 187-8 (s) 4,18,130,198,201, Berlin Papyrus 141 Calculation on the lines 184-90 206-15 Bern 50 Calculator 45 Confidential weight numerals Bernelinus 123 Calculi (see Counters) 85 Beyer, Johann Hartmann 175 Calculus 95 Congius 217 Logistica decimalis 175 Calendar computations 20 Control unit 210 Bhaskara I 108 Camayocs 51-2 Coptic (alphabet/numerals) Bhojadeva (inscriptions) 86, 112 Cambodia 87, 1It 97 Bible 30, 34, 57-9, 65, 68-9, 75 Cambridge 200-1 , 207 Cornish 31-2, 62-3 Billion 74 Candra, Hema 72 Counters 45, 176, 178-80, 183, Binary arithmetic 18,213-14 Capital tesserae 45 185-90 , 196 Binary numbers/system 17-18, Cards of the variables 203 Counting bank 180 213-15 Cash register (cipherisation) Counting board 106, 176-84, 186, Bishop Wulfila 5,57,65,68-9 79,95 191-2 Black Sea 65 Cassiodorus 182 Counting by pairs 9, 13 Bodhisattva, Gautama 72, 106 (ic/s) 29,31,53,62-5,70,74 Counting systems (general) 1-3, Body counting 36-7 Census (1980) 205 214 Boethius 128-9, 182, 218 Centum 49, 62, 70, 99 Court of the Exchequer 45 Boeotian numerals 179 Ceylon (see Sri Lanka) Crafte of Nombrynge 125 Bolivian Indians 52 Chaap Wurrong 33 Creation ritual 55 Bone handle from Ishango Chalkos 177-8 Crete 88 38-9 Chang 170 Croatian 63 Bonfils, Rabbi Immanuel 172 Charlemagne 17, 182 Crusades 185 Boochose, William de 47 Charrua 15 Cube roote table 156 Book of Ballymote, the 55 Chillioi 71-2, 177 Cuneiform writing 80-2 Boole, George 208,213 Chimpu 52 Cutter 45 An Investigation into the Laws of Chhih 170 Cycle (see Loop) Thought 213 Chhin Chiu-Shao 86 Cyril, St 97 The Mathematical Analysis of Su Shu Chiu Chang 86 Cyrillic (alphabet/numerals) Logic 213 Chhin Shih Huang Ti 170 97 Borel of Barcelona, Count 119 Chiffre 127 Czech 62-3, 65, 70 Bororo 34 China 49,63,87, 101 , 104, 181 Boullier 195 Chinese 29, 36, 44 Dandapani, Prince 72 Brahrnagupta 106, 113, 164-5, Chinese abacus/counting board Danish 29-31,62-3,66,74 168-9 (see Suan pan) Darius the Great 12, 179 Brtihma-sphuta-siddhiinta 106, Chinese decimal fractions 170 Darius vase 179-80 164 Chinese fractions 163-5 Data processing 212 Brahrni alphabet/numerals 104- Chinese Han sticks 49 Dead languages 63 6, no, 117 Chinese numerals 84-7, 102 Decad 69 Breton 31-2, 62-3, 65-6, 74 Chiu Chang Suan Shu 163, 192 Decem 49, 70 Briggs 175 Chronograms 107 Decimal calculation 174 British Exchequer tallies (see Chuquet, Nicolas 73 Decimal count(ing)/system 2, 5­ Exchequer tallies) Triparty 73 6, It, 17,24-5,27-9,68,71, British Isles 65 Cipher 127 214 British Museum (BM) 151 Cipherisation 79-80, 92, 95-6 Decimal fractions 4, 170-5 Brueghel, Pieter the Elder 40 Ciphra 126-7 Decimal number notation Bulgarian 63, 65 Circulus 127 (numerals) 6, 157 Burgi, Jost 174-5 Circumscribed (regular) polygon Decimal (place-value system) 87, Arithmetica 174 160 214-15 Burgos 117 Clavius 174-5 Decision point 211 Bushmen 7, 11 Astrolabium 174 Deka 68-70, 72 Index 223

*Dekm 67-70 Electronic Numeral Integrator and Brhat-ksetra-samasa 111 numerals 78-9 Calculator (EN lAC) 206-8 Ganita 166-7 Dene-Dinje 36 Elevation 188 Garibaldi 204 Descartes 175 Ell 60 Gaulish 63 Dialogus de Scaccario 47 English 58, 62-3, 65-71 Gender 58-9 Differences 201 English tally sticks 50 Gerard of Cremona 114 Difference Engine 201-2 Epicycles 166 Liber algebrae et almucabala Diffusion hypothesis 2-3,9, 12, Epitriton 159 114 55 Ep-ogdo-on 159 Gerlandus 123 Digamma 88 Erastosthenes 159 Gerbert abacus/counting board Digital device(s) 209 Esk imos 29 119, 122, 182 Digital Electronic Universal Euclid 158-9, 169 Gerbert numerals 119 Computing Engine Elements 158-9, 169 Gerbert of Aurillac 118-23, 126, (DEUCE) 207 Euclidean algorithm 163,218 128, 182-3 Diophantus 158,161,218 Eutocius 89 German(s) 31,40-1 ,69-70 A rithmetikalArithmetic 158, Excenters 167 German numerals 128 218 Excess 17 Germanic (languages) 2,29,50, Dominican Order 189 Exchequer tallies 44-8, 51 56,61-7, 74 Double tally sticks 42-3, 45 Germanic phonetic shift 67-8, Doubling 78, 134 Fa 170 70-1 Doubling and halving 188-9 False assumption, method of 140 Germany 65,67, 73 Dozen 16-17,75 Fen 170 Giro 128 Dual 59 Fermat 175 Glagolitic alphabet/numerals 97 Duality 12 Ferranti Mark I Computer 207-8 Gobar numerals 116-18, 183 place-value system Festuca 41 Gotaland 65 130 Figi Island club 39 Gothic alphabet 97 Duplation 180 Figura nihili 127 Gothic 5, 57-71 Durer 117 Figure(s) 127 Goths 65,96 Dusseldorf 61 Fiji Islands 39 Great hundred 71, 75 Dutch(man) 40,56,61,62-3,65-6 Finger and body counting 33-7 Great year 106 Dyadic system (see Binary system) Finger and toe count(ing) 1, 24, Greek 2,5,59-60,62-8,70-1 Dyche 127 34 Greek algebra/equations 169 New General English Finger counting 33-7 Greek alphabet 97 Dictionary 127 Finnish 64 Greek arithmetic 169 Finnish tally stick 42-3, 48 Greek astronom(y/ers) 83, 106, Early Greek numerals 88 Five 66 167 Early Hindu numerals 102-5 Five-group 179 Greek calculation 89 East Arabia 116 Foil 45,57 Greek counting board 177 East Arabic numerals 114-17 Florence 128 Greek fractions 95, 158-62, 165­ East Germanic 62, 64-5 Forgery 47 6 East Goths (see Ostrogoths) Fractions 3,131-75,214 Greek numerals 88-95, 102, 112, East Slavic 65 France 57, 65 115 Edward I 47 Franci 41 Greek texts 112, 149 Egyptian(s) 60, 94, 131 Franco-Prussian War 74 Greek trigonometry 167 Egyptian arithmetic 133 136 Frankfurt (-am-Main) 41, 128, Greeks 63,69, 72, 89, 92-6, 103, Egyptian calculation 4, 133 175 129, 161, 167 Egyptian division 134-5, 137 Frankfurt-on-the-Oder 61 Greenland 29 Egyptian fractions 4, 12, 131-43, Frankish Empire 119 Grouping 39 165 French 5, 29-30, 32, 56-8, 62-3 Gwalior (inscription) 86, 111-12, Egyptian multiplication 91, 134 French tally sticks 50 117 Egyptian numerals/system 14-15, Frisian 63, 65 76-80,83,85,88,90,95-6,102 Fully-cipherised place-value Haab 100 Electromechanical devices 206 system 129 Hammurabi (dynasty) 82, 151, Electronic computer (see 156 Computer) Gael ic 31, 63 Han dynasty 49 Electronic Delay Storage Galatians 65 Han Van 171 Automatic Computer Gallic 31 Hao 170 (EDSAC) 207 Gal}i, Jinabhadra 111 Harappa 102 224 Index

Hanseatic League 185 Indic 63-5 Kerbholz 40, 50 Hardware 211 Indo-China 87, 112 Kerfstok 50 Harz Mountains 186 Indo-European language(s), 2,6, Keta ul /93 Hau-calculations (see Aha- 30, 32-3, 50, 56, 59-60, 62-4, Khai-Yuan Chan Ching 87, 111 calculations) 66-8, 71, 75, 214 alphabet/numerals Hebrew alphabet/numerals 97 Indonesia 87, 181 104-5 Hebrews 96 Indus culture 102-3 Khoresm 113 Hekaton 68, 70 Inequalities 160 Kmtom 68-9 Henry I 44 Inflection 58 Knots 50-4 Herodianic Numerals (see Attic Input devices/unit 202, 210 Kobel 190 numerals) Inscribed {regular) polygon 160 Kodel 128 Herodotus 51 Integrated circuits 206 Koppa 88 History 51 International Business Machine Kurdish 63 Hieratic numerals 78-9,90-6 (IBM) Corporation 206 Hieroglyph(s/ics) 76-7,88,90, Ionian system/numerals 88, 90, Lal 148 131 95-6 Lalitavistara 72, 106 High German 56-8,61-3, 65 Iran 65 Lao-tse 52 Hindu(s) 93, 116 Iranian 63-5 Lata/Latadeva 109 Hindu algebra 169 Ireland 53 Latin 2, 5, 30, 32, 57-62, 65-8, Hindu arithmetic 169 Irish 29,31,62-3 70-1, 112 Hindu astronomy/astronomers Irish gravestone from Aglish 54-5 Least common divisor 163 106, 108, 166 Iron division 183-4 Least common multiple 138 Hindu counting board 106 Isidore of Seville 116 Leather roll (see Mathematical Hindu fractions 166-7, 158-70 Origines 116 leather roll) Hindu numerals 102-16, 193 Islamic civilisation/Empire 112- Leibniz 175, 197-200 Hindu texts/treatises 112, 162 13, 171 Leibniz calculating machine 197, Hindu-Arabic numerals/decimal Italian 5-6, 41, 50, 56-7, 71 199-200 system 116-30, 182-3, 214 Italic 63-5 Leonardo of Pisa 126, 128 Hippolytus 92 Italy 57,65, 73, 128 Liber abaci 128 Refutation ofall Heresies 92 Lettish 63 Hittite(s) 5, 63, 65 Jacquard, J.M. 203 Li 170 Holland 30, 72 Jacquard looms 203-4,210 Liber algorizmi 125 Hollerith, Hermann 205 Japan(ese) 36, 181 Life tables 201 Hollerith machine 205 Japanese abacus (see Soroban) Limit of counting 60 Homer 17, 72, 158 Johannes de Sacrobosco (see John Line board 185, 187-8, 196 Iliad 17, 72, 158 of Halifax) Linguistics 56 Homogeneity 94 John of Halifax/Hollywood 125, Lipta 167 Horizontal reckoning table 185 127 Lithuanian 62-3, 66-71 Horner 192 Algorismus vulgaris/Common Liu Hui 170 Hsiao-hsieh 85 Algorism 125, 127 Local value (principle) 93 Hu 170 John of Seville 114, 123 Logarithms 174, 197 Huku 15 Algoritmi de numero indorum Logic circuits 214 Hund 68-70 114 Logical algebra 213-14 Hunderod 69-70 Liber algorismi de practica Logical aspects/design 207-8 Hundred 68-70, 75 arithmetice 114 Logistic 91 Hundred-thousand 100 Logos 158 Huygens 175 Kaccayana 72 Lombard 128 Pali Grammar 72 Long count 100-1 Ib-si 148 Kahun Papyrus 77 Long hundred 75 Iceland 65, 75 Kanheri inscription 111 Loom(s) (see Jacquard loom) Icelandic 62-3, 66 Kapitaltesseln 44 Loop 205, 210 IlIyrian 63 Karvstock 40, 50 Lorraine 119 51-2 Kassel 61 Lovelace , Ada Augusta Countess Independent invention hypothesis Katapayadi systems 108 of 205, 209-10 2-3, 9, 12,55,214 Katun 100-1 Low German 61-3,65 India 63, 65, 96, 101, 103 Kendra 167 Lulu 15 Indian (see Hindu) Kentum languages 62-7, 69-71 Luther, Martin 58, 186 Indian towers 73 Kepler 92, 175 Luwian 63 Index 225

Lyons 203 Multiplication table(s) 89, 124-5, Old Surya-Siddhanta 108-9 Lyte , Henry 173 146-7, 152 Operation cards 203 Muris , Johannes de (see Meurs) Opus Palatinum 172 Ma tzu (see Confidential weight Musical chords 159 Oracle-bone numerals 84-7 numerals) Myriad 88 Origin of counting 12, 55 Macedonians 65 Myrioi 72 Original Indo-European 2, 60-72 Magdeburg 61 Ostrogoths 65 Magini of Bologna 174-5 Nagari alphabet/numerals 104, Ouden 107 Magister A 124 110, 117 Output devices/unit 202, 210 Book Introducing Algorism Nama 7 123-4 Named place-value system 86, Pacioli, Luca 5, 73 Magyar 64 100, 102, 105 Summa 5,73 Mahavira 165, 168-9 Nanaghat 105 Pairwise concept 12 Manx 63 Napier, John 174-5,197 Paper money 48 Massel 53 Mirifici logarithmorum canonis Pappus 94 Mathematical leather roll 132-3 constructio 175 Papuans 34, 36 Maya(n/s) 2,19-21,29 Mirifici logarithmorum canonis Partes proportionales 174 Mayan calendar 100 descriptio 174, 197 Pascal 175, 197 Mayan Empire 100 Napoleon 44, 195 Pascal calculating machine 197-8 Mayan numerals/system 91, National Physical Laboratory 207 Pebbles 187-90 100-1 Natural fractions 131 "Penque 66 Mayayuga (see Great year) Nemorarius, Jordanus 125, 189 Pepys, Samuel 196-7 Mecca 112 Neo-2-count 10-11 ,14 Diary 196-7 Mechanical desk calculator 197 Neo-Confucians 86 Persia 195 Mediation 188 Neumann, John von 207, 209 Pente 66 Medieval coin board 196 New Irish 31-2 Persian(s) 63, 96, 116 Medieval line board (see Line Newton 175 Persian Empire 65 board) Nichomachus of Gerasa 89, 149 Persian numerals 12-13 Medina 112 Arithmetical Introduction Persian texts 112 Menabrea, L. F. 204 149 Peru 51 Mensarius 180 Nicobar Islands 41 Peruvian Indians 52 Meurs, Jean de 172 Nine Chapters (see Chiu Chang Peurbach, Georg 172 Miao 170 Suan Shu) Pes 60 Middle Ages 99, 119, 123, 125, Nirvana 93 Petrus de Dacia 125 169, 179, 185, 196 Nonillion 74 Phillipine sword 39 Middle (High) German 57-8,61, North Germanic 62, 65 Phillipines 39 66 Norton, Robert 173,218 Phoenician(s) 104 Middle German phonetic shift 61, Norwegian 63 Phoenician alphabets 105 67-8,70 Nulla figura 127 Phonetic shift 67 Middle Persian 63 Number mysticism 169 Phrygia 65 Mihira, Varaha 106, 108-11 , 167 Number words 56-75 Phrygian 63 Pancasiddhantika 106, 109, 167 Numerals 3, 76-130 Piedmontese 30 Milk stick(s) 42, 50 Numeration 187 Place value (system) 82-3, 96, Mille 49, 71, 99 100-2, 105-10, 116 Miller's knots 52-3 Obol 177 Planudes, Maximos 116, 179 Milliard 74 Odo 123 Psephophoria kat'Indous 179 Million(s) 68,73-4,99-100 Official numerals 84-5 Plattdeutsch (see Low German) Mixed fractions 4, 159, 161-2, Ogham 53-5 Pliny 99 170 Old Babylonian (texts) 82 Historia Naturalis 99 Mocovi 15 Old Bulgarian 67 Pocket calculator 198 Modulus 94 Old Church Slavonic 66, 71 Poetic numbers 108 Mohammed 112 Old High German 57, 66, 70 Polish 63, 65 Mohenjo-daro 102 Old Irish 32, 67 Polybius 178-9 Monastic abacus 181-4 Old Nordic/Norse 62-3,66,69,75 Poncelet 195 Moorish Spain 116 Old Persian 2, 62-3, 65 Portugal 30 Morocco 115 Old Prussian 63 Portuguese 63, 75 Mu 163 Old Saxon 63, 67, 69-71 Positional arrangement/notation/ Mueller, Johann 172 Old Slavic 70 principle 90, 95 226 Index

Primitive counting 7-29 Roman coins 99 Sester knot 53 Processing unit (see Arithmetic Roman abacus/counting board astronomy 156-8 unit) 180-2,192, 196 Sexagesimal fractions 4, 144, 146, Program(ming) 205,209-12 Roman measures 17 152, 156-8, 170-2 Proto-Sumerian 83 Roman milestone 50, 98 Sexagesimal notation 151 Psephos/psephoi 123, 178-80 /system 37, 48­ Sexagesimal number words 75 Pseudo-Boethius 119-20, 123 50,85,88,90-1,98-100,102, Sexagesimal (place-value) system Geometry II 119, 123 128, 186 26-7,83, 144, 157 Ptolemaic forms 94 Romance language(s) 5,29,32-3, Sextarius 217 Ptolemy 83, 93, 106 56-7, 74 Shakespeare 30,48, 127, 186 Punched cards 203,205 Romania 57 Henry VI, Part 2 48 Pure place-value concept/system Romanian 56-7, 63 King Lear 127 102, 116, 123 Rome 181 Merry Wives of Windsor 30 Pure 2-count (see 2-count) Root 92 The Winter's Tale 186 Purification (see Elevation) Rosary 50 Shang oracle bone 87 Pythagoras 92, 128-9, 182, 218 Rotula (see Sipos) Shulgi, King 144 Pythmen (see Root) Royal Society 201 Sicilian 30, 75 Rudolff, Christoff 172 Sicily 116 Quadratic equations 149, 169 Example Booklet 172 Simultaneous equation 154 Quadrillion 74 Ruffini 192 Sine(s) 108, 174, 218 Quadrivium 119 Rules of the Cord (see Sulba-sutra) Sipos 123 Quillion 74 Russia(n) 63, 65, 70, 181 Sirsaprahelikd 72 Quinque 49,99 Russian abacus (see Stet) Slavic (languages) 63-5 51-2 Ryuku Islands 52 Slide rule 197 Slovak 63 91 Saka era 87 Slovenian 63, 65 Radulph of Laon 123 Salamis tablet 177-9, 196 Software 211 Raetia 57,217 Salem Monastery 125 Soroban 192-6 Ranks 102 Salic law 42 Soto, Hernando de 52 Rathjo 69 Samarkand 171 South Slavic 65 Ratio 95, 218 Sampi 88 Spain 29,57,65, 115-17, 128 Reciprocal(s) 146-7, 170 San Millan de la Cogolla 117 Spanish 2,50,56-7,63 Reciprocal table(s) 145-6 Sangi numerals (see Stick Spanish tally sticks 50 Reckoning boards 185 numerals) Square roots 149,167-8,170,172 Recorde, Robert 186 Sankheda inscription 111 Sri Lanka 105 The Grounde of Artes 186 Sanskrit 2, 60-3, 65-6, 68, 71-2, Sridhara 165, 168-9 Regiomontanus (see Mueller) 103, 107, 126 Ssmt 137 Register(s) 209 Satem 62 St Albans Book of Computations Reichenau, Hermann von der 123 Satem language(s) 62-7,69-70, 186 Reisch , Gregor 128-9 75 Staircase 21-3,25 Margarita philosophicalPearl of Savage, Richard 127 Standard mumerals 84-5 Philosophy 128-9 The Bastard 127 Statute book from Basel 43 Remnantsofancientcounting 30-3 Savasorda 123 Stepped 197, 200 Repetition 76 Book on Measurements 123 Stevin, Simon 94-5, 170, 173, Resolution 188-9 Scandinavia(n) 65-6, 75 175,219 Rhaeto-Romanic 63 Sceran 47 De Thiende (La Disme) 173, Rhabdas, Nikolaos 161 Stet 195-6 175 Rheims 119 Scheutz , Georg 201 Stick numerals 84-6, 191 Rhind Papyrus 77-8, 132-4, 136, Schiller 40 Stock 45 138,141-2,161,217 Wallenstein 's Camp 40 Store 201, 209-11 Richer 119 Score 41,47 Strong finger 69 Riese, Adam 189-90 Scottish (Gaelic) 29, 31, 63 Suan pan 190-2, 194-6 Ritual origin of counting 214 Sebocht, Severus 112 Subandu 111 Robert of Chester 114 Second(s) 172, 174 Vdsavadattd 111 Rod numerals (see Stick numerals) Seleucids 145, 151 Sui Shu 170 Romaka City 109 Semitic (languages) 59,96 Sulba-sutra 103 Romaka-siddhiinta 167 Serbo-Croatian 63, 65 Sumer(ians) 24,26,37, 102 Roman(s) 29, 57, 60, 103 Sester 53 Sumerian arithmetic 156 Index 227

Sumerian number system/ Thusundig 69 Vulgar fractions 95 numerals 12,14-16,26-7, Tibetan prayer strings 50 . Vulgar Latin 57 56, 76, 80-1, 129 Tickes 43 Sumerian tables 147-8 Toba 11, 13-14 Wagner, Ulrich 128 Sumerian tablets 144 Tokharian 61-6, 71 Wales 53,71 Sun Tzu Suan Ching 191- 2 Trade routes 171 Weaver's Statutes from Frankfurt- Sun ya 126 Trapezites 180 am-Main 41 Surya-Siddhanta t08-tO, 167 Trejes 67 Wedge 12 Sweden 30, 65 Trente, William 47 Welsh 31-2,63 Swedish 62-3,66, 74 Triakonta 69 West Arabic numerals 114-17 Swiss 40 Trigonometric calculations 172 West Germanic 62,64-5 Swiss tally sticks 48 Trigonometry 167 West Goths (see Visigoths) Switzerland 42, 44 Triton 160 West Slavic 65 Sylvester II, Pope (see Gerbert of Tshun 170 Westminster Abbey 44 Aurillac) Tun 100-1 Wolf bone from Vestonice 37-8 Syntithenai 179 Turkestan 63 Wotjobaluk 36 Syrians 96 Turkey 171 Written numbers (see Numerals) Systemless counting 7 Two-sided concept 12 Wulfila (see Bishop Wulfila) Tylft 75 Tabular arrangement 90 Tzolkin 100 Yale Babylonian Collection 149 Taglia 50 Tzu, Lao 163, 191 Yucatan 100 Tao Te Ching 191 Ta-hsieb 85 Yugoslavia 65 Taihun 68-70 Taille 40, 50 Vinal 100-1 Talea 50 Ukrainian 65 Zefiro/zefirum/zefro/zevero 126 Talent(a/on) 177-8, 180 Ulugh-Beg 171 Zero 85-7,95, ior -z. 110-12, Tallies and knots 37-55 Unit fractions 4, 132, 134, 159, 114,117,126-30,157,191 Tally stick(s)/tallying 14, 37-50, 161 ,165,170 Zodiac/Zodiacal circle 106,156-7 214 Unus 49,99 Tarja 50 2-count/system 1, 9-11, 13-14, Ten 68,70-1 Varga , Garcilaso de la 52 18, 27-9, 33, 37 Tenth 127, 173 Vau (see Digamma) 2:n table 139, 142 Tessele 41 Veda 103 4-count/system 1, 27-9 Thebes 132 Vedic 63,65-7 5-count 22, 33 Theca 127 Venetian 126, 128 5-1O-count/system 1,19,21-2, Theon of Alexandria 157 VietalViete, Francois 173 27-9 Thomas, C. 198,200 Canon mathematicus 173 5-20-count/system 11, 19,21-2, Thomas' commercial calculating number words 75 24-5,27-9,33-4 machine 198, 200 Villedieu , Alexandre de 125-6 6-count 27 Thousand 68-9, 71 Carmen de algorismo/Song of to-count/system (see Decimal Thrakian 63 Algorism 125-6 count) Thrako-Illyrian 63-5 Visigoths 65 to-6O-system 2,24-5 Thrako-Phrygian(s) 63-5 Visperterminen 44, 50 12-count 15, 75 Thumb 60,69 Vlacq 175 20-count/system 27-31, 33, 36 Thumb counting 34 Vorder-Asiatische Texte (VAT) 148 40-count 27