INDEX

Abacus 47. 58 Abraham 183 Abscissa 241. 242 Abulfaradj 278 Academia 148. 193. 194. 196. 229. 290 Academy 90. 179.202.229 Acropolis 106 Adaptation 140 Adelhard of Bath 58 Adrastus 233 Aetius 129 Aeschylus 106. 136 Agatharchus 136 Agesilaus 179 Agni-pudna 55 Agrimensor 277 Aha-calculations 27-29 Ahmes 15 Akhmen 36.48 Akkad 38 Akousmata 107 Al-Batdni 57. 265 Al-Biruni 226. 228 Alcaius 84 Alcmaeon 102. 128 Al~dtt62.82.85. 148. 190.201.202.

210 Al~dtt of Aphrodisias 131. 176 Alexandtt Polyhistor 101. 102 Alexandria 196. 200. 201. 208. 230. 237.

2f>7. 270. 276. 290 Algebra 6. 35. 37. 57. 63-73. 80. 116.

118-125.140.141.173.198.247.248. 265. 266. 280. 281. 282

Algebra. arabic 57. 66. 72. 264. 267. 280. 281

Algebra. geometric 71.118-125.126. 140. 172. 198. 199. 247. 265

Algebra. indian 279. ~. 292 Algebra. italian 267. 281 Algebra. rhetorical 282 Algebraic symbolism 280. 281 Algorithms. 58. 115 Aljabr 282 Alkhwarizmi 58. 59. 265. 280. 281. 282 Allman 100. 132 Al~est50.201.258.267.272.275.290 AI-Ma'mun 57

Al-Mansar 57 Almuklbala 282 Altunate interior angles 117 Alyattes 83 Amasis 83 Amicable numbers 98 Amyclas 90. 190 Analysis 90. 212. 286 Anaphorai 270 Anastasi I. papyrus 17 Anatolius of Al~dria 278 Anaxagoras 82. 90. 106. 128. 129. 130.

136. 139. 147. 201 Anaximandtt 82. 128 Anaximenes 82 Anderhub 143 Angle. division of 191 Angle. trisection of 225. 236. 263 Angles. vtttical 87 Angular velocity 239 Antanairesis 127. 176. 177. 178. 179 Anthemius 291 Anthologia Palatina 278. 280 Anthyphairesis 177 Antiphon 130. 131. 132 AO 6484 77 AO 6770 73. 97 AO 8862 63. 64 Apagoge 143 Apices 52. 58 Apogee 240 ApoUodorus 179 ApoUonius 6. 46. 56. 82, 119, 123, 139,

146, 183, 196.200,201. 207. 234, 235, 237 -263. 286, 290

Apotome 169 Applica:ion of areas 118-123, 145. 198,

199. 247 Applied mathematics 29. 31, 80, 277 Approximation 32. 44, 45, 127. 128, 205 Apuleius 210 Aqueduct 102 A-ra 42.44 Arabic numerals 51-59 Arachne 182 Aratus 182 Arbelos 225. 286 Arc. trisection of 225

294 INDEX

Archesilaus 193. 229 Archibald 199. 208 Archimedes 6. 7. 9. 47. 49. 82. 91.115.119.

200. 201. 202. 204. 205. 208-228. 230. 236.241.242.263.264.266.268.277. 280.286

Archimwes. book of lemmas 225 Archimedes. Desphaera et cylindro 186. 196.

220-222.267 Archimedes. Lemma of 178. 186. 221 Archimedes. measurement of the circle 204-

206 Archimedes. Method 137. 208. 212-215.

264 Archimedes. On conoids and spheroids 223.

224. 243 Archimedes. On equihbria of plane figures

213.214.227 Archimedes. On floating bodies 227 Archimedes. On spicals 208. 223. 224 Archimedes. postulate of 186 Archimedes. Q.uadratura parabolae 186.216-

219 Archimedes. Sand-counter 47. 202.. 227 Architecture 102. 106 Architecture. naval 202 Archyus6. 9. 46. 82. 90. 92. 107. 110-114

118. 138. 139. 140. 149-159. 161. 179 Area of a circle 32. 75. 89. 184. 192 Area of a cone 221 Area of a cylinder 221 Area of a hemisphere 33. 34 Area of a quadrangle 32. 89 Area of a rectangle 32. 277 Area of a sphere 215. 221 Area of a spherical segment 222 Area of a trapezoid 32. 68. 72. 73 Area of a triangle 32. 72. 228 Areas. propositions on 254 Aries 270 Aristaeus 234.241.270 Aristarchus 7. 82. 202-204. 211 Ariston 229 Aristophanes 130. 201 Aristotle 7. 16. 57. 59. 82. 85. 107. 109.

125. 131. 148. 155. 157. 176. 179. 180. 183. 189. 197. 201. 202. 232

Aristotle. Anal. post. 177. 178. 179 Aristotle. Metaphysics 15. 16.96. 101. 102.

109. 125. 152

Aristotle. Physica 131 Aristotle. Politics 149 Aristotle. Topica 176 Aristoxenos 107 Arithmetic progression 77 Al'ithmetica 61. 108. 196. 278. 279 Arithmos 125. 281 Arjuna 52 Artemis 102 ~abhata 55. 271. 279 Aryabhatiya 271. 280 Aryans 51 As 41 Asia Minor 83 Asoka 53 Assumption. wrong 27. 29. 67. 284 Assurbanipal 62 Assyria 62. 83. 84 Astrolabe 182 Astrology 276 Astronomy. arabic 265 Astronomy. babylonian 31. 62. 84. 86. 101.

102.270 Astronomy. egyptian 31 Astronomy. greek 31. 85-87. 101. 102.

104.107.129.180-184.193-195.201. 211. 237-240. 264. 270-273

Astronomy. indian 54. 55. 56. 57. 58. 265 Astronomy. modem 5. 265 Asymptote 235. 258 Athenaeus 90 Athens 106. 131. 165. 179.201. 220. 290 Atomists 82. 106 Augustus 82 Autolycus 82. 193-195. 229 Auxiliaries 26. 30 Axioms 183 Axis. principal 257. 287 Ayuta 52

Babylon 56. 57. 62. 84. 85. 94. 100. 125 Babylonian calculating 37-45 Babylonian geometry 72-76. 81 Babylonian influence 7. 35, 36. SO. 51. 56.

57. 83-88. 94. 96. 97. 116. 119. 171. 198. 199. 200. 270. 280. 281

Baghdad 57 Ba.illet 48 Ba·si 44 Basket 33

INDEX 295

Becker 7. 45. 108. 124. 132. 139. 152. 155. 177. 178

Beer 29.31 Belt of signs of zodiac 129 Berlin papyrus 6619 28. 29 Beta 228 Bh1skara 55 Bhattotpala 55 Bhudda 52 Bible 33. 83. 85 Bidez 103 Binomial 169-172 Bisector 204. 205 Blass 208 BM 10250: 20 BM 13901 : 69. 70 BM 34568 : 74. 76. 79 BM 85194 : 68. 75. 76 BM 85196: 76 BM 85200: 71 BM 85210 : 76 Boethius 60. 61. 111. 114. 153 Bombelli 267 Bonaparte. Jerome 210 Brahmagupta 279 Brahmi digits 52. 53. 54 Braunmiihl. von 275 Bread 23. 29. 31. 32 Briareus 209 Bruins 74. 75 BUr 66 Burgess 54 Bushel 28. 29. 41 Byzantium 131

Caesar 264 Calculating. Babylonian 37-45 Calculating. Egyptian 17-30 Calculating. Greek 45-50 Calculating. Hindu 51-58 Calculating. Sumerian 37-45 Calculating. Western-European 47. 58-60 Calculation with fractions 19-29. 48-51.

57 Calculations. sekem 26 Calendar periods 62. 129 callipus82. 181. 190. 194 Callisthenes 85 Cambyses 94 Cantor 6. 8. 99. 119. 159. 205. 277. 278 Cassina 35

Cassites 62 Catalogue of geometers. Proclus 90.91. 100.

135. 183. 190. 196 Cattle problem 208. 280 Cauchy 200 Cavalieri 138 Celts 41 Center 248. 287 Center of gravity 213-216 Central equation 251-252 Cercesura 177 Chace 32 Chaldeans 84. 85. 94 Charmides 48 Chiera 38 Chinese mathematics 278. 280 Chordal quadnlateral 206 Chords 88. 203-207 Chords. tables of 206.207.263.271-276 Chronological summary 15. 62, 82 Chronology 201. 228. 230 Chrysippus 229 Cicero 211. 276 Cifra 59 Cilicia 83 Circle. 87. 131-135. 194. 195. 199. 269 Circle. area of 32. 75. 89. 184. 192 Circle. circumference of 81. 192. 205. 263 Circle. measurement of 204-207 Circle. quadrature of 123. 132. 192. 286 Circle. segment of 132 Circle. squaring of 132. 192. 286 Cissoid 268 Clark 55 Cleostratus 84 Cnidos 179 Cobbler's knife 225. 286 Cochlias 208 Cochloid 235 Column. cylindrical 287 Commandinus 233 Commensurable 142. 166. 168 Commensurable. potentially 167 Commentaries 267 Compasses 263 "Concave on one side" 220 Conchoid 235 Cone 75. 81. 137. 138. 150. 186. 190.

221. 242-245 Cone. area of 221 Congruence 87

296 INDEX

Conic sections 119. 190. 191. 200. 226. 234. 240-261. 286. 287

Cornca 200. 240 Conjugate diameters 248. 249 Conjugate hyperbolas 248. 249 Conoids 223. 224 Conon 208. 260 Constantinople 212. 291 Continuity 152 Copernicus 6. 203. 265 Corinth 165 Cosmic solids 90 Counters of steps 230 Counting board. Salamis 47. 48 Courtier. 202. 229. 264 Crasis 146 Croesus 83. 84. 86 Cross-ratio 289 Croton 84. 85 Cube 100. 136. 156. 166. 173. 270. 281 Cube. duplication of 136. 139-141. 149-

151. 156. 159-165. 230-231. 236-237. 263. 268

Cube roots 44. 71 Cubic equations 44. 71.124.141.222.266 Cubic numbers 166 Cubocube 281 Cuneiform writing 38. 39. 40 Curves 146. 160. 220. 286 Cylinder 137. 150. 151. 211. 214. 215.

220-222 Cylinder. area of 221 Cylinder. segment of 216 Cylindrical segment 216 Cynics 229 Cypher 39 Cyprus 83 Cynllus 290 Cyrus 62. 84. 85 Cyzicus 179.202 Czwalina 212

Daedalus 193 Damascius 291 Damascus 291 Dandapani 52 Darius 48. 62. 85 Data 121. 198-199 Datta 51. 53. 54 Day. hours of the 84 Decay 202. 264-291

Decimal fraction 39. 51 Declination 275. 276 Deficiency 123. 140. 247 Degree. measurement of 230 Degrees 50. 270 Delatte 93. 94 Delian problem 9. 136. 1.)9-141. 149. 159.

190.228.230.267 DeIians 160. 161 Delivery. rapid 263 Democedes 85 Democracy 106. 1 I Democritus 15. 16. 35. 83. 91. 106. 128.

129. 136. 137. 138. 139. 147. 210 Desargues 264. 288 Descartes 249. 259. 264. 267 Description of heavens 182. 193. 202. 228 Determined 198 Diagonal 77 Diagonal numbers 126-127 Dialectics 148 Diameters. conjugate 248 Dickson 98 Diels 92. 96. 102. 109. 132. 149. 152 Digits. Gobar 52. 58 DigIts. indian 51-58 Dijksterhuis 9. 92. 119. 132. 138. 175.

177. 184. 197. 208. 212. 241. 245 Dinostratus 82. 90. 146. 191-193 Diodes 139. 264. 267-268 Diodenan 191 Diodorus 208 Diogenes 229 Diogenes Laertius 86. 88. 101. 141. 149.

180 Dionysius ISO. 278 Dionysus 93 Diophantine equations 279 Diophantus 278-286. 290 Dioptra 103. 104. 276 Diorism 261 Directrix 259 Dirhem 282 Distance of ships 87 Distances of sun and moon 203 Divisibility 11 0 Division 19. 22-26. 48. SO Division of angle 191 Division. Egyptian 19. 22-26 . .30 Division of figures 199 Division. fixed 262

INDEX 297

Dodecahedron 100. 107. 173. 175. 269. 270

Dodecahedron. Etruscan 100 Double equality. method of 283 "Double equations" 279. 285 Duplatio 18 Duplicate ratio 159 Duplication of the cube 136. 139-141.

149-151. 156. 159-165. 230-231. 236-237. 263. 268

Duplication of unit fractions 22-26 Dynamics 166. 281

Earth's measurement 230 Earth's orbit 202 Easter 59 Eclipse. lunar 128. 274 Eclipse. solar 86. 276. 286 Eclipses. explanation of 128 Eclipses. observation of 203 Eclipses. prediction of 86. 87 Ecliptic 128. 129. 180. 195. 228. 276 Ecphantus 107 Edfu 32 Education. free form of 90. 104 Egypt 15-36. 89. 94. 137. 179. 208 Egyptian calculating 17-30 Egyptian calculation with fractions 19-28 Egyptian division 19. 22-26. 30 Egyptian geometry 15. 31-35 Egyptian multiplication 18. 30. 47 Ekpetasmata 137 Eleatics 82. 84 Element 190 Elements 90-92. 135. 147. 196 E1eusis 93 Elimination 70. 73 E1leipsis 123 Ellipse 190. 223. 241-256. 287 Ellipsoid 215. 223 Empedocles 95 Enlightenment 106 Epaminondas 82 Ephesus 102 Eplcharmus 109 Epicurus 229 Epicycle 56. 201. 238-240. 265 Epigones 267 Epimoric 111. 158 Epinomis 9. 83. 112. 139. 140. 155-159 Epirtiton 50

Epogdone 50 Epsilon 237 Epsilontics 136. 220 Equality. double 283 Equations. central 251. 252 Equations. cubic 44. 71. 124. 222. 266 Equations. diophantine 279 Equations. double 279. 285 Equations. linear 66. 73. 116. 124. 198 Equations. normalized 124 Equations. Pell 208 Equations. quadratic 69-71. 73. 80. 101.

121-124. 140. 170. 284 Equations. systems of 63-67. 70. 72. 78.

80. 116. 120-124. 140. 170. 198. 199 Equator 276 Eratosthenes 49. 82. 91. 139. 160. 161. 162.

200. 201. 208. 228-234 Eratosthenes. sieve of 231 Etruscan dodecahedron 100 Euclid 7. 46. 57. 59. 72. 82. 88. 90. 91.190.

195-200. 202. 237. 265. 267. 287 Euclid. Catoptrica 200. 204 Euclid. Conica 200 Euclid. Data 121. 198. 199 Euclid. Divisiop of figures 199 Euclid. Elements 91.92. 196. 197.290 Euclid. Elements 1100.117.129.135.136.

183. 184. 196. 291 Euclid. Elements II 118. 118-121. 135.

136. 173. 184. 196 Euclid. Elements III and IV 135. 184. 196 Euclid. Elements V 175.179.183.184.187

-189. 196. 197 Euclid. Elements VI 184. 189. 197 Euclid. Elements VII 49. 110-115. 127.

143. 147. 155. 168. 176. 177. 197 Euclid. Elements VIII 140. 149. 150. 153-

155. 159. 168. 197 Euclid. Elements IX 97.98. 108. 109. 197 Euclid. Elements X 110. 123. 127. 142. 143.

146 159.167-172.173.178.179.184. 197

Euclid. Elements Xl 139. 184 Euclid. Elements XII 132. 184. 186-187.

197 Euclid. Elements XlII 100. 173. 174. 184.

197 Euclid. Elements XIV 263. 269. 270 Euclid. Elements XV 291 Euclid. Phaenomena 195. 196.200

298 INDEX

Euclid. Porismata 200 Euclid of Megara 165 Eudemus 82. 87-89.91. 92. 118. 123. 125.

129. 131. 132. 150. 151. 180. 291 Eudoxus 7. 82. 83. 90.114. 136. 139. 148.

160. 161. 163. 164. 179-190. 194. 196. 202. 218. 220. 221. 232

Eudoxus. Phaenomena 182 Euergetes 201 Euler 136 Eupalinus 102. 103 Eureka 209 Euripides 106 Eurytus 107 Eusebius 107 Eutocius 6. 47. 136. 139. ISO. 159. 163.

191. 222. 263. 268 Even and odd 96. 108-110. 116 Evolute 261 Excenter 201. 238-240. 265 Excess 123. 140. 247 Exhaustion method 184-187. 192. 216-

225 Expressible lines 168 Ezechias 103

Fabricius 103 Fibonacci 60 Fifth 95. 96. 107. 157 Figurate numbers 98-100 Figures 234 Firmicus Matemus 271 Flower of Thymaridas 116. 147.280 Foci 259 Forms. standard 80 Formula of Heron 228. 277 Formulas 68. 120 Fossey 38 Fotheringham 84 Fourth 95. 96. 107. 157 Fourth proportional 185. 192 Fractions. calculation with 19·29. 48-51. 57 Fractions. decimal 39. 51 Fractions. egyptian 19-29 Fractions. greek 19. 48-SO. 115. 116 Fractions. indian 49 Fractions. natural 19. 20. 21 Fractions. roman 41 Fractions. sexagesimal 37-44. SO. 51 Fractions. splitting 48 Fractions. sumerian 40

Fractions. unit 19. 20 Frank 107 Free form of education 90. 104 Freudenthal 9. 56. 73. 277 Fritz. von 93 Frustrum of pyramid 34. 75. 76. 81

Gallus 211 Gandz 15. 33. 66. 80 Gani 55 Gaudentius 95 Gautama 51 Gellius 149 Gelon 202.208 Geminus 85 General theorems 90. 183 General understandings 183 Geographers 202 Geometria 108 Geometric algebra 71. 118-126. 141. 173.

198-199. 247. 265 Geometrical loci 90. 234. 259. 260 Geometrical progression 77. 111. 112. 153.

219 Geometry. babylonian 72-76. 81 Geometry. egyptian 15. 31-35 Geometry. greek 15. 87 Geometry. plane 140 Geometry. projective 258. 259. 287. 288 Geometry. solid 136-141. 156 Geometry. spherical 194. 195 Gerbert 58. 60 German numerals 59 Germanic peoples 41 Gerstinger 49 Gillain 31 Given in position 198 Given in shape 198 Glaucus 160 Gnomon 84.99. 104. 121. 145 Gnomic numbers 99 Gobar digits 52. 58. 59 God 158. 194 Goethert 210 Golden Age 106-147 Golden proportionality 94 Golden section 90. 101. 121. 173. 184 Gonda 54 Gopa 52 Gorgias 116 Gravity. center of 213-216

INDEX 299

Greater ratio 188 Greatest common divison 49, 113, 115, 127 Greek astronomy 31. 85-87, 101.102, 104,

107, 129, 180-184, 193-195,201. 211. 237-240,264,270-273

Greek calculating 45-50 Greek multiplication 47, 48 Griffith 17 Grone Heinrich 59 Guldin 287 Gur 66 Gyges 83

Halley 261 Halsted 56 Halys 84, 86 Hammurabi 38, 62, 63 Harmonic mean 94, 107, 156, 157,232,233 Harmonic position 250 Harmonica 108 Harmony, theory of 93, 95-96, 107, 111,

157, 158 Harpedonaptai 15 Heath88,92, 111.159, 161.164, 180, 197,

200,203,205,212,237,241,262,281, 286

Heavens, description of 182, 193,202,228 Hee, van 278 Hegira 57 Heiberg 132, 159, 200, 212, 213 Heinrich, Grone 59 Helicon 161 Heliopolis 179 Heliocentric system 202 Helix 263 Hellas 83-85, 202 Hellenism 148, 201. 202 Hemiolion 157 Hemisphere, area of 33, 34 Heptagon, regular 226, 127 Heradides 82, 94, 194 Heraclitus 82, 93, 95 Herculaneum 210 Hermotimus 90 Herodotus 15, 16, 83, 84, 86, 102, 106,

201 Heron 45, 49, 77, 82, 103, 104, 139, 228,

239, 276-278 Hicetas 82, 107 Hieratic 18 Hiero 202, 208, 209

Hieroglyphics 17, 18 Hieronymus 210 Hindu calculating 51-58 Hipparchus 51. 82, 85,182,183,195,201,

207,211, 265,271,273,274 Hippasus 82, 106, 107, 147 Hippias 90, 146, 191 Hippocrates 82, 90, 92, 114, 131-136,

139, 141. 144, 147, 155,160,161. 184, 196, 226, 291

Hippolytus 128 Hippopede 181 Histaeus 85 Homer SO, 201 Hopfner 94 Horace 276 Horizon 194 Horologium 104 Horus 32 Hours of the day 84 Hultsch 195, 200, 233 Hydraulic screw 208 Hyksos 16 Hypatia 290 Hyperbola 162, 190, 191, 222, 226, 236,

241-260 Hyperbolas, conjugate 249 Hyperbolas, opposite 248 Hyperbole 123 Hyperboloid 216, 223, 224 Hypomnemata, pythagorean 101 "Hypotheses" 203 Hypotirosis 291 Hypsicles 85, 263, 269, 270

lamblichus 56, 91, 94, 97, 116, 141. 210 Ib-si 44 Icosahedron 173-175, 269, 270 Ideas 148 Inclination of planes 31. 32, 68 Incommensurable 142, 159, 166, 168 Indian algebra 279, 280, 282 Indian astronomy 54, 55, 56, 57,58, 265 Indian digits 51-58 Indian fractions 49 Indivisibility of unity 115 Induction, mathematical 126 Inequalities 136, 185, 187, 204, 220, 239 Initiation 93, 107 Integral 214, 224, 225 Integration 264

300

Interval 158 Inverses. table of 43. 44 Involution 289 Ionia 83. 84

INDEX

lrrational90. 110. 117. 125. 141, 159. 160.

Leon 90. 196 Lepsius 32 Lesson·text 73. 74 Lever 213. 216. 217 Uvy93

166.168-172. 178. 179.190.264 Irrationalities. unordered 263 Isidore the great 291 Isidore of Milete 291 Isocrates 94 Isoperimetric 268. 269 Italian algebra 267. 281 Italy 84. 107

Jerusalem 103. 213 Julian the apostate 278 Junge 169 Jupiter 181 Justinian 291

Kahun papyrus 17 KaDkara 52 Karpinski 280 Kaye 54 Keller 59 Kendra 56 Kepler 5. 6. 265 Kharosti numbers 53 Kingdom. middle 16 Kingdom. old 16.30 Klein. J. 116 "Knowledge without intellect" 93 KOdel59 Konig 85 Koti 52 Kugler 85

Lal45 La1itavistara 51 Langdon 84 Larsa 72 Lasos 107 Lateral numbers 126. 127 Latus rectum 248 Latus transversum 248 Least common multiple 49. 112-115 Leather scroll 2n Leimma 96 Lemma 267 Lemma of Archimedes 178. 186. 221 Leodamas 90

ubrary 201. 202. 229. 264. 267. 290 Labra 270 Limit 184. 219. 220 Lindemann 100 Line. tangent 249-256 Linear equations 66. 73. 116. 124. 198 Lines. expressJble 168. 169 Literature. roman 276 Livy 209 Loci. geometrical 90. 234. 259. 260 Logic 154. 155 Logistics 115. 116 "Long year" 129 Lucian 95 Lunar eclipse 128. 274 Lunules 131-136 Lyceum 229 Lydia 83 Lysimachia 230

Machines 149. 202. 209. 277 Magi 94.96 Magnitude 177. 187. 188 Major 172 Mamercus 90 Mana "II Mani\ius 271 Manitius 182. 206. 291 Marcellus 163. 209. 210 Mathemata 108 Mathematical induction 126 Mathematical proof 7. 88. 89. 90 Mathematics. applied 29. 31. 80. 277 Mathematikoi 107 Maurice de Nassau 172 Mean and extreme ratio 102. 121. 173 Mean. harmonic 94. 107. 156. 157. 232. 233 Mean proportionals 90. 118. 119. 134. 136.

139.150.154.158.159-163.231-234. 268

Means 94. 231-234 Measurement of circle 204-207 Measuring instruments 104.277 Mechanical method 212. 213 Mechanics 149. 163. 212. 227. 277. 287 Medes 83. 84

INDEX 301

Medieties 231-235 Meissner 38 Menaechnus 82. 90. 139. 161. 162. 190-

191. 241. 245 Menelaus 82. 271. 274-276 Menge 200 Menninger 48. 51. 52. 53 Mesotetes 232. 234 Method 137. 208. 212-215. 264 Meton 130 Metrics 277 Michel 183 Michigan papyrus 620. 280. 281 Middle kingdom 16 Milete 85 Minor 172 Minos 160. 161 Mishnat ha-Middot 33 Mixtures 146 MKT63 MohaIllllled 57 Money 41 Monochord 95 Moon 128. 180. 181.203.208.237.238 Moon in second quarter 203 Moscow papyrus 33. 34 Muhammed ben Musa 57 Muller 205 Multiplication 18.30.39.42.43.46.47.48 Multiplication. egyptian 18. 30. 47 Multiplication. greek 47. 48 Multiplication tables 42. 43. 46. 59 Musaeon 201 Music. theory of 94.95. 107. 111. 157. 158 Musical scales 96. 149. 157 Myriad 46 Mysteries 93. 94 Mysticism 94. 157. 158 Mysticism. number 93-100

Nabonassar 62 Nabopo1assar 84 Natural science 194 Naucratis 83 Naval architecture 202 Nebukadnezar 84 Nectanebus 179 Nehemiah 33 Neoclides 90 Neo-Pythagoreans 82. 91. 96. 97. 116 Neugebauer 6.7.20.28.31. 34. 38. 41. 42.

44.63.64.70.73.74.75.76.77.78.79. 85.183.241.252.257.259.276

Neusis 134. 226. 235. 236. 237. 263. 286 Newton 5. 6. 258 Nicomachus 97. 98. 99.109.124.231.232 Nicomedes 82. 139. 146. 191. 235-237 Nicoteles 249. 260 Nile floods 15 Nineveh 84 Nipsius 87 Niyuta 52 Nodes. recession of 180 Nonnalized equations 124 Nonnalized problems 122. 198 Normals 260 Notation. positional 38-41. 53-61 Number mysticism 93-100 Number symbols 52. 53 Number systems 17.37 "Numbers" lOB. 265 Numbers. auxiliary 26. 30 Numbers. figurate 98-100 Numbers. gnomic 99 Numbers. lateral 126. 127 Numbers. parallelopiped 98 Numbers. pentagonal 98. 99 Numbers. perfect 97. 98. 109 Numbers. poetic 54. 55 Numbers. prime 115. 231 Numbers. rational 169 Numbers. similar 140. 154. 158 Numbers. spatial 140. 154. 156. 158 Numbers. square 98. 99. 154. 166 Numbers. triangular 95. 99 Numerals 51-58 Numerals. arabic 51-59 Numerals. babylonian 38. 39. 40. 56 Numerals. egyptian 17. 18. 29. 30 Numerals. german 59 Numerals. greek 45. 46. 47 Numerals. indian 51-58 Numerals. roman 17.38. 39. 59. 60 Numerals. sumerian 48

Octahedron 173 Octave 95. 96. 107. 158 Oenopides 82. 90. 129. 130 Old kingdom 16.30 d'Ooge 97. 99 Opposite hyperbolas 248 Oppositio 282

.302 INDEX

Oracle 160, 161 Ordinate 241. 242 Orestes 290 Orgies 107 Orient 83, 84 Orphic prophets 93 Ovid 276

Palimpsest 213 Pamphile 88 Pannekoek 85 Pappus 6, 47, 82, 110, 125, 139, 146. 159,

231. 267. 286-290 Pappus, Collectio 1-46, 191. 196,200.232,

259,261.264.269.286.287 Pappus. commentary on Almagest 286 Pappus. commentary on Book X 169 Pappus, theorem of 287. 289 Papyrus, Akhmen 36,48 Papyrus, Anastasi I 17 Papyrus, Berlin 6619 28, 29 Papyrus, Kahun 17 Papyrus, Michigan 62Q 280, 281 Papyrus. Moscow 33. ·34 Papyrus. Rhind 16-32 Papyrus. Vindobon 49. 56 Parabola 1-46. 162. 190, 191,213,216-.219

241-259 Parabola. quadrature of 216·-219 Parabole 123 Parabolic segment 213, 216-219 Paraboloid 215. 223, 227 ParaIldogram 121. 122 ParaIIdopiped numbers 98 Paramekepipedon 45 Pargiter 55 Part 49, 114 Pascal 264, 290 Put 16.33 PdI equation 208 Pdoponnesian war 82, 106 Pentagon, star 100. 117 Pentagonal numbers 98, 99 Pentagramma 101 Pentathlos 228 Perfect numbers 97, 98, 109 Pericles 82, 106, 128 Perigee 240 Perimeter of circle 81. 192, 205, 263 Period of Saturn 129 Perpendicular 129

Perseus 146 Persia 83-85 Persian wars 82, 106 Persius 210 Perspective 129, 136, 137, 287 Pesu 29 Phaenomena (Eudoxus) 182 Phaedias 208 Philebus 180 Phillipus 90. 91. 155 Philolaus 107 Philon 139 Philoponius 131 Philosophy 106. 148, 149, 179, 193, 228,

229,2.30 Pindar 201 Piraeus 179 Pisces 270 Pisthetaerus 1.30 Plane geometry 140 Planetarium 211 Planets 128. 129. 180, 238 Plato 6.7,8.49.82.90.91. 92, 101. 102,

106. 107. 109. 110, 125, 129. 138-142. 144, 148, 149. 152, 161. 162, 163, 1M. 165.166.173.180,194.202,231,232, 279

Plato. Epinomis 9.83,112,139.140,155-159

Plato. Gorgias 116 Plato, Philebus 180 Plato. Republic 115. 126. 138, 144. 148,

193 Plato. Theaetetus 141. 165. 166 Plato. Timaeus 112. 173 P1atonicus 160. 161. 162. 163. 164, 228,

232.233 Pleasure 179 Plutarch 96.100, 117. 138. 161. 162, 163,

164. 209. 210. 212 Poetic numbers 54. 55 Polar 258 Pole 235.258 Polemon 229 Polos 84 Polybius 47, 209 Polycrates 102 Polygons. regular 117. 135. 136, 146, 184,

204, 205. 206, 228, 269 Polyhedra. regular 90, 100, 138. 173-175,

286,291

INDEX 303

Polyhedra. semi-regular 228 Porism 200. 287. 288 Position. given in 198 Position. harmonic 250 Positional notation 38-41. 53-61 Postulate of Archimedes 186 Potentially commensurable 167 Power proposition 258 Powers 153 Pressure. upward 209. 227 Prime numbers 115. 231 Principal axis 257. 287 Prism 137. 186 Problem. tangency 262 Problems. normalized 122. 198 Proclus 6. 82. 90. 91. 104. 129. 190. 290 Proclus catalogue 90. 91, 100.135.183.190.

196 Proclus. commentary on Euclid I 87. ~. 100.

117. 123. 129. 146. 196. 197.236.291 Proclus. commentary on The Republic 126 Proclus. Hypotiposis 291 Progressions. arithmetic 77 Progressions. geometric 77. 111. 112. 153.

219 Progressions 77. 80 Projective geometry 258. 259. 287. 288 Prophets. orphic 93 Proportional. fourth 185. 192 Proportionality 72. 73. 110-116. 132. 146.

147.175-179.184.187-189.231-234 264.266

Proportionality. golden 94 Proportionals. mean 90. 118. 119. 134. 136.

139. ISO. 154. 158. 159-163. 231-234.268 Proposition on areas 254 Psammetichus 83 Psellus 278 Psephoi 47. 48 Ptolemy I 91, 190. 196. 201 Ptolemy II 201. 209 Ptolemy III 161. 162.201. 202. 229. 230 Ptolemy. Claudius 6. 39. SO. 56. 57. 82. 85.

183.201.202.206.207.238.265.267. 271-273. 290

Ptolemy. Euergetes 201 PuliSa. -Siddh1nta 55 Pulu 62 Pyramid 34. 35. 91, 137.138.174.186.187 Pyramid. truncated 34. 35. 75. 81 Pythagoras 6.7.61. 30. 82. 83. 84. 90. 91,

92-102. 104. 106. 107. 117. 118. 125. 147.279

Pythagoras. theorem of 6.76.77.79. 100. 117. 118. 124. 136. 143. 147. 197.207. 237.273.286

Pythagoras. the tradition of 116. 117. 118 Pythagorean triangle 78. 79. 100. 285 Pythagorearu. 7. 9. 77. 80. 82. 91-94.106-

127. 129. 138. 141. 147. 149. ISO. 152. 155. 157-159. 168. 232

Pythmen 47. 279

Quadrangle. area of 32. 89 Q!!adrangle. complete 288 Quadratic equation 69-71. 73. 80. 101.

121-124. 140. 170. 284 Quadratrix 146. 191, 192 Quadrature of circle 130. 132. 192. 286 Quadrature of parabola 216-219 Quadrilataal. .:bordal 206 ~ellen und Studien 7 Quintilian 268

Rama 54 Raphael 210 Rapid delivery 263 Ratio 125. 187 Ratio. duplicate 159 Ratio. greater 188. 189 Ratio. mean and extreme 101. 121. 173 Rational numbers 169 Raven 101 Recession of the nodes 180 Rectangle 63. 118. 125 Rectangle. area of 32. 277 Rectangular numbers 98. 99 Rehm 183 Reidemeister 7. 159 Reisch 60 Renaissance 267 Republic 115. 126. 138. 144. 148. 193 Restauratio 282 Reuter 59 Revolution. solid of 287 Rhabdas 46. 48 Rhetorical algebra 282 Rhind papyrus 16-32 Riemann 223 Riese 61 Right triangle 8. 76. 77. 79-80. 99. 100.

136. 197. 285

INDEX

Rising 270 Robbins 280 Roman abacus 47. 48 Roman empire 264. 276 Roman literature 276 Roman numerals 17. 38. 39. 59. 60 Romaffi99. 102.209.264 Rome 274 Rome. A. 50. 286. 290 Rome. Paul 274 Root. cube 44. 71 Root. square 29. 37. 44-45. 123. 205 Rope stretchers 8. 15. 137 Rosen 282 Rostagni 93. 94. 110 Royal scribes 17 Rudio 131. 132

Sabians 57 Sachs. A. 44. 63. 70. 78. 79 Sachs. Eva 7. 139 Salamis counting board 48 Samos 102-104 Sanhenb83 San Sepulchri 212 Sar 40 SAROO Sargon II 62. 83 "Saros" 86 Sasi54 Saturn. period of 129 Saykad 31 Scales. musical 96. 149. 157 Schack.Schackenburg 29 Schiapardli 180 Scholium 167. 173. 187 Schoy. C. 226 Schwarz. E. 229 Schwarz. H. A. 269 Score 41 Screw. hydraulic 208 Scnbes. royal 17 Scroll. leather 20 Section 90. 173. 184 Section. conic 119. 190. 191. 200. 226.

234.240-261.286.287 Section. golden 90. 101. 121. 173. 184 Segment. parabolic 213. 216-219 Segment. spherical 222 Segments. unreasonable 169 Sekem calcuu.tions 26

Seleucia 202 Seleucids 42.62. 76. 85 Senkereh 63 Serapis 290 Sethe 20 Setting of stars 195 Severus ~bokht 58 Sexagesimal fractiOffi 37-44. 50. 51 Sexagesimal system 37-41. 50 Shape. given in 198 Siddhinta 55. 57 Sieve of Eratosthenes 231 Si8ffi of the zodiac 182. 270 Sil& 00 Silver mines 208 Similar numbers 140. 154. 158 SimWity 135 Simplicius 85. 131. 132. 133. 180. 291 Sindhind 57 Sines. tables of 54. 206. 271 Singh 51. 53. 54 Sinope 83 Sinus 204. 206. 207. 271 Siren 209 Skd 31 Socrates 82. 130. 141. 148. 165. 100.201.

232 Socrates. church history 290 Solar eclipse 86. 276. 286 Solid geometry 1.36-141. 156 Solid of revolution 287 Solstice 202 Sophists 106. 117. 202 Sophocles 106. 201 Sosigenes 180 Sound 152 Spain 58 Sparta 104. 106 Spatial numbers 140. 154. 156. 158 Sphaerica 195.274 Sphere 107.180-182.186.190.194. 195.

211. 214-215. 220-223. 269 Sphere. area of 215. 221 Sphere. geometry of 194 Spherical sector 222 Spherical segment 216. 222 Spherical trigonometry 274-276 Spheroid 215. 216. 223. 224 Spider 182. 183 Spiral 146. 223. 286 Splittin~ of fractions 48

INDEX 305

Sporus 139. 192 Square 110.117.119.141. 142. 165-172 Square numbers 98. 99. 154. 166 Square root 29. 37. 44-45. 123. 205 Squares. table of 44 Squaring the circle 130. 132. 192. 286 Stadia 230 Standard forms 80 Star-pentagon 100. 101. 117 Stars. setting of 195 Steele 263 Stein 227 Stenzel 7 Steps. counters of 230 Stereographic projection 182. 183 Stesichorus 90 Stevin 51. 172. 267 Stoa 102.229 Stobaeus 196 Stoicism 229 Stoics 82 Strabo 180 String 95. 96 Struve 33 Stylus 38 Subjects of study lOB. 196 Suda 130. 174 Suidas 130. 174. 290 Sulgi 42 Sumerian calculating 37-45 Sumerian tables 43. 44 Sumerians 37-43 Sun 128. 202. 2OB. 211 Surveyors 15. 99. 104 Slirya-Siddh1nta 54. 55. 56. 265. 271 Syene 230 Sylvester II 58 Symbolism. algebraic 281. 282 Symptom 146. 191.241.242.268 Synesius 290 Syntaxis mathematica 201 Syracosia 209 Syracuse 150. 202. 208-211 Syria 83 Systems of equations 63-67. 70. 72. 78. 80.

116. 120-124. 140. 170. 198. 199 System. sexagesimal 37-41. 50

Tabit ben Q!rra 57. 226 Tables of chords 206. 207. 263. 271-276 Tables of inverses 43. 44

Tables of multiplication 42. 43. 46. 59 Tables of sines 54. 206. 271 Tables of square roots 44. 47 Tables of squares 44 Tables. sumerian 43. 44 Talent 41 Talmud 33 Tangency problem 262 Tangent 204, 271 Tangent inequality 204 Tangent line 249, 250 Tannery 46,87, 111. 117, 132, 173,207,

234 Taranto 150 Taras 9, 90, 107. 139, 149 Tarentum 179 Taylor 138 T elephanes 85 T erpsion 165 Tetractys 95, 158 Tetrahedron 100, 137, 173. 174 Thaer 92, 197 Thales 7, 9, 82, 84, 85-89, 90, 104 Theaetetus 7, 9, 82, 90, 91, 100, 141, 142.

144, 148, 165-179, 184, 189. 190, 197. 265

Theaetetus (dialogue) 141, 165, 166 Themistius 131 Theodorus 9, 82, 90, 110, 141-146, 147,

165, 169 Theodosius 195 Theon of Alexandria 50, 183. 269, 290 Theon of Smyrna 126. 129, 161. 231. 232-

234 Theorem of Guldin 287 Theorem of Pappus 287. 289 Theorem of Pythagoras 6. 7. 61. 80. 82, 83.

84,90.91,92-102. 104. 106, 107. 117, 118. 125. 147. 279

Theory of harmony 93. 95-96, 107, 111. 157. 158

Theory of music 94,95. 107. 111, 157, 158 Theory of numbers. babylonian 77-80 Theory of numbers, greek 96-100, 108-

116. 153-155, 167, 231 Theudius 90. 196 Thibaut 55 Third. major 157 Third. minor 157 Thomson. 86. 169 Thucydides 82. 106, 202

306 INDEX

Thureau-Dangin 41. 45. 63. 73. 74. 75. 84 Thymaridas 116. 147. 280 Timaeus 112. 173 Toeplitz 7. 139. 155. 205 Tomb 160 Topoi200.234 Tradition of Pythagoras 116. 117. 118 Tradition. written 266 Transmigration of souls 92. 93 Transversal proposition 275. 276 Trapezoid 32. 68. 72. 73. 133. 200 Trapezoid. area of 32. 68. 72. 73 Trapezus 83 Triangle 32. 72. 95. 117. 271. 272 Triangle. area of 32. 72. 228 Triangle. isosceles 87 Triangle. right 8. 76. 77. 79-80. 99. 100.

136.197.285 Triangular numbers 95 Trigonometry 195.203.263.264.271-276 Trigonometry. plane 271-274 Trigonometry. spherical 274-276 Trisection 191. 225. 236. 263 Tropfke 226. 227 Truth 148 Tunnel 102-104 (2:n) table 23-26 Two-tangents theorem 252-256

Uncia 41 Unit fraction. duplication of 22-26 Unity. indivisibility of 115 Universe 93 Unreasonable segments 169 Upward pressure 209. 227 Uruk 73. 84 Usener 132

Varaha-Mihira 55 VAT 6598: 65 VAT 8389: 66 VAT 8512: 72 Velocity. angular 239 VerEecke 195. 212. 237. 241. 286. 291 Vergi1276 Vertical angles 88 Vettius Valens 238. 271 Vieta 267 Vindobon papyrus 49. 56 Virgo 270

Vitruvius32. 100. 102. 104. 136. 137. 149. 182. 183. 209. 276

Vivara 52 Vogel 36. 46. 47. 49. 205 Volten 31 Volume of cone 137. 138. 147. 186 Volume of ellipsoid 223 Volume of frustrum of cone 75. 81 Volume of frustrum of pyramid 34. 75. 81 Volume of pyramid 34. 137. 138. 147. 186 Volume of solid of revolution 287 Volume of sphere 214. 221 Volume of spherical sector 222 Volumes 34-36. 75. 277 Vorticial motion 128. 129

Waerden. van der 31. 38. 85. 87. 95. 108. 112. 153. 155

Wagner. Ulrich 61 Wallis 212 Weidner 84 Weierstrass 269 Weights 41 Wertheimer 280 Western european calculating 47. 58-60 Wilamowitz. von 160. 161 Wings. perspective 136 Winter 210 Writing. cuneiform 38 Written tradition 266

Xenocrates 229 Xenophanes 86. 92 Xenophon 130 Xerxes 85

Yale 44 YBC 7289 ; 45 YBC 10529 ; 44 Year. long : 129

Zalmoxis 92 Zaratas 94 Zeno 229 Zenodorus 264. 268. 269 Zero 39. SO. 55. 56. 58. 59. 60 Zeuthen 168.177.241.245.249.254.255.

259.265 Zodiac 84. 129. 182. 195. 270. 271