A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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. ftx i £> ', INDETERMINATE EQUATIONS 469 470 DIOPHANTUS OF ALEXANDRIA 1 . Double equation of the first degree. The equations are either a x + a = u 2 , + b — w 2 . or (2) the absolute term must vanish, that is, Diophantus has one general method taking slightly different forms according to the nature of the coefficients. £> 4 - 2 p 2 2 (a + b) + (a- b) = 0, or (a) First method of solution. { p — 2 (a 2 + b) } = 4 ab t This depends upon the identity so that ab must be a square number. {i(p+q)V'-{i(p-q)V- = pqif ocn 2 = /3m 2 , since, if oc x -f a is a square, (otx + a) If the difference between the two expressions in x can be separated into two factors p, q, the expressions themselves are equated to {i(p + 2 q}} and { \ (p — 2 q) } respectively. As ' Diophantus himself says in II. 11, we equate either the square of half the difference of the two factors to the lesser of the ocm 2 x + a = w 2] an 2 x + b — w 2 Multiply by n 2 , m Diophantus's point of view, remembering equations that the final quadratic in x must always reduce to a n 2 x+an 2 = u' 2 ocm single equation. oc m Subtracting, we have (oc — ft) x + (a — b) = u — n 2 x + bm = 2 w' 2 \ 2 w 2 . Separate the difference, an Separate (oc — ft) x + (a — — bm 2 , b) into the factors put u f ± iv' = p, p, {(a-ft)x + (a-b)} /p. u'+w' = We q\ write accordingly (oi—ft)x + (a-b) therefore u' = n + w — — \(p + 2 q) , w' = 2 \(p — -, P and a m 2 n 2 x + an 2 2 = %(p +
• INDETERMINATE EQUATIONS 471 Ex. from Diophantus 65- 6x = u 2 ) /t IV ^ t - 32 9 ; ( v ) 65-24x = w 2 therefore 200 - 24 x = u' 2 G5 — 2AX = i
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A HISTORY OF GREEK MATHEMATICS BY S
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CONTENTS vii Vlll CONTENTS Geminus
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CONTENTS XI XXI. COMMENTATORS AND B
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ARISTARCHUS OF SAMOS 3 contemporary
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Now whence ARISTARCHUS OF SAMOS BH:
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ARISTARCHUS OF SAMOS 11 while Y, Z
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ARISTARCHUS OF SAMOS 15 But AB.BG <
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MECHANICS 19 likely that he discove
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• LIST OF EXTANT WORKS 23 24 ARCH
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THE TEXT OF ARCHIMEDES 27 It was ma
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Now THE METHOD HA:AN=A'A:AN = KA:AQ
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ON THE SPHERE AND CYLINDER, I 35 ti
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ON THE SPHERE AND CYLINDER, I 39 wh
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ON THE SPHERE AND CYLINDER, II 43 T
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ON THE SPHERE AND CYLINDER, II 47 (
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MEASUREMENT OF A CIRCLE 51 sides co
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a MEASUREMENT OF A CIRCLE 55 be The
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ON CONOIDS AND SPHEROIDS 59 of the
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so that ON CONOIDS AND SPHEROIDS 63
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Then D : E > BM : MO, > OB : BT, ON
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ON SPIRALS 71 Let OF meet the spira
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ON SPIRALS 75 Lastly, it' E be the
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ON PLANE EQUILIBRIUMS, I, II 79 Thi
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THE SAND-RECKONER 83 Archimedes has
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curve in E 1 , R THE QUADRATURE OF
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THE QUADRATURE OF THE PARABOLA 91 t
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ON FLOATING BODIES, I, II 95 96 ARC
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ON SEMI-REGULAR POLYHEDRA 99 angula
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THE LIBER ASSUMPTORUM 103 Lastly, w
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MEASUREMENT OF THE EARTH 107 a, or
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DISCOVERY OF THE CONIC SECTIONS 111
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MENAECHMUS'S PROCEDURE 115 For, let
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ARISTAEUS'S SOLID LOCI 119 the same
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CONIC SECTIONS IN ARCHIMEDES 123 In
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THE TEXT OF THE CONICS 127 The edit
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THE CONICS 131 diorismi. Nicoteles
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THE C0NIC8, BOOK I 135 the centre o
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PL is THE CONICS, BOOK I 139 called
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THE CONIGS, BOOK I 143 (2) in the h
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It follows that THE CONICS, BOOK I
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THE CONICS, BOOK III 151 152 APOLLO
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To prove (1) R'l 2 : IW-H'Q 2 : QH
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THE CONICS, BOOKS IV-V 159 and, if
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THE CONICS, BOOK V 163 if P' be any
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THE CONICS, BOOKS V, VI 167 tively
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THE CONICS, BOOK VII 171 But A'Q*:A
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THE GONIGS, BOOK VII 175 As we have
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ON THE CUTTING-OFF OF A RATIO 179 F
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ON CONTACTS OR TANGENCIES 183 solve
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PLANE LOCI 187 other extremity will
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NET2EI2 (VERGINGS OR INCLINATIONS)
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OTHER LOST WORKS 195 Astronomy. We
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NICOMEDES 199 tators, and especiall
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DIOCLES. PERSEUS 203 1 Proclus on E
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ISOPERIMETRIC FIGURES. ZENODORUS 20
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ZENODORUS 211 Now, by hypothesis, D
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HYPSICLES 215 natural to divide eac
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DIONYSODORUS 219 generates the tore
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GEMINUS 223 An upper limit for his
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GEMINUS 227 Attempt to prove the Pa
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GEMINUS 231 and make equal angles w
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236 SOME HANDBOOKS XVI SOME HANDBOO
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THEON OF SMYRNA 239 edited by E. Hi
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THEON OF SMYRNA 243 to the seven he
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THEODOSIUS'S SPHAERIGA 247 (Books X
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THEODOSIUS'S SPHAERICA 251 Now, the
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HIPPARCHUS 255 that the lengths of
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HIPPARCHUS 259 in his Commentary, h
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MENELAUS'S SPHAERICA 263 already ap
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MENELAUS'S SPIIAERICA 267 For, if A
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^ ' MENELAUS'S SPHAERICA 271 272 TR
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' PTOLEMY'S SYNTAXIS 275 276 TRIGON
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PTOLEMY'S SYNTAXIS 279 The proposit
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PTOLEMY'S SYNTAXIti 283 284 TRIGONO
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THE ANALEMMA OF PTOLEMY 287 288 TRI
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THE ANALEMMA OF PTOLEMY 291 or tan
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THE OPTICS OF PTOLEMY 295 but the t
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CONTROVERSIES AS TO HERON'S DATE 29
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GEOMETRY 311 Of this class are the
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' ' concave ' and THE DEFINITIONS 3
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MENSURATION 319 Heiberg puts side b
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PROOF OF THE FORMULA OF HERON' 323
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THE REGULAR POLYGONS 327 Geom. 102
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height SEGMENT OF A CIRCLE 331 The
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iJ MEASUREMENT OF SOLIDS 335 descri
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DIVISIONS OF FIGURES 339 two opposi
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: —— Since (DG) : (FB) = m : No
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THE MECHANICS 347 the first chapter
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Supports '. ON THE CENTRE OF GRAVIT
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356 PAPPUS OF ALEXANDRIA Date of Pa
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THE COLLECTION 359 sizes and distan
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THE COLLECTION. BOOK III 363 Sectio
Page 189 and 190: Now THE COLLECTION. BOOK III 367 EA
Page 191 and 192: THE COLLECTION. BOOK IV 371 we may
Page 193 and 194: THE COLLECTION. BOOK IV 375 Therefo
Page 195 and 196: THE COLLECTION. BOOK IV 379 We have
Page 197 and 198: THE COLLECTION. BOOK IV 383 a certa
Page 199 and 200: THE COLLECTION. BOOK IV 387 length
Page 201 and 202: THE COLLECTION. BOOK V 391 the same
Page 203 and 204: of THE COLLECTION. BOOK V 395 the s
Page 205 and 206: THE COLLECTION. BOOKS VI, VII 399 T
Page 207 and 208: THE COLLECTION. BOOK VII 403 ' util
Page 209 and 210: THE COLLECTION. BOOK VII 407 Two pr
Page 211 and 212: PG THE COLLECTION. BOOK VII 411 Now
Page 213 and 214: The problem is THE COLLECTION. BOOK
Page 215 and 216: i.e. (if DA . THE COLLECTION. BOOK
Page 217 and 218: THE COLLECTION. BOOK VII 423 Since
Page 219 and 220: THE COLLECTION. BOOKS VII, VIII 427
Page 221 and 222: THE COLLECTION. BOOK VIII 431 432 P
Page 223 and 224: THE COLLECTION. BOOK VIII 435 is le
Page 225 and 226: THE COLLECTION. BOOK VIII 439 Then
Page 227 and 228: EPIGRAMS IN THE GREEK ANTHOLOGY 443
Page 229 and 230: HERONIAN INDETERMINATE EQUATIONS 44
Page 231 and 232: RELATION OF WORKS 451 Relation of t
Page 233 and 234: TRANSLATIONS AND EDITIONS 455 unfor
Page 235 and 236: NOTATION AND DEFINITIONS 459 When t
Page 237 and 238: DETERMINATE EQUATIONS 463 equation
Page 239: INDETERMINATE EQUATIONS 467 (ft) wh
Page 243 and 244: INDETERMINATE EQUATIONS 475 a squar
Page 245 and 246: METHOD OF APPROXIMATION TO LIMITS 4
Page 247 and 248: NUMBERS AS THE SUMS OF SQUARES 483
Page 249 and 250: ' (I. 35. x = my, y 2 = nx. 1 1. 36
Page 251 and 252: INDETERMINATE ANALYSIS 491 IV. 33.
Page 253 and 254: INDETERMINATE ANALYSIS 495 III. 14.
Page 255 and 256: INDETERMINATE ANALYSIS 499 IV. 29.
Page 257 and 258: INDETERMINATE ANALYSIS 503 Let the
Page 259 and 260: INDETERMINATE ANALYSIS 507 508 DIOP
Page 261 and 262: INDETERMINATE ANALYSIS 511 [The aux
Page 263 and 264: THE TREATISE ON POLYGONAL NUMBERS 5
Page 265 and 266: SERENUS 519 520 COMMENTATORS AND BY
Page 267 and 268: SERENUS 523 Observing that the sum
Page 269 and 270: THEON OF ALEXANDRIA 527 pp. 58-63).
Page 271 and 272: PROCLUS 531 viated form usual with
Page 273 and 274: PROCLUS 535 second Book \ But at th
Page 275 and 276: DOMNINUS. SIMPLICIUS 539 sophy at A
Page 277 and 278: , . a ANTHEMIUS 543 the focus to th
Page 279 and 280: PSELLUS. PACHYMERES. PLANUDES 547 R
Page 281 and 282: MOSCHOPOULOS. RHABDAS 551 of Smyrna
Page 283 and 284: PEDIASIMUS. BARLAAM. ARGYRUS 555 or
Page 285 and 286: APPENDIX 559 But (arc ASP) = OT,by
Page 287 and 288: , vddcov ' 564 INDEX OF GREEK WORDS
Page 289 and 290: 1 INDEX OF GREEK WORDS 567 568 INDE
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approximations = ENGLISH INDEX 571
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works measurements indeterminate On
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ENGLISH INDEX 579 530 ENGLISH INDEX
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ENGLISH INDEX 583 584 ENGLISH INDEX
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A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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