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

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APPENDIX 561 approaches that ratio without limit as R approaches P. But the proof does not enable us to say that RF' : (chord PR), which is > RF f : PG, is also always less than PM : MO. At first sight, therefore, it would seem that the proof must fail. Not so, however; Archimedes is nevertheless able to prove that, if PFand not PT is the tangent at P to the" spiral, an absurdity follows. For his proof establishes that, if PVis the tangent and OF' is drawn as in the proposition, then F'0:RO < OR': OP, or F'O < OR', which l is impossible '. Why this is impossible does not appear in Props. 18 and 20, but it follows from the argument in Prop. 13, which proves that a tangent to the spiral cannot meet the curve again, and in fact that the spiral is everywhere concave towards the origin. Similar remarks apply to the proof by Archimedes of the impossibility of the other alternative supposition (that the tangent at P meets OT at a point U nearer to than T is). Archimedes's proof is therefore in both parts perfectly valid, in spite of any appearances to the contrary. The only drawback that can be urged seems to be that, if we assume the tangent to cut OT at a point very near to T on either side, Archimedes's construction brings us perilously near to infinitesimals, and the proof may appear to hang, as it were, on a thread, albeit a thread strong enough to carry it. But it is remarkable that he should have elaborated such a difficult proof by means of Props. 7, 8 (including the ' solid ' vevcris of Prop. 8), when the figures of Props. 6 and 7 (or 9) themselves suggest the direct proof above given, which is independent of any vevcns. P. Tannery, 1 in a paper on Pappus's criticism of the proof as unnecessarily involving ' solid ' methods, has given another proof of the subtangent-property based on ' plane ' methods only ; but I prefer the method which I have given above because it corresponds more closely to the preliminary propositions actually given by Archimedes. 1 Tannery, Memoires scientifques, i, 1912, pp. 800-16. 1523.3
, vddcov ' 564 INDEX OF GREEK WORDS INDEX OF GREEK WORDS [The pages are those of the first volume except where otherwise stated.] d(3a^, dfiuKiov 47. ayEcoperprjTOS, -ov : aye copeTprjTOS p^- 8e\s etViVo) (Plato) iii. 355. dftiegiTrjTos, -ov, that cannot be gone through, i.e. infinite 343. ddvvaros, -ov 11. 462 : anayayr) eis dhvvarov, &C. 372. atrrjfxa, postulate 373. UKOVO-fJiaTlKoi 11. dXoyos, -ov, irrational 84, 90 : nepi dXdycov yfjappoov K(t\ viuttwv (DeniOeritus) 156-7, 181 : dXoyoi toanep ypappai (Plato) 157. dvd\r)fx/JL(i 11. 287. dvdXoyov, proportional : used as adjective 85. dvaXvopevos (tottos), Treasury of Analysis 421-2, ii. 399, 400, ii. 426. dvdnaXiv, inversely 385 : dmrraXiv \vo~is ii. 400. dvno-Tfje^avTi (dvao-Tpe
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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
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Now THE COLLECTION. BOOK III 367 EA
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THE COLLECTION. BOOK IV 371 we may
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THE COLLECTION. BOOK IV 375 Therefo
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THE COLLECTION. BOOK IV 379 We have
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THE COLLECTION. BOOK IV 383 a certa
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THE COLLECTION. BOOK IV 387 length
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THE COLLECTION. BOOK V 391 the same
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of THE COLLECTION. BOOK V 395 the s
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THE COLLECTION. BOOKS VI, VII 399 T
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THE COLLECTION. BOOK VII 403 ' util
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THE COLLECTION. BOOK VII 407 Two pr
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PG THE COLLECTION. BOOK VII 411 Now
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The problem is THE COLLECTION. BOOK
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i.e. (if DA . THE COLLECTION. BOOK
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THE COLLECTION. BOOK VII 423 Since
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THE COLLECTION. BOOKS VII, VIII 427
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THE COLLECTION. BOOK VIII 431 432 P
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THE COLLECTION. BOOK VIII 435 is le
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THE COLLECTION. BOOK VIII 439 Then
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EPIGRAMS IN THE GREEK ANTHOLOGY 443
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HERONIAN INDETERMINATE EQUATIONS 44
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RELATION OF WORKS 451 Relation of t
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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 and 240: INDETERMINATE EQUATIONS 467 (ft) wh
Page 241 and 242: • INDETERMINATE EQUATIONS 471 Ex.
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: APPENDIX 559 But (arc ASP) = OT,by
Page 289 and 290: 1 INDEX OF GREEK WORDS 567 568 INDE
Page 291 and 292: approximations = ENGLISH INDEX 571
Page 293 and 294: works measurements indeterminate On
Page 295 and 296: ENGLISH INDEX 579 530 ENGLISH INDEX
Page 297 and 298: 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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