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

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HYPATIA. PORPHYRY 529 only to the first six Books, in which case the fact that Hypatia wrote a commentary on them may account for the survival of these Books while the rest of the thirteen were first forgotten and then lost. It will be convenient to take next the series of Neo- Platonist commentators. It does not appear that Ammonius Saccas (about a.d. 175-250), the founder of. Neo-Platonism, or his pupil Plotinus (a.d. 204-69), who first expounded the doctrines in systematic form, had any special connexion with mathematics, but Porphyry (about 232-304), the disciple of Plotinus and the reviser and editor of his works, appears to have written a commentary on the Elements. This we gather from Proclus, who quotes from Porphyry comments on Eucl. I. 14 and 26 and alternative proofs of I. 18, 20. It is possible that Porphyry's work may have been used later by Pappus in writing his own commentary, and Proclus may have got his references from Pappus, but the form of these references suggests that he had direct access to the original commentary of Porphyry. Iamblichus (died about a.d. 330) was the author of a commentary on the Introductio arithmetica of Nicomachus, and of other works which have already been mentioned. He was a pupil of Porphyry as well as of Anatolius, also a disciple of Porphyry. But the most important of the Neo-Platonists to the historian of mathematics is Proclus (a.d. 410-85). Proclus received his early training at Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one Heron (of course a different Heron from the ' mechanicus Hero' of the Metrica, &c). He afterwards went to Athens, where he learnt the Neo-Platonic philosophy from Plutarch, the grandson of • Nestorius, and from his pupil Syrianus, and became one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contemporaries, as we learn from Marinus, his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the 1523.2 M m 530 COMMENTATORS AND BYZANTINES number of books which he wrote, including a large number of commentaries, mostly on the dialogues of PJato (e.g. the Timaeus, the Republic, the Parmenides, the Cratylus). He was an acute dialectician and pre-eminent among his contemporaries in the range of his learning; he was a competent mathematician ; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries, and a devout worshipper of divinities both Greek and Oriental. He was much more a philosopher than a mathematician. In his commentary on the Timaeus, when referring to the question whether the sun occupies a middle place among the planets, he speaks as no real mathematician could have spoken, rejecting the view of Hipparchus and Ptolemy because 6 Qzovpyos (sc. the Chaldean, says Zeller) thinks otherwise, whom it is not lawful to disbelieve '. Martin observes too, ' rather neatly, that for ' Proclus the Elements of Euclid had the good fortune not to be contradicted either by the Chaldean Oracles or by the speculations of Pythagoreans old and new '. Commentary on Euclid, Book I. For us the most important work of Proclus is his commentary on Euclid, Book I, because it is one of the main sources of our information as to the history of elementary geometry. Its great value arises mainly from the fact * that Proclus had access to a number of historical and critical works which are now lost except for fragments preserved by Proclus and others. (a) Sources of the Commentary. The historical work the loss of which is most deeply to be deplored is the History of Geometry by Eudemus. There appears to be no reason to doubt that the work of Eudemus was accessible to Proclus at first hand. For the later writers Simplicius and Eutocius refer to it in terms such as leave no doubt that they had it before them. Simplicius, quoting Eudemus as the best authority on Hippocrates's quadratures of lunes, says he will set out what Eudemus says * word for word ', adding only a little explanation in the shape of references to Euclid's Elements 'owing to the memorandum-like style of Eudemus, who sets out his explanations in the abbre-
PROCLUS 531 viated form usual with ancient writers. Now in the second book of the history of geometry he writes as follows '} In like manner Eutocius speaks of the paralogisms handed down in connexion with the attempts of Hippocrates and Antiphon to square the circle, 'with which I imagine that all persons are accurately acquainted who have examined {kTrz
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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
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 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: THEON OF ALEXANDRIA 527 pp. 58-63).
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
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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