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

More documents

Recommendations

Info

MENELAUS'S SPHAERICA 273 But, by III. 1 applied to the triangle BC\A l cut by the transversal PC A.., sinJ.jJ.3 sin C^C 3 sin PA sin BA 3 sin BC 3 T sin PC\ sin A, A- sin PA sinBA., sin PA sin PC sin C 1 C 3 sin PC Y sin BC 3 sin PC 1 sin PC. 3 Br 7\? 1'3 from above, Part 2 of the proposition proves that, if PC' 2^. 2 be drawn such that sin 2 PC 2 = sin PA 2 . sin PC, or r 2 2 = Rr (where r2 is the radius of the parallel circle through (7 2 ), BC 2 —BA 2 is a maximum, while Parts 3, 4 discuss the limits to the value of the ratio between the arcs A^A 3 and Cfi^. Nothing is known of the life of Claudius Ptolemy except that he was of Alexandria, made observations between the years a.d. 125 and 141 or perhaps 151, and therefore presumably wrote his great work about the middle of the reign of Antoninus Pius (a.d. 138-61). A tradition handed down by the Byzantine scholar Theodorus Meliteniota (about 1361) states that he was born, not at Alexandria, but at Ptolemais 17 'Epfieiov. Arabian traditions, going back probably to Hunain b. Ishaq, say that he lived to the age of 78, and give a number of personal details to which too much weight must not be attached. The MaOrj/iaTLKT) o-vvTagts (Arab. Almagest). Ptolemy's great work, the definitive astronomy, bore the title MaOrj/xaTiKrjs achievement of Greek Zwrdgem /3i/3\ia i-y, the Mathematical Collection in thirteen Books. By the time of the commentators who distinguished the lesser treatises on astronomy forming an introduction to Ptolemy's work as fXLKpb? dcrTpoyo/jiov/iepo? (7677-09), the 'Little Astronomy', the book came to be called the ' Great Collection ', fieydXrj
' PTOLEMY'S SYNTAXIS 275 276 TRIGONOMETRY (died 901), is extant in part, as well as the version by Nasiraddin at-TusI (1201-74). The first edition to be published was the Latin translation made by Gherard of Cremona from the Arabic, which was finished in 1175 but was not published till 1515, when it was brought out, without the author's name, by Peter Liechtenstein at Venice. A translation from the Greek had been made about 1160 by an unknown writer for a certain Henricus Aristippus, Archdeacon of Catania, who, having been sent by William I, King of Sicily, on a mission to the Byzantine Emperor Manuel I. Comnenus in 1158, brought back with him a Greek manuscript of the Syntaxis as a present; this translation, however, exists only in manuscripts in the Vatican and at Florence. The first Latin translation from the Greek to be published was that made by Georgius of Trebizond ' ' for Pope Nicolas V in 1451 ; this was revised and published by Lucas Gauricus at Venice in 1528. The editio princeps of the Greek text was brought out by Grynaeus at Basel in 1538. The next complete edition was that of Halma published 1813-16, which is now rare. All the more welcome, therefore, is the definitive Greek text of the astronomical works of Ptolemy edited by Heiberg (1899-1907), to which is now added, so far as the Syntaxis is concerned, a most valuable supplement in the German translation (with notes) by Manitius (Teubner, 1912-13). bo given here. Book I : Indispensable preliminaries to the study of the Ptolemaic system, general explanations of the different motions of the heavenly bodies in relation to the earth as centre, propositions required for the preparation of Tables of Chords, the Table itself, some propositions in spherical geometry leading to trigonometrical calculations of the relations of arcs of the equator, ecliptic, horizon and meridian, a ' Table of Obliquity ', for calculating declinations for each degree-point on the ecliptic, and finally a method of finding the right ascensions for arcs of the ecliptic equal to one-third of a sign or 10°. Book II: The same subject continued, i.e. problems on the sphere, with special reference to the differences between various latitudes, the length of the longest day at any degree of latitude, and the like. Book III On the length of the year and the motion of the sun on the eccentric and epicycle hypotheses. Book IV : The length of the months and the theory of the moon. Book V : The construction of the astrolabe, and the theory of the moon continued, the diameters of the sun, the moon and the earth's shadow, the distance of the sun and the dimensions of the sun, moon and earth. Book VI : Conjunctions and oppositions of sun and moon, solar and lunar eclipses and their periods. Books VII and VIII are about the fixed stars and the precession of the equinoxes, and Books IX-XIII are devoted to the movements of the planets. Summary of Contents. Trigonometry in Ptolemy. The Syntaxis is most valuable for the reason that it contains very full particulars of observations and investigations by Hipparchus, as well as of the earlier observations recorded by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy based himself very largely upon Hipparchus, e.g. in the preparation of a Table of Chords (equivalent to sines), the theory of eccentrics and epicycles, &c. ; and it is questionable whether he himself contributed anything of great value except a definite theory of the motion of the five planets, for which Hipparchus had only collected material in the shape of observations made by his predecessors and himself. A very short indication of the subjects of the different Books is all that can T 2 What interests the historian of mathematics is the trigonometry in Ptolemy. It is evident that no part of the trigonometry, or of the matter preliminary to it, in Ptolemy was new. What he did was to abstract from earlier treatises, and to condense into the smallest possible space, the minimum of propositions necessary to establish the methods and formulae used. Thus at the beginning of the preliminaries to the Table of Chords in Book I he says 1 We will first show how we can establish a systematic and speedy method of obtaining the lengths of the chords based on the uniform use of the smallest possible number of propositions, so that we may not only have the lengths of the chords
Page 1 and 2:
A HISTORY OF GREEK MATHEMATICS BY S
Page 3 and 4:
CONTENTS vii Vlll CONTENTS Geminus
Page 5 and 6:
CONTENTS XI XXI. COMMENTATORS AND B
Page 7 and 8:
ARISTARCHUS OF SAMOS 3 contemporary
Page 9 and 10:
Now whence ARISTARCHUS OF SAMOS BH:
Page 11 and 12:
ARISTARCHUS OF SAMOS 11 while Y, Z
Page 13 and 14:
ARISTARCHUS OF SAMOS 15 But AB.BG <
Page 15 and 16:
MECHANICS 19 likely that he discove
Page 17 and 18:
• LIST OF EXTANT WORKS 23 24 ARCH
Page 19 and 20:
THE TEXT OF ARCHIMEDES 27 It was ma
Page 21 and 22:
Now THE METHOD HA:AN=A'A:AN = KA:AQ
Page 23 and 24:
ON THE SPHERE AND CYLINDER, I 35 ti
Page 25 and 26:
ON THE SPHERE AND CYLINDER, I 39 wh
Page 27 and 28:
ON THE SPHERE AND CYLINDER, II 43 T
Page 29 and 30:
ON THE SPHERE AND CYLINDER, II 47 (
Page 31 and 32:
MEASUREMENT OF A CIRCLE 51 sides co
Page 33 and 34:
a MEASUREMENT OF A CIRCLE 55 be The
Page 35 and 36:
ON CONOIDS AND SPHEROIDS 59 of the
Page 37 and 38:
so that ON CONOIDS AND SPHEROIDS 63
Page 39 and 40:
Then D : E > BM : MO, > OB : BT, ON
Page 41 and 42:
ON SPIRALS 71 Let OF meet the spira
Page 43 and 44:
ON SPIRALS 75 Lastly, it' E be the
Page 45 and 46:
ON PLANE EQUILIBRIUMS, I, II 79 Thi
Page 47 and 48:
THE SAND-RECKONER 83 Archimedes has
Page 49 and 50:
curve in E 1 , R THE QUADRATURE OF
Page 51 and 52:
THE QUADRATURE OF THE PARABOLA 91 t
Page 53 and 54:
ON FLOATING BODIES, I, II 95 96 ARC
Page 55 and 56:
ON SEMI-REGULAR POLYHEDRA 99 angula
Page 57 and 58:
THE LIBER ASSUMPTORUM 103 Lastly, w
Page 59 and 60:
MEASUREMENT OF THE EARTH 107 a, or
Page 61 and 62:
DISCOVERY OF THE CONIC SECTIONS 111
Page 63 and 64:
MENAECHMUS'S PROCEDURE 115 For, let
Page 65 and 66:
ARISTAEUS'S SOLID LOCI 119 the same
Page 67 and 68:
CONIC SECTIONS IN ARCHIMEDES 123 In
Page 69 and 70:
THE TEXT OF THE CONICS 127 The edit
Page 71 and 72:
THE CONICS 131 diorismi. Nicoteles
Page 73 and 74:
THE C0NIC8, BOOK I 135 the centre o
Page 75 and 76:
PL is THE CONICS, BOOK I 139 called
Page 77 and 78:
THE CONIGS, BOOK I 143 (2) in the h
Page 79 and 80:
It follows that THE CONICS, BOOK I
Page 81 and 82:
THE CONICS, BOOK III 151 152 APOLLO
Page 83 and 84:
To prove (1) R'l 2 : IW-H'Q 2 : QH
Page 85 and 86:
THE CONICS, BOOKS IV-V 159 and, if
Page 87 and 88:
THE CONICS, BOOK V 163 if P' be any
Page 89 and 90:
THE CONICS, BOOKS V, VI 167 tively
Page 91 and 92: THE CONICS, BOOK VII 171 But A'Q*:A
Page 93 and 94: THE GONIGS, BOOK VII 175 As we have
Page 95 and 96: ON THE CUTTING-OFF OF A RATIO 179 F
Page 97 and 98: ON CONTACTS OR TANGENCIES 183 solve
Page 99 and 100: PLANE LOCI 187 other extremity will
Page 101 and 102: NET2EI2 (VERGINGS OR INCLINATIONS)
Page 103 and 104: OTHER LOST WORKS 195 Astronomy. We
Page 105 and 106: NICOMEDES 199 tators, and especiall
Page 107 and 108: DIOCLES. PERSEUS 203 1 Proclus on E
Page 109 and 110: ISOPERIMETRIC FIGURES. ZENODORUS 20
Page 111 and 112: ZENODORUS 211 Now, by hypothesis, D
Page 113 and 114: HYPSICLES 215 natural to divide eac
Page 115 and 116: DIONYSODORUS 219 generates the tore
Page 117 and 118: GEMINUS 223 An upper limit for his
Page 119 and 120: GEMINUS 227 Attempt to prove the Pa
Page 121 and 122: GEMINUS 231 and make equal angles w
Page 123 and 124: 236 SOME HANDBOOKS XVI SOME HANDBOO
Page 125 and 126: THEON OF SMYRNA 239 edited by E. Hi
Page 127 and 128: THEON OF SMYRNA 243 to the seven he
Page 129 and 130: THEODOSIUS'S SPHAERIGA 247 (Books X
Page 131 and 132: THEODOSIUS'S SPHAERICA 251 Now, the
Page 133 and 134: HIPPARCHUS 255 that the lengths of
Page 135 and 136: HIPPARCHUS 259 in his Commentary, h
Page 137 and 138: MENELAUS'S SPHAERICA 263 already ap
Page 139 and 140: MENELAUS'S SPIIAERICA 267 For, if A
Page 141: ^ ' MENELAUS'S SPHAERICA 271 272 TR
Page 145 and 146: PTOLEMY'S SYNTAXIS 279 The proposit
Page 147 and 148: PTOLEMY'S SYNTAXIti 283 284 TRIGONO
Page 149 and 150: THE ANALEMMA OF PTOLEMY 287 288 TRI
Page 151 and 152: THE ANALEMMA OF PTOLEMY 291 or tan
Page 153 and 154: THE OPTICS OF PTOLEMY 295 but the t
Page 155 and 156: CONTROVERSIES AS TO HERON'S DATE 29
Page 161 and 162: GEOMETRY 311 Of this class are the
Page 163 and 164: ' ' concave ' and THE DEFINITIONS 3
Page 165 and 166: MENSURATION 319 Heiberg puts side b
Page 167 and 168: PROOF OF THE FORMULA OF HERON' 323
Page 169 and 170: THE REGULAR POLYGONS 327 Geom. 102
Page 171 and 172: height SEGMENT OF A CIRCLE 331 The
Page 173 and 174: iJ MEASUREMENT OF SOLIDS 335 descri
Page 175 and 176: DIVISIONS OF FIGURES 339 two opposi
Page 177 and 178: : —— Since (DG) : (FB) = m : No
Page 179 and 180: THE MECHANICS 347 the first chapter
Page 181 and 182: Supports '. ON THE CENTRE OF GRAVIT
Page 183 and 184: 356 PAPPUS OF ALEXANDRIA Date of Pa
Page 185 and 186: THE COLLECTION 359 sizes and distan
Page 187 and 188: 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 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 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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?