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

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THEODOSIUS'S SPHAERIGA 249 tions. A particular small circle is the circle which is the limit of the stars which do not set, as seen by an observer at a particular place on the earth's surface ; the pole of this circle is the pole in the heaven. A great circle which touches this circle and is obliquely inclined to the ' parallel circles ' is the circle of the horizon ; the parallel circles of course represent the apparent motion of the fixed stars in the diurnal rotation, and have the pole of the heaven as pole. A second great circle obliquely inclined to the parallel circles is of course the circle of the zodiac or ecliptic. The greatest of the ' parallel circles ' is naturally the equator. All that need be said of the various propositions (except two which will be mentioned separately) is that the sort of result proved is like that of Props. 12 and 13 of Euclid's Phaenomena to the effect that in the half of the zodiac circle beginning with Cancer (or Capricornus) equal arcs set (or rise) in unequal times ; those which are nearer the tropic circle take a longer time, those further from it a shorter; those which take the shortest time are those adjacent to the equinoctial points ; those which are equidistant from the equator rise and set in equal times. In like manner Theodosius (III. 8) in effect takes equal and contiguous arcs of the ecliptic all on one side of the equator, draws through their extremities great circles touching the circumpolar ' parallel ' circle, and proves that the corresponding arcs of the equator intercepted between the latter great circles are unequal and that, of the said arcs, that corresponding to the arc of the ecliptic which is nearer the tropic circle is the greater. The successive great circles touching the circumpolar circle are of course successive positions of the horizon as the earth revolves about its axis, that is to say, the same length of arc on the ecliptic takes a longer or shorter time to rise according as it is nearer to or farther from the tropic, in other words, farther from or nearer to the equinoctial points. It is, however, obvious that investigations of this kind, which only prove that certain arcs are greater than others, and do not give the actual numerical ratios between them, are useless for any practical purpose such as that of telling the hour of the night by the stars, which was one of the fundamental problems in Greek astronomy ; and in order to find 250 TRIGONOMETRY the required numerical ratios a new method had to be invented, namely trigonometry. No actual trigonometry in Theodosius. It is perhaps hardly correct to say that spherical triangles are nowhere referred to in Theodosius, for in III. 3 the congruence-theorem for spherical triangles corresponding to Eucl. I. 4 is practically proved ; but there is nothing in the book that can be called trigonometrical. The nearest approach is in III. 11, 12, where ratios between certain straight lines are compared with ratios between arcs. ACc (Prop. 11) is a great circle through the poles A, A' ; CDc, CD are two other great circles, both of which are at right angles to the plane of ACc, but CDc is perpendicular to AA\ while CD is inclined to it at an acute angle. Let any other great circle AB'BA' through A A' cut CD in any point B between C and D, and CD in B'. Let the ' parallel ' circle EB'e be drawn through B\ .and let Cc r be the diameter of the ' parallel ' circle touching the great circle CD. Let L, K be the centres of the ' parallel ' circles, and let R, p be the radii of the ' parallel ' circles CDc, Cc f respectively. It is required to prove that 2R:2p> (arc CB) : (arc CB r ). Let CO, Ee meet in N, and join NB'. Then B'N, being the intersection of two planes perpendicular to the plane of ACCA f , is perpendicular to that plane and therefore to both Ee and CO.
THEODOSIUS'S SPHAERICA 251 Now, the triangle NLO being right-angled at L, NO > NL. Measure NT along NO equal to NL, and join TB'. Then in the triangles B'NT, B f NL two sides B'N, NT are equal to two sides B'N, NL, and the included angles (both being right) are equal ; respects, and lNLB' = LNTB'. Now If a', therefore the triangles are equal in all 2R:2p = OC':C'K = ON:NL = ON:NT [= tun NTB'itim NOB'] > A NTS': I NOB' > INLB':/_N0B' > IC0B:IN0B' > (arc.BC):(arc.B' oc > /3. tan oc
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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
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: THEODOSIUS'S SPHAERIGA 247 (Books X
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 and 142: ^ ' MENELAUS'S SPHAERICA 271 272 TR
Page 143 and 144: ' PTOLEMY'S SYNTAXIS 275 276 TRIGON
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
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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
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NOTATION AND DEFINITIONS 459 When t
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DETERMINATE EQUATIONS 463 equation
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INDETERMINATE EQUATIONS 467 (ft) wh
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• INDETERMINATE EQUATIONS 471 Ex.
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INDETERMINATE EQUATIONS 475 a squar
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METHOD OF APPROXIMATION TO LIMITS 4
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NUMBERS AS THE SUMS OF SQUARES 483
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' (I. 35. x = my, y 2 = nx. 1 1. 36
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INDETERMINATE ANALYSIS 491 IV. 33.
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INDETERMINATE ANALYSIS 495 III. 14.
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INDETERMINATE ANALYSIS 499 IV. 29.
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INDETERMINATE ANALYSIS 503 Let the
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INDETERMINATE ANALYSIS 507 508 DIOP
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INDETERMINATE ANALYSIS 511 [The aux
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THE TREATISE ON POLYGONAL NUMBERS 5
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SERENUS 519 520 COMMENTATORS AND BY
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SERENUS 523 Observing that the sum
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THEON OF ALEXANDRIA 527 pp. 58-63).
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PROCLUS 531 viated form usual with
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PROCLUS 535 second Book \ But at th
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DOMNINUS. SIMPLICIUS 539 sophy at A
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, . a ANTHEMIUS 543 the focus to th
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PSELLUS. PACHYMERES. PLANUDES 547 R
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MOSCHOPOULOS. RHABDAS 551 of Smyrna
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PEDIASIMUS. BARLAAM. ARGYRUS 555 or
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APPENDIX 559 But (arc ASP) = OT,by
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, vddcov ' 564 INDEX OF GREEK WORDS
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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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