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

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THE GONIG& BOOK VII 173 conjugate diameters in an ellipse or conjugate hyperbolas, and if the tangents at their extremities form the parallelogram LL'MM', then the parallelogram LL'MM' — rect. AA' . The proof is interesting. BB' Let the tangents at P, D respectively meet the major or transverse axis in T, T'. Now (by VII. 4) PT 2 : GD 2 = NT:GN\ therefore 2 A GPT : 2 A T'DC = NT : CN. 174 APOLLONIUS OF PERGA GN r a = PN.^:GN (1.37,39) 2AGPT: (GL) = V(CN. NT) : = PN.GT:GT.GN .^ CA = 2AGPT.CA.GB; therefore (GL) = GA . GB. The remaining propositions of the Book trace the variations of different functions of the conjugate diameters, distinguishing the maximum values, &c. The functions treated are the following p', the parameter of the ordinates to PP' in the hyperbola, according as A f is (1) not less than^j, the parameter corresponding to A A', (2) less than p but not less than -|p, (3) less than \p (VII. 33-5). PP f^p\ as compared with AA' ^p> in the hyperbola (VII. 36) or the ellipse (VII. 37). PP'+p' „ „ AA f +p in the hyperbola (VII. 38-40) or the ellipse (VII. 41). PP'.p'* „ „ AA'.p in the hyperbola (VII. 42) or the ellipse (VII. 43). But 2AGPT:(GL) = PT:CD, = OP : DT' , by similar triangles, = (CL):2AT'DG. That is, (OX) is a mean proportional between 2AGPT and 2ArDC. Therefore, since >/(NT . OiV) is a mean proportional between iVT and CUV, PP' 2 +p' 2 „ „ AA' 2 +p 2 in the hyperbola, according as (1) A A' is not less than p, or (2) ^4J/ or II. 51).
THE GONIGS, BOOK VII 175 As we have said, 'Book VIII is lost. The nature of its contents can only be conjectured from Apollonius's own remark that it contained determinate conic problems for which Book VII was useful, particularly in determining limits of possibility. Unfortunately, the lemmas of Pappus do not enable us to form any clearer idea. But it is probable enough that the Book contained a number of problems having for their object the finding of conjugate diameters in a given conic such that certain functions of their lengths have given values. It was on this assumption that Halley attempted a restoration of the Book. If it be thought that the above account of the Gonics is disproportionately long for a work of this kind, it must be remembered that the treatise is a great classic which deserves to be more known than it is. What militates against its being read in its original form is the great extent of the exposition (it contains 387 separate propositions), due partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbrousness of the enunciations of complicated propositions in general terms (without the help of letters to denote particular points) and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout. Other works by Apollonius. Pappus mentions and gives a short indication of the contents of six other works of Apollonius which formed part of the Treasury of Analysis. 1 Three of these should be mentioned in close connexion with the Conies. (a) On the Gutting-off of a Ratio (Xoyou a7roTOfirj), two Books. This work alone of the six mentioned has survived, and that only in the Arabic ; it was published in a Latin translation by Edmund Halley in 1706. + It deals with the general problem, Given two straight ' lines, parallel to one another or intersecting, and a fixed point on each line, to draw through 1 Pappus, vii, pp. 640-8, 660-72. 176 APOLLONIUS OF PERGA a given point a straight line which shall tut off segments from each line (measured from the fixed points) bearing a given ratio to one another! Thus, let A, B be fixed points on the two given straight lines A C, BK, and let be the given point. It is required to draw through a straight line cutting the given straight lines in points M, N respectively such that AM is to BN in a given ratio. The two Books of the treatise discussed the various possible cases of this problem which arise according to the relative positions of . the given straight lines and points, and also the necessary conditions and limits of possibility in cases where a solution is not always possible. The first Book begins by supposing the given lines to be parallel, and discusses the different cases which arise ; Apollonius then passes to the cases in which the straight lines intersect, but one of the given points, A or B, is at the intersection of the two lines. Book II proceeds to the general case shown in the above figure, and first proves that the general case can be reduced to the case in Book I where one of the given points, A or B, is at the intersection of the two lines. The reduction is easy. For join OB meeting AG in B', and draw B'N' parallel to BN to meet OM in N'. the ratio B'N' : BN, being equal to the ratio OB' : Then OB, is constant. Since, therefore, BN: AM is a given ratio, the ratio B'N' : AM is also given. Apollonius proceeds in all cases by the orthodox method of analysis and synthesis. Suppose the problem solved and OMN drawn through in such a way that B'N : given ratio = A, say. AM is a
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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
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: THE CONICS, BOOK VII 171 But A'Q*:A
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 and 142: ^ ' 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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Page 161 and 162:
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
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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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