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

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• By THE CONIGS, BOOK II 149 angle (II. 50), making a given angle with the diameter through the point of contact (II. 51, 53) II. 52 contains a Siopio-pos for ; so that ACQT= CU'R'F'. 150 APOLLONIUS OF PERGA the portions of the asymptotes which include between them the last problem, proving that, if the tangent to an ellipse at a branch of the conjugate hyperbola (II. 11), the constancy of any point P meets the major axis in T, the angle GPT is not the rectangle contained by the straight lines drawn from any greater than the angle ABA', where B is one extremity of the point of the curve in fixed directions to meet the asymptotes minor axis. (equivalent to the Cartesian equation with reference to the Book III begins with a series of propositions about the asymptotes, xy = const.) (II. 12), and the fact that the curve equality of certain areas, propositions of the same kind as, and and the asymptotes proceed to infinity and approach con- easily derived from, the propositions (I. 41-50) by means of which, as already shown, the transformation of coordinates is the diameters through Q, P respectively in E, T, then points on two branches the property of II. 8. II. 1 7 shows that AOPT = A0QE (III. 1, 4) ; and, if P, Q be points on adjacent 'conjugate opposites' (two conjugate double-branch hyperbolas) have the same asymptotes. Propositions follow about With the same notation, if R be any other point on the conic, branches of conjugate hyperbolas, AGPE = ACQT (III. 13.). and if we draw BU parallel to the tangent at Q meeting the the diameters through Q, P in F, W, then AHQF = quadrilateral HTUR (III. 2. ; 6) this is proved at once from the fact that ABMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6 above) by subtracting or adding the area HRMQ on each of the asymptotes (II. 21) if a chord Qq in one branch of side. Next take any other point B', and draw B'U', F'H'B'W ; a hyperbola meet the asymptotes in R, r and the conjugate in the same way as before ; it is then proved that, if BU, R'W hyperbola in Q', q', then Q'Q.Qq' = 2 CD 2 (II. 23). Of the meet in I and B'U', R W in J, the quadrilaterals F'IBF, IUU'R' rest of the propositions in this part of the Book the following are equal, and also the quadrilaterals FJB'F', JU'TJR (III. 3, may be mentioned : if TQ, TQ' are two tangents to a conic 7, 9, 10). The proof varies according to the actual positions and V is the middle point of QQ', TV is a diameter (II. 29, of the points in the figures. 30, 38) ; if tQ, tQ' be tangents to opposite branches of a hyperbola, RR' the chord through t parallel to QQ', v the middle In Figs. 1, 2 AHFQ = quadrilateral HTUR, point of QQ', then vR, vR' are tangents to the hyperbola (II. 40) ; in a conic, or a circle, or in conjugate hyperbolas, if AH'F'Q = H'TU'R'. two chords not passing through the centre intersect, they do not bisect each other (II. 26, 41, 42). II. 44-7 show how to find subtraction, FHH'F'= IUU'R + (IB); a diameter of a conic and the centre of a central conic, the axis of a parabola and the axes of a central conic. The Book whence, if IE be added or subtracted, F'IRF = IUU'R', concludes with problems of drawing tangents to conies in and again, if IJ be added to both, FJR'F' = JU'UR. certain ways, through any point on or outside the curve (II. 49), making with the axis an angle equal to a given acute In Fig. 3 AR'U'W = A CF'W - A CQT, tinually nearer to one another, so that the distance separating them can be made smaller than any given length (II. 14). II. 15 proves that the two opposite branches of a hyperbola have the same asymptotes and II. 16 proves for the chord connecting conjugate hyperbolas; any tangent tcPthe conjugate hyperbola will meet both branches of the original hyperbola and will be bisected at the point of contact (II. 19); if Q be any point on a hyperbola, and GE parallel to the tangent at Q meets the conjugate hyperbola in E, the tangent at E will be parallel to GQ and GQ, GE will be conjugate diameters (II. 20), while the tangents at Q, E will meet on one effected. We have first the proposition that, if the tangents at any points P, Q of a conic meet in 0, and if they meet diameter through P^ in U and the diameter through Q in M, and RW parallel to the tangent at P meeting QT in H and
THE CONICS, BOOK III 151 152 APOLLONIUS OF PERGA Adding the quadrilateral CF'H'T, we have AH'F'Q = H'TU'R', E F 1 F and similarly By subtraction, AHFQ = HTUR. F'H'HF= H'TU'R' -HTUR. Adding H'IRH to each side, we have F'IRF == IUU'R'. T U l)' P If each of these quadrilaterals is subtracted from //, FJR'F' = JU'UR. Fig. 1. Fig. 2. The corresponding results are proved in III. 5, 11, 12, 14 for the case where the ordinates through RR' are drawn to a secondary diameter, and in III. 15 for the case where P, Q are on the original hyperbola and R, R' on the conjugate hyperbola. The importance of these propositions lies in the fact that they are immediately used to prove the well-known theorems about the rectangles contained by the segments of intersecting chords and the harmonic properties of the pole and polar. The former question is dealt with in III. 16-23, which give a great variety of particular cases. We will give the proof of one case, to the effect that, if OP, OQ be two tangents to any conic and Rr, R'r' be any two chords parallel to them respectively and intersecting in J, an internal or external point, then RJ . Jr : R f J . Jr' = OP 2 : OQ 2 = (const.). We have RJ. Jr = RW ^JW 2 2 , and RW 2 : JW = ARUW 2 : therefore RJ.Jr: RW 2 = (RW 2 - JW 2 ) : RW But RW 2 : OP 2 = ARUW: AOPT; AJU'W; = 2 JU'UR : ARUW. Fig. 3. therefore, ex aequali, RJ.Jr: OP = 2 JU'UR : A OPT.
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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
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: It follows that THE CONICS, BOOK I
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
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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
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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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