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

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ON THE CUTTTNG-OFF OF A RATIO 177 Draw OG parallel to BN or B'N' to meet AM in G. D on AM such that OC:AB =X = B'N' : J.lf, Then AM : ^D = .S'iV' : 00 = B'M:CM; therefore MB:AB = B'G : or CM .MB = AB . BV, CM, a given rectangle. Take Hence the problem is reduced to one of applying to GB a rectangle (GM . MB) equal to a given rectangle (AB . B'G) but falling short by a square figure. In the case as drawn, whatever be the value of A, the solution is always possible because the given rectangle AB . GB' is always less than CA . AB, and therefore always less than J- GB 2 ; one of the positions of M falls between A and B because GM . MB
ON THE CUTTING-OFF OF A RATIO 179 Further, if we put for A the ratio between the lengths of the two fixed tangents, then if h, k be those lengths, k h y h + x-2\/hx which can easily be reduced to ©*+©-' the equation of the parabola referred to the two fixed tangents as axes. 180 APOLLONIUS OF PERGA treatment of this problem in all its particular cases with their Swpio-fxoi could present no difficulty to Apollonius. If the two straight lines are parallel, the solution of the problem gives a means of drawing any number of tangents to an ellipse when two parallel tangents, their points of contact, and the length of the parallel semi-diameter are given (see Conies, III. 42). In the case of the hyperbola (III. 43) the intercepts made by any tangent on the asymptotes contain a constant rectangle. Accordingly the drawing of tangents depends upon the particular case of our problem in which both fixed points are the intersection of the two fixed lines. •(/?) On the cutting-off of an area (\coptov ccttoto/jltj), two Books. This work, also in two Books, dealt with a similar problem, with the difference that the intercepts on the given straight lines measured from the given points are required, not to have a given ratio, but to contain a given rectangle. Halley included an attempted restoration of this work in his edition of the De sectione rationis. The general case can here again be reduced to the more special one in which one of the fixed points is at the intersection of the two given straight lines. Using the same figure as before, but with D taking the position shown by (D) in the figure, we take that point such that or OC . AD — the given rectangle. We have then to draw ON'M through B'N' .AM=OC.AD, B'N':OC=AD:AM. But, by parallels, B''N' : OC = B'M: CM; therefore so that AM :CM=AD: B'M = MD:B'C, B'M .MD = AD. B'C. such that Hence, as before, the problem is reduced to an application of a rectangle in the well-known manner. The complete n 2 (y) On determinate section (SLoopicr/ievr] rofirj), two Books. The general problem here is, Given four points A, B, G, D on a straight line, to determine another point P on the same straight line such that the ratio AP . CP : BP . DP has a given value. It is clear from Pappus's account x of the contents of this work, and from his extensive collection of lemmas to the different propositions in it, that the question was very exhaustively discussed. To determine P by means of the equation AP.GP=\.BP.DP, where A, B, C, D, A are given, is in itself an easy matter since the problem can at once be put into the form of a quadratic equation, and the Greeks would have no difficulty in reducing it to the usual application of areas. If, however (as we may fairly suppose), it was intended for application in further investigations, the complete discussion of it would naturally include not only the finding of a solution, but also the determination of the limits of possibility and the number of possible solutions for different positions of the point-pairs A, C and B, D, for the cases in which the points in either pair coincide, or in which one of the points is infinitely distant, and so on. This agrees with what we find in Pappus, who makes it clear that, though we do not meet with any express mention of series of point-pairs determined by the equation for different values of A, yet the treatise contained what amounts to a com- 1 Pappus, vii, pp. 642-4.
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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
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: THE GONIGS, BOOK VII 175 As we have
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
Page 143 and 144: ' 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 159 and 160:
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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