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

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THE COLLECTION. BOOK VIII 433 with its extremities on AC, AB and so that AC : BD is a given ratio, then the centre of gravity of the triangle ADC will lie on a straight line. Take E, the middle point of AC, and Fa, point on BE such that DF = 2 FE. Also let H be a point on BA such that BH=2HA. Draw FG parallel to AC. Then AG = J AD, and AH=^AB; therefore #G = § 5Z). Also .TO = § ,4# = § ,4C. Therefore, since the ratio AC:BD is given, the ratio GH: GF is given. And the angle FGH (= A) is given ; therefore the triangle FGH is given in species, and consequently the angle GHF is given. And if is a given point. * Therefore HF is a given straight line, and it contains the centre of gravity of the triangle A DC. The inclined plane,. Prop. 8 is on the construction of a plane at a given inclination to another plane parallel to the horizon, and with this Pappus leaves theory and proceeds to the practical part. Prop. 9 (p. 1054. 4 sq.) investigates the problem 'Given a weight which can be drawn along a plane parallel to the horizon by a given force; and a plane inclined to the horizon at a given angle, to find the force required to draw the weight upwards on the. inclined plane'. This seems to be the first or only attempt in ancient times to investigate motion on an inclined plane, and as such it is curious, though of no value. Let A be the weight which can be moved by a force C along a horizontal plane. Conceive a sphere with weight equal to A placed in contact at L with the given inclined plane ; the circle OGL represents a section of the sphere by a vertical plane passing through E its centre and LK the line of greatest slope drawn through the point L. Draw EGH horizontal and therefore parallel to MN in the plane of section, and draw LF perpendicular to EH. Pappus seems to regard the plane as rough, since he proceeds to make a system in equilibrium 1523.2 Y f 434 PAPPUS OF ALEXANDRIA about FL as if L were the fulcrum of a lever. Now the weight A acts vertically downwards along a straight line through E. To balance it, Pappus supposes a weight B attached with its centre of gravity at G. Then A:B=GF:EF = (EL-EF):EF [= (l-sin0):sin0, where IKMN = $]; and, since LKMN is given, the ratio EF: EL, and therefore the ratio (EL-EF) : given ; thus B is found. EF, is Now, says Pappus, if D is the force which will move B along a horizontal plane, as C is the force which will move A along a horizontal plane, the sum of C and D will be the force required to move the sphere upwards on the inclined plane. He takes the particular case where 6 = 60°. Then sin 6 is approximately y§£ (he evidently uses \ . ff and A\B— 16:104. for \ \/3), Suppose, for example, that A = 200 talents; then B is 1300 talents. Suppose further that C is 40 man-power ; then, since D:C = B: A, D = 260 man-power ; and it will take D + C, or 300 man-power, to move the weight up the plane Prop. 10 gives, from Heron's Barulcus, the machine consisting of a pulley, interacting toothed wheels, and a spiral screw working on the last wheel and turned by a handle Pappus merely alters the proportions of the weight to the force, and of the diameter of the wheels. At the end of the chapter (pp. 1070-2) he repeats his construction for the finding of two mean proportionals. Construction of a conic through Jive points. Chaps. 13-17 are more interesting, for they contain the solution of the problem of constructing a conic through five given points. The problem arises in this way. Suppose we are given a broken piece of the surface of a cylindrical column such that no portion of the circumference of either of its base
THE COLLECTION. BOOK VIII 435 is left intact, and let it be required to find the diameter of a circular section of the cylinder. We take any two points A, B on the surface of the fragment and by means of these we find five points on the surface all lying in one plane section, in general oblique. This is done by taking n\e different radii and drawing pairs of circles with A, B as centres and with each of the five radii successively. These pairs of circles with equal radii, intersecting at points on the surface, determine five points on the plane bisecting AB at right angles. The five points are then represented on any plane by triangulation. Suppose the points are A, B, C, D, E and are such that ^ no two of the lines connecting the different pairs cr are parallel. 436 PAPPUS OF ALEXANDRIA to this diameter. Then R is determined by means of the relation in this way. RG.GD.BG.GA = RE.EB:FE.EE (1) Join DB, RA, meeting EF in K, L respectively. Then, by similar triangles, RG.GD.BG.GA = (RE : EL) . (BE : EK) = RE.EB:KE.EL Therefore, by ( 1 ), FE.EE = KE . EL, whence EL is determined, and therefore L. The intersection of AL, BE determines R. Next, in order to find the extremities P, P / of the diameter through V, W, we draw ED, RF meeting FP / in M, JS T respectively. Then, as before, FW. WE:P'W. WP = FE.EE-.RE.ED, by the ellipse, = FW.WE-.NW.WM, by similar triangles. This case can be reduced to the construction of a conic through the five points A, B, D, E, F where EF is parallel to AB. This is shown in a subsequent lemma (chap. 16). For, if EF be drawn through E parallel to A B, and if CD meet AB in and EF in 0', we have, by the well-known proposition about intersecting chords, C0.0D:A0.0B = CO' . O'D : EC . O'F, whence O'F is known, and F is determined. We have then (Prop. 13) to construct a conic through A, B, D, E, F, where EF is parallel to AB. Bisect AB, EF at V, W ; then VW produced both ways is a diameter. Draw DR, the chord through D parallel F f 2 Q! Therefore P' W. WP = NW. WM ; and similarly we can find the value of P'V . VP. Now, says Pappus, since P'W. WP and P'V.VP are given areas and the points V, W are given, P, P' are given. His determination of P, P' amounts (Prop. 14 following) to an elimination of one of the points and the finding of the other by means of an equation of the second degree. Take two points Q, Q' on the diameter such that P'V.VP=WV.VQ, (a) P f W.WP = VW.WQ'\ (13) Q, Q' are thus known, while P, P' remain to be found. By (a) P'V: VW= QV: VP, whence P' W : VW= PQ : P V. Therefore, by means of (/?), PQ:PV=Q / W:WP,
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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
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It follows that THE CONICS, BOOK I
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THE CONICS, BOOK III 151 152 APOLLO
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To prove (1) R'l 2 : IW-H'Q 2 : QH
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THE CONICS, BOOKS IV-V 159 and, if
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THE CONICS, BOOK V 163 if P' be any
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THE CONICS, BOOKS V, VI 167 tively
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THE CONICS, BOOK VII 171 But A'Q*:A
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THE GONIGS, BOOK VII 175 As we have
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ON THE CUTTING-OFF OF A RATIO 179 F
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ON CONTACTS OR TANGENCIES 183 solve
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PLANE LOCI 187 other extremity will
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NET2EI2 (VERGINGS OR INCLINATIONS)
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OTHER LOST WORKS 195 Astronomy. We
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NICOMEDES 199 tators, and especiall
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DIOCLES. PERSEUS 203 1 Proclus on E
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ISOPERIMETRIC FIGURES. ZENODORUS 20
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ZENODORUS 211 Now, by hypothesis, D
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HYPSICLES 215 natural to divide eac
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DIONYSODORUS 219 generates the tore
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GEMINUS 223 An upper limit for his
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GEMINUS 227 Attempt to prove the Pa
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GEMINUS 231 and make equal angles w
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236 SOME HANDBOOKS XVI SOME HANDBOO
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THEON OF SMYRNA 239 edited by E. Hi
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THEON OF SMYRNA 243 to the seven he
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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
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: THE COLLECTION. BOOK VIII 431 432 P
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
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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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