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

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BE ED THE COLLECTION. BOOK VII 405 17-20 deal with three straight lines a, b, c in geometrical progression, showing how to mark on a straight line containing a, b, c as segments (including the whole among 'segments'), lengths equal to a + c ± 2 V(ac) ; the lengths are of course equal to a 4- c + 2 b respectively. These lemmas are preliminary to the problem (Prop. 21), Given two straight lines AB, BC (C lying between A and B), to find a point D on BA produced such that BD:DA=CD: (AB + BC-2 VABTBC). This is, of course, equivalent to the quadratic equation (a + x):x = (a — c + x):(a + c — 2 Vac), and, after marking off AE along AD equal to the fourth term of this proportion, Pappus solves the equation in the usual way by application of areas. (fi) Lemmas to the ' Determinate Section ' of Apollonius. The next set of Lemmas (Props. 22-64, pp. 704-70) belongs to the Determinate Section of Apollonius. As we have seen (pp. 180-1, above), this work seems to have amounted to a Theory of Involution. Whether the application of certain of Pappus's lemmas corresponded to the conjecture of Zeuthen or not, we have at all events in this set of lemmas some remarkable applications of ' geometrical algebra '. They may be divided into groups as follows I. Props. 22, 25, 29 If in the figure AD. DC = BD . DE\ then 406 PAPPUS OF ALEXANDRIA Props. 32, 34 are alternative proofs based on other lemmas The algebraic equivalent may be (Props. 31, 33 respectively). stated thus : if ax = by, then - = , III. Props. 35, 36. V ~ —- • (x + y)(a-y) If AB.BE = CB.BD, then AB : BE = DA . AC :CE . ED, and CB:BD = AC .CE-.AD.DE, results equivalent to the following : if ax = by, then a __ (ct — y) (a-b) b_ (a — b)(b-x) x" (b-x) (y-x) IV. Props. 23, 24, 31, 57, 58. If A § 1 ' i n y" (a-y)(y-x)' C E D 1 1 AB = CD, and E is any point in CD, AC.CD = AE.ED + BE.EC, and similar formulae hold for other positions of E. If E is between B and C, AC . CD = AE . ED-BE is on AD produced, BE . EC = AE . + BD . .EC; and if E DC. V. A small group of propositions relate to a triangle ABC with two straight lines AD, AE drawn from the vertex A to points on the base BC in conditions (a) (b) accordance with one or other of the that the angles BAC, DAE are supplementary, that the angles BAE, DAC are both right angles or, as we BD:DE = AB.BC:AE. EC. A QPE B The proofs by proportions are not difficult. Prop. 29 is an alternative proof by means of Prop. 26 (see below). The algebraic equivalent may be expressed thus : if ax = by, then II. Props. 30, 32, 34. If in the same figure BD : b (a + b)(b + x) y " (*+y){x+y) AD.DE= BD. DC, then DC = AB . : EC . CA . may add from Book VI, Prop. 12, (c) that the angles BAD, EAC are equal. The theorems are : In case (a) BC.CD:BE.ED = CA 2 : AE\ (b) BC.CE:BD.DE= CA 2 : AD\ (c) DC.CE:EB.BD = AC-.AB 2 .
THE COLLECTION. BOOK VII 407 Two proofs are given of the first theorem. We will give the first (Prop. 26) because it is a case of theoretical analysis followed by synthesis. Describe a circle about ABD : produce EA, CA to meet the circle again in F, G, and join BF, FG. Substituting GC . CA for BC . CD and FE . EA for BE. ED, we have to inquire whether GC . CA : CA 2 = FE . EA : AE 2 , i. e. whether t GC:CA = FE: EA, i.e. whether GA : AC = FA : AE, i.e. whether the triangles GAF, CAE are similar or, in other words, whether GF is parallel to BC. But GF is parallel to BC, because, the angles BAC, DAE being supplementary, Z DAE = Z GAB — Z GFB, while at the same time Z DAE = suppt. of Z FAD = Z FBD. The synthesis is obvious. An alternative proof (Prop. 27) dispenses with the circle, and only requires EKH to be drawn parallel to CA to meet AB, AD in H, K. Similarly (Prop. 28) for case (b) it is only necessary to draw FG through D parallel to AC meeting BA in F and AE produced in G. 408 PAPPUS OF ALEXANDRIA VI. Props. 37, 38. If AB:BC = AD 2 : DC than AD, then 2 ,- whether AB be "greater or less AB.BG = BD 2 . \E in the figure is a point such that ED = CD.] A (E) D C B i 1 1 1 A C B P (f) i i The algebraical equivalent is: If - = .,"" L t then ac — b 2 . These lemmas are subsidiary to the next (Props. 39, 40), being used in the first proofs of them. Props. 39, 40 prove the following: If AGDEB be a straight line, and if BA . AE: BD.DE = AC 2 : CD 2 , 2 ; then AB.BD:AE.ED = BG 2 : GE if, again, AG .CB.AE . EB = CD 2 :DE 2 , then EA .AC:CB.BE = AD 2 : DB 2 . If AB = a, BG = b, BD = c, BE— d, the algebraic equivalents are the following. a (a-d) (a-b) 2 ac b 2 c(c-d) ' ' (b-cf ' (a-d) (c-d) " (6-d) 2 ' , .„ (a — b)b (b — 2 c) .. (a — 2 c£)( then (a — d)d (c — df bd c 2 Then, I FAG, Z ADF (= IDAC) being both right angles, FD . DG = DA 2 . Therefore CA 2 : AD = 2 CA 2 :FD.DG = (CA : FD) . (CA : DG) = (BC:BD).(CE:DE) = BC.CE:BD.DE. In case (c) a circle is circumscribed to ADE cutting AB in F and AC in G. Then, since Z FAD = L GAE, the arcs DF, EG are equal and therefore FG is parallel to DE. The proof is like that of case (a). VII. Props. 41, 42, 43. If AD.DC=BD.DE, suppose that in Figures (1) and (2) 0) O A $ D E B (2) A E D C B° " : 1 1 (3) A E B C ,p k = AE+GB, and in Figure (3) k = AE-BG, then k.AD = k.GD = BA.AEt BG.GE, k.BD = AB.BC, k.DE=AE.EC.
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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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Page 159 and 160: CONTROVERSIES AS TO HERON'S DATE 30
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
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: THE COLLECTION. BOOK VII 403 ' util
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 and 222: THE COLLECTION. BOOK VIII 431 432 P
Page 223 and 224: THE COLLECTION. BOOK VIII 435 is le
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
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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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