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

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ON SPIRALS 65 then two simple geometrical propositions, followed by propositions (5-9) which are all of one type. Prop. 5 states that, given a circle with centre 0, a tangent to it at A, and c, the Fig. 1. circumference of any circle whatever, it is possible to draw a straight line OPF meeting the circle in P and the tangent in F such that FP:OP < (arc AP) : Archimedes takes D a straight line greater than c, c. draws OH parallel to the tangent at A and then says ' let PH be placed equal to D verging (vevovcra) towards A '. This is the usual phraseology of the type of problem known as vevcris where a straight line of given length has to be placed between two lines or curves in such a position that, if produced, it passes through a given point (this is the meaning of verging) Each of the propositions 5-9 depends on a vevcris of this kind, Fig. 2. which Archimedes assumes as * possible ' without showing how it is effected. Except in the case of Prop. 5, the theoretical solution cannot be effected by means of the straight line and circle ; it depends in general on the solution of an equation of the fourth degree, which can be solved by means of the 1523.2 F 66 ARCHIMEDES points of intersection of a certain rectangular hyperbola and a certain parabola. It is quite possible, however, that such problems were in practice often solved by a mechanical method, namely by placing a ruler, by trial, in the position of the required line : for it is only necessary to place the ruler so that it passes through the given point and then turn it round that point as a pivot till the intercept becomes of the given length. In Props. 6-9 we have a circle with centre 0, a chord AB less than the diameter in it, OM the perpendicular from on AB, BT the tangent at B, OT the straight line through parallel to AB ; B E as the case may be, than the ratio BM : : is any ratio less or greater, MO. Props. 6, 7 (Fig. 2) show that it is possible to draw a straight line OFP Fig. 3. : meeting AB in F and the circle in P such that FP : PB=D E (OP meeting AB in the case where D:E < BM:M0, and meeting AB produced when D : E MO). In Props. 8, 9 > BM : (Fig. 3) it is proved that it is possible to draw a straight line OFP meeting AB in F, the circle in P and the tangent at B in G, such that FP:BG = D:E (OP meeting AB itself in the case where D : E < BM: MO, and meeting AB produced in the case where D:E > BM : MO). We will illustrate by the constructions in Props. 7, 8, as it is these propositions which are actually cited later. Prop. 7. If D : E is any ratio > BM : MO, it is required (Fig. 2) to draw 0P F / / meeting the circle in P f and AB produced in F' so that F P / / :P B = / D:E. Draw OT parallel to AB, and let the tangent to the circle at B meet OT in T.
Then D : E > BM : MO, > OB : BT, ON SPIRALS 67 by hypothesis, by similar triangles. Take a straight line P'H' (less than BT) such that D : = 05 : P'H', and place P'fT' between the circle and OT 1 verging towards ' B (construction assumed). Then F'P' : P'B = OP' : P'H' Prop. 8. If D : £ = OB : P'H' = D:E. E is any given ratio < BM: MO, it is required to draw OFPG meeting AB in F, the circle in P, and the tangent at B to the circle in G so that FP : BG = D : E. 68 ARCHIMEDES Let QGO meet the original circle in P and AB in F. OFPG is the straight line required. For CG.GT=OG.GQ = OG. BK. But OF:OG = BT: GT, by parallels, whence OF.GT=OG.BT. Therefore CG . GT : OF . GT = OG . BK : OG whence CG:OF= BK : BT = BG:OB = P BT, and OK is perpendicular to OT, it is possible to place QG [between the circle TKC and BO] equal to BK and verging towards ' (construction assumed). ' it seems to be a grave error into which geometers fall whenever any one discovers the solution of a plane problem by means of conies or linear (higher) curves, or generally solves it by means of a foreign kind, as is the case e.g. (1) with the problem in the fifth Book of the Conies of Apollonius relating to the parabola, and (2) when Archimedes assumes in his work on the spiral a vedais of a "solid" character with reference to a circle ; for it is possible without calling in the aid of anything solid to find the proof of the theorem given by Archimedes, that is, to prove that the circumference of the circle arrived at in the first revolution is equal to the straight line drawn at right angles to the initial line to meet the tangent to the spiral (i.e. the subtangent).' There is, however, this excuse for Archimedes, that he only assumes that the problem can be solved and does not assume the actual solution. Pappus 1 himself gives a solution of the particular vevcris by means of conies. Apollonius wrote two Books of vevo-61?, and it is quite possible that by Archimedes's time there may already have been a collection of such problems to which tacit reference was permissible. Prop. 10 repeats the result of the Lemma to Prop. 2 of On 1 Pappus, iv, pp. 298-302.
Page 1 and 2: A HISTORY OF GREEK MATHEMATICS BY S
Page 3 and 4: CONTENTS vii Vlll CONTENTS Geminus
Page 5 and 6: CONTENTS XI XXI. COMMENTATORS AND B
Page 7 and 8: ARISTARCHUS OF SAMOS 3 contemporary
Page 9 and 10: Now whence ARISTARCHUS OF SAMOS BH:
Page 11 and 12: ARISTARCHUS OF SAMOS 11 while Y, Z
Page 13 and 14: ARISTARCHUS OF SAMOS 15 But AB.BG <
Page 15 and 16: MECHANICS 19 likely that he discove
Page 17 and 18: • LIST OF EXTANT WORKS 23 24 ARCH
Page 19 and 20: THE TEXT OF ARCHIMEDES 27 It was ma
Page 21 and 22: Now THE METHOD HA:AN=A'A:AN = KA:AQ
Page 23 and 24: ON THE SPHERE AND CYLINDER, I 35 ti
Page 25 and 26: ON THE SPHERE AND CYLINDER, I 39 wh
Page 27 and 28: 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: so that ON CONOIDS AND SPHEROIDS 63
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 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
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
Page 145 and 146:
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
Page 151 and 152:
THE ANALEMMA OF PTOLEMY 291 or tan
Page 153 and 154:
THE OPTICS OF PTOLEMY 295 but the t
Page 155 and 156:
CONTROVERSIES AS TO HERON'S DATE 29
Page 157 and 158:
Page 159 and 160:
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
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: —— Since (DG) : (FB) = m : No
Page 179 and 180:
THE MECHANICS 347 the first chapter
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Supports '. ON THE CENTRE OF GRAVIT
Page 183 and 184:
356 PAPPUS OF ALEXANDRIA Date of Pa
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THE COLLECTION 359 sizes and distan
Page 187 and 188:
THE COLLECTION. BOOK III 363 Sectio
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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 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
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
Page 273 and 274:
PROCLUS 535 second Book \ But at th
Page 275 and 276:
DOMNINUS. SIMPLICIUS 539 sophy at A
Page 277 and 278:
, . a ANTHEMIUS 543 the focus to th
Page 279 and 280:
PSELLUS. PACHYMERES. PLANUDES 547 R
Page 281 and 282:
MOSCHOPOULOS. RHABDAS 551 of Smyrna
Page 283 and 284:
PEDIASIMUS. BARLAAM. ARGYRUS 555 or
Page 285 and 286:
APPENDIX 559 But (arc ASP) = OT,by
Page 287 and 288:
, vddcov ' 564 INDEX OF GREEK WORDS
Page 289 and 290:
1 INDEX OF GREEK WORDS 567 568 INDE
Page 291 and 292:
approximations = ENGLISH INDEX 571
Page 293 and 294:
works measurements indeterminate On
Page 295 and 296:
ENGLISH INDEX 579 530 ENGLISH INDEX
Page 297 and 298:
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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