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

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THE CATTLE-PROBLEM 97 (ii) if sQ 1 Q 2 :AM\ the solid will not rest with its base touching the surface of the fluid in one point only, but in a position with the base entirely out of the fluid and the axis making with the surface an angle greater than U ; (iiia) if s = Q X Q 2 \ AM 2 , there is stable equilibrium with one point of the base touching the surface and AM inclined to it at an angle equal to U; 2 (iiib) if s = P l P : 5 AM 2 , there is stable equilibrium with one point of the base touching the surface and with AM inclined to it at an angle equal to T ; (iv) iis>P 1 P 2 :AM 2 but
ON SEMI-REGULAR POLYHEDRA 99 angular, but not similar, polygons ; those discovered by Archimedes were 13 in number. If we for convenience designate a polyhedron contained by m regular polygons of oc sides, n regular polygons of /? sides, &c, by (m a , %...), the thirteen Archimedean polyhedra, which we will denote by jF}, P 2 ...P IZ , are as follows: Figure with 8 faces: P x = (4,, 4 G ). Figures with 14 faces: P 2 = (8 3 , 6 4 ), P 3 = (6 4 , 8 6 ), P 4 = (83J 6 8 ). Figures with 26 faces : P 5 = (8 3 , 18 4 ), P 6 = (12 4 , 8 6 , 6 8 ). Figures with 32 faces: P 7 = (20 3 , 12 5 ), P 8 = (12 5 , 20 6 ), P 9 = (20lf 12 10 ). Figure with 38 faces: P 10 = (32 3 , 6 4 ). Figures with 62 faces: P n = (20 3 , 30 4 , 12 5 ), P ]2 EE(30 4 ,20 G ,12 10 ). Figure with 92 faces: P 13 = (80 3 , 12 5 ). Kepler 1 showed how these figures can be obtained. A method of obtaining some of them is indicated in a fragment of a scholium to the Vatican MS. of Pappus. If a solid angle of one of the regular solids be cut off symmetrically by a plane, i.e. in such a way that the plane cuts off the same length from each of the edges meeting at the angle, the section is a regular polygon which is a triangle, square or pentagon according as the solid angle is formed of three, four, or five plane angles. If certain equal portions be so cut off from all the solid angles respectively, they will leave regular polygons inscribed in the faces of the solid ; this happens (A) when the cutting planes bisect the sides of the faces and so leave in each face a polygon of the same kind, and (B) when the cutting planes cut off a smaller portion from each angle in such a way that a regular polygon is left in each face which has double the number of sides (as when we make, say, an octagon out of a square by cutting off the necessary portions, 1 Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6. H 2 100 ARCHIMEDES symmetrically, from the corners). We have seen that, according to Heron, two of the semi-regular solids had already been discovered by Plato, and this would doubtless be his method. The methods (A) and (B) applied to the five regular solids give the following out of the 13 semi-regular solids. We obtain (1) from the tetrahedron, P 1 by cutting off angles so as to leave hexagons in the faces ; (2) from the cube, P 2 by leaving squares, and P 4 by leaving octagons, in the faces (3) from the octahedron, P 2 by leaving triangles, and P 3 by leaving hexagons, in the faces ; (4) from the icosahedron, Bj by leaving triangles, and P g by leaving hexagons, in the faces; (5) from the dodecahedron, P 7 by leaving pentagons, and P 9 by leaving decagons in the faces. Of the remaining six, four are obtained by cutting off all the edges symmetrically and equally by planes parallel to the edges, and then cutting off angles. Take first the cube. (1) Cut off from each four parallel edges portions which leave an octagon as the section of the figure perpendicular to the edges ; then cut off equilateral triangles from the corners (see Fig. 1) ; this gives P 5 containing 8 equilateral triangles and 18 squares. (P 5 is also obtained by bisecting all the edges of P 2 and cutting off corners.) (2) Cut off from the edges of the cube a smaller portion so as to leave in each face a square such that the octagon described in it has its side equal to the breadth of the section in which each edge is cut; then cut off hexagons from each angle (see Fig. 2); this ""r "jr Fig. 1. Fig. 2. gives 6 octagons in the faces, 12 "" \*Y~ • ! l— , squares under the edges and exactly 8 hexagons at the corners; that is, we have P 6 . An
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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 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: ON FLOATING BODIES, I, II 95 96 ARC
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
Page 149 and 150:
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
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 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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