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8 ARISTARCHUS OF SAMOS The first part follows from the relation found in Prop. 4, namely BG : for BA < 1:45, EF = 2 BC. The second part requires the use of the circle drawn with centre A and radius AC. This circle is that on which the ends of the diameter of the ' dividing circle ' move as the moon moves in a circle about the earth. If r is the radius AC of this circle, a chord in it equal to r subtends at the centre A an angle of %R or 60°; and the arc CD subtends at A an angle of 2°. or But (arc subtended by CD) : (arc subtended by r) < CD:r, 2:60 < CD:r; that is, CD:CA > 1:30. And, by similar triangles, CL:CA = CB:BA, or Therefore FE: BA > 1 : 30. CD.CA = 2CB: BA = FE.BA. The proposition which is of the greatest interest on the whole is Prop. 7, to the effect that the distance of the sun from the earth is greater than 18 times, but less than 20 times, the distance of the moon from the earth. This result represents a great improvement on all previous attempts to estimate the relative distances. The first speculation on the subject was that of Anaximander {circa 611-545 B.C.), who seems to have made the distances of the sun and moon from the earth to be in the ratio 3 : 2. Eudoxus, according to Archimedes, made the diameter of the sun 9 times that of the moon, and Phidias, Archimedes's father, 1 2 times ; and, assuming that the angular diameters of the two bodies are equal, the ratio of their distances would be the same. Aristarchus's proof is shortly as follows. A is the centre of the sun, B that of the earth, and C that of the moon at the moment of dichotomy, so that the angle ACB is right. is a square, and AE is ABEF a quadrant of the sun's circular orbit. Join BF, and bisect the angle FBE by BG, so that « IGBE= \R or 22|°.
; ARISTARCHUS OF SAMOS 9 I. Now, by Hypothesis 4, /.ABC = 87°, so that Z HBE = Z 5ZIC = 3° therefore Z.GBE:lHBE=iR: 3 \R = 15:2, so that GE:HE[ = tan ££# : tan #£#] > Z QBE : Z #£# > 15 :2. (1) The ratio which has to be proved > 18:1 is AB:BC or FE:EH. Now FG:GE=FB:BE, whence FG 2 : GE = 2 FB 2 : BE and FG:GE = V2:1 > 7:5 2 =2:1, (this is the approximation to V2 mentioned by Plato and known to the Pythagoreans).
Page 5 and 6: A HISTORY OF GREEK MATHEMATICS VOLU
Page 7 and 8: A HISTORY OF GREEK MATHEMATICS BY S
Page 9 and 10: CONTENTS OF VOL II XII. ARISTARCHUS
Page 11 and 12: CONTENTS vii Geminus pages 222-234
Page 13 and 14: CONTENTS ix The Synagoge or Collect
Page 15: CONTENTS XI XXI. COMMENTATORS AND B
Page 18 and 19: ' 2 ARISTARCHUS OF SAMOS stamping t
Page 20 and 21: : 4 ARISTARCHUS OF SAMOS Archimedes
Page 22 and 23: 6 ARISTARCHUS OF SAMOS (3) that the
Page 26 and 27: ; 10 ARISTARCHUS OF SAMOS Therefore
Page 28 and 29: 12 ARISTARCHUS OF SAMOS (2) ON: (di
Page 30 and 31: . 14 ARISTARCHUS OF SAMOS Prop. 14
Page 32 and 33: ; XIII ARCHIMEDES The siege and cap
Page 34 and 35: 18 ARCHIMEDES the making of spheres
Page 36 and 37: 20 ARCHIMEDES birth to the calculus
Page 38 and 39: : 22 ARCHIMEDES on the Quadrature o
Page 40 and 41: 24 • ARCHIMEDES " works on the le
Page 42 and 43: 26 ARCHIMEDES private hands. In 149
Page 44 and 45: 28 ARCHIMEDES Mechanical Theorems,
Page 46 and 47: 30 ARCHIMEDES the whole segment of
Page 48 and 49: V 32 ARCHIMEDES circles are section
Page 50 and 51: 34 ARCHIMEDES Therefore AX: AG = (A
Page 52 and 53: 3 36 ARCHIMEDES that, if we continu
Page 54 and 55: ,' 38 ARCHIMEDES Therefore B is not
Page 56 and 57: 40 ARCHIMEDES less than a hemispher
Page 58 and 59: 42 ARCHIMEDES ttt 2 .2 sin -^- ~2n
Page 60 and 61: , 44 ARCHIMEDES values for each of
Page 62 and 63: 46 ARCHIMEDES the curves touch at t
Page 64 and 65: 48 ARCHIMEDES Draw QMN through M pa
Page 66 and 67: 50 ARCHIMEDES To return to Archimed
Page 68 and 69: 52 ARCHIMEDES and Zeuthen) is that
Page 70 and 71: , 54 ARCHIMEDES Now the triangles A
Page 72 and 73: 56 ARCHIMEDES case of all, where we
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58 ARCHIMEDES first of the three pr
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60 ARCHIMEDES by planes obliquely i
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, 62 ARCHIMEDES II. In the case of
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} 64 ARCHIMEDES The conclusion, con
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66 ARCHIMEDES points of intersectio
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: 68 ARCHIMEDES Let QGO meet the or
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70 ARCHIMEDES touches it at one poi
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72 ARCHIMEDES Let OF'G meet the spi
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74, ARCHIMEDES And OB, OP, OQ, . .
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76 ARCHIMEDES proofs. We do not fin
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. 78 ARCHIMEDES Hence the centre of
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. 80 ARCHIMEDES area. Lastly (Prop.
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82 ARCHIMEDES of the universe, has
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; 84 ARCHIMEDES the sun when it has
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86 ARCHIMEDES generally), then howe
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88 ARCHIMEDES that is, it takes the
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90 ARCHIMEDES Taking the limit, wc
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: ; 92 ARCHIMEDES deduced from Post
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; ; 94 ARCHIMEDES method attributed
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; 96 ARCHIMEDES where k is the axis
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98 ARCHIMEDES Secondly, it is requi
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; 100 ARCHIMEDES symmetrically, fro
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102 ARCHIMEDES of the two smaller s
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104 ERATOSTHENES Long as the presen
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106 ERATOSTHENES geometric mean bet
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; 108 ERATOSTHENES out of 83 contai
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XIV CONIC SECTIONS. APOLLONIUS OF P
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112 CONIC SECTIONS a section throug
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114 CONIC SECTIONS But, by similar
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116 CONIC SECTIONS at right angles)
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: ' 118 CONIC SECTIONS areas, manip
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120 CONIC SECTIONS Proof from Pappu
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; ; 122 CONIC SECTIONS chords drawn
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124 CONIC SECTIONS only : p is simp
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126 APOLLONIUS OF PERGA for the oth
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128 APOLLONIUS OF PERGA Gregory, ho
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130 APOLLONIUS OF PERGA minima and
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132 APOLLONIUS OF PERGA Preface to
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134 APOLLONIUS OF PERGA straight li
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136 APOLLONIUS OF PERGA as particul
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138 APOLLONIUS OF PERGA Therefore Q
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; 140 APOLLONIUS OF PERGA ' If in a
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142 APOLLONIUS OF PERGA /
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; 144 APOLLONIUS OF PERGA Therefore
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146 APOLLONIUS OF PERGA (2) In the
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148 APOLLONIUS OF PERGA It has been
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150 APOLLONIUS OF PERGA the last pr
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152 APOLLONIUS OF PERGA Adding the
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154 APOLLONIUS OF PERGA The general
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156 APOLLONIUS Ot PERGA L, 1/ and M
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; 158 APOLLONIUS OF PERGA tively. T
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) 160 APOLLONIUS OF PERGA Next Apol
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162 APOLLONIUS OF PERGA Similarly,
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: 164 APOLLONIUS OF PERGA (2) if A
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166 APOLLONIUS OF PERGA The proposi
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168 APOLLONIUS OF PERGA each make e
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: 170 APOLLONIUS OF PERGA Secondly,
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172 APOLLONIUS OF PERGA A number of
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: A 174 APOLLONIUS OF PERGA 2AGPT:
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176 APOLLONIUS OF PERGA a given poi
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178 APOLLONIUS OF PERGA the value o
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180 APOLLONIUS OF PERGA treatment o
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182 APOLLONIUS OF PERGA a circle, (
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184 APOLLONIUS OF PERGA HG meeting
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186 AP0LL0N1US OF PERGA of a line.
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.; 188 APOLLONIUS OF PERGA cut off
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190 APOLLONIUS OF PERGA III. Given
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192 APOLLONIUS OF PERGA But, by sim
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194 APOLLONIUS OF PERGA (A) On the
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196 APOLLONIUS OF PERGA another hyp
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198 SUCCESSORS OF THE GREAT GEOMETE
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; 200 SUCCESSORS OF THE GREAT GEOME
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; 202 SUCCESSORS OF -THE GREAT GEOM
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Page 222 and 223:
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Page 228 and 229:
Page 230 and 231:
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Page 236 and 237:
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' 224 SUCCESSORS OF THE GREAT GEOME
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Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
; 236 SOME HANDBOOKS the first cent
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238 SOME HANDBOOKS larger than the
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; 240 SOME HANDBOOKS numbers, plane
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242 SOME HANDBOOKS the earth, repre
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: 244 SOME HANDBOOKS We next have (
Page 262 and 263:
246 TRIGONOMETRY dosius was of Bith
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; 248 TRIGONOMETRY circle from its
Page 266 and 267:
250 TRIGONOMETRY the required numer
Page 268 and 269:
252 TRIGONOMETRY We may contrast wi
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254 TRIGONOMETRY The work of Hippar
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\ 256 TRIGONOMETRY Improved Instrum
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258 TRIGONOMETRY in relation to one
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' 260 TRIGONOMETRY by ¥Jo from §
Page 278 and 279:
262 TRIGONOMETRY translation by Mau
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264 TRIGONOMETRY Eucl. I. 16, 32 ar
Page 282 and 283:
266 TRIGONOMETRY (a) ' Menelaus s t
Page 284 and 285:
268 TRIGONOMETRY But, by the propos
Page 286 and 287:
270 TRIGONOMETRY It follows that th
Page 288 and 289:
272 TRIGONOMETRY (1) If * 1 >P 1 >2
Page 290 and 291:
274 TRIGONOMETRY the superlative /x
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: : 276 TRIGONOMETRY bo given here.
Page 294 and 295:
. ; 278 TRIGONOMETRY The constructi
Page 296 and 297:
280 TRIGONOMETRY The equation in fa
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; 282 TRIGONOMETRY greater, then sh
Page 300 and 301:
284 TRIGONOMETRY the sines obtained
Page 302 and 303:
286 TRIGONOMETRY , Thus AC is found
Page 304 and 305:
288 TRIGONOMETRY the diagram is one
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, ; 290 TRIGONOMETRY both in the pl
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292 TRIGONOMETRY seeing that Diodor
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; 294 TRIGONOMETRY on the mirror wh
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296 TRIGONOMETRY I. To prove I. 28,
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XVIII MENSURATION: HERON OF ALEXAND
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300 HERON OF ALEXANDRIA Pappus goes
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' 302 HERON OF ALEXANDRIA with Hero
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304 HERON OF ALEXANDRIA And first w
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306 HERON OF ALEXANDRIA the surface
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: 308 HERON OF ALEXANDRIA education
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310 HERON OF ALEXANDRIA Besthorn an
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312 HERON OF ALEXANDRIA triangle wh
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: ; 314 HERON OF ALEXANDRIA Now the
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316 HERON OF ALEXANDRIA or any angl
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318 HERON OF ALEXANDRIA formed by t
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320 HERON OF ALEXANDRIA in an Archi
Page 338 and 339:
322 HERON OF ALEXANDRIA chap. 30 of
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. , ^ the square) is . ' 26J-| inst
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326 HERON OF ALEXANDRIA is given as
Page 344 and 345:
; 328 HERON OF ALEXANDRIA Heron. As
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, 330 HERON OF ALEXANDRIA (£) Segm
Page 348 and 349:
332 HERON OF ALEXANDRIA view of the
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334 HERON OF ALEXANDRIA The method
Page 352 and 353:
336 HERON OF ALEXANDRIA Book III. D
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338 HERON OF ALEXANDRIA area'. Thes
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340 HERON OF ALEXANDRIA lines ', an
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8 342 HERON OF ALEXANDRIA and, solv
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344 HERON OF ALEXANDRIA Quadratic e
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346 HERON OF ALEXANDRIA that to mee
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348 HERON OF ALEXANDRIA case suppos
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' 350 HERON OF ALEXANDRIA between t
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1 ' 352 HERON OF ALEXANDRIA do grea
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354 HERON OF ALEXANDRIA reversing t
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356 PAPPUS OF ALEXANDRIA Date of Pa
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358 PAPPUS OF ALEXANDRIA however, a
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360 PAPPUS OF ALEXANDRIA who is men
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362 PAPPUS OF ALEXANDRIA a problem
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364 PAPPUS OF ALEXANDRIA the proble
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366 PAPPUS OF ALEXANDRIA triangle w
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368 PAPPUS OF ALEXANDRIA the other,
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370 PAPPUS OF ALEXANDRIA contained
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372 PAPPUS OF ALEXANDRIA There is,
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374 PAPPUS OF ALEXANDRIA But theref
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376 PAPPUS OF ALEXANDRIA IV. We now
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378 PAPPUS OF ALEXANDRIA With as ce
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380 PAPPUS OF ALEXANDRIA principii.
Page 398 and 399:
382 PAPPUS OF ALEXANDRIA curve, des
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384 PAPPUS OF ALEXANDRIA » Now (se
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386 PAPPUS OF ALEXANDRIA means of a
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388 PAPPUS OF ALEXANDRIA Measure DA
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390 PAPPUS OF ALEXANDRIA ambrosia i
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39.2 PAPPUS OF ALEXANDRIA Draw AE a
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394 PAPPUS OF ALEXANDRIA which have
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396 PAPPUS OF ALEXANDRIA synthetica
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398 PAPPUS OF ALEXANDRIA in length
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400 PAPPUS OF ALEXANDRIA book, prac
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402 PAPPUS OF ALEXANDRIA Eratosthen
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404 PAPPUS OF ALEXANDRIA If the pas
Page 422 and 423:
406 PAPPUS OF ALEXANDRIA Props. 32,
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408 PAPPUS OF ALEXANDRIA VI. Props.
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410 PAPPUS OF ALEXANDRIA but also w
Page 428 and 429:
412 PAPPUS OF ALEXANDRIA Therefore
Page 430 and 431:
414 PAPPUS OF ALEXANDRIA I need onl
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416 PAPPUS OF ALEXANDRIA solution *
Page 434 and 435:
: 418 PAPPUS OF ALEXANDRIA This is
Page 436 and 437:
' 420 PAPPUS OF ALEXANDRIA opposite
Page 438 and 439:
i 4?2 PAPPUS OF ALEXANDRIA Props. 1
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424 PAPPUS OF ALEXANDRIA And, since
Page 442 and 443:
; 426 PAPPUS OF ALEXANDRIA is fond
Page 444 and 445:
428 PAPPUS OF ALEXANDRIA Historical
Page 446 and 447:
' 430 PAPPUS OF ALEXANDRIA in bette
Page 448 and 449:
432 PAPPUS OF ALEXANDRIA We have ne
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! ; 434 PAPPUS OF ALEXANDRIA about
Page 452 and 453:
436 PAPPUS OF ALEXANDRIA to this di
Page 454 and 455:
438 PAPPUS OF ALEXANDRIA Take point
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XX ALGEBRA: DIOPHANTUS OF ALEXANDRI
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442 ALGEBRA: DIOPHANTUS OF ALEXANDR
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} 444 ALGEBRA: DIOPHANTUS OF ALEXAN
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446 ALGEBRA: DIOPHANTUS OF ALEXANDR
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;: 448 ALGEBRA: DIOPHANTUS OF ALEXA
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450 DIOPHANTUS OF ALEXANDRIA betwee
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* 452 DIOPHANTUS OF ALEXANDRIA catt
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454 DIOPHANTUS OF ALEXANDRIA Greek.
Page 472 and 473:
456 DIOPHANTUS OF ALEXANDRIA litera
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: x a 458 DIOPHANTUS OF ALEXANDRIA
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460 DIOPHANTUS OF ALEXANDRIA Attach
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;' ' 462 DIOPHANTUS OF ALEXANDRIA T
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1 464 DIOPHANTUS OF ALEXANDRIA equa
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: :; 466 DIOPHANTUS OF ALEXANDRIA (
Page 484 and 485:
# 468 DIOPHANTUS OF ALEXANDRIA subs
Page 486 and 487:
i £> ', n 470 DIOPHANTUS OF ALEXAN
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: 472 DIOPHANTUS OF ALEXANDRIA or,
Page 490 and 491:
. 474 DIOPHANTUS OF ALEXANDRIA The
Page 492 and 493:
: 476 DIOPHANTUS OF ALEXANDRIA More
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; — 478 DIOPHANTUS OF ALEXANDRIA
Page 496 and 497:
480 DIOPHANTUS OF ALEXANDRIA 2. If
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1 482 DIOPHANTUS OF ALEXANDRIA hypo
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484 DIOPHANTUS OF ALEXANDRIA or fra
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486 DIOPHANTUS OF ALEXANDRIA I. 12.
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' . 488 DIOPHANTUS OF ALEXANDRIA IV
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i 490 DIOPHANTUS OF ALEXANDRIA (v)
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492 DIOPHANTUS OF ALEXANDRIA II. 1
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I III. 494 DIOPHANTUS OF ALEXANDRIA
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; 496 DIOPHANTUS OF ALEXANDRIA afte
Page 514 and 515:
498 DIOPHANTUS OF ALEXANDRIA Simila
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500 DIOPHANTUS OF ALEXANDRIA V. 6.
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502 DIOPHANTUS OF ALEXANDRIA (V. 24
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^ ~ } 504 DIOPHANTUS OF ALEXANDRIA
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506 DIOPHANTUS OF ALEXANDRIA 9 A; 2
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— - 508 DIOPHANTUS OF ALEXANDRIA
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) 510 DIOPHANTUS OF ALEXANDRIA Lemm
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- £ 512 DIOPHANTUS OF ALEXANDRIA s
Page 530 and 531:
514 DIOPHANTUS OF ALEXANDRIA term i
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2 516 DIOPHANTUS OF ALEXANDRIA 4. T
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XXI COMMENTATORS AND BYZANTINES We
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520 COMMENTATORS AND BYZANTINES pro
Page 538 and 539:
522 COMMENTATORS AND BYZANTINES (Pr
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. 524 COMMENTATORS AND BYZANTINES u
Page 542 and 543:
; 526 COMMENTATORS AND BYZANTINES A
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; 528 COMMENTATORS AND BYZANTINES a
Page 546 and 547:
530 COMMENTATORS AND BYZANTINES num
Page 548 and 549:
: 532 COMMENTATORS AND BYZANTINES U
Page 550 and 551:
534 COMMENTATORS AND BYZANTINES cas
Page 552 and 553:
536 COMMENTATORS AND BYZANTINES reg
Page 554 and 555:
538 COMMENTATORS AND BYZANTINES com
Page 556 and 557:
540 COMMENTATORS AND BYZANTINES mad
Page 558 and 559:
542 COMMENTATORS AND BYZANTINES By
Page 560 and 561:
• 544 COMMENTATORS AND BYZANTINES
Page 562 and 563:
; 546 COMMENTATORS AND BYZANTINES a
Page 564 and 565:
548 COMMENTATORS AND BYZANTINES The
Page 566 and 567:
550 COMMENTATORS AND BYZANTINES ins
Page 568 and 569:
; ; ; ; 552 COMMENTATORS AND BYZANT
Page 570 and 571:
— 554 COMMENTATORS AND BYZANTINES
Page 572 and 573:
' APPENDIX On Archimedes's 'proof o
Page 574 and 575:
558 APPENDIX the small increases of
Page 576 and 577:
560 APPENDIX Next, for the point Q'
Page 579 and 580:
, vddcov INDEX OF GREEK WORDS [The
Page 581 and 582:
: INDEX OF GREEK WORDS 565 of unkno
Page 583 and 584:
1 : : INDEX OF GREEK WORDS 567 vuaa
Page 585 and 586:
INDEX OF GREEK WORDS 569 jutp?;?, v
Page 587 and 588:
: ENGLISH INDEX 571 Anthemius of Tr
Page 589 and 590:
ENGLISH INDEX 573 Babylonians : civ
Page 591 and 592:
: ENGLISH INDEX 575 448 : works and
Page 593 and 594:
:: Ghetaldi, Marino, ii. 190. Girar
Page 595 and 596:
: Jan, C. 444. Joachim Catnerarius
Page 597 and 598:
: ENGLISH INDEX 581 of Euclid and A
Page 599 and 600:
: ENGLISH INDEX 583 Gizeh, and Medu
Page 601 and 602:
: ENGLISH INDEX 58; numbers) 77 : a
Page 604:
PRINTED IN ENGLAND AT THE OXFORD UN
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