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. ; 278 TRIGONOMETRY The construction in fact easily leads to the results EO = \a( a/5-1), EC = %a
; PTOLEMY'S SYNTAXIS 279 The proposition giving the required formula depends upon a lemma, which is the famous Ptolemy's theorem ' '. Given a quadrilateral ABCD inscribed in a circle, the diagonals being AG, BD, to prove that AC.BD = AB.DC+AD.BC. The proof is well known. Draw BE so that the angle ABE is equal to the angle DBG, and let BE meet AG in E. Then the triangles A BE, DBG are equiangular, and therefore AB:AE=BD:DC, or AB.DC= AE.BD. (1) Again, to each of the equal angles ABE, DBG add the angle EBD ; then the angle ABD is equal to the angle EBC, and the triangles ABD, EBC are equiangular therefore BC : or CE = BD: DA, AD.BC = CE.BD. (2) By adding (1) and (2), we obtain AB.DC+AD.BC=AC. BD. Now let AB, AC be two arcs terminating at A, the extremity of the diameter. AD of a circle, and let AG (= oc) be greater than AB (=/?;. Suppose that (crd. AC) and (crd. AB) are given : it is required to find (crd. BC). Join BD y CD. Then, by the above theorem, AC .BD = BC . AD + AB .CD. Now AB, AC are given"; therefore BD = crd. (180° -AB) and CD = crd. (180° — A C) are known. And AD is known. Hence the remaining chord BC (crd. BC) is known.
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A HISTORY OF GREEK MATHEMATICS VOLU
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A HISTORY OF GREEK MATHEMATICS BY S
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CONTENTS OF VOL II XII. ARISTARCHUS
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CONTENTS vii Geminus pages 222-234
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CONTENTS ix The Synagoge or Collect
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CONTENTS XI XXI. COMMENTATORS AND B
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' 2 ARISTARCHUS OF SAMOS stamping t
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: 4 ARISTARCHUS OF SAMOS Archimedes
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6 ARISTARCHUS OF SAMOS (3) that the
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8 ARISTARCHUS OF SAMOS The first pa
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; 10 ARISTARCHUS OF SAMOS Therefore
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12 ARISTARCHUS OF SAMOS (2) ON: (di
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. 14 ARISTARCHUS OF SAMOS Prop. 14
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; XIII ARCHIMEDES The siege and cap
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18 ARCHIMEDES the making of spheres
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20 ARCHIMEDES birth to the calculus
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: 22 ARCHIMEDES on the Quadrature o
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24 • ARCHIMEDES " works on the le
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26 ARCHIMEDES private hands. In 149
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28 ARCHIMEDES Mechanical Theorems,
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30 ARCHIMEDES the whole segment of
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V 32 ARCHIMEDES circles are section
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34 ARCHIMEDES Therefore AX: AG = (A
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3 36 ARCHIMEDES that, if we continu
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,' 38 ARCHIMEDES Therefore B is not
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40 ARCHIMEDES less than a hemispher
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42 ARCHIMEDES ttt 2 .2 sin -^- ~2n
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, 44 ARCHIMEDES values for each of
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46 ARCHIMEDES the curves touch at t
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48 ARCHIMEDES Draw QMN through M pa
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50 ARCHIMEDES To return to Archimed
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52 ARCHIMEDES and Zeuthen) is that
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, 54 ARCHIMEDES Now the triangles A
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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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' 224 SUCCESSORS OF THE GREAT GEOME
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Page 244 and 245: 228 SUCCESSORS OF THE GREAT GEOMETE
Page 252 and 253: ; 236 SOME HANDBOOKS the first cent
Page 254 and 255: 238 SOME HANDBOOKS larger than the
Page 256 and 257: ; 240 SOME HANDBOOKS numbers, plane
Page 258 and 259: 242 SOME HANDBOOKS the earth, repre
Page 260 and 261: : 244 SOME HANDBOOKS We next have (
Page 262 and 263: 246 TRIGONOMETRY dosius was of Bith
Page 264 and 265: ; 248 TRIGONOMETRY circle from its
Page 266 and 267: 250 TRIGONOMETRY the required numer
Page 268 and 269: 252 TRIGONOMETRY We may contrast wi
Page 270 and 271: 254 TRIGONOMETRY The work of Hippar
Page 272 and 273: \ 256 TRIGONOMETRY Improved Instrum
Page 274 and 275: 258 TRIGONOMETRY in relation to one
Page 276 and 277: ' 260 TRIGONOMETRY by ¥Jo from §
Page 278 and 279: 262 TRIGONOMETRY translation by Mau
Page 280 and 281: 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
Page 292 and 293: : : 276 TRIGONOMETRY bo given here.
Page 296 and 297: 280 TRIGONOMETRY The equation in fa
Page 298 and 299: ; 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
Page 306 and 307: , ; 290 TRIGONOMETRY both in the pl
Page 308 and 309: 292 TRIGONOMETRY seeing that Diodor
Page 310 and 311: ; 294 TRIGONOMETRY on the mirror wh
Page 312 and 313: 296 TRIGONOMETRY I. To prove I. 28,
Page 314 and 315: XVIII MENSURATION: HERON OF ALEXAND
Page 316 and 317: 300 HERON OF ALEXANDRIA Pappus goes
Page 318 and 319: ' 302 HERON OF ALEXANDRIA with Hero
Page 320 and 321: 304 HERON OF ALEXANDRIA And first w
Page 322 and 323: 306 HERON OF ALEXANDRIA the surface
Page 324 and 325: : 308 HERON OF ALEXANDRIA education
Page 326 and 327: 310 HERON OF ALEXANDRIA Besthorn an
Page 328 and 329: 312 HERON OF ALEXANDRIA triangle wh
Page 330 and 331: : ; 314 HERON OF ALEXANDRIA Now the
Page 332 and 333: 316 HERON OF ALEXANDRIA or any angl
Page 334 and 335: 318 HERON OF ALEXANDRIA formed by t
Page 336 and 337: 320 HERON OF ALEXANDRIA in an Archi
Page 338 and 339: 322 HERON OF ALEXANDRIA chap. 30 of
Page 340 and 341: . , ^ the square) is . ' 26J-| inst
Page 342 and 343: 326 HERON OF ALEXANDRIA is given as
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; 328 HERON OF ALEXANDRIA Heron. As
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, 330 HERON OF ALEXANDRIA (£) Segm
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332 HERON OF ALEXANDRIA view of the
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334 HERON OF ALEXANDRIA The method
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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.
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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
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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
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412 PAPPUS OF ALEXANDRIA Therefore
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414 PAPPUS OF ALEXANDRIA I need onl
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416 PAPPUS OF ALEXANDRIA solution *
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: 418 PAPPUS OF ALEXANDRIA This is
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' 420 PAPPUS OF ALEXANDRIA opposite
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i 4?2 PAPPUS OF ALEXANDRIA Props. 1
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424 PAPPUS OF ALEXANDRIA And, since
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; 426 PAPPUS OF ALEXANDRIA is fond
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428 PAPPUS OF ALEXANDRIA Historical
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' 430 PAPPUS OF ALEXANDRIA in bette
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432 PAPPUS OF ALEXANDRIA We have ne
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! ; 434 PAPPUS OF ALEXANDRIA about
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436 PAPPUS OF ALEXANDRIA to this di
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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.
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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 (
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# 468 DIOPHANTUS OF ALEXANDRIA subs
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i £> ', n 470 DIOPHANTUS OF ALEXAN
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: 472 DIOPHANTUS OF ALEXANDRIA or,
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. 474 DIOPHANTUS OF ALEXANDRIA The
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: 476 DIOPHANTUS OF ALEXANDRIA More
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; — 478 DIOPHANTUS OF ALEXANDRIA
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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
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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
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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
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522 COMMENTATORS AND BYZANTINES (Pr
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. 524 COMMENTATORS AND BYZANTINES u
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; 526 COMMENTATORS AND BYZANTINES A
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; 528 COMMENTATORS AND BYZANTINES a
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530 COMMENTATORS AND BYZANTINES num
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: 532 COMMENTATORS AND BYZANTINES U
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534 COMMENTATORS AND BYZANTINES cas
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536 COMMENTATORS AND BYZANTINES reg
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538 COMMENTATORS AND BYZANTINES com
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540 COMMENTATORS AND BYZANTINES mad
Page 558 and 559:
542 COMMENTATORS AND BYZANTINES By
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• 544 COMMENTATORS AND BYZANTINES
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; 546 COMMENTATORS AND BYZANTINES a
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548 COMMENTATORS AND BYZANTINES The
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550 COMMENTATORS AND BYZANTINES ins
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; ; ; ; 552 COMMENTATORS AND BYZANT
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— 554 COMMENTATORS AND BYZANTINES
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' APPENDIX On Archimedes's 'proof o
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558 APPENDIX the small increases of
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560 APPENDIX Next, for the point Q'
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, vddcov INDEX OF GREEK WORDS [The
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: INDEX OF GREEK WORDS 565 of unkno
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1 : : INDEX OF GREEK WORDS 567 vuaa
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INDEX OF GREEK WORDS 569 jutp?;?, v
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: ENGLISH INDEX 571 Anthemius of Tr
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ENGLISH INDEX 573 Babylonians : civ
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: ENGLISH INDEX 575 448 : works and
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:: Ghetaldi, Marino, ii. 190. Girar
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: Jan, C. 444. Joachim Catnerarius
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: ENGLISH INDEX 581 of Euclid and A
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: ENGLISH INDEX 583 Gizeh, and Medu
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: ENGLISH INDEX 58; numbers) 77 : a
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PRINTED IN ENGLAND AT THE OXFORD UN
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