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172 APOLLONIUS OF PERGA A number of other ratios are expressed in terms of the straight lines terminating at A, A', H, H', M, M' as follows (VII. 14-20). In the ellipse A A'* : PP' 2 * DD' = 2 A'H : 2 CM, and in the hyperbola or ellipse ord mates to PP') (if p be the parameter of the and AA' 2 : (PP' AA' 2 :p 2 = A'H.MH': ME 2 , A A' 2 : (PP' + pf = A'H . MH' : AA' 2 :PP'.p = A'H:MH, 2 ±p 2 ) = A'H . MH':(MH' (MH + MH') 2 , 2 + MH 2 ). Apollonius is now in a position, by means of all these relations, resting on the use of the auxiliary points H, H', M, to compare different functions of any conjugate diameters with the same functions of the axes, and to show how the former vary (by way of increase or diminution) as P moves away from A. The following is a list of the functions compared, where for brevity I shall use a, b to represent A A'', BB'; a', V to represent PP', DD' ; and p, p' to represent the parameters of the ordinates to AA', PP' respectively. In a hyperbola, according as a > or < b, a' > or < b', and the ratio a':b' decreases or increases as P moves from A on either side; also, if a = b, a' — b' (VII. 21-3); in an ellipse a:b > a':b', and the latter ratio diminishes as P moves from A to B (VII. 24). In a hyperbola or ellipse a + b
. THE GONIG& BOOK VII 173 conjugate diameters in an ellipse or conjugate hyperbolas, and if the tangents at their extremities form the parallelogram LL'MM', then the parallelogram LL'MM' — rect. AA' . BB' The proof is interesting. Let the tangents at P, D respectively meet the major or transverse axis in T, T'. Now (by VII. 4) PT 2 : GD 2 = NT:GN\ therefore 2 A GPT : 2 A T'DC = NT : CN. But 2AGPT:(GL) = PT:CD, = OP : DT' , by similar triangles, = (CL):2AT'DG. That is, (OX) is a mean proportional between 2AGPT and 2ArDC. Therefore, since >/(NT . OiV) is a mean proportional between iVT and CUV,
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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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Page 140 and 141: 124 CONIC SECTIONS only : p is simp
Page 142 and 143: 126 APOLLONIUS OF PERGA for the oth
Page 144 and 145: 128 APOLLONIUS OF PERGA Gregory, ho
Page 146 and 147: 130 APOLLONIUS OF PERGA minima and
Page 148 and 149: 132 APOLLONIUS OF PERGA Preface to
Page 150 and 151: 134 APOLLONIUS OF PERGA straight li
Page 152 and 153: 136 APOLLONIUS OF PERGA as particul
Page 154 and 155: 138 APOLLONIUS OF PERGA Therefore Q
Page 156 and 157: ; 140 APOLLONIUS OF PERGA ' If in a
Page 158 and 159: 142 APOLLONIUS OF PERGA /
Page 160 and 161: ; 144 APOLLONIUS OF PERGA Therefore
Page 162 and 163: 146 APOLLONIUS OF PERGA (2) In the
Page 164 and 165: 148 APOLLONIUS OF PERGA It has been
Page 166 and 167: 150 APOLLONIUS OF PERGA the last pr
Page 168 and 169: 152 APOLLONIUS OF PERGA Adding the
Page 170 and 171: 154 APOLLONIUS OF PERGA The general
Page 172 and 173: 156 APOLLONIUS Ot PERGA L, 1/ and M
Page 174 and 175: ; 158 APOLLONIUS OF PERGA tively. T
Page 176 and 177: ) 160 APOLLONIUS OF PERGA Next Apol
Page 178 and 179: 162 APOLLONIUS OF PERGA Similarly,
Page 180 and 181: : 164 APOLLONIUS OF PERGA (2) if A
Page 182 and 183: 166 APOLLONIUS OF PERGA The proposi
Page 184 and 185: 168 APOLLONIUS OF PERGA each make e
Page 186 and 187: : 170 APOLLONIUS OF PERGA Secondly,
Page 190 and 191: : A 174 APOLLONIUS OF PERGA 2AGPT:
Page 192 and 193: 176 APOLLONIUS OF PERGA a given poi
Page 194 and 195: 178 APOLLONIUS OF PERGA the value o
Page 196 and 197: 180 APOLLONIUS OF PERGA treatment o
Page 198 and 199: 182 APOLLONIUS OF PERGA a circle, (
Page 200 and 201: 184 APOLLONIUS OF PERGA HG meeting
Page 202 and 203: 186 AP0LL0N1US OF PERGA of a line.
Page 204 and 205: .; 188 APOLLONIUS OF PERGA cut off
Page 206 and 207: 190 APOLLONIUS OF PERGA III. Given
Page 208 and 209: 192 APOLLONIUS OF PERGA But, by sim
Page 210 and 211: 194 APOLLONIUS OF PERGA (A) On the
Page 212 and 213: 196 APOLLONIUS OF PERGA another hyp
Page 214 and 215: 198 SUCCESSORS OF THE GREAT GEOMETE
Page 216 and 217: ; 200 SUCCESSORS OF THE GREAT GEOME
Page 218 and 219: ; 202 SUCCESSORS OF -THE GREAT GEOM
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222 SUCCESSORS OF THE GREAT GEOMETE
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' 224 SUCCESSORS OF THE GREAT GEOME
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; 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 (
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246 TRIGONOMETRY dosius was of Bith
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; 248 TRIGONOMETRY circle from its
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250 TRIGONOMETRY the required numer
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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 §
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262 TRIGONOMETRY translation by Mau
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264 TRIGONOMETRY Eucl. I. 16, 32 ar
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266 TRIGONOMETRY (a) ' Menelaus s t
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268 TRIGONOMETRY But, by the propos
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270 TRIGONOMETRY It follows that th
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272 TRIGONOMETRY (1) If * 1 >P 1 >2
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274 TRIGONOMETRY the superlative /x
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: : 276 TRIGONOMETRY bo given here.
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. ; 278 TRIGONOMETRY The constructi
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280 TRIGONOMETRY The equation in fa
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; 282 TRIGONOMETRY greater, then sh
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284 TRIGONOMETRY the sines obtained
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286 TRIGONOMETRY , Thus AC is found
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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
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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
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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
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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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