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: 170 APOLLONIUS OF PERGA Secondly, Apollonius proves that, if PN be a principal ordinate in a parabola, p the principal parameter, p' the parameter of the ordinates to the diameter through P, then p' = p + 4Al¥ (VII. 5); this is proved by means of the same property as VII. 4, namely \p' - :PT' OP : PE. Much use is made in the remainder of the Book of two points Q and where AQ is drawn parallel to the conjugate My diameter CD to meet the curve in Q, and M is the foot of the principal ordinate at Q ; since the diameter GP bisects both A A! and QA, it follows that A'Q is parallel to GP. Many ratios between functions of PP', DD' are expressed in terms of AM, A'M, MH, MH', AH, A'H,&c. The first propositions of the Book proper (VII. 6, 7) prove, for instance, that PP' 2 : DD' = 2 ME': MH. For PT 2 : CD 2 = NT:GN = AM: A'M, by similar triangles. Also GP 2 :PT 2 = A'Q 2 :AQ 2 . Therefore, ex aequali, 2 ) GP 2 : GD 2 = (AM : A'M) x (A'Q 2 : AQ = (AM: A'M) x (A'Q 2 : A'M. MH') x (A'M.MH': AM. MH) x (AM.MH : AQ 2 ) = (AM: A'M) x (AA': AH') x (A'M: AM) x (MH':MH) x (A'H:AA'), by aid of VII. 2, 3. Therefore PP' 2 : DD' 2 = MH' : MH. Next (VII. 8, 9, 10, 11) the following relations are proved, namely (\)AA' 2 :(PP' + DD'f = A'H.MH':{MH'+V(MH.MH')} 2 , (2) AA' 2 : PP' .DD' = A'H : V(MH.MH'), (3) A A' 2 : (PP' 2 + DD' 2 ) = A'H MH± : MH'. The steps by which these results are obtained are as follows. First, A A' 2 : PP' = A'H 2 : MH' (oc) (This is proved thus = A'H.MH':MH' 2 . AA' 2 :PP' =GA 2 2 :GP 2 = GJST.CT:CP 2 = A'M. A'A : A'Q 2 .
# ' ; THE CONICS, BOOK VII 171 But A'Q*:A'M.MH'= AA'.AH' (VII. 2, 3) = AA':A'H = A'M. AA': A'M. A'H, so that, alternately, A'M. AA': A'Q = 2 A'M. A'H : A'M = A'H:MH'.) Next, PP' 2 : DD' . ME 2 = MH': MH, as above, (0) = MH' 2 :MH.MH', whence PP': DD' = ikf^T r :
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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
Page 136 and 137: 120 CONIC SECTIONS Proof from Pappu
Page 138 and 139: ; ; 122 CONIC SECTIONS chords drawn
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 188 and 189: 172 APOLLONIUS OF PERGA A number of
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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220 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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