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: 472 DIOPHANTUS OF ALEXANDRIA or, since pq = (xd 2 — @c 2 , p 2 x 2 -2x((xd 2 + l3c 2 ) + q 2 -4e 2 d 2 = 0. In order that this may reduce to a simple equation, as Diophantus requires, the absolute term must vanish, so that The method therefore only gives one solution, since q = 2cd. q is restricted to the value 2cd. Ex. from Diophantus Sx + 4 =u 2 ) 6^ + 4 = w 2 ) /TTr (IV. 39) Difference .2 x; q necessarily taken to be 2 a/4 or 4; factors therefore \x, 4. Therefore Sx + 4 = \ {\x + 4) 2 , and a; = 112. (/?) Second method of solution of a double equation of the first degree. There is only one case of this in Diophantus, the equations being of the form hx + n 2 = u 2 ) (h+f)x + n 2 =w 2 ) Suppose hx + n 2 = (y + n) 2 ; therefore hx = y 2 + 2 ny, •f and (h +f ) x + n 2 — (y + n) 2 + r (y 2 + 2 ny). It only remains to make the latter expression a square, which is done by equating it to {jiy — nf. The case in Diophantus is the same as that last mentioned (IV. 39). Where I have used y, Diophantus as usual contrives to use his one unknown a second time. 2. Double equations of the second degree. The general form is Ax 2 +Bx +C = u 2 \ A'x 2 + B'x + C'=iv 2 \ ' but only three types appear in Diophantus, namely . . p 2 x 2 + (xx + a — u 2 (1) J- p 2 x 2 + px + b — w l ) ) . , where, except in one case, a = 6. •
INDETERMINATE EQUATIONS 473 x 2 +
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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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; 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
Page 438 and 439: i 4?2 PAPPUS OF ALEXANDRIA Props. 1
Page 440 and 441: 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
Page 450 and 451: ! ; 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
Page 456 and 457: XX ALGEBRA: DIOPHANTUS OF ALEXANDRI
Page 458 and 459: 442 ALGEBRA: DIOPHANTUS OF ALEXANDR
Page 460 and 461: } 444 ALGEBRA: DIOPHANTUS OF ALEXAN
Page 462 and 463: 446 ALGEBRA: DIOPHANTUS OF ALEXANDR
Page 464 and 465: ;: 448 ALGEBRA: DIOPHANTUS OF ALEXA
Page 466 and 467: 450 DIOPHANTUS OF ALEXANDRIA betwee
Page 468 and 469: * 452 DIOPHANTUS OF ALEXANDRIA catt
Page 470 and 471: 454 DIOPHANTUS OF ALEXANDRIA Greek.
Page 472 and 473: 456 DIOPHANTUS OF ALEXANDRIA litera
Page 474 and 475: : x a 458 DIOPHANTUS OF ALEXANDRIA
Page 476 and 477: 460 DIOPHANTUS OF ALEXANDRIA Attach
Page 478 and 479: ;' ' 462 DIOPHANTUS OF ALEXANDRIA T
Page 480 and 481: 1 464 DIOPHANTUS OF ALEXANDRIA equa
Page 482 and 483: : :; 466 DIOPHANTUS OF ALEXANDRIA (
Page 484 and 485: # 468 DIOPHANTUS OF ALEXANDRIA subs
Page 486 and 487: i £> ', n 470 DIOPHANTUS OF ALEXAN
Page 490 and 491: . 474 DIOPHANTUS OF ALEXANDRIA The
Page 492 and 493: : 476 DIOPHANTUS OF ALEXANDRIA More
Page 494 and 495: ; — 478 DIOPHANTUS OF ALEXANDRIA
Page 496 and 497: 480 DIOPHANTUS OF ALEXANDRIA 2. If
Page 498 and 499: 1 482 DIOPHANTUS OF ALEXANDRIA hypo
Page 500 and 501: 484 DIOPHANTUS OF ALEXANDRIA or fra
Page 502 and 503: 486 DIOPHANTUS OF ALEXANDRIA I. 12.
Page 504 and 505: ' . 488 DIOPHANTUS OF ALEXANDRIA IV
Page 506 and 507: i 490 DIOPHANTUS OF ALEXANDRIA (v)
Page 508 and 509: 492 DIOPHANTUS OF ALEXANDRIA II. 1
Page 510 and 511: I III. 494 DIOPHANTUS OF ALEXANDRIA
Page 512 and 513: ; 496 DIOPHANTUS OF ALEXANDRIA afte
Page 514 and 515: 498 DIOPHANTUS OF ALEXANDRIA Simila
Page 516 and 517: 500 DIOPHANTUS OF ALEXANDRIA V. 6.
Page 518 and 519: 502 DIOPHANTUS OF ALEXANDRIA (V. 24
Page 520 and 521: ^ ~ } 504 DIOPHANTUS OF ALEXANDRIA
Page 522 and 523: 506 DIOPHANTUS OF ALEXANDRIA 9 A; 2
Page 524 and 525: — - 508 DIOPHANTUS OF ALEXANDRIA
Page 526 and 527: ) 510 DIOPHANTUS OF ALEXANDRIA Lemm
Page 528 and 529: - £ 512 DIOPHANTUS OF ALEXANDRIA s
Page 530 and 531: 514 DIOPHANTUS OF ALEXANDRIA term i
Page 532 and 533: 2 516 DIOPHANTUS OF ALEXANDRIA 4. T
Page 534 and 535: XXI COMMENTATORS AND BYZANTINES We
Page 536 and 537: 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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PRINTED IN ENGLAND AT THE OXFORD UN
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