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Vlll CONTENTS XVIII. MENSURATION: HERON OF ALEXANDRIA, pages 298-354 Controversies as to Heron's date . Character of works .... List of treatises Geometry (a) Commentary on Euclid's Elements (/3) The Definitions .... Mensuration The' Metrica, Geometrica, Stereometrica, Geodaesia, Mensurae ..... Contents of the Metrica Book I. Measurement of areas (a) Area of scalene triangle Proof of formula A = \/{s(s — a) (s — b (s —•')) < (3) Method of approximating to the square root of a non-square number .... Quadrilaterals (y) (8) Regular polygons with 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12 sides (?) The circle (£) Segment o'f a circle (rj) Ellipse, parabolic segment, surface of cylinder, right cone, sphere and segment of sphere . Measurement of volumes (a) Cone, cylinder, parallelepiped(prism),pyramid and frustum ...... Book II. (3) Wedge-shaped solid (/3cojuio-ko? or (T
CONTENTS ix The Synagoge or Collection pages 357-439 (a) Character of the work ; wide range . . . 357-358 (j3) List of authors mentioned ..... 358-360 (y) Translations and editions 360-361 (8) Summary of contents . . . . . . 361-439 Book III. Section (1). On the problem of the two mean proportionals ...... 361-362 Section (2). The theory of means . . . 363-365 ' Section (3). The Paradoxes ' of Erycinus . .365-368 Section (4). The inscribing of the five regular solids in a sphere ...... 368-369 Book IV. Section (1). Extension of theorem of Pythagoras 369-371 Section (2). On circles inscribed in the apftrjXos ('shoemaker's knife') . . _ . . .371-377 Sections (3), (4). Methods of squaring the circle and trisecting any angle ..... 377-386 (a) The Archimedean spiral 377-379 (#) The conchoid of Nicomecles .... 379 ( y ) The Qnadratrix \ 379-382 (b) Digression: a spiral on a sphere . . . 382-385 Trisection (or division in any ratio) of any angle 385-386 Section (5). Solution of the vevais of Archimedes, On Spirals, Prop. 8, by means of conies . . 386-388 Book V. Preface on the sagacity of Bees . . 389-390 Section (1). Isoperimetry after Zenodorus . . 390-393 Section (2). Comparison of volumes of solids having their surfaces equal. Case of sphere . . . 393-394 Section (3). Digression on semi-regular solids of Archimedes........ 394 Section (4). Propositions on the lines of Archimedes, On the Sphere and Cylinder..... 394 395 Section (5). Of regular solids with surfaces equal, that is greater which has more faces . . . 395-396 Book VI .... 396-399 Problem arising out of Euclid's Optics . . . 397-399 Book VII. On the Treasury of Analysis ' ' . . 399-427 Definition of Analysis and Synthesis . . . 400-401 List of works in the Treasury of Analysis' ' . . 401 Description of the treatises ..... 401-404 Anticipation of Guldin's Theorem . . . 403 Lemmas to the different treatises .... 404-426 (a) Lemmas to the Sectio ration Is and Sectio spatii of Apollonius ....... 404-405 (13) Lemmas to the Determinate Section of Apollonius 405-412 (y) Lemmas on the Nt-vtreis of Apollonius . . 412 416 (h) Lemmas on the On Contacts of Apollonius . 416-417 (?) Lemmas to the Plane Loci of Apollonius . 417-419 (() Lemmas to the Porisms of Euclid . . . 419-424 (17) Lemmas to the Comes of Apollonius . .424-425 (6) Lemmas to the Surface Loci of Euclid . . 425-426 • (t) An unallocated lemma . . . . . 426-427 430-433 Book VIII. Historical preface 427-429 The object of the Book 429-430 On the centre of gravity .....
Page 5 and 6: A HISTORY OF GREEK MATHEMATICS VOLU
Page 7 and 8: A HISTORY OF GREEK MATHEMATICS BY S
Page 9 and 10: CONTENTS OF VOL II XII. ARISTARCHUS
Page 11: CONTENTS vii Geminus pages 222-234
Page 15: CONTENTS XI XXI. COMMENTATORS AND B
Page 18 and 19: ' 2 ARISTARCHUS OF SAMOS stamping t
Page 20 and 21: : 4 ARISTARCHUS OF SAMOS Archimedes
Page 22 and 23: 6 ARISTARCHUS OF SAMOS (3) that the
Page 24 and 25: 8 ARISTARCHUS OF SAMOS The first pa
Page 26 and 27: ; 10 ARISTARCHUS OF SAMOS Therefore
Page 28 and 29: 12 ARISTARCHUS OF SAMOS (2) ON: (di
Page 30 and 31: . 14 ARISTARCHUS OF SAMOS Prop. 14
Page 32 and 33: ; XIII ARCHIMEDES The siege and cap
Page 34 and 35: 18 ARCHIMEDES the making of spheres
Page 36 and 37: 20 ARCHIMEDES birth to the calculus
Page 38 and 39: : 22 ARCHIMEDES on the Quadrature o
Page 40 and 41: 24 • ARCHIMEDES " works on the le
Page 42 and 43: 26 ARCHIMEDES private hands. In 149
Page 44 and 45: 28 ARCHIMEDES Mechanical Theorems,
Page 46 and 47: 30 ARCHIMEDES the whole segment of
Page 48 and 49: V 32 ARCHIMEDES circles are section
Page 50 and 51: 34 ARCHIMEDES Therefore AX: AG = (A
Page 52 and 53: 3 36 ARCHIMEDES that, if we continu
Page 54 and 55: ,' 38 ARCHIMEDES Therefore B is not
Page 56 and 57: 40 ARCHIMEDES less than a hemispher
Page 58 and 59: 42 ARCHIMEDES ttt 2 .2 sin -^- ~2n
Page 60 and 61: , 44 ARCHIMEDES values for each of
Page 62 and 63:
46 ARCHIMEDES the curves touch at t
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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
Page 80 and 81:
} 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, . .
Page 92 and 93:
76 ARCHIMEDES proofs. We do not fin
Page 94 and 95:
. 78 ARCHIMEDES Hence the centre of
Page 96 and 97:
. 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
Page 104 and 105:
88 ARCHIMEDES that is, it takes the
Page 106 and 107:
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
Page 132 and 133:
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:
Page 192 and 193:
176 APOLLONIUS OF PERGA a given poi
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178 APOLLONIUS OF PERGA the value o
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180 APOLLONIUS OF PERGA treatment o
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182 APOLLONIUS OF PERGA a circle, (
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184 APOLLONIUS OF PERGA HG meeting
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186 AP0LL0N1US OF PERGA of a line.
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.; 188 APOLLONIUS OF PERGA cut off
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190 APOLLONIUS OF PERGA III. Given
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192 APOLLONIUS OF PERGA But, by sim
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194 APOLLONIUS OF PERGA (A) On the
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196 APOLLONIUS OF PERGA another hyp
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198 SUCCESSORS OF THE GREAT GEOMETE
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; 200 SUCCESSORS OF THE GREAT GEOME
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; 202 SUCCESSORS OF -THE GREAT GEOM
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Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
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' 224 SUCCESSORS OF THE GREAT GEOME
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Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
; 236 SOME HANDBOOKS the first cent
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238 SOME HANDBOOKS larger than the
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; 240 SOME HANDBOOKS numbers, plane
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242 SOME HANDBOOKS the earth, repre
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: 244 SOME HANDBOOKS We next have (
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246 TRIGONOMETRY dosius was of Bith
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; 248 TRIGONOMETRY circle from its
Page 266 and 267:
250 TRIGONOMETRY the required numer
Page 268 and 269:
252 TRIGONOMETRY We may contrast wi
Page 270 and 271:
254 TRIGONOMETRY The work of Hippar
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\ 256 TRIGONOMETRY Improved Instrum
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258 TRIGONOMETRY in relation to one
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' 260 TRIGONOMETRY by ¥Jo from §
Page 278 and 279:
262 TRIGONOMETRY translation by Mau
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
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: : 276 TRIGONOMETRY bo given here.
Page 294 and 295:
. ; 278 TRIGONOMETRY The constructi
Page 296 and 297:
280 TRIGONOMETRY The equation in fa
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
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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
Page 326 and 327:
310 HERON OF ALEXANDRIA Besthorn an
Page 328 and 329:
312 HERON OF ALEXANDRIA triangle wh
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: ; 314 HERON OF ALEXANDRIA Now the
Page 332 and 333:
316 HERON OF ALEXANDRIA or any angl
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318 HERON OF ALEXANDRIA formed by t
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320 HERON OF ALEXANDRIA in an Archi
Page 338 and 339:
322 HERON OF ALEXANDRIA chap. 30 of
Page 340 and 341:
. , ^ the square) is . ' 26J-| inst
Page 342 and 343:
326 HERON OF ALEXANDRIA is given as
Page 344 and 345:
; 328 HERON OF ALEXANDRIA Heron. As
Page 346 and 347:
, 330 HERON OF ALEXANDRIA (£) Segm
Page 348 and 349:
332 HERON OF ALEXANDRIA view of the
Page 350 and 351:
334 HERON OF ALEXANDRIA The method
Page 352 and 353:
336 HERON OF ALEXANDRIA Book III. D
Page 354 and 355:
338 HERON OF ALEXANDRIA area'. Thes
Page 356 and 357:
340 HERON OF ALEXANDRIA lines ', an
Page 358 and 359:
8 342 HERON OF ALEXANDRIA and, solv
Page 360 and 361:
344 HERON OF ALEXANDRIA Quadratic e
Page 362 and 363:
346 HERON OF ALEXANDRIA that to mee
Page 364 and 365:
348 HERON OF ALEXANDRIA case suppos
Page 366 and 367:
' 350 HERON OF ALEXANDRIA between t
Page 368 and 369:
1 ' 352 HERON OF ALEXANDRIA do grea
Page 370 and 371:
354 HERON OF ALEXANDRIA reversing t
Page 372 and 373:
356 PAPPUS OF ALEXANDRIA Date of Pa
Page 374 and 375:
358 PAPPUS OF ALEXANDRIA however, a
Page 376 and 377:
360 PAPPUS OF ALEXANDRIA who is men
Page 378 and 379:
362 PAPPUS OF ALEXANDRIA a problem
Page 380 and 381:
364 PAPPUS OF ALEXANDRIA the proble
Page 382 and 383:
366 PAPPUS OF ALEXANDRIA triangle w
Page 384 and 385:
368 PAPPUS OF ALEXANDRIA the other,
Page 386 and 387:
370 PAPPUS OF ALEXANDRIA contained
Page 388 and 389:
372 PAPPUS OF ALEXANDRIA There is,
Page 390 and 391:
374 PAPPUS OF ALEXANDRIA But theref
Page 392 and 393:
376 PAPPUS OF ALEXANDRIA IV. We now
Page 394 and 395:
378 PAPPUS OF ALEXANDRIA With as ce
Page 396 and 397:
380 PAPPUS OF ALEXANDRIA principii.
Page 398 and 399:
382 PAPPUS OF ALEXANDRIA curve, des
Page 400 and 401:
384 PAPPUS OF ALEXANDRIA » Now (se
Page 402 and 403:
386 PAPPUS OF ALEXANDRIA means of a
Page 404 and 405:
388 PAPPUS OF ALEXANDRIA Measure DA
Page 406 and 407:
390 PAPPUS OF ALEXANDRIA ambrosia i
Page 408 and 409:
39.2 PAPPUS OF ALEXANDRIA Draw AE a
Page 410 and 411:
394 PAPPUS OF ALEXANDRIA which have
Page 412 and 413:
396 PAPPUS OF ALEXANDRIA synthetica
Page 414 and 415:
398 PAPPUS OF ALEXANDRIA in length
Page 416 and 417:
400 PAPPUS OF ALEXANDRIA book, prac
Page 418 and 419:
402 PAPPUS OF ALEXANDRIA Eratosthen
Page 420 and 421:
404 PAPPUS OF ALEXANDRIA If the pas
Page 422 and 423:
406 PAPPUS OF ALEXANDRIA Props. 32,
Page 424 and 425:
408 PAPPUS OF ALEXANDRIA VI. Props.
Page 426 and 427:
410 PAPPUS OF ALEXANDRIA but also w
Page 428 and 429:
412 PAPPUS OF ALEXANDRIA Therefore
Page 430 and 431:
414 PAPPUS OF ALEXANDRIA I need onl
Page 432 and 433:
416 PAPPUS OF ALEXANDRIA solution *
Page 434 and 435:
: 418 PAPPUS OF ALEXANDRIA This is
Page 436 and 437:
' 420 PAPPUS OF ALEXANDRIA opposite
Page 438 and 439:
i 4?2 PAPPUS OF ALEXANDRIA Props. 1
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 488 and 489:
: 472 DIOPHANTUS OF ALEXANDRIA or,
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
Page 538 and 539:
522 COMMENTATORS AND BYZANTINES (Pr
Page 540 and 541:
. 524 COMMENTATORS AND BYZANTINES u
Page 542 and 543:
; 526 COMMENTATORS AND BYZANTINES A
Page 544 and 545:
; 528 COMMENTATORS AND BYZANTINES a
Page 546 and 547:
530 COMMENTATORS AND BYZANTINES num
Page 548 and 549:
: 532 COMMENTATORS AND BYZANTINES U
Page 550 and 551:
534 COMMENTATORS AND BYZANTINES cas
Page 552 and 553:
536 COMMENTATORS AND BYZANTINES reg
Page 554 and 555:
538 COMMENTATORS AND BYZANTINES com
Page 556 and 557:
540 COMMENTATORS AND BYZANTINES mad
Page 558 and 559:
542 COMMENTATORS AND BYZANTINES By
Page 560 and 561:
• 544 COMMENTATORS AND BYZANTINES
Page 562 and 563:
; 546 COMMENTATORS AND BYZANTINES a
Page 564 and 565:
548 COMMENTATORS AND BYZANTINES The
Page 566 and 567:
550 COMMENTATORS AND BYZANTINES ins
Page 568 and 569:
; ; ; ; 552 COMMENTATORS AND BYZANT
Page 570 and 571:
— 554 COMMENTATORS AND BYZANTINES
Page 572 and 573:
' APPENDIX On Archimedes's 'proof o
Page 574 and 575:
558 APPENDIX the small increases of
Page 576 and 577:
560 APPENDIX Next, for the point Q'
Page 579 and 580:
, vddcov INDEX OF GREEK WORDS [The
Page 581 and 582:
: INDEX OF GREEK WORDS 565 of unkno
Page 583 and 584:
1 : : INDEX OF GREEK WORDS 567 vuaa
Page 585 and 586:
INDEX OF GREEK WORDS 569 jutp?;?, v
Page 587 and 588:
: ENGLISH INDEX 571 Anthemius of Tr
Page 589 and 590:
ENGLISH INDEX 573 Babylonians : civ
Page 591 and 592:
: ENGLISH INDEX 575 448 : works and
Page 593 and 594:
:: Ghetaldi, Marino, ii. 190. Girar
Page 595 and 596:
: Jan, C. 444. Joachim Catnerarius
Page 597 and 598:
: ENGLISH INDEX 581 of Euclid and A
Page 599 and 600:
: ENGLISH INDEX 583 Gizeh, and Medu
Page 601 and 602:
: ENGLISH INDEX 58; numbers) 77 : a
Page 604:
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