A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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X VA> a^K Now suppose that, in Heron's formulae, we put a = X , 0*1 = .A^, AjOL x = x\, and so on. We then have A = i ' a + - ) = -J (A + x ), #, = — = , „ - or A/a — x Q , that is, Xj, ^! are, respectively, the arithmetic and harmonic means between A , x ; 2 , ic 2 are the arithmetic and harmonic means between x Xy , , and so on, exactly as in AlexeiefFs formulae. x Let us now try to apply the method to Archimedes's case, «/3, and we shall see to what extent it serves to give what we want. Suppose we begin with 3 >y / 3 > 1. We then have J(3 + l)>v/ 3>3/^(3 + l), or 2> v / 3>:j, and from this we derive successively i ^ V ° ^ 7> 56^V d/ 9 7 > 10864 -^ V ° ^ 188IT* But, if we start from f, obtained by the formula «+ ; za + 1 we obtain the following approximations by excess, 1 (3. _Lii\ — 20 I (-Ail 4. 4_5\ _ 13Jl1 " (a 2 + b), The second process then gives one of Archimedes's results, VWn bat neither of the two processes gives the other, fff, directly. The latter can, however, be obtained by using the formula that, if -
THE REGULAR POLYGONS 327 Geom. 102 (21, 14, Heib.). 2 lb. 102 (21, 16, 17, Heib.). In these cases (chaps. 22, 24; the Table of Chords (i e. 328 HERON OF ALEXANDRIA now 3 + xjt = || = i (f£)> so that the approximation used by Heron for \/3 is here Heron. As a matter of fact, however, 6 . ff For the side 10, the method gives (§ + jq) = -^3- exactly, and only the Metrica gives the more accurate calculation. the same result as above, for §§ . 100 = 43J. The regular pentagon is next taken (chap. 18). Heron The regular heptagon. premises the following lemma. Heron assumes (chap. 20) that, if a be the side and r the Let ABC be a right-angled triangle, with the angle A equal radius of the circumscribing circle, a — |r, being approximately equal to the perpendicular from the centre of the to §i*. Produce AC to so that CO = AC If now AO is divided in extreme and circle to the side of the regular hexagon inscribed in it mean ratio, AB | is equal to the greater is the approximate value of \ \^3). This theorem is quoted by segment. (For produce AB to D so that Jordanus Nemorarius (d. 1237) as an 'Indian rule'; he probably obtained it from Abu'l Wafa (940-98). The Metrica, ,4D = AO, and join 50, DO. Then, since J. 2)0 is isosceles and the angle at A—^R, shows that it is of Greek origin, and, if Archimedes really I ADO = AA0D = ±R, and, from the wrote a book on the heptagon in a circle, it may be clue to equality of the triangles ABC, 0BC, Z.A0B = LBA0 = fiS. It follows that him. If then p is the perpendicular from the centre of the circle on the side (a) of the inscribed heptagon, r/(-|a) = 8/3-§ the triangle J.D0 is the isosceles triangle of Eucl. IV. 10, and or 16/7, AD p 2 /(^a) 2 = -iw'i and 'p/\ a — (approximately) 14|/7 or 43/21. Consequently the area of the is divided in extreme and mean ratio in B.) Therefore, says Heron, (BA+ACf = 5 AC 2 . [This is Eucl. XIII. 1.] heptagon = 7 . \pa — 7 . fftr — fi^2 - Now, since LB0C — %R, if BC be produced to E so that CE = BC, BE subtends at an angle equal to ~R, and The regular octagon, decagon and dodecagon. therefore BE is the side of a regular pentagon inscribed in the In these cases (chaps. 21, 23, 25) Heron finds p by drawing circle with as centre and 0B as radius. (This circle also the perpendicular 00 from 0, the centre of the passes through D, and BD is the side of a regular decagon in circumscribed circle, on a side AB, and then making the same circle.) If now B0 — AB = r, 0C = p, BE = a, the angle OAD equal to the angle AOD. we have from above, (r + p) — 2 5p 2 , whence, since V5 is For the octagon, approximately J, we obtain approximately r = %p, and I la = %p, so that p — ADO = ±R, and \a. Hence \pa — p ^ Ja(l + V2) = Ja(l + \o?, and f|) the area of the pentagon = fa 2 . Heron adds that, if we take a closer or \a . f§ approximately. approximation to a/5 than |, we shall obtain the area still For the decagon, more exactly. In the Geometry 1 the formula is given as \^-a 2 . ZADC = f R, and AD : DC =5:4 nearly (see preceding page) The regular hexagon (chap. 19) is simply 6 times the hence AD : AC =5:3, and equilateral triangle with the same side. If A p = \a (§ + be the area § ) — fa. of the equilateral triangle with side a, Heron has proved For the dodecagon, that A = 2 T3 ga 4 (Metrica I. 17), hence (hexagon) 2 = $?-a\ If, Z ADC = | £, and p = |a (2 + V3) = \a (2 + J) = ^-a e.g. a = 10, (hexagon) = 2 67500, and (hexagon) = 259 nearly. approximately. In the Geometry 2 the formula is given as ^-a', while another ' book' is quoted as giving 6(J + T^)a 2 ; it is added that the Accordingly A 8 = ^g-a 2 , A 10 = -^tt 2 , A2 = -4 4 5 ' - a 2 where a is > latter formula, obtained from the area of the triangle, (J + ^) a 2 , represents the more accurate procedure, and is fully set out by the side in each case. The regular enneagon and hendecagon. 1
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A HISTORY OF GREEK MATHEMATICS BY S
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CONTENTS vii Vlll CONTENTS Geminus
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CONTENTS XI XXI. COMMENTATORS AND B
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ARISTARCHUS OF SAMOS 3 contemporary
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Now whence ARISTARCHUS OF SAMOS BH:
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ARISTARCHUS OF SAMOS 11 while Y, Z
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ARISTARCHUS OF SAMOS 15 But AB.BG <
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MECHANICS 19 likely that he discove
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• LIST OF EXTANT WORKS 23 24 ARCH
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THE TEXT OF ARCHIMEDES 27 It was ma
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Now THE METHOD HA:AN=A'A:AN = KA:AQ
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ON THE SPHERE AND CYLINDER, I 35 ti
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ON THE SPHERE AND CYLINDER, I 39 wh
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ON THE SPHERE AND CYLINDER, II 43 T
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ON THE SPHERE AND CYLINDER, II 47 (
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MEASUREMENT OF A CIRCLE 51 sides co
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a MEASUREMENT OF A CIRCLE 55 be The
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ON CONOIDS AND SPHEROIDS 59 of the
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so that ON CONOIDS AND SPHEROIDS 63
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Then D : E > BM : MO, > OB : BT, ON
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ON SPIRALS 71 Let OF meet the spira
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ON SPIRALS 75 Lastly, it' E be the
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ON PLANE EQUILIBRIUMS, I, II 79 Thi
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THE SAND-RECKONER 83 Archimedes has
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curve in E 1 , R THE QUADRATURE OF
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THE QUADRATURE OF THE PARABOLA 91 t
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ON FLOATING BODIES, I, II 95 96 ARC
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ON SEMI-REGULAR POLYHEDRA 99 angula
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THE LIBER ASSUMPTORUM 103 Lastly, w
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MEASUREMENT OF THE EARTH 107 a, or
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DISCOVERY OF THE CONIC SECTIONS 111
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MENAECHMUS'S PROCEDURE 115 For, let
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ARISTAEUS'S SOLID LOCI 119 the same
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CONIC SECTIONS IN ARCHIMEDES 123 In
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THE TEXT OF THE CONICS 127 The edit
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THE CONICS 131 diorismi. Nicoteles
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THE C0NIC8, BOOK I 135 the centre o
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PL is THE CONICS, BOOK I 139 called
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THE CONIGS, BOOK I 143 (2) in the h
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It follows that THE CONICS, BOOK I
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THE CONICS, BOOK III 151 152 APOLLO
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To prove (1) R'l 2 : IW-H'Q 2 : QH
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THE CONICS, BOOKS IV-V 159 and, if
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THE CONICS, BOOK V 163 if P' be any
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THE CONICS, BOOKS V, VI 167 tively
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THE CONICS, BOOK VII 171 But A'Q*:A
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THE GONIGS, BOOK VII 175 As we have
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ON THE CUTTING-OFF OF A RATIO 179 F
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ON CONTACTS OR TANGENCIES 183 solve
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PLANE LOCI 187 other extremity will
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NET2EI2 (VERGINGS OR INCLINATIONS)
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OTHER LOST WORKS 195 Astronomy. We
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NICOMEDES 199 tators, and especiall
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DIOCLES. PERSEUS 203 1 Proclus on E
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ISOPERIMETRIC FIGURES. ZENODORUS 20
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ZENODORUS 211 Now, by hypothesis, D
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HYPSICLES 215 natural to divide eac
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DIONYSODORUS 219 generates the tore
Page 117 and 118: GEMINUS 223 An upper limit for his
Page 119 and 120: GEMINUS 227 Attempt to prove the Pa
Page 121 and 122: GEMINUS 231 and make equal angles w
Page 123 and 124: 236 SOME HANDBOOKS XVI SOME HANDBOO
Page 125 and 126: THEON OF SMYRNA 239 edited by E. Hi
Page 127 and 128: THEON OF SMYRNA 243 to the seven he
Page 129 and 130: THEODOSIUS'S SPHAERIGA 247 (Books X
Page 131 and 132: THEODOSIUS'S SPHAERICA 251 Now, the
Page 133 and 134: HIPPARCHUS 255 that the lengths of
Page 135 and 136: HIPPARCHUS 259 in his Commentary, h
Page 137 and 138: MENELAUS'S SPHAERICA 263 already ap
Page 139 and 140: MENELAUS'S SPIIAERICA 267 For, if A
Page 141 and 142: ^ ' MENELAUS'S SPHAERICA 271 272 TR
Page 143 and 144: ' PTOLEMY'S SYNTAXIS 275 276 TRIGON
Page 145 and 146: PTOLEMY'S SYNTAXIS 279 The proposit
Page 147 and 148: PTOLEMY'S SYNTAXIti 283 284 TRIGONO
Page 149 and 150: THE ANALEMMA OF PTOLEMY 287 288 TRI
Page 151 and 152: THE ANALEMMA OF PTOLEMY 291 or tan
Page 153 and 154: THE OPTICS OF PTOLEMY 295 but the t
Page 155 and 156: CONTROVERSIES AS TO HERON'S DATE 29
Page 161 and 162: GEOMETRY 311 Of this class are the
Page 163 and 164: ' ' concave ' and THE DEFINITIONS 3
Page 165 and 166: MENSURATION 319 Heiberg puts side b
Page 167: PROOF OF THE FORMULA OF HERON' 323
Page 171 and 172: height SEGMENT OF A CIRCLE 331 The
Page 173 and 174: iJ MEASUREMENT OF SOLIDS 335 descri
Page 175 and 176: DIVISIONS OF FIGURES 339 two opposi
Page 177 and 178: : —— Since (DG) : (FB) = m : No
Page 179 and 180: THE MECHANICS 347 the first chapter
Page 181 and 182: Supports '. ON THE CENTRE OF GRAVIT
Page 183 and 184: 356 PAPPUS OF ALEXANDRIA Date of Pa
Page 185 and 186: THE COLLECTION 359 sizes and distan
Page 187 and 188: THE COLLECTION. BOOK III 363 Sectio
Page 189 and 190: Now THE COLLECTION. BOOK III 367 EA
Page 191 and 192: THE COLLECTION. BOOK IV 371 we may
Page 193 and 194: THE COLLECTION. BOOK IV 375 Therefo
Page 195 and 196: THE COLLECTION. BOOK IV 379 We have
Page 197 and 198: THE COLLECTION. BOOK IV 383 a certa
Page 199 and 200: THE COLLECTION. BOOK IV 387 length
Page 201 and 202: THE COLLECTION. BOOK V 391 the same
Page 203 and 204: of THE COLLECTION. BOOK V 395 the s
Page 205 and 206: THE COLLECTION. BOOKS VI, VII 399 T
Page 207 and 208: THE COLLECTION. BOOK VII 403 ' util
Page 209 and 210: THE COLLECTION. BOOK VII 407 Two pr
Page 211 and 212: PG THE COLLECTION. BOOK VII 411 Now
Page 213 and 214: The problem is THE COLLECTION. BOOK
Page 215 and 216: i.e. (if DA . THE COLLECTION. BOOK
Page 217 and 218: THE COLLECTION. BOOK VII 423 Since
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THE COLLECTION. BOOKS VII, VIII 427
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THE COLLECTION. BOOK VIII 431 432 P
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THE COLLECTION. BOOK VIII 435 is le
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THE COLLECTION. BOOK VIII 439 Then
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EPIGRAMS IN THE GREEK ANTHOLOGY 443
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HERONIAN INDETERMINATE EQUATIONS 44
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RELATION OF WORKS 451 Relation of t
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TRANSLATIONS AND EDITIONS 455 unfor
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NOTATION AND DEFINITIONS 459 When t
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DETERMINATE EQUATIONS 463 equation
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INDETERMINATE EQUATIONS 467 (ft) wh
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• INDETERMINATE EQUATIONS 471 Ex.
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INDETERMINATE EQUATIONS 475 a squar
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METHOD OF APPROXIMATION TO LIMITS 4
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NUMBERS AS THE SUMS OF SQUARES 483
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' (I. 35. x = my, y 2 = nx. 1 1. 36
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INDETERMINATE ANALYSIS 491 IV. 33.
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INDETERMINATE ANALYSIS 495 III. 14.
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INDETERMINATE ANALYSIS 499 IV. 29.
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INDETERMINATE ANALYSIS 503 Let the
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INDETERMINATE ANALYSIS 507 508 DIOP
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INDETERMINATE ANALYSIS 511 [The aux
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THE TREATISE ON POLYGONAL NUMBERS 5
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SERENUS 519 520 COMMENTATORS AND BY
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SERENUS 523 Observing that the sum
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THEON OF ALEXANDRIA 527 pp. 58-63).
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PROCLUS 531 viated form usual with
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PROCLUS 535 second Book \ But at th
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DOMNINUS. SIMPLICIUS 539 sophy at A
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, . a ANTHEMIUS 543 the focus to th
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PSELLUS. PACHYMERES. PLANUDES 547 R
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MOSCHOPOULOS. RHABDAS 551 of Smyrna
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PEDIASIMUS. BARLAAM. ARGYRUS 555 or
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APPENDIX 559 But (arc ASP) = OT,by
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, vddcov ' 564 INDEX OF GREEK WORDS
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1 INDEX OF GREEK WORDS 567 568 INDE
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approximations = ENGLISH INDEX 571
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works measurements indeterminate On
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ENGLISH INDEX 579 530 ENGLISH INDEX
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ENGLISH INDEX 583 584 ENGLISH INDEX
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A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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