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

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RHABDAS 553 or or Rhabdas solves the equation 1- - = a, practically as we should, by multiplying up to get rid of fractions, obtains x = mna/(m + n). equations x + y = a, tux = ny ; y m m n whence he Again he solves the simultaneous also the pair of equations x n x + — = y + - — a. Of course, m, n, a ... have particular numerical values in all cases. Rhabdas 's Rule for approximating to the square root of a non-square number. We find in Rhabdas the equivalent of the Heronian formula for the approximation to the square root of a non-square number A = a 2 + b, namely b &c; (2) by using one or other of the following approximations, where a 2 < A < (a+1) 2 , and A = a 2 + b = (a+l) 2 -c, a + — » a + — - b > namely, b b 1 2a 2a + 2 a 2(a+l) v 7 2(a+l) or a combination of two of these with (3) the formula that, if T < -= , then b a a ma + nc c b mb + nd d 2(a+l) It is clear that it is impossible to deny to the Greeks the knowledge of these simple formulae. Three more names and we have done. Ioannes Pediasimus, also called Galenus, was Keeper of the Seal to the Patriarch of Constantinople in the reign of Andronicus III (1328-41). Besides literary works of his, some notes on difficult points in arithmetic and a treatise on the duplication of the cube by him are said to exist in manuscripts. His Geometry, which was edited by Friecllein in 1866, follows very closely the mensuration of Heron. BaulaAM, a monk of Calabria, was abbot at Constantinople and later Bishop of Geraci in the neighbourhood of Naples; he died in 1348. He wrote, in Greek, arithmetical demonstrations of propositions in Euclid, Book II, 1 and a Logistic in six Books, a laborious manual of calculation in whole numbers, 1 Edited with Latin translation by Dasypodius in 1564, and included in Heiberg and Menge's Euclid, vol. v, ad fin.
PEDIASIMUS. BARLAAM. ARGYRUS 555 ordinary fractions and sexagesimal fractions (printed at Strassburg in 1592 and at Paris in 1600). Barlaam, as we have seen, knew the Heronian formulae for finding successive approximations to square roots, and was aware that they could be indefinitely continued. Isaac Argyrus, a monk, who lived before 1368, was one of a number of Byzantine translators of Persian astronomical works. In mathematics he wrote a Geodaesia and scholia to the first six Books of Euclid's Elements. The former is contained in the Paris MS. 2428 and is called 'a method of geodesy or the measurement of surfaces, exact and shortened ' the introductory letter addressed to one Colybos is followed by a compilation of extracts from the Geometrica and Stereometrica of Heron. He is apparently the author of some further additions to Rhabdas's revision of the Manual of Planudes contained in the same manuscript. A short tract of his On f the discovery of the square roots of non-rational square numbers ' is mentioned as contained in two other manuscripts at Venice and Rome respectively (Codd. Marcianus Gr. 333 and Vaticanus Gr. 1058), where it is followed by a table of the square roots of all numbers from 1 to 102 in sexagesimal fractions (e.g.
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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
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GEMINUS 223 An upper limit for his
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GEMINUS 227 Attempt to prove the Pa
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GEMINUS 231 and make equal angles w
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236 SOME HANDBOOKS XVI SOME HANDBOO
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THEON OF SMYRNA 239 edited by E. Hi
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THEON OF SMYRNA 243 to the seven he
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THEODOSIUS'S SPHAERIGA 247 (Books X
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THEODOSIUS'S SPHAERICA 251 Now, the
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HIPPARCHUS 255 that the lengths of
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HIPPARCHUS 259 in his Commentary, h
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MENELAUS'S SPHAERICA 263 already ap
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MENELAUS'S SPIIAERICA 267 For, if A
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^ ' MENELAUS'S SPHAERICA 271 272 TR
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' PTOLEMY'S SYNTAXIS 275 276 TRIGON
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PTOLEMY'S SYNTAXIS 279 The proposit
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PTOLEMY'S SYNTAXIti 283 284 TRIGONO
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THE ANALEMMA OF PTOLEMY 287 288 TRI
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THE ANALEMMA OF PTOLEMY 291 or tan
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THE OPTICS OF PTOLEMY 295 but the t
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CONTROVERSIES AS TO HERON'S DATE 29
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GEOMETRY 311 Of this class are the
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' ' concave ' and THE DEFINITIONS 3
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MENSURATION 319 Heiberg puts side b
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PROOF OF THE FORMULA OF HERON' 323
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THE REGULAR POLYGONS 327 Geom. 102
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height SEGMENT OF A CIRCLE 331 The
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iJ MEASUREMENT OF SOLIDS 335 descri
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DIVISIONS OF FIGURES 339 two opposi
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: —— Since (DG) : (FB) = m : No
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THE MECHANICS 347 the first chapter
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Supports '. ON THE CENTRE OF GRAVIT
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356 PAPPUS OF ALEXANDRIA Date of Pa
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THE COLLECTION 359 sizes and distan
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THE COLLECTION. BOOK III 363 Sectio
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Now THE COLLECTION. BOOK III 367 EA
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THE COLLECTION. BOOK IV 371 we may
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THE COLLECTION. BOOK IV 375 Therefo
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THE COLLECTION. BOOK IV 379 We have
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THE COLLECTION. BOOK IV 383 a certa
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THE COLLECTION. BOOK IV 387 length
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THE COLLECTION. BOOK V 391 the same
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of THE COLLECTION. BOOK V 395 the s
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THE COLLECTION. BOOKS VI, VII 399 T
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THE COLLECTION. BOOK VII 403 ' util
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THE COLLECTION. BOOK VII 407 Two pr
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PG THE COLLECTION. BOOK VII 411 Now
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The problem is THE COLLECTION. BOOK
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i.e. (if DA . THE COLLECTION. BOOK
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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
Page 231 and 232: RELATION OF WORKS 451 Relation of t
Page 233 and 234: TRANSLATIONS AND EDITIONS 455 unfor
Page 235 and 236: NOTATION AND DEFINITIONS 459 When t
Page 237 and 238: DETERMINATE EQUATIONS 463 equation
Page 239 and 240: INDETERMINATE EQUATIONS 467 (ft) wh
Page 241 and 242: • INDETERMINATE EQUATIONS 471 Ex.
Page 243 and 244: INDETERMINATE EQUATIONS 475 a squar
Page 245 and 246: METHOD OF APPROXIMATION TO LIMITS 4
Page 247 and 248: NUMBERS AS THE SUMS OF SQUARES 483
Page 249 and 250: ' (I. 35. x = my, y 2 = nx. 1 1. 36
Page 251 and 252: INDETERMINATE ANALYSIS 491 IV. 33.
Page 253 and 254: INDETERMINATE ANALYSIS 495 III. 14.
Page 255 and 256: INDETERMINATE ANALYSIS 499 IV. 29.
Page 257 and 258: INDETERMINATE ANALYSIS 503 Let the
Page 259 and 260: INDETERMINATE ANALYSIS 507 508 DIOP
Page 261 and 262: INDETERMINATE ANALYSIS 511 [The aux
Page 263 and 264: THE TREATISE ON POLYGONAL NUMBERS 5
Page 265 and 266: SERENUS 519 520 COMMENTATORS AND BY
Page 267 and 268: SERENUS 523 Observing that the sum
Page 269 and 270: THEON OF ALEXANDRIA 527 pp. 58-63).
Page 271 and 272: PROCLUS 531 viated form usual with
Page 273 and 274: PROCLUS 535 second Book \ But at th
Page 275 and 276: DOMNINUS. SIMPLICIUS 539 sophy at A
Page 277 and 278: , . a ANTHEMIUS 543 the focus to th
Page 279 and 280: PSELLUS. PACHYMERES. PLANUDES 547 R
Page 281: MOSCHOPOULOS. RHABDAS 551 of Smyrna
Page 285 and 286: APPENDIX 559 But (arc ASP) = OT,by
Page 287 and 288: , vddcov ' 564 INDEX OF GREEK WORDS
Page 289 and 290: 1 INDEX OF GREEK WORDS 567 568 INDE
Page 291 and 292: approximations = ENGLISH INDEX 571
Page 293 and 294: works measurements indeterminate On
Page 295 and 296: ENGLISH INDEX 579 530 ENGLISH INDEX
Page 297 and 298: 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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