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

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LIST OF TREATISES 309 by B. Baldi (Augsburg, 1616), Thevenot (Vet. math.), Kochly and Riistow (1853) and by Wescher (Pollorcetique des Grecs, 1867, the first critical edition). VI. The Cheirobalistra ('Hpcovos ^ipofiaWicrTpas KaracrKevrj Kal
GEOMETRY 311 Of this class are the different cases of I. 35, 36, III. 7, 8 (where the chords to be compared are drawn on different sides of the diameter instead of on the same side), III. 12 (which is not Euclid's at all but Heron's own, adding the case of external to that of internal contact in III. 11 \ VI. 19 (where the triangle in which an additional line is drawn is taken to be the smaller of the two), VII. 19 (where the particular case is given of three numbers in continued proportion instead of four proportionals). (3) Alternative proofs. It appears to be Heron who first introduced the easy but uninstructive semi-algebraical method of proving the propositions II. 2-10 which is now so popular. On this method the propositions are proved ' without figures ' as consequences of II. 1 corresponding to the algebraical formula a (b + c + d + . . .) = ab + ac -f ad + . Heron explains that it is not possible to prove II. 1 without drawing a number of lines (i. e. without actually drawing the rectangles), but that the following propositions up to II. 10 can be proved by merely drawing one line. He distinguishes two varieties of the method, one by dissolutio, the other by compositio, by which he seems to mean splitting-up of rectangles and squares and combination of them into others. But in his proofs he sometimes combines the two varieties. Alternative proofs are given (a) of some propositions of Book III, namely III. 25 (placed after III. 30 and starting from the arc instead of the chord), III. 10 (proved by means of III. 9), III. 13 (a proof preceded by a lemma to the effect that a straight line cannot meet a circle in more than two points). A class of alternative proof is (6) that which is intended to meet a particular objection (eWracriy) which had been or might be raised to Euclid's constructions. Thus in certain cases Heron avoids producing a certain straight line, where Euclid produces it, the object being to meet the objection of one who should deny our right to assume that there is any space available. Of this class are his proofs of I. 11, 20 and his note on I. 16. Similarly in I. 48 he supposes the right-angled . 312 HERON OF ALEXANDRIA triangle which is constructed to be constructed on the same side of the common side as the given triangle is. A third class (c) is that which avoids reductio ad absurdum, e.g. a direct proof of I. 19 (for which he requires and gives a preliminary lemma) and of I. 25. (4) Heron supplies certain converses of Euclid's propositions e.g. of II. 12, 13 and VIII. 27. (5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e. g. the construction of isosceles and scalene triangles in a note on I. 1 and the construction of two tangents in III. 17. The most important extension is that of III. 20 to the case where the angle at the circumference is greater than a right angle, which gives an easy way of proving the theorem of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of the one equal to two sides of the other and with the included angles supplementary are equal in area, and compares the areas where the sum of the included angles (one being supposed greater than the other) is less or greater than two right angles, and on I. 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of Euclid's proposition (see next page), the straight lines AL, BG, GE meet in a point. This last proof is worth giving. First come the lemmas. (1) If in a triangle ABG a straight line DE be drawn parallel to the base BG cutting the sides AB, AC or those sides produced in D, E, and if F be the middle point of BG, then the straight line AF (produced if necessary) will also bisect DE. (HK is drawn through A parallel to DE, and HDL, REM through D, E parallel to AF meeting the base in L, M respectively. Then the triangles ABF, AFC between the same parallels are equal. So are the triangles DBF, EFC. Therefore the differences, the triangles ADF, AEF, are equal and so therefore are the parallelograms HF, KF. Therefore LF = FM, or DG = GE.) (2) is the converse of Eucl. 1. 43. If a parallelogram is
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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
Page 109 and 110: ISOPERIMETRIC FIGURES. ZENODORUS 20
Page 111 and 112: ZENODORUS 211 Now, by hypothesis, D
Page 113 and 114: HYPSICLES 215 natural to divide eac
Page 115 and 116: 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 157 and 158: CONTROVERSIES AS TO HERON'S DATE 30
Page 159: CONTROVERSIES AS TO HERON'S DATE 30
Page 163 and 164: ' ' concave ' and THE DEFINITIONS 3
Page 165 and 166: MENSURATION 319 Heiberg puts side b
Page 167 and 168: PROOF OF THE FORMULA OF HERON' 323
Page 169 and 170: THE REGULAR POLYGONS 327 Geom. 102
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
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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
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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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