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

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PORISMS AND PROPOSITIONS ASSUMED 481 In fact YZ±a = {(m+l)(2m+l) + 2a} 2 , ZX±a = {m(2m+ l) + 2a} 2 , XY±a= {.m(m+l) + a} 2 . (3) If X = m 2 + 2, F=(m+l)*+2, Z = 2{m 2 + (m + 1) 2 + 1 ] + 2, then the six expressions F^_(F+if), #X-(#+X), XY-(X+Y) } YZ-X, ZX-Y, XY-Z are all squares (V. 6). In fact YZ- (Y + Z) = {2m 1 + 3m + 3) 2 , F^-I=(2m 2 + 3m + 4) 2 ; &c. 2. The second class is much more important, consisting of propositions in the Theory of Numbers which we find first stated or assumed in the Arithmetica. It was in explanation or extension of these that Fermat's most famous notes were written. How far Diophantus possessed scientific proofs of the theorems which he assumes must remain largely a matter of speculation. (a) Theorems on the composition of numbers as the sum of two squares. (1) Any square number can be resolved into two squares in any number of ways (II. 8). (2) Any number which is the sum of two squares can be resolved into two other squares in any number of ways (II. 9). (It is implied throughout that the squares may be fractional as well as integral.) (3) If there are two whole numbers each of which is the sum of two squares, the product of the numbers can be resolved into the sum of two squares in two ways. In fact (a 2 + b 2 ) (c2 + d 2 ) = (ac ± bd) 2 + {ad + be) 2 . This proposition is used in III. 19, where the problem is to find four rational right-angled triangles with the same 1523.2 I i 482 DIOPHANTUS OF ALEXANDRIA hypotenuse. The method is this. Form two right-angled triangles from (a, b) and (c, d) respectively, by which Diophantus means, form the right-angled triangles (a 2 + b 2 , a 2 -b 2 , 2ab) and (c 2 + d 2 , c 2 -d 2 , 2cd). Multiply all the sides in each triangle by the hypotenuse of the other; we have then two rational right-angled triangles with the same hypotenuse (a 2 + b 2 ) (c 2 + d 2 ). Two others are furnished by the formula above; for we have only to form two right-angled triangles ' ' from (ac + bd, ad — be) and from (ac — bd, ad + be) respectively. The method fails if certain relations hold between a, b, c, d. They must not be such that one number of either pair vanishes, i.e. such that ad = be or ac = bd, or such that the numbers in either pair are equal to one another, for then the triangles are illusory. In the case taken by Diophantus a 2 + b 2 = 2 2 + 2 = 5, c 2 + d 2 = 3 2 + 2 2 = 1 3, and the four right-angled triangles are (65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33). On this proposition Fermat has a long and interesting note as to the number of ways in which a prime number of the form 4 n + 1 and its powers can be (a) the hypotenuse of a rational right-angled triangle, (b) the sum of two squares. He also extends theorem (3) above : If a prime number which ' is the sum of two squares be multiplied by another prime number which is also the sum of two squares, the product will be the sum of two squares in two ways ; if the first prime be multiplied by the square of the second, the product will be the sum of two squares in three ways ; the product of the first and the cube of the second will be the sum of two squares in four ways, and so on ad infinitum' Although the hypotenuses selected by Diophantus, 5 and 13, are prime numbers of the form 4 n + 1 , it is unlikely that he was aware that prime numbers of the form 4 n + 1 and numbers arising from the multiplication of such numbers are the only classes of numbers which are always the sum of two squares ; this was first proved by Euler. (4) More remarkable is a condition of possibility of solution prefixed to V. 9, 'To divide 1 into two parts such that, if
NUMBERS AS THE SUMS OF SQUARES 483 a given number is added to either part, the result will be a square.' The condition is in two parts. There is no doubt as to the first, 'The given number must not be odd' [i.e. no number of the form 4n + 3 or 4 w, — 1 can be the sum of two squares] ; the text of the second part is corrupt, but the words actually found in the text make it quite likely that corrections made by Hankel and Tannery give the real meaning of the original, ' nor must the double of the given number plus 1 be measured by any prime number which is less by 1 than a multiple of 4 '. This is tolerably near the true condition stated by Fermat, The given number must not be odd, and ' the double of it increased by 1 , when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4 n — 1 .' (ft) On numbers which are the sum of three squares. In V. 11 the number 3
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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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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
Page 219 and 220: THE COLLECTION. BOOKS VII, VIII 427
Page 221 and 222: THE COLLECTION. BOOK VIII 431 432 P
Page 223 and 224: THE COLLECTION. BOOK VIII 435 is le
Page 225 and 226: THE COLLECTION. BOOK VIII 439 Then
Page 227 and 228: EPIGRAMS IN THE GREEK ANTHOLOGY 443
Page 229 and 230: 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: METHOD OF APPROXIMATION TO LIMITS 4
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 and 282: MOSCHOPOULOS. RHABDAS 551 of Smyrna
Page 283 and 284: PEDIASIMUS. BARLAAM. ARGYRUS 555 or
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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