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

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THE COLLECTION. BOOKS IV, V 389 390 PAPPUS OF ALEXANDRIA Book V. Preface on the Sagacity of Bees. It is characteristic of the great Greek mathematicians that, whenever they were free from the restraint of the technical language of mathematics, as when for instance they had occasion to write a preface, they were able to write in language of the highest literary quality, comparable with that of the philosophers, historians, and poets. We have only to recall the introductions to Archimedes's treatises and the prefaces to the different Books of Apollonius's Conies. Heron, though severely practical, is no exception when he has any general explanation, historical or other, to give. We have now to note a like case in Pappus, namely the preface to Book V of the Collection. The editor, Hultsch, draws attention to the elegance and purity of the language and the careful writing the latter is illustrated by the studied avoidance of hiatus. 1 The subject is one which a writer of taste and imagination would naturally find attractive, namely the practical intelligence shown by bees in selecting the hexagonal form for the cells in the honeycomb. Pappus does not disappoint us ; the passage is as attractive as the subject, and deserves to be reproduced. ' It is of course to men that God has given the best and most perfect notion of wisdom in general and of mathematical science in particular, but a partial share in these things he allotted to some of the unreasoning animals as well. To men, as being endowed with reason, he vouchsafed that they should do everything in the light of reason and demonstration, but to the other animals, while denying them reason, he granted that each of them should, by virtue of a certain natural instinct, obtain just so much as is needful to support life. This instinct may be observed to exist in very many other species of living creatures, but most of all in bees. In the first place their orderliness and their submission to the queens who rule in their state are truly admirable, but much more admirable still is their emulation, the cleanliness they observe in the gathering of honey, and the forethought and housewifely care they devote to its custody. Presumably because they know themselves to be entrusted with the task of bringing from the gods to the accomplished portion of mankind a share of 1 Pappus, vol. iii, p. 1233. ambrosia in this form, they do not think it proper to pour it carelessly on ground or wood or any other ugly and irregular material ; but, first collecting the sweets of the most beautiful flowers which grow on the earth, they make from them, for the reception of the honey, the vessels which we call honeycombs, (with cells) all equal, similar and contiguous to one another, and hexagonal in form. And that they have contrived this by virtue of a certain geometrical forethought we may infer in this way. They would necessarily think that the figures must be such as to be contiguous to one another, that is to say, to have their sides common, in order that no foreign matter could enter the interstices between them and so defile the purity of their produce. Now only three rectilineal figures would satisfy the condition, I mean regular figures which are equilateral and equiangular; for the bees would have none of the figures which are not uniform. . . . There being then three figures capable by themselves of exactly filling up the space about the same point, the bees by reason of their instinctive wisdom chose for the construction of the honeycomb the figure which has the ,most angles, because they conceived that it would contain more honey than either of the two others. ' Bees, then, know just this fact which is of service to themselves, that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material used in constructing the different figures. We, however, claiming as we do a greater share in wisdom than bees, will investigate a problem of still wider extent, namely that, of all equilateral and equiangular plane figures having an equal perimeter, that which has the greater number of angles is always greater, and the greatest plane figure of all those which have a perimeter equal to that of the polygons is the circle.' Book V then is devoted to what we may call isoperimetry', including in the term not only the comparison of the areas of different plane figures with the same perimeter, but that of the contents of different solid figures with equal surfaces. Section (1). lsoperimetry after Zenodorus, The first section of the Book relating to plane figures (chaps. 1-10, pp. 308-34) evidently followed very closely the exposition of Zenodorus wepl la-ouerpcou crxv yLOLTQ&v (see pp. 207-13, above) ; but before passing to solid figures Pappus inserts the proposition that of all circular segments having
THE COLLECTION. BOOK V 391 the same circumference the semicircle is the greatest, with some preliminary lemmas which deserve notice (chaps. 15, 16). (1) ABC is a triangle right-angled at B. With C as centre and radius CA describe the arc AD cutting CB produced in D. To prove that (R denoting a right angle) (sector CAD) : (area ABD) > R./BCA. Draw AF at right angles to CA meeting CD produced in F, and draw BH perpendicular to AF. With A as centre and AB as radius describe the arc GBE. « (area EBH) > (area EBF) : (sector ABE), Now (area EBF) : and, componendo, AFBH: (EBH) > AABF: (ABE). But (by an easy lemma which has just preceded) whence and AFBH: (EBH) = AABF: (ABD), AABF: (ABD) > AABF: (ABE), (ABE) > (ABD). Therefore (ABE) : (ABG) > (ABD) : (ABG) > (ABD): A ABC, a fortiori. Therefore Z BAF: Z BA C > (ABD) : A ABC, whence, inversely, AABC:(ABD) > Z BAG: Z BAF. and, componendo, (sector ACD) : (ABD) > R : Z BCA. [If a. be the circular measure of /.BCA, this gives (if AC=b) 39.2 PAPPUS OF ALEXANDRIA Draw AE at right angles to AC. With A as centre and AC as radius describe the circle FCE meeting AB produced in Fand AE in E. Then, since I ACD > ICAE, (sector ACD) > (sector AGE). Therefore (ACD) : A ABC > (ACE) : AABC Inversely, > (ACE) : (ACF), > LEAC-.LCAB. a fortiori, AABC: (ACD) < lCAB:lEAC, and, componendo, (ABD) : (ACD) < Z EAB : Inversely, (A CD) : (ABD) > I EAC: /.EAB >R:LACD. Z EAC. We come now to the application of these lemmas to the proposition comparing the area of a semicircle with that of other segments of equal circumference (chaps. 17, 18). A semicircle is the greatest of all segments of circles which have the same circumference. Let ABC be a semicircle with centre G, and DEF another segment of a circle such that the circumference DEF is equal %otb 2 :(^ocb — 2 -J sin a cos a . 6 2 ) >^7r:a, or 2a:(2a — sin2a) > tt:2oc; that is, 0/(0 — sin 0) > n/0, where < < tt.] (2) ABC is again a triangle right-angled at B. With C as AD meeting BC pro- centre and CA as radius draw a circle duced in D. To prove that (sector CAD) : (area ABD) > R : /.ACD. to the circumference ABC. I say that the area of ABC is greater than the area of DEF. Let H be the centre of the circle DEF. Draw EHK, BG at right angles to DF, AC respectively. Join DH, and draw LHM parallel to DF.
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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
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 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: THE COLLECTION. BOOK IV 387 length
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 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
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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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