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

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ON CONOIDS AND SPHEROIDS 57 of the hyperboloid the portion within the solid of the line joining the vertex of the enveloping cone to the vertex of the segment and produced, and in the case of the spheroids the line joining the points of contact of the two tangent planes parallel to the base of the segment. Definitions are added of a segment of a cone ' ' (the figure cut off towards the vertex by an elliptical, not circular, section of the cone) and a frustum ' of a cylinder' (cut off by two parallel elliptical sections). Props. 1 to 1 8 with a Lemma at the beginning are preliminary to the main subject of the treatise. The Lemma and Props. 1, 2 are general propositions needed afterwards. They include propositions in summation, 2 {a + 2a + 3a+ ... + na} > n.na > 2{a + 2a+ ... + (n—l)a} (this is clear from S n = ^n(n + I) a) (Lemma) (n + 1) (na) 2 +-a(a + 2a + 3a + ... + na) = 3 {a 2 + (2a) 2 + (3a) 2 + ... + (no) 2 } (Lemma to Prop. 2) whence (Cor.) 3 {a 2 + (2a) 2 + (3a) 2 + ... +(na) 2 } > n(na) 2 > 3{a 2 + (2a) 2 + ... + (n-la) 2 } ; . lastly, Prop. 2 gives limits for the sum of n terms of the series ax + x 2 , a.2x + (2x) 2 , a . 3 x + (3 x) 2 , . . , in the form of inequalities of ratios, thus n{a.nx + (nx) 2 } : 2/'" 1 {a.rx + (rx) 2 } > (a + nx) : (\a + \nx) > n { a . nx + (nx) 2 } : 2 X W { a . rx + (rx) 2 } Prop. 3 proves that, if QQ' be a chord of a parabola bisected at Fby the diameter PV, then, if PV be of constant length, the areas of the triangle PQQ' and of the segment PQQ f are also constant, whatever be the direction of QQ' ; to prove it Archimedes assumes a proposition proved in the conies ' ' and by no means easy, namely that, if QD be perpendicular to PV, and if p, pa be the parameters corresponding to the ordinates parallel to QQ' and the principal ordinates respectively, then Props. 4-6 deal with the area of an ellipse, which is, in the 58 ARCHIMEDES first of the three propositions, proved to be to the area of the auxiliary circle as the minor axis to the major ; equilateral polygons of 4 n sides are inscribed in the circle and compared with corresponding polygons inscribed in the ellipse, which are determined by the intersections with the ellipse of the double ordinates passing through the angular points of the polygons inscribed in the circle, and the method of exhaustion is then applied in the usual way. Props. 7, 8 show how, given an ellipse with centre C and a straight line CO in a plane perpendicular to that of the ellipse and passing through an axis of it, (1) in the case where OC is perpendicular to that axis, (2) in the case where it is not, we can find an (in general oblique) circular cone with vertex such that the given ellipse is a section of it, or, in other words, how we can find the circular sections of the cone with vertex which passes through the circumference of the ellipse ; similarly Prop. 9 shows how to find the circular sections of a cylinder with CO as axis and with surface passing through the circumference of an ellipse with centre C, where CO is in the plane through an axis of the ellipse and perpendicular to its plane, but is not itself perpendicular to that axis. Props. 11-18 give simple properties of the conoids and spheroids, easily derivable from the properties of the respective conies; they explain the nature and relation of the sections made by planes cutting the solids respectively in different ways (planes through the axis, parallel to the axis, through the centre or the vertex of the enveloping cone, perpendicular to the axis, or cutting it obliquely, respectively), with especial reference to the elliptical sections of each solid, the similarity of parallel elliptical sections, &c. Then with Prop. 19 the real business of the treatise begins, namely the investigation of the volume of segments (right or oblique) of the two conoids and the spheroids respectively. The method is, in all cases, to circumscribe and inscribe to the segment solid figures made up of cylinders or ' frusta of cylinders ', which can be made to differ as little as we please from one another, so that the circumscribed and inscribed figures are, as it were, compressed together and into coincidence with the segment which is intermediate between them. In each diagram the plane of the paper is a plane through the axis of the conoid or spheroid at right angles to the plane
ON CONOIDS AND SPHEROIDS 59 of the section which is the base of the segment, and which is a circle or an ellipse according as the said base is or is not at right angles to the axis ; the plane of the paper cuts the base in a diameter of the circle or an axis of the ellipse as the case may be. 60 ARCHIMEDES by planes obliquely inclined to the axis. The base of the segment is an ellipse in which BB' is an axis, and its plane is at right angles to the plane of the paper, which passes through the axis of the solid and cuts it in a parabola, a hyperbola, or an ellipse respectively. The axis of the segment is cut into a number of equal parts in each case, and planes are drawn through each point of section parallel to the base, cutting the solid in ellipses, similar to the base, in which PP', QQ', &c, are axes. Describing frusta of cylinders with axis AD and passing through these elliptical sections respectively, we draw the circumscribed and inscribed solids consisting of these frusta. It is evident that, beginning from A, the first inscribed frustum is equal to the first circumscribed frustum, the second to the second, and so on, but there is one more circumscribed frustum than inscribed, and the difference between the circumscribed and inscribed solids is equal to the last frustum of which BB' is the base, and ND is the axis. Since ND can be made as small as we please, the difference between the circumscribed and inscribed solids can be made less than any assigned solid whatever. Hence we have the requirements for applying the method of exhaustion. Consider now separately the cases of the paraboloid, the hyperboloid and the spheroid. I. The 'paraboloid (Props. 20-22). The frustum the base of which is the ellipse in which PP' is an axis is proportional to PP' 2 or PN 2 , i.e. proportional to AX. Suppose that the axis AD (= c) is divided into n equal parts. Archimedes compares each frustum in the inscribed and circumscribed figure with the frustum of the whole cylinder BF cut oft' by the same planes. Thus (first frustum in BF) : (first frustum in inscribed figure) The nature of the inscribed and circumscribed figures will be seen from the above figures showing segments of a paraboloid, a hyperboloid and a spheroid respectively, cut off Similarly (second frustum in BF) : 2 = BD 2 : PN = AD:AN = BD : TK (second in inscribed figure) = HN:3M, and so on. The last frustum in the cylinder BF has none to
Page 1 and 2: A HISTORY OF GREEK MATHEMATICS BY S
Page 3 and 4: CONTENTS vii Vlll CONTENTS Geminus
Page 5 and 6: CONTENTS XI XXI. COMMENTATORS AND B
Page 7 and 8: ARISTARCHUS OF SAMOS 3 contemporary
Page 9 and 10: Now whence ARISTARCHUS OF SAMOS BH:
Page 11 and 12: ARISTARCHUS OF SAMOS 11 while Y, Z
Page 13 and 14: ARISTARCHUS OF SAMOS 15 But AB.BG <
Page 15 and 16: MECHANICS 19 likely that he discove
Page 17 and 18: • LIST OF EXTANT WORKS 23 24 ARCH
Page 19 and 20: THE TEXT OF ARCHIMEDES 27 It was ma
Page 21 and 22: Now THE METHOD HA:AN=A'A:AN = KA:AQ
Page 23 and 24: ON THE SPHERE AND CYLINDER, I 35 ti
Page 25 and 26: ON THE SPHERE AND CYLINDER, I 39 wh
Page 27 and 28: ON THE SPHERE AND CYLINDER, II 43 T
Page 29 and 30: ON THE SPHERE AND CYLINDER, II 47 (
Page 31 and 32: MEASUREMENT OF A CIRCLE 51 sides co
Page 33: a MEASUREMENT OF A CIRCLE 55 be The
Page 37 and 38: so that ON CONOIDS AND SPHEROIDS 63
Page 39 and 40: Then D : E > BM : MO, > OB : BT, ON
Page 41 and 42: ON SPIRALS 71 Let OF meet the spira
Page 43 and 44: ON SPIRALS 75 Lastly, it' E be the
Page 45 and 46: ON PLANE EQUILIBRIUMS, I, II 79 Thi
Page 47 and 48: THE SAND-RECKONER 83 Archimedes has
Page 49 and 50: curve in E 1 , R THE QUADRATURE OF
Page 51 and 52: THE QUADRATURE OF THE PARABOLA 91 t
Page 53 and 54: ON FLOATING BODIES, I, II 95 96 ARC
Page 55 and 56: ON SEMI-REGULAR POLYHEDRA 99 angula
Page 57 and 58: THE LIBER ASSUMPTORUM 103 Lastly, w
Page 59 and 60: MEASUREMENT OF THE EARTH 107 a, or
Page 61 and 62: DISCOVERY OF THE CONIC SECTIONS 111
Page 63 and 64: MENAECHMUS'S PROCEDURE 115 For, let
Page 65 and 66: ARISTAEUS'S SOLID LOCI 119 the same
Page 67 and 68: CONIC SECTIONS IN ARCHIMEDES 123 In
Page 69 and 70: THE TEXT OF THE CONICS 127 The edit
Page 71 and 72: THE CONICS 131 diorismi. Nicoteles
Page 73 and 74: THE C0NIC8, BOOK I 135 the centre o
Page 75 and 76: PL is THE CONICS, BOOK I 139 called
Page 77 and 78: THE CONIGS, BOOK I 143 (2) in the h
Page 79 and 80: It follows that THE CONICS, BOOK I
Page 81 and 82: THE CONICS, BOOK III 151 152 APOLLO
Page 83 and 84: To prove (1) R'l 2 : IW-H'Q 2 : QH
Page 85 and 86:
THE CONICS, BOOKS IV-V 159 and, if
Page 87 and 88:
THE CONICS, BOOK V 163 if P' be any
Page 89 and 90:
THE CONICS, BOOKS V, VI 167 tively
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THE CONICS, BOOK VII 171 But A'Q*:A
Page 93 and 94:
THE GONIGS, BOOK VII 175 As we have
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ON THE CUTTING-OFF OF A RATIO 179 F
Page 97 and 98:
ON CONTACTS OR TANGENCIES 183 solve
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PLANE LOCI 187 other extremity will
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NET2EI2 (VERGINGS OR INCLINATIONS)
Page 103 and 104:
OTHER LOST WORKS 195 Astronomy. We
Page 105 and 106:
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
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
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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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Page 159 and 160:
Page 161 and 162:
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
Page 221 and 222:
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
Page 229 and 230:
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
Page 259 and 260:
INDETERMINATE ANALYSIS 507 508 DIOP
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INDETERMINATE ANALYSIS 511 [The aux
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THE TREATISE ON POLYGONAL NUMBERS 5
Page 265 and 266:
SERENUS 519 520 COMMENTATORS AND BY
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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
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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
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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