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OSMANIA UNIVERSITY LIBRARY
Call No. '5f* ? #42 & Accession No. ft ^ %
Author #e*J / ' *"' '
This book should be returned on or before the date last marked below,
A HISTORY
OF
GREEK MATHEMATICS
VOLUME II
OF
GREEK MATHEMATICS
HY
SIR THOMAS H^A*
K.C.B., K.C.V.O., K.R.S.
SC.J). C'AMH. ; HON. D.SC. OXFORD
HONORARY FKLf.OW (KOIIMKHLV I'ELLOw) Ol 1 TRINITY (OLIFOK,
*. . , All independent world,
Treated out ot pure intelligence.'
WORDSWORTH.
VOLUME H
FROM AltlSTAU( 1 lI(lS TO DIOrilANTUS
O X F R i)
AT THE CLARENDON PRESS
OXFORD UNIVERSITY PRESS
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne (ape Town
Bombay Caleutta Madras Shanghai
HUMPHREY M1LFORI)
Publisher to the University
CONTENTS OF VOL II
XII. ARISTARCHUS OF SAMOS PAGES 1-15
XIII. AKCH1MEDKS 16-101
Traditions
(() Astronomy 17-lX
(3) Mechanics 18
Summary of main achievements . ... 19- 20
Character of treatises 20 22
List of works still extant 22-23
Traces of lost works 23 25
The text of Archimedes 25-27
Contents of The Method 27-34
On the fyhere and Cylinder. I, IT 34-50
Cubic equation arising out of 1 1. 4 .... 43-4(5
(ij Archimedes's own solution ..... 45-46
(ii) Dionysodorus's solution 40
(lii) DioeWs solution of original problem . 47-49
Measurement of a Circle ....... 50 56
On Cvnoidt* and Hplietoid* ...... 5664
On ^jiirals ......... 64-75
On I'htn** K<iu;iibrhnH*,\,\\ 75-81
The Hand-reel* oner (Psdnnnitcs or Arcnartnx} . . . 81-K5
The Quadrature of the Parabola 85-91
On Floating Bodic*. I, II . 91 97
The problem of the crown . . . 92-94
Other works
M The Cattle-Problem 97-98
(.1) On semi regular polyhedra ..... 98-101
(y) The Liber Asunmptortnn 101 10->
(ft) Formula for area of triangle . . . . 103
Kratosthenes 104-109
Measurement of the Karth 106 1 OS
XlV f . CONIC SUCTIONS. APOLLONIU8 OF PKRUA . . 110 196
A. HISTORY OF CONICS UP TO APOLLOMUS . . 110 126
Discoveiy of the ronie sections by Menaeehmus . 110-111
Menaeehmus\s probable procedure . . . 111-116
Woiks by Aristaeus and Kuclid . . . 116-117
4 Solid loci 'and 4 solid problems' . . . 117-118
Aristaeus's Solid Loci US 119
Focus-directrix property known to Kuclid . . 119
Proof from Pappus 120121
Propositions included in Kuclid's Conies . . 121-12*2
Conic sections in Archimedes .... 122-126
vi CONTENTS
XIV. CONTINUED.
B. APOLLONIUS OF PERGA .... PAGES 126-196
The text of the Conies 126-128
Apollonius's own account of the Con lets . . . 128-133
Extent of claim to originality .... 182-133
Great generality of treatment .... 133
Analysis of the Con ics 133-175
Book I 133-148
Conies obtained in the most general way from
oblique cone 134-138
New names, ' parabola ', * ellipse ', * hyperbola ' . 138-139
Fundamental properties equivalent to Cartesian
equations 139-141
Transition to new diameter and tangent at its
extremity 141-147
First appearance of principal axrs . . .147-148
Hook 11 148-150
13ookJIl 150 lf>7
Book IV 157-158
BookV 158 167
Normals us maxima and minima .... 159-163
Number of normals from a point .... 163 164
Propositions leading immediately to detei munition
of pvolutt' of conic ...... 164-166
Construction of normals ..... 166 167
Hook VI 167-168
Hook Vll 168-174
Other works by Apollonius 175-194
(<i) On the Cutting off' of <t Ratio (\nyov fin-oro/ir}),
two Books .* 175-179
(ft) On the Cutting- off of an Area (^w/uou (ITTOTO^I/),
two Books .' 179 180
(y) On Determinate Section (Sico/mr/uri; TO/LU}), two
Books 1SO-181
(ft) On Contacts or Tanyencies (oratfwu), two Hooks . 181-1x5
(f) Plane Loci, two Books 185-189
(C) Neuo-ns- (Vcrgimjx or lm'linatfoHs\ two Books . 189-192
(r;) Comparison of dodecahedron ;/'/'// ieo^ahedron . 192
(ff) General Treatise 192-193
(t) On the Cochlia* l<j:;
(K) On Unordered Irrational* 193
(X) On the Jhtrtiiny-mirror . . . . . 19 1
(/Ll) '&KVTUKIV ........ 194
Astronomy [ 195-196
XV. TIIK SUCCESSORS OF TIIK CJRKAT aEOMKTKRS . 197 234
Nicomedes 199
Diodes .""..' 200-203
Perseus 203-206
Isoperimetrie figures. Kenodorus 206-213
liypsieles 213-218
Dionysodorus 218-219
Posidonius 219-222
CONTENTS vii
Ueminus PAGES 222-234
Attempt to prove the Parallel-Postulate . . . 227-230
On Meteoroloyica of Posidonius 231-232
Introduction to the Phaeuomena attributed to deminus 232-234
XVI. SOME HANDBOOKS 235-244
Cleomedes, De motu circular! 235-238
Nicomachus ... 238
Theon of Smyrna, Kjrpositio rerutn muthemu tit-arum ad
teyenduiH Platonem ut Hi it in 238-244
XVII. TRIGONOMETRY: HIPPARCIIUS, M EN E LA US, PTO-
LEMY 245-297
Theodosius 245 216
Works by Theodosius ....... 246
Contents of the *S)>//'/w'<7/ 246-252
No actual trigonometiy in Theodosius . . . 250-2*52
The beginnings of trigonometiy 252-253
Hipparchus 25:5-260
The work of Hipparchus 254-256
First systematic use of trigonometis .... 257-259
Table of chords . . . . ' . . . . 259-260
Menelaus 260-273
The Sphft erica of Menclaus 261-273
(n) l Menelau&'b theorem ' for tin* spheie . . 266 -26S
(,i) Deductions from MenelauVs theorem . . 268-269
(y) Anharmonic properly of four ^reat circles
tin ou^h one point' 269270
(fi) Propositions ana logons to Kucl. VI. -\ . 270
Claudius Ptolemy 27:> 297
The M<i0/piT<K/ (nW.i^if (Arab. Ahmnjpxt) . . . 273 286
Commentaries ....... 274
Translations and editions .... 274 275
Summary of content^ ...... 275 276
Tiigonometry in Ptolemy 276 286
() Lemma for finding MH Is and sin ))<> . . 277 27s
(,:*) K(|iiivalent of sin'tf-j cus'^? _- 1 . . . 278
(y) * Ptolemy's theorem \ "iving the equivaJent of
sin (^ -</>) = sin cos c/> - cos & sin c/> . . . 278 280
(fl) Kquivalent of sin" \ti = \ (1 -co^) . . . 2SO-281
(t ) Iviuivalent of cos(^-l </) eos^eosr/; sin^sin 281
() Method of interpolation based on formula
sin a/sin ^ < u/tf (In > a > ji) . . . 2^1 2^2
d;) Table of chords ." . 283
(O) Further USD of proportional increase . . 283-284
(t) Plane trigonometry in effect used . . . 284
Spherical trigonometry: formulae in solution of
spherical triangles 284-286
VliGAnalciunta 286292
The FltiHis2>hacriuni 292 293
The Optics 293-295
A mechanical work, Ue/ii (Htntav ..... 295
Attempt to prove the Parallel-Postulate . . . 295-297
viii CONTENTS
XVIII. MENSURATION: HKRON OF ALEXANDRIA. PAGES 298-354
Controversies as to Heron's date 298-306
Character of works 307-308
List of treatises 308-310
Geometry
(a) Commentary on Euclid's Elements . . . 310 314
() The Definitions 314 316
Mensuration 316-344
The Metrica, Geometrica, Stereomdrica, Geoduexia,
Mensure 316 320
Contents of the Mt tried 320-344
Book I. Measurement of areas .... 320331
(a) Area of scalene triangle .... 320-321
Proof of formula A - J{s(8-)(8-b (s-c)\ 321-323
(/3) Method of approximating to the square root
of a non-square number .... 323-326
(7) Quadrilaterals 326
(6) Regular polygons with 3, 4, 5, 6, 7, 8, *J, 10,
11, or 12 sides 326-329
(0 The circle :',29
(f) Segment of a circle 330-331
(17) Kllipse,parabolic segment, surface of cylinder,
right cone, sphere and segment of sphere . 331
Book II. Measurement of volumes . . . 331-335
(a) Cone, cylinder, parallelepipedf prism), pyramid
and frustum ...... 332
(#) Wedge-shaped solid (#o>/n'<rK09 or or/^iWn) . 332-334
(y) Frustum of cone, sphere, and segment thereof 334
(8) Anchor-ring or tore 334-335
( f ) The two special solids of Arch imcdes's 'Method ' 335
(C) The five regular solids 335
Book III. Divisions of figures 336-343
Approximation to the cube root of a non-cube
number 341-342
Quadratic equations solved in Heron . . . 344
Indeterminate problems in the Geometrica . . 344
The Dioptm 345-346
The Mechanic* 346-352
Aristotle's Wheel 347-348
The parallelogram of velocities . . . .348-349
Motion on an inclined piano 349- 350
On the centre of gravity .... 350-351
The five mechanical powers ..... 351
Mechanics in daily life : queries and answers . 351-352
Problems on the centre of gravity, &e. . . . 352
The Catoptrica . . 352-354
Heron's proof of equality of angles of incidence and
reflection 353-354
XIX. PAPPUS OF ALEXANDRIA 355 439
Date of Pappus :}:>6
Works (commentaiies) other than the Collection . . 356-357
CONTENTS ix
The Synayoye or Collection .... PAGES 357-439
() Character of the work ; wide range . . . 357-358
(/:$) List of authors mentioned ..... 358-360
(y) Translations and editions ..... 360-361
(6) Summary of contents ...... 361-439
Book 111. Section (1). On the problem of the two
mean proportionals . . . . . .361 362
Section (2). The theory of means . . .363-365
Section (:5). The * Paradoxes' of Krycinus . . 365-368
Section (4). The inscribing of the five regular
solids in a sphere ...... 308-369
Book IV. Section (1). Kxtension of theorem of
Pythagoras 369 371
Section (2). On circles inscribed in the up0rj\ns
(' shoemaker's knife') 371 377
Sections (*), (4). Methods of squaring the circle
and trisecting any angle ..... 377-386
(n) The Archimedean spiral .... 377 379
l#) The conchoid of Nicomedes .... 379
(y) The Vita(?9-atri.r ... . 379 -382
(h) Digression: ;i. spiral on a sphere . . . 382 3S5
Triscction (or division in any ratio) of any angle 385-386
Section (5). Solution of the ufOcy of Archimedes,
On Sj)tfft/tt, Pro 1 1. S. by means of conies . . 386-388
Book V. Pief.ii-o on the sagacity of Hees . . 389-390
Section (1). Isoperimetry after Zeiiodoius . . 390 - 39.
Section (2). Compaiisoii of volumes of solids havin<c
their surfaces equal. Case of sphere . . . -9-l 394
Section (3). Di^re-sion on semi-regulai solids of
Archimedes ........ 394
Section (4) Propositions on the lines of Archimedes,
On the Sphere unit Cyfiiufer . . . . .394 395
Section (5). Of regular solids with surfaces equal,
that is greater which has more faces . . . 395-390
Hook VI 396-399
Problem arising out of Km-lid's O/tttr* . . 397-399
Book VI L On the 'Treasury of Analysis ' . 309-427
Dei] nit ion of Analysis and Synthesis . . . 400 401
List of works in the ' Treasury of Analysis' . 4U1
Description of the treatises .... .-401 404
Anticipation of (luldin's Theorem . . . 403
Lemmas to the different treatises .... 404 426
(ft) Lemmas to the Sect to nttioni* and Scctio
spat ii of Apollonius ..... 404 405
(ii) Lemmas to ihe J)tft rintinttt Srrtion of
Apollonius ....... 405 4P2
(y) Lemmas on the Nti'a^v of Apollonius . . tP2 410
(h) Lemmas on the On Contacts of Apollonius . 41(> 417
(f) Lemmas to the Wane Loci of Apollonius . 417 419
(f) Lemmas to the L'orisn^ of Kuclid . . . 419 424
\rj) Lemmas to the Conies of Apollonius . . 424 425
(^') Lemmas to the Surfac Loci of Kuelid . . 425-426
(t) An unallocated lemma, ..... 426 427
Hook VIII. Historical preface 427-429
The object of the Book 429-430
On the centre of gravity ..... 430-433
x ..CONTENTS
XIX. CONTINUED.
Book VIII (continued)
The inclined plane .... PAGES 433 434
Construction of a conic through five points . . 434-437
Given two conjugate diameters of an ellipse, to find
the axes 437 438
Problem of seven hexagons in a circle . . . 438 439
Construction of toothed wheels and indented screws 439
XX. ALGEBRA: DIOPHANTUS OF ALEXANDRIA . . 410 517
Beginnings learnt from Egypt ...... 440
* Hau '-calculations 440 441
Arithmetical epigrams in the Greek Anthology . . 441 443
Indetei urinate equations of first degree .... 443
Indeterminate equations of second degree beforo Dio-
phantus 443 444
Indeterminate equations in Heronian collections . . 444 447
Numerical solution of quadratic equations . . . 418
Works of Diophantus 44S 450
The Arithmetics 449 514
The seven lost Books and their place . . . 449 450
Relation of ' Porisms 1 to Arithmetica .... 451 452
Commentators from Ilypatia downwaids . . . 453
Translations and editions ...... 453 455
Notation and definitions 455461
Sign for unknown ( = x) and its origin . . . 450 457
Signs for powers of unknown &c 458 459
The sign (/1\) for minus and its meaning . . . 459-4GO
The methods of Diophantus 462-479
I. Diophantus's treatment of equations . . . 462 476
(A) Determinate equations
(1) Pure determinate equations . . . 462 463
(2) Mixed quadratic equations .... 463 465
(3) Simultaneousequationsinvolving quadratics 465
(4) Cubic equation 465
(B) Indeterminate equations
(a) Indeterminate equations of the second degree 466-473
(1) Single equation 466 468
(2) Double equation 468 473
1. Double equations of first degree . 469 472
2. Double equations of second degree 472-473
th) Indeterminate equations of degree higher
than second 473 476
(1) Single equations 473-475
(2) Double equations 475 476
II. Method of limits 476 477
III. Method of approximation to limits . , . . 477 479
Porisms and propositions in the Theory of Numbers . 479 4H4
(a) Theorems on tins composition of numbers as the
sum of two squares 481 483
(3) On numbers which are the sum of three squares . 483
(y) Composition of numbers as the sum of four squares 483-484
Conspectus of Arithmetica, with typical solutions . . 484-514
The treatise on Polygonal Numbers 514-517
CONTENTS xi
XXI. COMMENTATORS AND BYZANTINES . . PAGES 518 555
Sorenus 519 526
(a) On Mr Section of a Cylinder 519-522
(ft) On tJir Serf inn of a Cone 522-526
Theoii of Alexandria 526528
Commentary on the Si/nta^'fn ..... 526 527
Edition of Euclid's Elements 527528
Edition of the Oj/fics of Euclid 528
Ilypatia . 52K 529
Porphyry. lamblichus ....... 529
Proclus 529 537
Commentary on Euclid, Hook I . . . . . 530 535
(a) Sources of the Commentary .... 530 532
(tt) Character of the Commentary . . . .532535
Ift/potypotti* of Astronomical Hypothws . . . 535-536
Commentary on the Republic ..... 536 537
Marinus of Neapolis 537-538
Domninus of Larissa ..... . 538
Simplicius ......... 538 540
Extracts from Eudemus ...... 539
Eutocius ... 540 541
Antheming of Trail es ....... 541-543
On hurnhn/-niirront . . . . . . .541 543
The Papyrus of Akhimm . . . . . . .513 545
G< ortttcttiff of ' Heron the Younger '. . . . 545
Michael IVellux .... . . 545 546
lieorgius Pachymcres ....... 546
Maximus Planndes . . . - . . . . . 546 549
Extinction of the square root ..... 547-549
Two problems ........ 549
Manuel Moschopoulos ....... 549-550
Nii-ol.is Rhabdas 550 554
Rule for apiiroximalin^ to square root of a non-square
number ......... 553 554
loanne* Podiasinuis ........ 554
Barlaam . . . . . . . . . .554 555
Isaac Ar^yrus ......... 555
APPENDIX. On Archimedes*:* proof ot the subtungent-property
of a spiral .......... 556-561
INDEX OF GREEK WORDS 563-569
ENGLISH INDEX 570 586
XII
AB1STAUCHUS OF SAMOS
HISTORIANS of mathematics have, as a rule, given too little
attention to Aristarchus of Samos. Tlie reason is no doubt
that he was an astronomer, and therefore it might be supposed
that his work would have no sufficient interest for the mathe-
matician. The Greeks knew better; they called him Aristar-
ehus ' the mathematician ', to distinguish him from the host
of other Aristarchuses ; he is also included by Vitruvius
among the i'ew great men who possessed an equally profound
knowledge of all branches of science, geometry, astronomy,
music, &c.
' Men of this type- are rare, men such as were, in limes past,
Aristarchus of Samos, Philolaus and Arehytas of Tarentum,
Apollonius of Perga, Eratosthenes of Cyreiie, Archimedes and
Scopinas of Syracuse, who left to posterity many mechanical
and gnomonic* appliances which they invented and explained
on mathematical (lit. 'numerical') principles.' 1
That Aristarchus was a very capable geometer is proved by
his extant work On the shcs <in<l <li$t<t iiees of the Siai and
Moon which will be noticed later in this chapter: in the
mechanical line he is credited with the discovery of an im-
proved sun-dial, the so-called o-/c</>?;, which had, not a plane,
but a concave hemispherical surface, with a pointer erected
vertically in the middle throwing shadows and so enabling
the direction and the height of the sun to be read oft' by means
of lines marked on the surface of the hemisphere. He also
wrote on vision, light and colours. His views on the latter
subjects were no doubt largely influenced by his master, Strato
of Lampsacus; thus Strato held that colours were emanations
from bodies, material molecules, as it won*, which imparted to
the intervening air the same colour as that possessed by the
body, while Aristarchus said that colours are v shapes or forms
Vitruvius, DC architect lira, i. 1. 10.
1B28.2 Ji
2 ARISTARCHUS OF SAMOS
stamping the air with impressions like themselves, as it were ',
that ' colours in darkness have no colouring ', and that c light
is the colour impinging on a substratum '.
Two facts enable us to fix Aristarchus's date approximately.
In 281/280 B.C. he made an observation of the summer
solstice ; and a book of his, presently to be mentioned, was
published before the date of Archimedes's Psammites or Sand-
reckoner, a work written before 216 B.C. Aristarchus, there-
fore, probably lived circa 310-230 B.C., that is, he was older
than Archimedes by about 25 years.
To Aristarchus belongs the high honour of having been
the first to formulate the Copernican hypothesis, which was
then abandoned again until it was revived by Copernicus
himself. His claim to the title of ' the ancient Copernicus ' is
still, in my opinion, quite unshaken, notwithstanding the in-
genious and elaborate arguments brought forward by Schia-
parelli to prove that it was Heraclides of Pontus who first
conceived the heliocentric idea. Heraclides is (along with one
Ecphantus, a Pythagorean) credited with having been the iirst
to hold that the earth revolves about its own axis every 24
hours, and he was the first to discover that Mercury and Venus
revolve, like satellites, about the sun. But though this proves
that Heraclides came near, if he did not actually reach, the
hypothesis of Tycho Brahe, according to which the earth was
in the centre and the rest of the system, the sun with the
planets revolving round it, revolved round the earth, it does
not suggest that he moved the earth away from the centre.
The contrary is indeed stated by Acitius, who says that ' Hera-
clides and Ecphantus make the earth move, not in the sense of
translation, but by way of turning on an axle, like a wheel,
from west to east, about its own centre V None of the
champions of Heraclides have been able to moot this positive
statement. But we have conclusive evidence in favour of the
claim of Aristarchus ; indeed, ancient testimony is unanimous
on the point. Not only does Plutarch tell us that Cleanthes
held that Aristarchus ought to be indicted for the impiety of
' putting the Hearth of the Universe in motion ' 2 ; we have the
best possible testimony in the precise statement of a great
1 Aet. iii. 13. 3, Vors. i 3 , p. 341. 8.
2 Plutarch, De facie in orbe Iwiae, c. 6, pp. 922 F-923 A.
ARISTARCHUS OF SAMOS 3
contemporary, Archimedes. In the Sand-reckoner Archi-
medes has this passage.
1 You [King Celon] are aware that ''universe" is the name
given l>y most astronomers to the sphere the centre of which
is the cojitre of the earth, while its radius is equal to the
straight line between the centre of the sun and the centre of
the earth. This is the common account, as you have heard
from astronomers. But Aristarchus brought out a book con-
ninting of certain hypotheses, wherein it appears, as a conse-
quence of the assumptions made, that the universe is many
times greater than the " universe "just mentioned. His hypo-
theses are that the fixed stars and the HUH remain unmoved,
that the, earth revolves about the sun in the circumference of a
circle, the sun lying in the middle, of the orhit, and that the
sphere of the fixed stars, situated about the same centre as the
sun, is so great that the circle in which he supposes the earth
to revolve bears such a proportion to the distance of the fixed
stars as the centre of the sphere bears to its surface/
(The last statement is a variation of a traditional phrase, for
which there are many parallels (cf. Aristarchus' s Hypothesis 2
' that the earth is in the relation of a point and centre to the
sphere in which the moon moves'), and is a method of saying
that the * universe* is infinitely great in relation not merely to
the size of the sun but even to the orbit of the earth in its
revolution about it ; the assumption was necessary to Aris-
tarchus in order that he might not have to take account of
parallax.)
Plutarch, in the passage referred to above, also makes it
clear that Aristarchus followed Heraclides in attributing to
the earth the daily rotation about its axis. The bold hypo-
thesis of Aristarchus found few adherents. Seleucus, of
Seleucia on the Tigris, is the only convinced supporter of it of
whom we hear, and it was speedily abandoned altogether,
mainly owing to the groat authority of Hipparchus. Nor'do
we find any trace of the heliocentric hypothesis in Aris-
tarchus's extant work 0)i the sizes and distances of the
Hun and Moon. This is presumably because that work was
written before the hypothesis was formulated in the book
referred to by Archimedes. The geometry of the treatise
is, however, unaffected by the difference between the hypo-
theses.
B 2
4 ARISTARCHUS QF SAMOS
Archimedes also says that it was Aristarchus who dis-
covered that the apparent angular diameter of the sun is about
l/720th part of the zodiac circle, that is to say, half a degree.
We do not know how he arrived at this pretty accurate figure :
but, as he is credited with the invention of the cr/ca^T/, lie may
have used this instrument for the purpose. But here again
the discovery must apparently have been later than the trea-
tise On sizes ami distances, for the value of the subtended
angle is there assumed to be 2 (Hypothesis 6). How Aris-
tarchus came to assume a value so excessive is uncertain. As
the mathematics of his treatise is not dependent on the actual
value taken, 2 may have been assumed merely by way of
illustration ; or it may have been a guess at the apparent
diameter made before he had thought of attempting to mea-
sure it. Aristarchus assumed that the angular diameters of
the sun and moon at the centre of the earth are equal.
The method of the treatise* depends on the, just observation,
which is Aristarchus's third ' hypothesis ', that ' when the moon
appears to us halved, the great circle which divides the 1 , dark
and the bright portions of the moon is in the, direction of our
eye' ; the effect of this (since the moon receives its light from
the sun), is that at the time of the dichotomy the centres of
the sun, moon and earth form a triangle right-angled at the
centre of the moon. Two other assumptions were necessary :
first, an estimate of the size of the angle of the latter triangle,
at the centre of the earth at the moment of dichotomy : this
Aristarchus assumed (Hypothesis 4) to be, Mess than a quad-
rant by one-thirtieth of a quadrant', i.e. 87, again an inaccu-
rate estimate, the true value being 89 50' ; secondly, an esti-
mate of the breadth of the earth's shadow where the moon
traverses it: this he assumed to be 'the breadth of two
moons' (Hypothesis 5).
The inaccuracy of the assumptions docs not, however, detract
from the mathematical interest of the succeeding investigation.
Here we find the logical sequence of propositions and the abso-
lute rigour of demonstration characteristic of Greek geometry ;
the only remaining drawback would be, the practical difficulty
of determining the exact moment when the moon 'appears to
us halved'. The form and style, of the book are thoroughly
classical, as befits the period between Euclid and Archimedes ;
ARISTARCHUS OF SAMOS 5
the Greek is even remarkably attractive. The content from
the mathematical point of view is no less interesting, for we
have here the first specimen extant of pure geometry used
with a trigonometrical object, in which respect it is a sort of
forerunner of Archimedcs's Measurement of a Circle. Aristar-
chus does not actually evaluate the trigonometrical ratios
on which the ratios of the sixes and distances to be obtained
depend ; he finds limits between which they lie, and that by
means of certain propositions which he assumes without proof,
and which therefore must have been generally known to
mathematicians of his day. These propositions are the equi-
valents of the statements that,
(1) if a is what we call the circular measure of an angle
and a is less than \ TT, then the ratio sin y/oi decreases, and the
ratio tan a/cx increase*, as a increases from to TT ;
(2) if /3 lie the circular measure of another angle less than
\ TT, and a>/3, then
sin a 3 tan a
sin/^ < ft < tan/3'
Aristarchus of course deals, not with actual circular measures,
sines and tangents, but with angles (expressed not in degrees
but as fractions of right angles), arcs of circles and their
chords. Particular results obtained by Aristarchus are the
equivalent of the Following:
^ >sin3 > ,* , [Prop. 7]
4 \ >sinl> G 1 , [Prop, llj
1 >cosl > g, [Prop. 12J
1 >cosM> J-f. [Prop. 13]
The book consists of eighteen propositions. Beginning with
six hypotheses to the effect already indicated, Aristarchus
declares that he is now in a position to prove
(1) that the distance of the sun from the earth is greater than
eighteen times, but less than twenty times, the distance of the
moon from the earth ;
(2) that the diameter of the sun has the same ratio as afore-
said to the diameter of the moon;
6 ARISTARCHUS OF SAMOS
(3) that the diameter of the sun has to the diameter of the
earth a ratio greater than 19:3, but less than 43 : 6.
The propositions containing these results are Props. 7, 9
and 15.
Prop. 1 is preliminary, proving that two equal spheres are
comprehended by one cylinder , and two unequal spheres by
one cone with its vertex in the direction of the lesser sphere,
and the cylinder or cone touches the spheres in circles at
right angles to the line of centres. Prop. 2 proves that, if
a sphere be illuminated by another sphere larger than itself,
the illuminated portion is greater than a hemisphere. Prop. 3
shows that the circle in the moon which divides the dark from
the bright portion is least when the cone comprehending the
sun and the moon has its vertex at our eye. The ' dividing
circle ', as we shall call it for short, which was in Hypothesis 3
spoken of as a great circle, is proved in Prop. 4 to be, not
a great circle, but a small circle not perceptibly different
from a great circle. The proof is typical and is worth giving
along with that of some connected propositions (11 and 12).
B is the centre of the moon, A that of the earth, CD the
diameter of the ' dividing circle in the moon ', EF the parallel
diameter in the moon. BA meets the circular section of the
moon through A and EF in G, and CD in L. Gil, GK
are arcs each of which is equal to half the arc CE. By
Hypothesis 6 the angle CAD is ' one-fifteenth of a sign' = 2,
and the angle BAG = 1.
Now, says Aristarchus,
1:45[> tan 1: tan 45]
>BC:CA,
and, a fortiori,
BG:BA or BG:BA
< 1:45;
that is, BG< -
therefore, a fortiori,
Now
whence
ARISTARCHUS OF SAMOS
BIl :11A[ = sin HAB : sin 1IBA\
> LHAE-.LILEA,
L11AE< &LURA,
F D
and (taking the doubles) Z II A K < ^ Z II BK.
But Z HBK = Z EBG = ^ /* (where U is a right angle) ;
therefore Z //^i/f < ^ R.
But 'a magnitude (arc UK) seen under such an angle is
imperceptible to our eye ' ;
therefore, a fortiori, the arcs CE, DF are severally imper-
ceptible to our eye. Q. E. I).
An easy deduction from the same figure is Prop. 12, which
shows that the ratio of CD, the diameter of the c dividing
circle', to EF, the diameter of the moon, is < 1 but > f{J .
We have Z EBC = Z BAG = 1 ;
therefore (arc A T C f ) = / (arc AY?),
and accordingly (arc CO) : (arc fr'A') = 81) : 90.
Doubling the arcs, we have
(arc G(!D) : (arc EGF) = 89 : 90.
But CD : EF > (arc CGD) : (arc EGF)
[equivalent to sin a /sin /3 > a//3, where /.CBD = 2 a,
and 2 /S = TT] ;
therefore CW : EF [ = cos 1] > 89 : 1)0,
while obviously CD : EF < 1.
Prop. 11 finds limits to the ratio EF:BA (the ratio of the
diameter of the moon to the distance of its centre from
the centre of the earth) ; the ratio is < 2 : 45 but > 1 : 30.
8 AKISTARCHUS OF SAMOS
The first part follows from the relation found in Prop. 4,
namely BC:BA < 1 :45,
for EF=2BC.
The second part requires the use of the circle drawn with
centre A and radius AC. This circle is that on which the
ends of the diameter of the ' dividing circle ' move as the moon
moves in a circle about the earth. If r is the radius AC
of this circle, a chord in it equal to r subtends at the centre
A an angle of U or 60; and the arc CD subtends at A
an angle of 2.
But (arc subtended by CD) : (arc subtended by r)
< CD:r,
or 2 : GO < CD : r ;
that is, CD:CA > 1 :30.
And, by similar triangles,
GL:GA = OB:A, or GD:GA = 2CB-.HA = FKiBA.
Therefore FE:BA > 1 : 30.
The proposition which is of the greatest interest 011 the
whole is Prop. 7, to the effect that the distance of the xiin
from the earth is greater titan 18 tidies, but tess than k JO
times, the distance of the motiii from the earth. This result
represents a great improvement on all previous attempts to
estimate the relative distances. The first speculation on the
subject was that of Aiiaximander (circa 61 1-545 B.C.), who
seems to have made the distances of the sun and moon from
the earth to be in the ratio 3 : 2. Eudoxus, according to
Archimedes, made the diameter of the sun 9 times that of
the moon, and Phidias, Archimedes's father, 12 times; and,
assuming that the angular diameters of the; two bodies are
equal, the ratio of their distances would be the same.
Aristarchtis's proof is shortly as follows. A is the centre of
the sun, 7i that of the earth, and C that of the moon at the
moment of dichotomy, so that the angle ACIi is right. AllEF
is a square, and AE is a quadrant of the sun's circular orbit.
Join BF, and bisect the angle FJiK by EG, so that
ZG'BA'= \R or 22 J.
ARISTARCHUS OF SAMOS
I. Now, by Hypothesis 4, Z ABC = 87,
so that LUBE LBAC 3;
therefore Z GBE : L UBE =%R: -/ R
= 15:2,
A
so that (1E:11E[ = Ian (,'HE:tnn 1UM] > Z HUE: L HBE
> 15 :2. (1)
The ratio which lias lo bo proved > 18:1 is AB:B(J or
FK-.Kll.
Now FG:GE = FB:RE,
whence F({* : (!E~ = F1P : BE* = 2:1,
and FG : GE = V2 : 1
> 7:5
(this is the approximation to \/2 mentioned by Plato and
known to the Pythagoreans).
10 ARISTARCHUS OF SAMOS
Therefore FE : EG > 1 2 : 5 or 36 : 1 5.
Compounding this with (1) above, we have
FE<EH > 36:2 or 18:1.
II. To prove BA < 20 EC.
Let BH meet the circle AE in A and draw DK parallel
to EB. Circumscribe a circle about the triangle BK D, and
let the chord EL be equal to the radius (/) of the circle.
Now Z SDK = Z DBE = & R >
so that arc BK = ^ (circumference of circle).
Thus (arc BK) : (arc BL) = & : | ,
= 1 : 10.
And (arc BK ) : (arc BL) < BK : r
[this is equivalent to a//J < sin a/sin j8, where a < j8 < ^TT]*
so that r < lO^Jf,
and 7?D < 20 BK.
But BD:J3K = AB:BC',
therefore AB < 20 ^6 f . Q. E. 1).
The remaining results obtained in the treatise can be
visualized by means of the three figures annexed, which have
reference to the positions of the sun (centre A), the earth
(centre B) and the 1110011 (centre C) during an eclipse. Fig. I
shows the middle position of the moon relatively to the earth's
shadow which is bounded by the cone comprehending the sun
and the earth. ON is the arc with centre K along which
move the extremities of the diameter of the dividing circle in
the moon. Fig. 3 shows the same position of the moon in the
middle of the shadow, but on a larger scale. Fig. 2 shows
the moon at the moment when it has just entered the shadow ;
and, as the breadth of the earth's shadow is that of two moons
(Hypothesis 5), the moon in the position shown touches BN at
J^aiid BL at Z, where L is the middle point of the arc ON.
It should be added that, in Fig. 1, Z7Fis the diameter of the
circle in which the sun is touched by the double cone with B
as vertex, which comprehends both the sun and the moon,
ARISTARCHUS OF SAMOS
11
while Y, X are the points in which the perpendicular through
A, the centre of the sun, to BA meets the cone enveloping the
sun and the earth.
N
FIG. 1.
This being premised, the main results obtained are as
follows :
Prop. 13.
0)
but
ON : (diam. of moon) < 2 : 1
> 88:45.
12
ARISTARCHUS OF SAMOS
(2) 0-ZV:(diam. of sun) < 1 :9
but > 22 : 225.
(3) ON:YZ> 979:10125.
Prop. 14 (Fig. 3).
Prop. 15.
BC:CS> 675:1.
but
(Diaiu. of sun) : (diain. of earth) > 19 : 3
< 43:6.
B
Via. 2.
Prop. 17.
(Diam. of earth); (cliam. of moon) > 108 : 43
but <60:19.
It is worth while to show how these results are proved.
Prop. 13.
(1) In Fig. 2 it is clear that
ON < 2/,iVand, a fortiori, < 2/,P.
The triangles LON, CLN being similar,
therefore
ON:NL = NL : | LP
> 89 : 45. (by Prop. 1 2)
ARISTARCHUS OF SAMOS 13
Hence ON : LC = ON 2 : NL*
> 89 2 :45 2 ;
therefore ON: LP > 7921 : 4050
> 88 : 45, says Aristarchus.
[If '4050 * )(> Developed as a continued fraction, we easily
1 11 88 1
obtain 1 + ^, which is in tact -
(2) ON < 2 (diam. of moon).
But (diam. of moon) < - g (diam. of sun) ; (Prop. 7)
therefore ON < | (diam. of sun).
Again ON: (diam. of moon) > 88:45, from above,
and (diam. of moon) : (diam. of sun) > 1 : 20 ; (Prop. 7)
therefore, ex ae<jiiali,
ON: (diam. of sun) > 88 : 000
> 22 : 225.
(3) Siii<v tht* same conn comprehends the sun and the moon,
the triangle HUV (Fig. 1) and the triangle BLN (Fig. 2) are
similar, and
LN:LP = UV: (diam. of sun)
= WI T :UA
= UA:AS
< CA:AY.
But LN: LP > 89 : 1)0 ; (Prop. 12)
thrrefon*, </ fortiori, UA : A Y > 89 : 90.
And UA : A V = 2 [7^1 : F^T
= (diam. of sun) : YZ.
Jiut OiV: (diam. of sun) > 22 : 225 ; (Prop. 1 3)
therefore, ex a equal i,
ON: YZ > 89 x 22 : 90 x 225
> 979:10125.
14 ARISTARCHUS OF SAMOS
Prop. 14 (Fig. 3).
The arcs OM, ML, LP, PN are all equal ; therefore so are
the chords. EM, BP are tangents to the circle MQP, so that
CM is perpendicular to BM, while BM is perpendicular to ()L.
Therefore the triangles LOS, GMR are similar.
Therefore SO : MR = SL :
But 80 < 2 MR, since ON < 2MP: (Prop. 13)
therefore SL < 2 RC,
and, a fortiori, SR<2 RC, or SC < 3 RC,
that is, CR:CS>1 :3.
Again, MC : CR = BC : CM
> 45: 1 ; (see Prop. 11)
therefore, ex aequ-ali,
CM:C8> 15:1.
And BC:CM > 45 : 1 ;
therefore BCiVS > 675 : 1.
Prop. 15 (Fig. 1).
We have NO : (diam. of sun) < 1 : 9, (Prop. 13)
and, a fortiori, YZ : NO > 9 : 1 ;
therefore, by similar triangles, if Y(), ZN meet in X,
AXiXR > 9:1,
and convertendo, XA :AR< 9:8.
But AB > \$BC, and, a fortiori, > 18 BR,
whence AB > \%(AR-AB), or 19 AB > \%AR;
that is, AR:AB < 19:18.
Therefore, ex aequali,
XA:AB< 19:16,
and, convertendo, AX : XB > 19:3;
therefore (diam. of sun) : (diam. of earth) > 19:3.
Lastly, since OB : CR > 675 : 1, (Prop. 1 4)
OBiBR < 675:674.
ARISTA ECHUS OF SAMOS 15
But AB-.BC < 20:1;
therefore, ex aequali,
AB.BR < 13500: 674
< 6750:337,
whence, by inversion anr] componendo,
RAiAB > 7087:6750. (1)
But AXiXR = YZ-.NO
< 10125:979; (Prop. 13)
therefore, conrertendo,
XA :AR> 10125:9146.
From this and (1) we have, ex aerjiutli,
XA :AI1> 101 25 X 7087: 9146 X 6750
> 71755875 : 61735500
> 43 : 37, a fortiori.
[It is difficult not to see in 43 : 37 the expression 1 -h ,
6 -f- ft
which suggests that 43 : 37 was obtained by developing the
ratio as a continued fraction.]
Therefore, conrertendo,
XA :XB < 43:6,
whence (diaiu. of sun) : (diain. of earth) < 43 : 6. Q. E. D.
XIII
ARCHIMEDES
THE siege and capture of Syracuse by Marcellus during the
second Punic war furnished the occasion for the appearance of
Archimedes as a personage in history; it is witli this histori-
cal event that most of the detailed stories of him are con-
nected ; and the fact that he was killed in the sack of the city
in 212 B.C., when he is supposed to have been 75 years of age,
enables us to fix his date at about 287-212 B.C. He was the
son of Phidias, the astronomer, and was on intimate terms
with, if not related to, King Hicron and his son Gelon. It
appears from a passage of Diodorus that he spent some* time
in Egypt, which visit was the occasion of his discovery of the
so-called Archimedean screw as a means of pumping water. 1
It may be inferred that he studied at Alexandria with the
successors of Euclid. It was probably at Alexandria that he
made the acquaintance of COIIOTI of Samos (for whom he had
the highest regard both as a mathematician and a friend) and
of Eratosthenes of Gyrene,. To the former he was in the habit
of communicating his discourses before their publication ;
while it was to Eratosthenes that he sent The Method, with an
introductory letter which is of the, highest interest, as well as
(if we may judge by its heading) the famous Cattle-Problem.
Traditions.
It is natural that history or legend should say more of his
mechanical inventions than of his mathematical achievements,
which would appeal less to the average mind. His machines
were used with great effect against the Romans in the siege
of Syracuse. Thus he contrived (so we are told) catapults so
ingeniously constructed as to be equally serviceable at long or
1 Diodorus, v. 37. 3.
TRADITIONS 17
short range, machines for discharging showers of missiles
through holes made in the walls, and others consisting of
long movable poles projecting beyond the walls which either
dropped heavy weights on the enemy's ships, or grappled,
tkeir prows by means of an iron hand or a beak like that of
a crane, then lifted them into the air and let them fall again. 1
Marcelhis is said to have derided his own engineers with the
words, ' Shall we not make an end of fighting against this
geometrical Briareus who uses our ships like cups to ladle
water from the sea, drives off our sambuca ignominiously
witli cudgel-blows, and by the multitude of missiles that he
hurls at us all at once outdoes the hundred-handed giants of
mythology?'; but all to no purpose, for the Romans were in
such abject terror that, 'if they did but see a piece- of rope
or wood projecting abow the wall, they would cry "there it
is", declaring that Archimedes was setting some engine in
motion against them, and would turn their backs and run
away Y 2 These things, however, were merely the ' diversions
of geometry at play ', :? and Archimedes himself attached no
importance to them. According to Plutarch,
' though these inventions had obtained for him the renown of
more than human sagacity, he yet would not even deign to
leave behind him any written work on such subjects, but,
regarding as ignoble and sordid the business of mechanics and
every sort of art which is directed to use and profit, he placed
his whole ambition in those speculations the beauty and
subtlety of which is untainted by any admixture of the com-
mon needs of life/ 4
(a) Astronomy.
Archimedes did indeed write one mechanical book, On
Sphere-making, which is lost; this described the construction
of a spheiv to imitate the motions of the sun, moon and
planets/' Cicero saw this contrivance and gives a description
of it ; he says that it represented the periods of the moon
and the apparent motion of the sun with such accuracy that
it would even (over a short period) show the eclipses of the
sun and moon/ 1 As Pappus speaks of ' those who understand
1 Polybius, Hist. viii. 7, 8 ; Livy xxiv. 34 ; Plutarch, Marcellus, cc. 15-17.
* lh., c. 17. 3 /&., c. 14. 4 /&., c. 17.
5 Carpus in Pappus, viii, p. 1026. 9; Proclus on Eucl. I, p. 41. 16.
6 Cicero, De rep. i. 21, 22, Tu*c. i. 63, De nat* rfeor. ii. 88.
1628.2 C
18 ARCHIMEDES
the making of spheres and produce a model of the heavens by
means of the circular motion of water', it is possible that
Archirnedes's sphere was moved by water. In any case Archi-
medes was much occupied with astronomy. Livy calls him
' unicus spectator caeli siderumque '.* Hipparchus says, ' From
these observations it is clear that the differences in the years
are altogether small, but, as to the solstices, I almost think
that Archimedes and I have both erred to the extent of a
quarter of a day both in the observation and in the deduction
therefrom'. 2 Archimedes then had evidently considered the
length of the year. Macrobius says he discovered the dis-
tances of the planets, 3 and he himself describes in his Sand-
reckoner the apparatus by which he measured the apparent
angular diameter of the sun.
(/3) Mechanic*.
Archimedes wrote, as wo shall see, on theoretical mechanics,
and it was by theory that he solved the problem To 'move a
given weight by a tjiven force, for it was in reliance ' on the
irresistible cogency of his proof 'that he declared to Hieron
that any- given weight could be moved by any given force;
(however small), and boasted that, fi if he were given a place to
stand on, he could move the earth ' (na /3o>, KOL KLV> rav ydv,
as he said in his Doric dialect). The story, told by Plutarch,
is that, 'when Hieron was struck with amazement and asked
Archimedes to reduce the problem to practiee and to give an
illustration of some great weight moved by a small force, he-
fixed upon a ship of burden with three masts from the king's
arsenal which had only been drawn up with great labour by
many men, and loading her with many passengers and a full
freight, himself the while sitting far off, with no great effort
but only holding the end of a compound pulley (rro\v(nra(TTos)
quietly in his hand and pulling at it, he drew the ship along
smoothly and safely as if she were moving through the sea.' 4
The story that Archimedes set the Roman ships on fire by
an arrangement of burning-glasses or concave mirrors is not
found in any authority earlier than Lucian ; but it is quite
1 Livy xxiv. 34. 2. 2 Ptolemy, Syntaxis, 111. 1, vol. i, p. 194. 23.
3 Maorobius, In Somn. Scip. ii. 3 ; cf. the figures in Tlippolytus, Refnt.,
p. 66. 52 sq., ed. Duncker.
4 Plutarch, Marcellus, c. 14.
MECHANICS 19
likely that he discovered some form of burning-mirror, e.g. a
paraboloid of revolution, which would reflect to one point all
rays falling on its concave surface in a direction parallel to
its axis.
Archimedes's own view of the relative importance of his
many discoveries is well shown by his request to his friends
and relatives that they should place upon his tomb a represen-
tation of a cylinder circumscribing a sphere, with an inscrip-
tion giving the ratio which the cylinder bears to the sphere ;
from which we may infer that he regarded the discovery of
this ratio as his greatest achievement. Cicero, when quaestor
in Sicily, found the tomb in a neglected state and repaired it 1 ;
but it has now disappeared, and no one knows whew, he was
buried.
Archimedes's entire preoccupation by his abstract studies is
illustrated by a number of stories. We are told that he would
forget all about his food and such necessities of life, and would
bo drawing geometrical figures in the ashes of the fire or, when
anointing himself, in the oil on his body. 2 Of the same sort
is the tale that, when lie discovered in a bath the solution of
the question referred to him by Hieron, as to whether a certain
crown supposed to haw been made of gold did not in fact con-
tain a certain proportion of silver, he ran naked through the
street to his home shouting tvpjjKa, tvprjKa* He was killed
in the sack of Syracuse by a Roman soldier. The story is
told in various forms; the, most picturesque is that found in
Txet/es, which represents him as saying to a Roman soldier
who found him intent on some diagrams which he had drawn
in the dust and came too close, ' Stand away, fellow, from my
diagram ', whereat the man was so enraged that he killed
him. 4
Summary of main achievements.
In geometry Archimedes's work consists in the main of
original investigations into the quadrature of curvilinear
plane figures and the, quadrature and cubature of curved
surfaces. These investigations, beginning where Euclid's
Book XII left off, actually (in the words of Chaslea) ' gave
1 Cicero, TM.W. v. fi4 sq. 2 Tlutarcli, MarctUus, c. 17.
"' Vitrnvius, De architectum, ix. 1. 9, 10.
4 Tzetzes, Chiliad, ii. 35. 135.
c 2
20 ARCHIMEDES
birth to the calculus of the infinite conceived and brought to
perfection successively by Kepler, Cavalieri, Fermat, Leibniz
and Newton'. He performed in fact what is equivalent to
integration in finding the area of a parabolic segment, and of
a spiral, the surface and volume of a sphere and a segment of
a sphere, and the volumes of any segments of the solids of
revolution of the second degree. In arithmetic he calculated
approximations to the value of TT, in the course of which cal-
culation he shows that he could approximate to the value of
square roots of large or small non-square numbers ; he further
invented a system of arithmetical terminology by which he
could express in language any number lip to that which wo
should write down with 1 followed by 80,000 million million
ciphers. In mechanics he not only worked out the principles of
the subject but advanced so far as to find the centre of gravity
of a segment of a parabola, a semicircle, a cone, a hemisphere,
a segment of a sphere, a right segment of a paraboloid and
a spheroid of revolution. His mechanics, as we shall see, lias
become more important in relation to his geometry since the
discovery of the treatise called The Method which was formerly
supposed to be lost. Lastly, he invented the whole science of
hydrostatics, which again he carried so far as to give a most
complete investigation of the positions of rest and stability of
a right segment of a paraboloid of revolution floating in a
fluid with its base either upwards or downwards, but HO that
the base is either wholly above or wholly below the surface of
the fluid. This represents a sum of mathematical achieve-
ment unsurpassed by any one man in the world's history.
Character of treatises.
The treatises are, without exception, monuments of mathe-
matical exposition; the gradual revelation of the plan of
attack, the masterly ordering of the propositions, the stern
elimination of everything not immediately relevant to the
purpose, the finish of the whole, are so impressive in their
perfection as to create a feeling akin to awe in the mind of
the reader. As Plutarch said, 'It is not possible to find in
geometry more difficult and troublesome questions or proofs
set out in simpler and clearer propositions'. 1 There is at the
1 Plutarch. Marcellus, c. 17.
CHARACTER OF TREATISES 21
same time a certain mystery veiling the way in which he
arrived at his results. For it is clear that they were not
discovered by the steps which lead up to them in the finished
treatises. If the geometrical treatises stood alone, Archi-
medes might seem, as Wallis said, ' as it were of set purpose
to have covered up the traces of his investigation, as if lie had
grudged posterity the secret of his method of inquiry, while
lie wished to extort from them assent to his results'. And
indeed (again in the words of Wallis) 'not only Archimedes
but nearly all the ancients so hid from posterity their method
of Analysis (though it is clear that they had one) that more
modern mathematicians found it easier to invent a new
Analysis than to seek out the old'. A partial exception is
now furnished by The Method of Archimedes, so happily dis-
covered by Heiberg. In this book Archimedes tells us how
h(3 discovered certain theorems in quadrature and cubature,
namely by the use of mechanics, weighing elements of a
figure against elements oi' another simpler figure the mensura-
tion of which was already known. At the same time he is
careful to insist on the difference between (1) the means
which may be sufficient to suggest the truth of theorems,
although not furnishing scientific proofs of them, and (2) the
rigorous demonstrations of them by orthodox geometrical
methods which must follow before they can be finally accepted
us established :
'certain things', lie says, 'first became clear to me by a
mechanical method, although they had to be demonstrated by
geometry afterwards because their investigation by the said
method did not furnish an actual demonstration. But it is
of course easier, when we have previously acquired, by the
method, some knowledge of the questions, to supply the proof
than it is to find it without any previous knowledge/ 'This',
he adds, 4 is a reason why, in the case of the theorems that
the, volumes of a cone and a pyramid are one-third of the
volumes of the cylinder and prism respectively having the
same base and equal height, the proofs of which Eudoxus was
the first to discover, no small share 1 of the credit should be
given to Democritus who was the first to state the fact,
though without proof.'
Finally, he says that the very first theorem which he found
out by means of mechanics was that of the separate treatise
22 ARCHIMEDES
on the Quadrature of the parabola, namely that the area of any
segment of a section of a right-angled cone (i.e. a parabola] is
four-thirds of that of the triangle which has the same base and
height. The mechanical proof, however, of this theorem in the
Quadrature of the Parabola is different from that in the
Method, and is more complete in that the argument is clinched
by formally applying the method of exhaustion.
List of works still extant.
The extant works of Archimedes in the order in which they
appear in Heiberg's second edition, following the order of the
manuscripts so far as the first seven treatises are concerned,
are as follows :
(5) On the Sphere and Cylinder: two Books.
(9) Meamreineiitufa Circle.
(7) On Conoids and fyheroidt*.
(6) On Mpiral*.
(1) On Plane Equilibrium*, Book I.
(3) Book II.
(10) The Hand- reckoner (Psammites).
(2) Quadrature of the Parabola.
(8) On Floating Bodlex: two Books.
? titovutchion (a fragment).
(4) The Method.
This, however, was not the order of composition; and,
judging (a) by statements in Archimedes's own prefaces to
certain of the treatises and (b) by the use in certain treatises
of results obtained in others, wo can make out an approxi-
mate chronological order, which I have indicated in the above
list by figures in brackets. The treatises On Floating Kodiw
was formerly only known in the Latin translation by William
of Moerbeke, but the Greek text of it has now been in great
r")
part restored by Heiberg from the. Constantinople manuscript
which also contains The Method and the fragment of the
Otowutchion.
Besides these works we have a collection of propositions
(Liber assumptorum) which has reached us through the
Arabic. Although in the title of the translation by Thabit b.
LIST OF EXTANT WORKS 23
Qurra the book is attributed to Archimedes, the propositions
cannot be his in their present form, since his name is several
times mentioned in them ; but it is quite likely that some
of them are of Archimedean origin, notably those about the
geometrical figures called ap/^Aoy ('shoemaker's knife') and
(rd\ivw (probably ' salt-cellar ') respectively and Prop. 8 bear-
ing on the trisection of an angle.
There is also the Cattle- Problem in epigrammatic form,
which purports by its heading to have been communicated by
Archimedes to the mathematicians at Alexandria in a letter
to Eratosthenes. Whether the epigrammatic form is due to
Archimedes himself or not, there is 110 sufficient reason for
doubting the possibility that the substance of it was set as a
problem by Archimedes.
Traces of lost works.
Of works which are lost we have the following traces.
1. Investigations relating to polyltnlra are referred to by
Pappus who, after alluding to the five regular polyhedra,
describes thirteen others discovered by Archimedes which are
semi-regular, being contained by polygons equilateral and
equiangular but not all similar. 1
2. Then; was a book of arithmetical content dedicated to
Xeuxippus. We learn from Archimedes himself that it dealt
with the nainiiuj of nuuilu'rx (/earoi/o/za*9 rS>v dpidfjL>v)'* and
expounded the system, which we find in the tfa ad-reckoner, of
expressing numbers higher than those which could be written
in the ordinary Greek notation, numbers in fact (as we have
said) up to the enormous tigure represented by 1 followed by
80,000 million million ciphers.
3. One or more works on mechanics are alluded to contain-
ing propositions not included in the extant treatise On Plane
KquttiLriwfUH. Pappus mentions a, work n Balu nces or Levers
(irtpl vy$>v) in which it was pro VIM I (as it also was in Philon's
and Heron's Mechanic*) that ' greater circles overpower lesser
circles when they revolve about the same centre V Heron, too,
speaks of writings of Archimedes * which bear the title of
1 Pappus, v, pp. 352 8.
2 Archimedes, vol. ii, pp. 1216. 18, 236. 17 22 ; rf. p. 220. 4.
3 Pappus, viii, p. 1068.
24 ARCHIMEDES
" works on the lever " f . 1 Simplicius refers to problems on Ike
centre of gravity, KevrpoftapiKd, such as the many elegant
problems solved by Archimedes and others, the object of which
is to show how to find the centre of gravity, that is, the point
in a body such that if the body is hung up from it, the body
will remain at rest in any position. 2 This recalls the assump-
tion in the Quadrature of the Parabola (6) that, if a body hangs
at rest from a point, the centre of gravity of the body and the
point of suspension are in the same vertical line. Pappus lias
a similar remark with reference to a point of support, adding
that the centre of gravity is determined as the intersection of
two straight lines in the body, through two points of support,
which straight lines are vertical when the body is in equilibrium
so supported. Pappus also gives the characteristic of the centre
of gravity mentioned by Simplicius, observing that this is
the most fundamental principle of the theory of the centre of
gravity, the elementary propositions of which are found in
Archimedes's On Equilibriums (Trtpl i<roppo7ri>i') and Heron's
Mechanics. Archimedes himself cites propositions which must
have been proved elsewhere, e.g. that the centre of gravity
of a cone divides the axis in the ratio 3:1, the longer segment
being that adjacent to the vertex 3 ; he also says that l it is
proved in the Equilibriums ' that the centre of gravity of any
segment of a right-angled conoid (i. e. paraboloid of revolution)
divides the axis in such *a way that the portion towards the
vertex is double of the remainder. 4 It is possible that there
was originally a larger work by Archimedes On KrjuilUtriuinH
of which the surviving books On Plane Equilibriums formed
only a part ; in that case irepl vyS>v and KtvTpofHapiKd may
only be alternative titles. Finally, Heron says that Archi-
medes laid down a certain procedure in a book bearing the
title ' Book on Supports *. 6
4. Theon of Alexandria quotes a proposition from a work
of Archimedes called Catoptrica (properties of mirrors) to the
effect that things thrown into water look larger and still
larger the farther they sink. G Olympiodorus, too, mentions
Heron, Mechanics, i. 32.
Simpl. on Arist. De caelo, ii, p. 508 a 30, Brandis ; p. 543. 24, Heib.
Method, Lemma 10. 4 On Floating Bodies, ii. 2.
Heron, Mechanics, i. 25.
Theon on Ptolemy's Syntaxis, i, p. 29, Halma.
TRACES OF LOST WORKS 25
that Archimedes proved the phenomenon of refraction 'by
means of the ring placed in the vessel (of water) '.* A scholiast
to the Pseudo-Euclid's Catoptrica quotes a proof, which he
attributes to Archimedes, of the equality of the angles of
incidence and of reflection in a mirror.
The text of Archimedes.
Heron, Pappus and Theon all cite works of Archimedes
which no longer survive, a fact which shows that such works
were still extant at Alexandria as late as the third and fourth
centuries A.D. But it is evident that attention came to be
concentrated on two works only, the Measurement of a Circle
and On the Sphere and Cylinder. Eutocius (fl. about A.D. 500)
only wrote commentaries on these works and on the Plane
Equilibriums, and he does not seem even to have been
acquainted with the (Quadrature of tfie Parahola or the work
On Spirals, although these have survived. Isidorus of Miletus
revised the commentaries of Eutocius on the Measurement
of (t (Circle and the two Books On the ti[)ltere and Cylinder,
and it would soem to have been in the school of Isidorus
that these treatises were turned from their original Doric
into the ordinary language, with alterations designed to make
them nurfe intelligible to elementary pupils. But neither in
Isidorus's time nor earlier was there, any collected edition
of Archimedes's works, so that those which were less read
tended to disappear.
In the ninth century Leon, who restored the University
of Constantinople, collected together all the works that he
could find at Constantinople, and had the manuscript written
(the archetype, lleiberg's A) which, through its derivatives,
was, up to the discovery of the Constantinople manuscript (C)
containing Tlie Method, the only source for the Greek text.
Leon's manuscript came, in the twelfth century, to the
Norman Court at Palermo, and thence passed to the House
of Hohenstaufen. Then, with all the library of Manfred, it
was given to the Pope by Charles of Anjou after the battle
of Benevento in 1266. It was in the Papal Library in the
years 1269 and 1311, but, some time after 1368, passed into
1 Olympiodoma on Arist. Meteorofayica, ii, p. 94, Ideler ; p. 211.18,
Busse.
26 ARCHIMEDES
private hands. In 1491 it belonged to Georgius Valla, who
translated from it the portions published in his posthumous
work De expetendis et fugiendis rebus (1501), and intended to
publish the whole of Archimedes with Eutocius's commen-
taries. On Valla's death in 1500 it was bought by Alberius
Pius, Prince of Carpi, passing in 1530 to his nephew, Rodolphus
Pius, in whose possession it remained till 1544. At some
time between 1544 and 1564 it disappeared, leaving no
trace.
The greater part of A was translated into Latin in 1269
by William of Moerbeke at the Papal Court at Viterbo. This
translation, in William's own hand, exists at Rome (Cod.
Ottobon. lat. 1850, Heiberg's B), and is one of our prime
sources, for, although the translation was hastily done qnd
the translator sometimes misunderstood the Greek, he followed
its wording so closely that his version is, for purposes of
collation, as good as a Greek manuscript. William used also,
for his translation, another manuscript from the same library
which contained works not included in A. This manuscript
was a collection of works on mechanics and optics ; William
translated from it the two Books On Floating Bodies, and it
also contained the Plane Equilibriums and the Quadrature
of the Parabola, for which books William used both manu-
scripts.
The four most important extant Greek manuscripts (except
C, the Constantinople manuscript discovered in 1906) were
copied from A* The earliest is E, the Venice manuscript
(Marcianus 305), which was written between the years 1449
ai\d 1472. The next is D, the Florence manuscript (Laurent.
XXVIII. 4), which was copied in 1491 for Angelo Poliziano,
permission having been obtained with some difficulty in con-
sequence of the jealousy with which Valla guarded his treasure.
The other two are G (Paris. 2360) copied from A after it had
passed to Albertus Pius, and H (Paris. 2361) copied in 1544
by Christopherus Auverus for Georges d'Armagnac, Bishop
of Rodez. These four manuscripts, with the translation of
William of Moerbeke (B), enable the readings of A to be
inferred.
A Latin translation was made at the instance of Pope
Nicholas V about the year 1450 by Jacobus Cremonensis.
THE TEXT OF ARCHIMEDES 27
It was made from A, which was therefore accessible to Pope
Nicholas though it does not seem to have belonged to him.
Regiomontanus made a copy of this translation about 1468
and revised it with the help of E (the Venice manuscript of
tlie Greek text) and a copy of the same translation belonging
to Cardinal Bessarion, as well as another 'old copy' which
seems to have been B.
The editio prinwps was published at Basel (apad Herva-
gium) by Thomas Gechauff Venatorius in 1544. The Greek
text was based on a Niirnberg MS. (Norirnberg. Cent. V,
app. 12) which was copied in the sixteenth century from A
but witli interpolations derived from B; the Latin transla-
tion was Regiomontamis's revision of Jacobus Cremonensis
(Norimb. Cent. V, 15).
A translation by F. Commandinus published at Venice in
1558 contained the Measurement of a Circle, On Spirals, the
Quadrature of the Parabola, On do no ids and fyke raids, and
the tia nd-reckoner. This translation was based] on the Basel
edition, but Commandinus also consulted E and other Greek
manuscripts.
Torelli's edition (Oxford, 1792) also followed the editio
{>ri itcep** in the main, but Tore Hi also collated E. The book
was brought out after Torelli's death by Abram Robertson,
who also collated five more manuscripts, including I), G
and II. The collation, however, was not well done, and the
edition was not properly corrected when in the press.
The second edition of Hei berg's text of all the works of
Archimedes with Eutocius's commentaries, Latin translation,
apparatus criticus, &c., is now available (1910-15) and, of
course, supersedes the first edition (1880-1) and all others.
It naturally includes The Method, the fragment of the tftoma-
'chion, and so much of the Greek text of the two Books On
FloatiiKj Bodies as could be restored from the newly dis-
covered Constantinople manuscript. 1
Contents of Tlw. Method.
Our description of the extant works of Archimedes
may suitably begin with The, Method (the full title is On
1 The Works of Archimedes, edited in modern notation by the present
writer in 1897, was based on Heiberg's first edition, and the Supplement
28 ARCHIMEDES
Mechanical Theorems, Method (communicated) to Eratosthenes).
Premising certain propositions in mechanics mostly taken
from the Plane Equilibriums, and a lemrna which forms
Prop. 1 of On Conoids and Spheroids, Archimedes obtains by
his mechanical method the following results. The area of any
segment of a section of a right-angled cone (parabola) is f of
the triangle with the same base and height (Prop. 1). The
right cylinder circumscribing a sphere or a spheroid of revolu-
tion and with axis equal to the diameter or axis of revolution
of the sphere or spheroid is 1^ times the sphere or spheroid
respectively (Props. 2, 3). Props. 4, 7, 8, 11 find the volume of
any segment cut off, by a plane at right angles to the axis,
from any right-angled conoid (paraboloid of revolution),
sphere, spheroid, and obtuse-angled conoid (hyperboloid) in
terms of the cone which has the same base as the segment and
equal height. In Props. 5, 6, 9, 10 Archimedes uses his ftiethod
to find the centre of gravity of a segment of a paraboloid of
revolution, a sphere, and a spheroid respectively. Props.
12-15 and Prop. 16 are concerned with the cubature of two
special solid figures. (1) Suppose a prism witji a square base
to have a cylinder inscribed in it, the circular bases of the
cylinder being circles inscribed in the squares which are
the bases of the prism, and suppose a plane drawn through
one side of one base of the prism and through that diameter of
the circle in the opposite base which is parallel to the said
side. This plane cuts off a solid bounded by two planes and
by part of the curved .surface of the cylinder (a solid shaped
like a hoof cut off by a plane); and Props. 12-15 prove that
its volume is one-sixth of the volume of the prism. (2) Sup-
pose a cylinder inscribed in a cube, so that the circular bases
of the cylinder are circles inscribed in two opposite faces of
the cube, and suppose another cylinder similarly inscribed
with reference to two other opposite faces. The two cylinders
enclose a certain' solid which is actually made up of eight
'hoofs' like that of Prop. 12. Prop. 16 proves that the
volume of this solid is two-thirds of that of the cube. Archi-
medes observes in his preface that a remarkable fact about
(1912) containing The Method, on the "original edition of Heiberg (in
THE METHOD 29
those solids respectively is that each of them is equal to a
solid enclosed by planes, whereas the volume of curvilinear
solids (spheres, spheroids, &c.) is generally only expressible in
terms of other curvilinear solids (cones and cylinders). In
accordance with his dictum that the results obtained by the
mechanical method are merely indicated, but not actually
proved, unless confirmed by the rigorous methods of pure
geometry, Archimedes proved the facts about the two last-
named solids by the orthodox method of exhaustion as
regularly used by him in his other geometrical treatises ; the
proofs, partly lost, were given in Props. 15 and 16.
We will first illustrate the method* by giving the argument
of Froj). 1 about the area of a parabolic segment.
Lot AB(! be the segment, BD its diameter, (-F the tangent
at (\ Lot P 1)0 any point on the segment, and lot AKF,
M
OPNM bo drawn parallel to BI). Join ( 1 B and produce it to
meet MO in N and FA in K, and lot KH bo made equal to
K(\
Now, by a proposition ' proved in a lemma ' (cf. Quadrature
of the Parabola, Prop. 5)
= CA :A()
= CK : KN
Also, by tho property of the parabola, EB = BD, so that
It follows that, if HC be regarded as tho bar of a balance,
a lino TG equal to PO and placed with its middle point at //
balances, about K, the straight line MO placed where it is,
i.e. with its middle point at N.
Similarly with all linos, as MO, PO, in the triangle CFA
and the segment CBA respectively.
And there are the same number of these lines. Therefore*
30 ARCHIMEDES
the whole segment of the parabola acting at H balances the
triangle CFA placed where it is.
But the centre of gravity of the triangle CFA is at TT,
where CW = 2 WK [and the whole triangle may be taken as
acting at W],
Therefore (segment ABC) : &CFA = WK : KH
= 1:3,
so that (segment ABC) = &CFA
Q.E.D.
It will be observed that Archimedes takes the segment and
the triangle to be made up of parallel lines indefinitely close
together. In reality they are made up of indefinitely narrow
strips, but the width (dx, as we might say) being the same
for the elements of the triangle and segment respectively,
divides out. And of course the weight of each element in
both is proportional to the area. Archimedes also, without
mentioning momenta, in effect assumes that the sum of the
moments of each particle of a figure, acting where it is, is
equal to the moment of- the whole figure applied as one mass
at its centre of gravity.
We will now take the case of any segment of a spheroid
of revolution, because that will cover several propositions of
Archimedes as particular cases.
The ellipse with axes AA', BB' is a section made by the
plane of the paper in a spheroid with axis A A'. It is require* 1
to find the volume of any right segment AD(! of the spheroid
in terms of the right cone with the same base and height.
Let DC be the diameter of the circular base of the segment.
Join AB y AB', and produce them to meet the tangent at A' to
the ellipse in K, K', and DC produced in E, F.
Conceive a cylinder described with axis AA f and base the
circle on KK f as diameter, and cones described with AG as
axis and bases the circles on EF, DC as diameters.
Let N be any point on AG, and let MOPQNQ'P'O'M' be
drawn through N parallel to BB' or DC as shown in the
figure.
Produce A' A to H so that HA = A A'.
Now
THE METHOD
HA : AN = A' A: AN
= KA:AQ
= MN:NQ
31
It is now necessary to prove that MN . NQ = NP 2 + NQ*.
H
M
W
G C
0'
B'
K A' K'
By the property of the ellipse,
AN.NA': NP 2 = (| A A')'* : (% BB'f
therefore
whence ^V7 )2 = MQ . QN.
Add NQ Z to each side, and we have
XQ* : Nl = A X' 2 :AN. NA'
Therefore, from above,
HA : AN = MN 2 : (NP* + NQ*). (1)
But MN*, NP 2 , NQ 2 are to one another as the areas of the
circles with MM', PP', QQ' respectively as diameters, and these
32 ARCHIMEDES
circles arc sections made by the plane though iV at right
angles to AA f in the cylinder, the spheroid and the cone AEF
respectively.
Therefore, if HA A' be a lever, and the sections of the
spheroid and cone be both placed witli their centres of gravity
at JET, these sections placed at II balance, about A, the section
MM' of the cylinder where it is.
Treating all the corresponding sections of the segment of
the spheroid, the cone and the cylinder in the same way,
we find that the cylinder with axis AG, where it is, balances,
about A y the cone AEF and the segment ADG together, when
both are placed with their centres of gravity at //; and,
if IF be the centre of gravity of the cylinder, i.e. the middle
point of AG t
HA :AW= (cylinder, axis A G) : (cone AEF+ segmt. A DC).
If we call V the volume* of the cone AEF, and >S f that of the
segment of the spheroid, we have
(cylinder) : ( K+ ti) = 3 V. ,, : ( V + ,S'),
while HA : A W = A A' : A G.
Therefore AA':%AG = 3V. :(V +
and
whence
Again, let V be the volume of the cone A DC.
Then V:V'= RG Z :DG*
But DG 2 :AG.GA'= BB' Z :AA' 2 .
Therefore F: V = AG 2 : AG . OA'
THE METHOD 33
It* follows that 8 = V
r , %AA'-AG
- A >Q
_ Y' v.-^-^' + A'G
A'G
which is the result stated by Archimedes in Prop. 8.
The result is the same for the segment of a sphere. The
proof, of course slightly simpler, is given in Prop. 7.
In the particular case where the segment is half the sphere
or spheroid, the relation becomes
>S' = 2 V, (Props. 2, 3)
and it follows that the volume of the whole sphere or spheroid
is 4 V , where V is the volume of the cone ABB' \ i.e. the
volume of the sphere or spheroid is two-thirds of that of the
circumscribing cylinder.
In order now to find th< centre of gravity of the segment
of a spheroid, we must have the segment acting where it is,
not at II.
Therefore formula (1) above will not serve. But we found
that MN . NQ = (NP* + iVQ 2 ),
whence J/iV 2 : (.VP 2 + .VQ 2 ) = (iVP 2 + .VQ 2 ) : JV r Q a ;
therefore HA : AN = (NP* + NQ*) : NQ*.
(This is separately proved by Archimedes for the sphere
in Prop. 9.)
From this we derive, as usual, that the cone AEF and the
segment ADC both acting where they (ire balance a volume
equal to the cone A EV placed with its centre of gravity at //.
Now the centre of gravity of the cone AEF is on the line
AG at a distance f AG from A. Let ^Y be the required centre
of gravity of the segment. Then, taking moments about A,
we have V. HA = ti. AX + V.%AG,
or
-l)AX, from above.
34 ARCHIMEDES
Therefore AX:AG = (AA'-%AG) : (%AA'-AG)
whence AXiXG = (*AA'-SAG):(2AA'-AG)
which is the result obtained by Archimedes in Prop. 9 for the
sphere and in Prop. 10 for the spheroid.
In the case of the hemi-spheroid or hemisphere the ratio
AX : XG becomes 5 : 3, a result obtained for the hemisphere in
Prop. 6.
The cases of tho paraboloid of revolution (Props. 4, 5) and
the hyperboloid of revolution (Prop. 1 1) follow the same course,
and it is unnecessary to reproduce them.
For the cases of the two solids dealt with at the ond of tho
treatise the reader must bo referred to the propositions them-
selves. Incidentally, in Prop. 13, Archimedes finds the centre
of gravity of the half of a cylinder cut by a piano through
the axis, or, in other words, the centre of gravity of a semi-
circle.
We will now take the other treatises in tho order in which
they appear in the editions.
On the Sphere and Cylinder, I, II.
The main results obtained in Book I are shortly stated in
a prefatory letter to Dositheus. Archimedes tells us that
they are new, and that he is now publishing them for the
first time, in order that mathematicians may be able to ex-
amine the proofs and judge of their value. The results are
(1) that the surface of a sphere is four times that of a great
circle of the sphere, (2) that the surface of any segment of a
sphere is equal to a circle the radius of which is equal to the
straight line drawn from the vertex of the segment to a point
on the circumference of the base, (3) that the volume of a
cylinder circumscribing a sphere and with height equal to the
diameter of the sphere is f of the volume of the sphere,
(4) that the surface of the circumscribing cylinder including
its bases is also f of the surface of the sphere. It is worthy
of note that, while the first and third of these propositions
appear in the book in this order (Props. 33 and 34 respec-
ON THE SPHERE AND CYLINDER, I 35
tively), this was not the order of their discovery; for Archi-
medes tells us in The Method that
' from the theorem that a sphere is four times as great as the
cone with a great circle of the sphere as base and wi^h height
equal to the radius of the sphere I conceived the notion that
the surface of any sphere is four times as great as a great
circle in it ; for, judging from the fact that any circle is equal
to a triangle with base equal to the circumference and height
equal to the radius of the circle, I apprehended that, in like
manner, any sphere is equal to a cone with base equal to the
surface of the sphere and height equal to the radius '.
Book I begins with definitions (of 'concave in the same
direction ' as applied to curves or broken lines and surfaces, of
{, 'solid sector* and a 'solid rhombus') followed by five
Assumptions, all of importance. Of all line* irhlch hare the
wniw extremities the xtraijfkt line /x the lea*t, and, if there are
two curved or bent lines in a plane having the same extremi-
ties and concave in the same direction, but one is wholly
included by, or partly included by and partly common with,
the other, then that which is included is the lesser of the two.
Similarly with plane surfaces and surfaces concave in the
same direction. Lastly, Assumption 5 is the famous 'Axiom
of Archimedes', which however was, according to Archimedes
himself, used by earlier geometers (Eudoxus in particular), to
the effect that Of unequal magnitudes the (jreatcr exceed*
the Jew fry such a inuynitiule as, when added to itself, can be
made to exceed tiny awiffued magnitude of the same kind]
the axiom is of course practically equivalent to Eucl. V, Def. 4,
and is closely connected with the theorem of Eucl. X. 1.
As, in applying the method of exhaustion, Archimedes uses
both circumscribed and inscribed figures with a view to com-
prcsshu) them into coalescence with the curvilinear figure to
be measured, he has to begin with propositions showing that,
given two unequal magnitudes, then, however near the ratio
of the greater to the less is to 1, it is possible to find two
straight lines such that the greater is to the less in a still less
ratio ( > 1), and to circumscribe and inscribe similar polygons to
a circle or sector such that the perimeter or the area of the
circumscribed polygon is to that of the inner in a ratio less
than the given ratio (Props. 2 6): also, just as Euclid proves
36 ARCHIMEDES
that, if we continually double the number of the sides of the
regular polygon inscribed in a circle, segments will ultimately be
left which are together less than any assigned area, Archimedes
has to supplement this (Prop. 6) by proving that, if wo increase
the number of the sides of a circumscribed regular polygon
sufficiently, we can make the excess of the area of the polygon
over that of the circle less than any given area. Archimedes
then addresses himself to the problems of finding the surface of
any right cone or cylinder, problems finally solved in Props. 1 3
(the cylinder) and 14 (the cone). Circumscribing and inscrib-
ing regular polygons to the bases of the cone and cylinder, he
erects pyramids and prisms respectively on the polygons as
bases and circumscribed or inscribed to the cone and cylinder
respectively. In Props. 7 and 8 he finds the surface of the
pyramids inscribed and circumscribed to the cone, and in
Props. 9 and 10 he proves that the surfaces of the inscribed
and circumscribed pyramids respectively (excluding the base)
are less and greater than the surface of the cone (excluding
the base). Props. 11 and 12 prove the same thing of the
prisms inscribed and circumscribed to the cylinder, and finally
Props. 13 and 14 prove, by the method of exhaustion, that the
surface of the cone or cylinder (excluding the bases) is equal
to the circle the radius of which is a mean proportional
between the 'side' (i.e. generator) of the cone or cylinder and
the radius or diameter of the base (i.e. is equal to TTTS in the
case of the cone and 2nrs in the case of the cylinder, where
r is the radius of the base and s a generator). As Archimedes
here applies the method of exhaustion for the first time, we
will illustrate by the case of the cone (Prop. 14).
Let A be the base of the cone, C a straight line equal to its
radius, D a line equal to a generator of the cone, E a mean
proportional to (7, D, and JB a circle with radius equal to E.
ON THE SPHERE AND CYLINDER, I 37
If S is the surface of the cone, we have to prove that $ = B.
For, if S is not equal to B, it must be either greater or less.
I. Suppose B < &
Circumscribe a regular polygon about B, and inscribe a similar
polygon in it, such that the former lias to the latter a ratio less
than S: B (Prop. 5). Describe about A a similar polygon and
set up from it a pyramid circumscribing the cone.
Thou (polygon about 4) : (polygon about B)
= (polygon about A) -.(surface of pyramid).
Therefore (surface of pyramid) = (polygon about B).
But (polygon about B) : (polygon in B) < S: JS;
therefore (surface of pyramid) : (polygon in B) < ti : B.
But this is impossible, since (surface of pyramid) > >S', while
(polygon in B) < B',
therefore B is not less than &
II. Suppose B > S.
Circumscribe and inscribe similar regular polygons to B
such that the former has to the latter a ratio < B :X. Inscribe
in A a similar polygon, and erect on A the inscribed pyramid.
Then (polygon in A) : (polygon in B) = (!* : E 2
= 0:7J
> (polygon in ^4) : (surface of pyramid).
(The latter inference is clear, because the ratio of C:D is
greater than the ratio of the perpendiculars from the centre of
A and from the vertex of the pyramid respectively on any
side of the polygon in A] in other words, if /? < & < -|TT,
sin a > sin /3.)
Therefore (surface of pyramid) > (polygon in B).
But (polygon about B) : (polygon in B) < B: 8,
whence (a fortiori)
(polygon about B) : (surface of pyramid) < B : 8,
which is impossible, for (polygon about B) > B, while (surface
of pyramid) < &
38
ARCHIMEDES
Therefore B is not greater than #.
Hence 8, being neither greater nor less than ti, is equal to B.
Archimedes next addresses himself to the problem of finding
the surface and volume of a sphere or a segment thereof, but
has to interpolate some propositions about 'solid rhombi'
(figures made up of two right cones, unequal or equal, with
bases coincident and vertices in opposite directions) the neces-
sity of which will shortly appear.
Taking a great circle of the sphere or a segment of it, he
inscribes a regular polygon of an even number of sides bisected
FIG. 1.
by the diameter AA', and approximates to the surface and
volume of the sphere or segment by making the polygon
revolve about A A' and measuring the surface and volume of
solid so inscribed (Props. 21-7). He then does the same for the
a circumscribed solid (Props. 28-32). Construct the inscribed
polygons as shown in the above figures. Joining BB', CC', ... ,
CB', DC' ... we see that BE', CO' ..v are all parallel, and so are
AB, CB', DC'....
Therefore, by similar triangles, BF:FA = A'B:BA, and
= B'F:FK
= E'I:IA' in Fig. 1
(= PM: MN in Fig. 2),
ON THE SPHERE AND CYLINDER, 1 39
whence, adding antecedents and consequents, we have
(Fig. 1) (BB' + QC' + . . . + EE') : A A' = A'B : BA, (Prop. 21)
(Fig. 2) (BB' + CC'+... + %PP'):AM=A'B:BA. (Prop. 22)
When we make the polygon revolve about AA ', the surface
of the inscribed figure so obtained is made up of the surfaces
of cones and frusta of cones; Prop. 14 has proved that the
surface of the cone ABB' is what we should write TT . AB . BF,
and Prop. 16 has proved that the surface of the frustum
BCC'J? is 7r.B(!(BF+CG). It follows that, since AB =
BC = . . . , the surface of the inscribed solid is
that is, TT . AB(BB' + CC'+ ... + EE') (Fig. 1), (Prop. 24)
or TT . AB (BB' + CC' + ... +%PP') (Fig. 2). (Prop. 35)
Hence, from above, the surface of the inscribed solid is
IT. A'B. A A' or ?r . A'B . A J/, and is therefore less than
TT . AA' 2 (Prop. 25) or TT . A' A . AM, that is, ?r . AP 2 (Prop. 37).
Similar propositions with regard to surfaces formed by the
revolution about AA' of regular circumscribed solids prove
that their surfaces aiv greater than tr.AA'* and n.AP 2
respectively (Props. 28-30 and Props. 39-40). The case of the
segment is more complicated because the circumscribed poly-
gon with its sides parallel to AB, J3C ... DP circumscribes
the sector POP'. Consequently, if the segment is less than a
semicircle, as CAC', the base of the circumscribed polygon
(cc') is on the side of GY" towards A, and therefore the circum-
scribed polygon leaves over a small strip of the inscribed. This
complication is dealt with in Props. 39-40. Having then
arrived at circumscribed and inscribed figures with surfaces
greater and less than n.AA'* and TT. AP* respectively, and
having proved (Props. 32, 41) that the surfaces of the circum-
scribed and inscribed figures are to one another in the duplicate
ratio of their sides, Archimedes proceeds to prove formally, by
the method of exhaustion, that the surfaces of the sphere and
segment are equal to these circles respectively (Props. 33 and
42); IT. A A'* is of course equal to four times the great circle
of the sphere. The segment is, for convenience, taken to be
40 ARCHIMEDES
less than a hemisphere, and Prop. 43 proves that the same
formula applies also to a segment greater than a hemisphere.
As regards the volumes different considerations involving
' solid rhombi ' come in. For convenience Archimedes takes,
in the case of the whole sphere, an inscribed polygon of 4?i
sides (Fig. 1). It is easily seen that the solid figure formed
by its revolution is made up of the following : first, the solid
rhombus formed by the revolution of the quadrilateral ABOB'
(the volume of this is shown to be equal to the cone with base
equal to the surface of the cone ABB' and height equal to p y
the perpendicular from on AB, Prop. 18); secondly, the
extinguisher-shaped figure formed by the revolution of the
triangle BOO about A A' (this figure is equal to the difference
between two solid rhombi formed by the revolution of TBOB'
and TCOC' respectively about AA', where T is the point of
intersection of CB> C B' produced with A' A produced, and
this difference is proved to be equal to a cone with base equal
to the surface of the frustum of a cone described by BC in its
revolution and height equal to p the perpendicular from on
BC, Prop. 20) ; and so on ; finally, the figure formed by the
revolution of the triangle COD about AA' is the difference
between a cone and a solid rhombus, which is proved equal to
a cone with base equal to the surface of the frustum of a cone
described by CD in its revolution and height p (Prop. 19).
Consequently, by addition, the volume of the whole solid of
revolution is equal to the cone with base equal to its whole
surface and height p (Prop. 26). But the whole of the surface
of the solid is less than 4 ?rr 2 , and p< r ; therefore the volume
of the inscribed solid is less than four times the cone with
base ?rr 2 and height r (Prop. 27).
It is then proved in a similar way that the revolution of
the similar circumscribed polygon of 4n sides gives a solid
the volume of which is greater than four times the same cone
(Props. 28-31 Cor.). Lastly, the volumes of the circumscribed
and inscribed figures are to one another in the triplicate ratio of
their sides (Prop. 32) ; and Archimedes is now in a position to
apply the method of exhaustion to prove that the volume of
the sphere is 4 times the cone with base ?rr 2 and height r
(Prop. 34).
Dealing with the segment of a sphere, Archimedes takes, for
ON THE SPHERE AND CYLINDER, I 41
convenience, a segment less than a hemisphere and, by the
same chain of argument (Props. 38, 40 Corr., 41 and 42), proves
(Prop. 44) that the volume of the sector of the sphere bounded
by the surface of the segment is equal to a cone with base
equal to the surface of the segment and height equal to the
radius, i.e. the cone with base 7T.AP 2 and height r (Fig. 2).
It is noteworthy that the proportions obtained in Props. 21,
22 (see p. 39 above) can be expressed in trigonometrical form.
If 471 is the number of the sides of the polygon inscribed in
the circle, and 2n the number of the sides of the polygon
inscribed in the segment, and if the angle AOP is denoted
by a, the trigonometrical equivalents of the proportions are
respectively
(1) sin-^ + sin ~ +...+ fiin(2?i 1) - = cot-^-;
v ' 2n 2/i ' 2 ti 4/i
(2) 2 } sin + sin + ... + sin (n~ !)-> +sin<x
x ' ( n n n)
= (1 cos a) cot
2/<,
Thus the two proportions give in effect a summation of the
series
sin0 + sin 20 + ... +Hin(*i 1) 0,
both generally where nd is equal to any angle a less than n
and in the particular case where n is even and = ir/n.
Props. 24 and 35 prove that the areas of the circles equal to
the surfaces of the solids of revolution described by the
polygons inscribed in the sphere and segment are the above
7T Oi
series multiplied by 4?rr 2 sin .and nr?' 2 . 2 sin respectively
4 ?t M tl
and are therefore 4?rr 2 cos and nrr 2 . 2 cos - (1 cos a)
4 n 2n- }
respectively. Archimedes's results for the surfaces of the
sphere and segment, 4?rr L ' and 2?rr 2 (l cos a), are the
limiting values of these expressions when n is indefinitely
increased and when therefore cos and cos - become
471, 2n
unity. And the two series multiplied by 47rr 2 sin and
42
ARCHIMEDES
?rr 2 . 2 sin respectively are (when n is indefinitely increased)
2 7i/
precisely what we should represent by the integrals
and
r
j .
Jo
Jo
or
, or 27rr 2 (l eosa).
Book 11 contains six problems and three theorems. Of the
theorems Prop. 2 completes the investigation of the volume of
any segment of a sphere, Prop. 44 of Book I having only
brought us to the volume of the corresponding sector. If
ABB' be a segment of a sphere cut off by a plane at right
angles to A A', we learnt in I. 44 that the volume of the sector
OBAB' is equal to the cone with base equal to the surface
of the segment and height equal to the radius, i.e. JTT . AB 2 . r,
where r is the radius. The volume of the segment is therefore
Archimedes wishes to express this as a cone with base the
same as that of the segment. Let AM, the height of the seg-
ment, = h.
Now AB 2 : BAP = A' A : A'M = 2r : (2r-h).
Therefore
That is, the segment is equal to the cone with the same
base as that of the segment and height Ti(^rh)/(2r h).
ON THE SPHERE AND CYLINDER, II 43
This is expressed by Archimedes thus. If 1IM is the height
of the required cone,
A'M):A'M, (1)
and similarly the cone equal to the segment A' BE' has the
height H'M, where
If'M : A'M = (OA + AM) : AM. (2)
His proof is, of course, not in the above form but purely
geometrical.
This proposition leads to the most important proposition in
the Book, Prop. 4, which solves the problem To cut a given
sphere by a jjlane in sack a any that the volumes of the
segments are to one another in a glren ratio.
Cubic equation arising out of II. 4.
If m : n be the given ratio of the cones which are equal to
the segments and the heights of which an; /t, h', we have
and, if we eliminate h' by means of the relation h + hf = 2r,
we easily obtain the following cubic equation in A,
_ .
m+ n
Archimedes in effect reduces the problem to this equation,
which, however, he treats as a particular case of the more
general problem corresponding to the equation
where b is a given length and c* 2 any given area,
or x 2 (a x) = be' 2 , where x = 2r h and 3?' = a.
Archimedes obtains his cubic equation with one unknown
by means of a geometrical elimination of 11, IV from the
7Ai/
equation UM = . //'J/, where HiW, H'M have the values
11
determined by the proportions (1) and (2) above, after which
the one variable point M remaining corresponds to the one
unknown of the cubic equation. His method is, first, to find
44 ARCHIMEDES
values for each of the ratios A'H' : H'M and H'H : A'H' which
are alike independent of H, H' and then, secondly, to equate
the ratio compounded of these two to the known value of the
ratio HH':H'M.
(a) We have, from (2),
A'H' : H'M = OA : (OA + AM). (3)
() From (1) and (2), separando,
AH:AM=OA':A'M, (4)
A'H':A'M = OA:AM. (5)
Equating the values of the ratio A'M : AM given by (4), (5),
we have OA' : AH = Air :OA
= OIF: OH,
whence HH' : OH' = OH' : A' 11', (since OA = OA')
or Elf. A' IF = 011'*,
so that HE' : A'H' = OH' 2 : A'H' Z . (6)
But, by (5), OA' : A'H' = AM: A'M,
and, componendo, OH' : A'H' = A A' : A'M.
By substitution in (6),
HH' : A' It' = A A' 2 : A'M' 2 . (7)
Compounding with (3), we obtain
HH' : H'M = (A A' z : A'M*) . (OA : OA + AM). (8)
[The algebraical equivalent of this is
m + n 4 r 3
, . , , , 4r l
which reduces to ----- = ~ - =- 3
m 3h 2 rh*
or 7^--3A 2 r+ - r 3 = 0, as above.]
J
Archimedes expresses the result (8) more simply by pro-
ducing OA to D so that OA = AD, and then dividing AD at
ON THE SPHERE AND CYLINDER, II 45
E %o that AD:DE = HH'iH'M or (m + n):n. We have
then OA = AD and OA + AM = MD, so that (8) reduces to
AD : DE = (A A'* : A'M*) .(AD: MD),
or MI):I)E= AA'*:A'M*.
Now, says Archimedes, D is {riven, since AD OA. Also,
yt/): /)/? being a given ratio, DE is given. Hence the pro-
blem reduces itself to that of dividing A'D into two parts at
^f such that
AID : (a given length) = (a given area) : A'M*.
That is, the generalized equation is of the form
x 2 (a x) = bo 2 , as above.
(i) Archimedes's own solution of the cubic.
Archimedes adds that, 'if the. problem is propounded in this
general form, it requires a Siopta-pos [i.e. it is necessary to
investigate tae limits of possibility], but if the conditions are
added which exist in the, present case [i.e. in the actual
problem of Prop. 4], it does not require a ^opjoyjoy' (in other
words, a solution is always possible). He then promises to
give ' at the end ' an analysis and synthesis of both problems
[i.e. the Siopia/jLos and the problem itself]. The promised
solutions do not appear in the treatise as we have it, but
Eutocius gives solutions taken from ' an old book ' which he
managed to discover after laborious search, iind which, since it
was partly written in Archimedes's favourite Doric, he with
fair reason assumed to contain the missing addendum by
Archimedes.
In the Archimedean fragment preserved by Eutocius the
above equation, x*(a x) = 6c 2 , i? solved by means of the inter-
section of a parabola and a rectangular hyperbola, the equations
of which may be written thus
<' 2
or = ;y, (ax) y = ab.
(Jb
The SiopicrfjLos takes the form of investigating the maximum
possible vplue of x 2 (a x), and it is proved that this maximum
value for a real solution is that corresponding to the value
x = a. This is established by showing that, if 6c* 2 = ^\a a ,
46
ARCHIMEDES
the curves touch at the point for which x = fa. If on the
other hand be 2 < ^a 3 , it is proved that there are two real
solutions. In the particular case arising in Prop. 4 it is clear
that the condition for a real solution is satisfied, for the
expression corresponding to be 2 is - 4r 3 , and it is only
4r 3 should be not greater than ^V 6 " or
necessary that ~-
J m + n
4 r 3 , which is obviously the case.
(ii) Solution of the cubic by Dionysodorus.
It is convenient to add here that Eutocius gives, in addition
to the solution by Archimedes, two other solutions of our
problem. One, by Dionysodorus, solves the cubic equation in
the less general form in which it is required for Archimedes's
proposition. This form, obtained from (8) above, by putting
A'M = .r, is
4r 2 :^ 2 = (3 r -,i-): - r,
in + 'ii
and the solution is obtained by drawing the parabola and
y
the rectangular hyperbola which we should represent by the,
equations
?'(3r x) = y 2 and - 2r~ = try,
referred to A' A and the perpendicular to it through A as axes
of x, y respectively.
(We make FA equal to OA, and draw the perpendicular
AH of such a length that
FA :AH CE : ED = (m + n) : n.)
ON THE SPHERE AND CYLINDER, II 47
(iii). Solution of the original problem of II. 4 by Diocles.
Diocles proceeded in a different manner, satisfying, by
a geometrical construction, not the derivative cubic equation,
but the three simultaneous relations which hold in Archi-
medes's proposition, namely
HA : h = r :h'
ll'A': h' = r:h
with the slight generalization that he substitutes for r in
these equations another length a.
H R
The problem is, given a straight line A A', a ratio m:n, and
another straight line AK (= a), to divide A A' at a point M
and at the same time to find two points 77, II' on AA'
produced such that the above relations (with a in place
of r) hold.
The analysis leading to the construction is very ingenious.
Place AK ( = a) at right angles to AA r , and draw A'K' equal
and parallel to it.
Suppose the problem solved, and the points M, //, 11' all
found.
Join KM, produce it, and complete the rectangle KG-EK'.
48 ARCHIMEDES
Draw QMN through M parallel to AK. Produce K'M to
meet KG produced in F.
By similar triangles,
FA : AM = K'A' : A'M, or FA : h = a : h',
whence FA = AH (k, suppose).
Similarly A'E = A'U' (k', suppose).
Again, by similar triangles,
(FA + AM) : (A'K' + A'M) = AM: A'M
= (AK + A M) :(EA' + A'M),
or (k + h) : (a + h') = (a + h) : (k' + h'),
i. e. (k + h) (k' + h') = (a + h) (n + h'). ( 1 )
Now, by hypothesis,
m : n = (k + h) : (k' + h')
= (a + h) (a + h') : (k' + h') 2 [by ( 1 )]. (2)
Measure AH, A'R' on A A' produced both ways equal to a.
Draw RP, R'P' at right angles to RR' as shown in the figure.
Measure along MN the length MV equal to MA' or k', and
draw PP' through V, A' to meet RP, R'P'.
Then QV=k' + h',
whence PV.P'V=2(a + k) (a, + h') ;
and, from (2) above,
2 w : n = 2 (a + h) (a + h') : (k' + h') *
= PV.P'V:QV\ (3)
Therefore Q is on an ellipse in which PP' is a diameter, and
QVis an ordinate to it.
Again, oGQNK is equal to dAA'K'K, whence
GQ.QN= AA'. A'K' =(h + h') a = 2ra, (4)
and therefore Q is on the rectangular hyperbol^ with KF,
KK' as asymptotes and passing through A'.
ON THE SPHERE AND CYLINDER, II 49
How this ingenious analysis was suggested it is not possible
to say. It is the equivalent of reducing the four unknowns
h> h' y />', k' to two, by putting h = r + x, hf == rx and k' = y,
and then reducing the given relations to two equations in x,y,
which are coordinates of a point in relation to Ox, Oy as axes,
where is the middle point of A A ', and Ox lies along OA',
while Oy is perpendicular to it.
Our original relations (p. 47) give
7/ ah' r x , ah r + x ^ m h + k
= A; = -=- = a - > A* = -,-/ = & ------ ' and = r7 r/'
A r + .i: A rx n h ' + k
We have at once, from the first two equations,
whence (
am 1 . (;/: -f r) (// + rt) = 2 m,
which is the rectangular hyperbola (-1) above.
m
whence we obtain a cubic CM [nation in ./',
Oil
/ . \9 / \ iiv / ,)
(r-f xY (r + a s) = - (r ,r)~
which mves
u . // a , V 4 /' .'' r + a-f.r
But - = -, whence - -- = ---
r x r + M r x
and the equation becomes
(y + r - a:) 2 = (r + a) 2 - ^ 2 ,
'/i/
wliich is the ellipse (3) above.
1523.2
50 ARCHIMEDES
To return to Archimedes. Book II of our treatise contains
further problems : To find a sphere equal to a given cone or
cylinder (Prop. 1), solved by reduction to the finding of two
mean proportionals; to cut a sphere by a plane into two
segments having their surfaces in a given ratio (Prop. 3),
which is easy (by means of I. 42, 43) ; given two segments of
spheres, to find a third segment of a sphere similar to one
of the given segments and having its surface equal to that of
the other (Prop. 6) ; the same problem with volume substituted
for surface (Prop. 5), which is again reduced to the finding
of two mean proportionals; from a given sphere to cut off
a segment having a given ratio to the cone with the same
base and equal height (Prop. 7). The Book concludes with
two interesting theorems. If a sphere be cut by a plane into
two segments, the greater of which has its surface equal to fl
and its volume equal to F, while $', V are the surface and
volume of the lesser, then V: V < A S' 2 : >S" 2 but >&'5:jS"J
(Prop. 8) : and, of all segments of spheres which haves their
surfaces equal, the hemisphere is the greatest in volume
(Prop. 9).
Measurement of a Circle.
The book on the Measurement of a Circle consists of throe
propositions only, and is not in its original form, having lost
(as the treatise On the Sphere and Cylinder also has) prac-
tically all trace of the Doric dialect in which Archimedes
wrote ; it may be only a fragment of a larger treatise 4 . The
three propositions which survive prove (1) that the area of
a circle is equal to that of a right-angled triangle in which
the perpendicular is equal to the radius, and the base to the
circumference, of the circle, (2) that the area of a circle is to
the square on its diameter as 11 to 14 (tho text of this pro-
position is, however, unsatisfactory, and it cannot have been
placed by Archimedes before Prop. 3, on which it depends),
(3) that the ratio of the circumference of any circle to Us
diameter (i.e. TT) is < 3^ but > 3^. Prop. 1 is proved by
the method of exhaustion in Archimedes's usual form : he
approximates to the area of the circle in both directions
(a) by inscribing successive regular polygons with a number of
MEASUREMENT OF A CIRCLE 51
sides % continually doubled, beginning from a square, (6) by
circumscribing a similar set of regular polygons beginning
from a square, it being shown that, if the number of the
sides of these polygons be continually doubled, more than half
of the portion of the polygon outside the circle will be taken
away each time, so that we shall ultimately arrive at a circum-
scribed polygon greater than the circle by a space less than
any assigned area.
Prop. 3, containing the arithmetical approximation to TT, is
the most interesting. The method amounts to calculating
approximately the perimeter of two regular polygons of 96
sides, one of which is circumscribed, and the other inscribed,
to the circle : and the calculation starts from a greater and
a lesser limit to the value of \/3, whicil Archimedes assumes
without remark as known, namely
265 ^ A / Q ^ 1 35 1
How did Archimedes arrive, at those particular approxi-
mations? No puzzle has exercised more fascination upon
writers interested in the history of mathematics. De Lagny,
Mollwcide, ttu/engeiger, Hauber, Zeuthen, P. Tannery, Heiler-
mann, Hultsch, Hunrath, Wertheim, Bobyniii : these are the
names of sonic* of the authors of different conjectures. The
simplest supposition is certainly that of Hunrath and Hultsch,
who suggested that the formula used was
where a~ is the nearest square number above or below (t~b t
as the case may be,. The use of the first part of this formula
by Heron, who made a number of such approximations, is
proved by a passage in his Metrica ', where a rule equivalent
to this is applied to \/720; the second part of the formula is
used by the Arabian Alkarkhi (eleventh century) who drew
from Greek sources, and one approximation in Heron may be
obtained in this way.- Another suggestion (that of Tannery
1 Heron, Metrica, i. 8.
2 Sterrom. ii, p. 184. 19, Hultsch; p. 154. 19, Heib. V^4 = 7J=7 1 V
instead of 7 /> 4 .
K2
52 ARCHIMEDES
and Zeuthen) is that the successive solutions in integers of
the equations
o; a -32/ 2 =l )
j5 2 -3y a = -2)
may have been found in a similar way to those of the
equations & 2 2 y 1 = +1 given by Theon of Smyrna after
the Pythagoreans. The rest of the suggestions amount for the
most part to the use of the method of continued fractions
more or less disguised.
Applying the above formula, we easily find
or
Next, clearing of fractions, we consider 5 as an approxi-
mation to \/3 . 3- or \/27, and we have
whence ff > ^ 3 > T
T-
Clearing of fractions again, and taking 20 as an approxi-
mation to \/3 . 15- or >/675, we have
28 ra > 15v/3 > 2 *>-5T>
which reduces to
1351 v. A /o -. 265
~T50- > V 5 > T ^.
Archimedes first takes the case of the circumscribed polygon.
Let CA be the tangent at A to a circular arc with centre 0.
Make the angle AOC equal to one-third of a right angle.
Bisect the angle AOG by OD, the angle AOD by OK, the
angle AOEby OF, and the angle AOF by OG. Produce GA
to .4.#, making ^l// equal to AG. The angle GOlf is them
equal to the angle FOA which is ^th of a right angle, so
that GH is the side of a circumscribed regular polygon with
96 sides.
Now UA:AC[= V3:l] > 265:153, (l)
and OC : CA = 2:1 = 306:153. (2)
MEASUREMENT OF A CIRCLE
And, since 01) bisects the angle CO A,
so that
or
Hence
53
(CO + OA): OA = CA : DA,
(CO + 04) :CA = OA:A D.
OA :AD> 571 : 153, by (1) and (2).
And Oir-:AD* = (OA~ + AD 2 ) :AD~
> (571-+153-): 153-
> 349450:23109.
Therefore, says Archimedes,
OD:DA > 591 1 : 153.
Next, just as we have found the limit of OD:AD
from 00: ('A ami the limit of OA : AC?, we find the limits
of OA:AK and OKiAE from the limits of OD:DA and
OA : AD, and so on. This gives ultimately the limit of
OA-.AG.
Dealing with the inscribed polygon, Archimedes gets a
similar series of approximations. ABC being a semicircle, the
angle JiAO\a made equal to one-third of a right angle. Then,
if the angle KAC is bisected by AD, the angle BAD by AK,
the angle BAK by AF, and the angle 7M.F by AG, the
straight line BG is the side of an inscribed polygon with
96 sides.
54
ARCHIMEDES
Now the triangles ADB, BDd, ACd are similar;
therefore AD:DB = BD:Dd = AC : Cd
= AB : Bd, since AD bisects Z BAG,
But AC:GB < 1351 : 780,
while BA : BC = 2 : 1 = 1560 : 780.
Therefore AD : DB < 2911 : 780.
Hence
AB 2 : BD* < (291 1' 2 + 7 80-) : 780-
< 9082321 : 608400,
and, says Archimedes,
AB:BD < 301 3;|: 780.
Next, just as a limit is found for AD:Dli and AB:BD
from AB:BC and the limit of AC:(JB, so we find limits tor
AE : EB and AB : BE from the limits of AB : BD and AD : DH,
and so on, and finally we obtain the limit of AB : BG.
We have therefore in both cases two series of terms a , a,,
<7 2 ... a n and 6 () , b l9 b. 2 ...b n , for which the rule of formation is
where ^ = v' (^^ + c 2 ), 6 2 = /(a/ + c 2 ) . . . ;
and in the first case
a = 265, 6 = 306, c = 153,
while in the second case
a a = 1351, b n = 1560, c = 780.
MEASUREMENT OF A CIRCLE 55
The series of values found by Archimedes are shown in the
following table :
a b c n a b c
265 306 153 1351 1560 780
571 > |V(571~-H53 2 )] 153 1 2911 < i/(2911 a + 780 2 ) 780
153 2/5924J-J ... 780|
1823 (<i/(1823 2 + 240 2 ) 240]
< 1838 T 9 T
2334}>[V r {(2334}) a +153 58 }] 153 3 3661 T 9 T ... 240f
> 2339J 1007 (<V(l 007' 2 + 66 2 ) 66
4673$ 153 4 2016JJ < ^{(2016j)- + 66* } 66
1 < 2017}
and, bearing in mind that in the first case the final ratio
4 : c is the ratio OA :AG = 2 OA : Gil, and in the second case
the final ratio fe 4 :c is the ratio AB : BG, while GH in the first
figure and BG in the second are the sides of regular polygons
of 96 sides circumscribed and inscribed respectively, we have
finally
96x153 96x66
4673J >?r> 2017} "
Archimedes simply infers from this that
35 >7T > 3#.
A r f i 96X153 Q 667 667-
As a matter of iact w jjr=* W7 ^ aiid ^ = 1.
1 1
It is also to be observed that 3^ = 3+ - , and it may
have been arrived at by a method equivalent to developing
f\ n o t*
the fraction ' * .in the form of a continued fraction.
2017}
It should be noted that, in the text as we have it, the values
of ftp 6 2 , 6 3 , /> 4 are simply stated in their final form without
the intermediate step containing the radical except in the first
* t Here the ratios of a to c are in the first instance reduced to lower
terms.
56 ARCHIMEDES
case of all, where we are told that OD*:AD* > 349450 : 23409
and then that OD:DA > 591|:153. At the points marked
* and f in the table Archimedes simplifies the ratio a 2 : c and
r* 3 : c before calculating & 2 , & 3 respectively, by multiplying each
term in the first case by T \ and in the second case by -J.
He gives no explanation bf the exact figure taken as the
approximation to the square root in each case, or of the
method by which he obtained it. We may, however, be sure
that the method amounted to the use of the formula (ce + 6) 2
= a 2 2ab + b 1 ) much as our method of extracting the square
root also depends upon it.
We have already seen (vol. i, p. 232) that, according to
Heron, Archimedes made a still closer approximation to the
value of 77.
On Conoids and Spheroids.
The main problems attacked in this treatise are, in Archi-
medes's manner, stated in his preface addressed to Dositheus,
which also sets out the premisses with regard to the solid
figures in question. These premisses consist of definitions and
obvious inferences from them. The figures are (1) the right-
angled conoid (paraboloid of revolution), (2) the obtuse-angled
conoid (hyperboloid of revolution), and (3) the spheroids
(a) the oblong, described by the revolution of an ellipse about
its 'greater diameter' (major axis), (6) the flat, described by.
the revolution of an ellipse about its * lesser diameter ' (minor
axis). Other definitions are those of the vertex and w/.s of the
figures or segments thereof, the vertex of a segment being
the point of contact of the tangent plane to the solid which
is parallel to the base of the segment. The centre is only
recognized in the case of the spheroid; what corresponds to
the centre in the case of the hyperboloid is the 'vertex of
the enveloping cone' (described by the revolution of what
Archimedes calls the * nearest lines to the section of the
obtuse-angled cone', i.e. the asymptotes of the hyperbola),
and the line between this point and the vertex of the hyper-
boloid or segment is called, not the axis or diameter, but (the
line) 'adjacent to the axis'. The axis of the segment is in
the case of the paraboloid the line through the vertex of the
segment parallel to the axis of the paraboloid, in the case
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 18 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 * t a + 2a + Sa+... +n,} > n . na > 2 \a + 2<t + ... + (/&!)}
(Lemma)
(this is clear from tf w = $ii(n+l)a) ;
+ 2a + 3a+ ... +na)
(Lemma to Prop. 2)
whence (Cor.)
*
)*} > n(na)
lastly, Prop. 2 gives limits for the sum of n terms of the
series ax + ,/r, a .2x + (2 ;i:) 2 , a . 3 x 4- (3 a) 2 , . . . , in t ho form of
inequalities of ratios, thus:
11 { a .nx + ( nx)' 2 \ : S 1 "- 1 [ a . rx + (rx) 2 }
> (a -f nx) : (\<t 4- -3 i*v)
> n \ a . nx -f (natf* \ : S, 7 ' { a . rx 4- (rx)- \ .
Prop. 3 proves that, if QQ' be a chord of a parabola bisected
at V by the diameter PV, then, if PV be of constant length,
the areas of the triangle PQQ? and of the segment PQQ' are
also constant, whatever be the direction of (J$ \ to prove it
Archimedes assumes a proposition ' proved in the conies ' and
by no means easy, namely that, if QI) be perpendicular to PV,
and if p, p a 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 parsing
through the circumference of an ellipse with centre 6', where
CO is in the plane through an axis of the ellipse and perpen-
dicular 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.
The nature of the inscribed and circumscribed figures will
be seen from the above figures showing segments of a para-
boloid, a hyperboloid and a spheroid respectively, cut oft*
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 t \ 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 BR'
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 fruvstum the base of which is the ellipse in which PP' is
an axis is proportional to PP'* or PN~, 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 off* by the same planes. Thus
(first frustum in BF) : (first frustum in inscribed figure)
= BD* : PN*
= BD : TN.
Similarly
(second frustum in BF) : (second in inscribed figure)
= HN:SM,
and so on. The last frustum in the cylinder BF has none to
ON CONOIDS AND SPHEROIDS 61
correspond to it in the inscribed figure, and we should write
the ratio as (BD : zero).
Archimedes concludes, by means of a lemma in proportions
forming Prop. 1, that
(frustum BF) : (inscribed figure)
where XO = k, so that BD = TI/J.
In like manner, lie concludes that
(frustum BF) : (circumscribed figure)
But, by the Lemma preceding Prop. 1,
whence
(Frustum BF) : (inscr. fig.) > 2 > (frustum BF) : (circumscr. fig.).
This indicates the desired result, which is then confirmed by
the method of exhaustion, namely that
(frustum BF) = 2 (segment of paraboloid),
or, if V be the volume of the ' segment of a cone ', with vertex
A and base the same as that of the segment, *
(volume of segment) = f V.
Archimedes, it will be seen, proves in ett'ect that, if k be
indefinitely diminished, and n indefinitely increased, while nk
remains equal to c t then
limit of k{k+2k + 3lc+...+(u- l)k\ = |<r,
that is, in our notation,
pc
xdx =
f
Jo
Prop. 23 proves that the volume is constant for a given
length of axis AD, whether the segment is cut oft" by a plane
perpendicular or not perpendicular to the axis, and Prop. 24
shows that the volumes of two segments are as the squares on
their axes.
62 ARCHIMEDES
II. In the case of the hyperboloid (Props. 25, 26) let the axis
AD be divided into n parts, each of length h, and let AA'*=a.
Then the ratio of the volume of the frustum of a cylinder on
the ellipse of which any double ordinate QQ' is an axis to the
volume of the corresponding portion of the whole frustum BF
takes a different form ; for, if AM = rh, we have
(frustum in BF) : (frustum on base QQ')
= BD* : QM*
= AD.A'D:AM.A'M
By means of this relation Archimedes proves that
(frustum BF) : (inscribed figure)
and
(frustum BF) : (circumscribed figure)
= n { a . nh + (nh) 2 } : S^* [ a . rh + (rh) 2 } .
But, by Prop. 2,
From these relations it is inferred that
(frustum BF) : (volume of segment) = (a + nh) : (%a + ^ nh),
or (volume of segment) : (volume of cone ABB')
= (AD +30 A): (AD + 20 A);
and this is confirmed by the method of exhaustion.
The result obtained by Archimedes is equivalent to proving
that, if h be indefinitely diminished while n is indefinitely
increased but nh remains always equal to 6, then
limit of
or limit of ti n = & 2 (%a + &),
76
where
ON CONOIDS AND SPHEROIDS 63
so thai
The limit of this latter expression is what we should write
,6
I
Jo
and Archimedes's procedure is the equivalent of this integration.
III. In the case of the spheroid (Props. 29, 30) we take
a segment less than half the spheroid.
As in the case of the hyperboloid,
(frustum in BF) : (frustum on ba v se QQ')
= BD 2 : QM 2
= AD.A'DiAM.A'M\
but, in order to reduce the summation to the same as that in
Prop. 2, Archimedes expresses AM . A'M in a different form
equivalent to the following.
Let AD (=b) be divided into n, equal parts of length h,
and suppose that A A' '= a, CD = \c.
Then AD.4'/) = Ja 2 -Je 2 ,
and AM . A'M = a 2 - (c + rh) 2 (DM = r/i)
Thus in this case we have
(frustum BF) : (inscribed figure)
= n (cb + b 2 ) : [n (cb + 6 2 ) - ^ {c.rk + (rh) 2 } ]
arid
(frustum BF) : (circumscribed figure)
= n (cb + b 2 ) : [n (cb + b 2 ) - S^" 1 J c . rh + (rh) 2 } ].
And, since b = nh, we have, by means of Prop. 2,
n(cb + b 2 ) : [n(cb + b 2 )-^ n {c . rh + (rh) 2 }]
64 ARCHIMEDES
The conclusion, confirmed as usual by the method of ex-
haustion, is that
(frustum BF) : (segment of spheroid) = (c + b) : {c + b - (|c + ^b) }
= (o + 6):fto + |6),
whence (volume of segment) : (volume of cone ABB')
= (3CA-AD):(2CA-AD), since GA = J
As a particular case (Props. 27, 28), half the spheroid is
double of the corresponding cone.
Props. 31, 32, concluding the treatise, deduce the similar
formula for the volume of the greater segment, namely, in our
figure,
(greater segmt.) : (cone or segmt.of cone with same base and axis)
On Spirals.
The treativse On Spirals begins with a preface addressed to
Dositheus in which Archimedes mentions the death of Conon
as a grievous loss to mathematics, and then summarizes the
main results of the treatises On the Sphere and Cylinder and
On Conoids and Spheroids, observing that the last two pro-
positions of Book II of the former treatise took the place
of two which, as originally enunciated to Dositheus, were
wrong; lastly, he states the main results of the treatise
On Spirals, premising the definition of a spiral which is as
follows :
1 If a straight line one extremity of which remains fixed be
made to revolve at a uniform rate in a plane until it returns
to the position from which it started, and if, at the same time
as the straight line is revolving, a point move at a uniform
rate along the straight line, starting from the fixed extremity,
the point will describe a spiral in the plane.'
As usual, we have a series of propositions preliminary to
the main subject, first two propositions about uniform motion,
ON SPIRALS
65
then .two simple geometrical propositions, followed by pro-
positions (5-9) which are all of one type. Prop. 5 states that,
given a circle with centre 0, a tangent to it at A, and c, the
FIG. 1.
circumference of any circle whatever, it is possible to draw
a straight line OPF meeting the circle in P and the tangent
in F such that
FP : OP < (arc AP) : c.
Archimedes takes I) a straight line greater than c> 9 draws
077 parallel to the tangent at A and then says * let PH be
placed equal to 7) verging (vtvovora} towards A '. This is the
usual phraseology of the type of problem known as i/eCo-*?
where a straight line of given length has to be placed between
two lines or curves in such a position that, if produced, it
passes through a given point (this is the meaning of verging).
Each of the propositions 5-9 depends on a vtvvis of this kind,
FIG. 2.
which Archimedes assumes as ' possible ' without showing how
it is effected. Except in the case of Prop. 5, the theoretical
solution cannot be effected by means of the straight line and
circle; it depends in general on the solution of an equation
of the fourth degree, which can be solved by means of tbe
66
ARCHIMEDES
points of intersection of a certain rectangular hyperbola
and a certain parabola. It is quite possible, however, that
such problems were in practice often solved by a mechanical
method, namely by placing a ruler, by trial, in the position of
the required line : for it is only necessary to place the ruler
so that it passes through the given point and then turn it
round that point as a pivot till the intercept becomes of the
given length. In Props. 6-9 we have a circle with centre 0,
a chord AB less than the diameter in it, OM the perpendicular
from on AB, BT the tangent at B y OT the straight line
through parallel to A B ; D : E is any ratio less or greater,
as the case may be, than the ratio BM : MO. Props. 6, 7
(Fig. 2) show that it is possible to draw a straight line OFP
FIG. 3.
meeting AB in F and the circle in P such that FP : PBD: E
(OP meeting AB in the case where D\E<BM:MO, and
meeting AB produced when D : E > BM : MO). In Props. 8, 9
(Fig. 3) it is proved that it is possible to draw a straight line
OFP meeting AB in F, the circle in P and the tangent at B in
<?, such that FP:BG=D:E (OP meeting AB itself in the case
where D:E<BM:MO, and meeting AB produced in the
case where D:E > BM : MO).
We will illustrate by the constructions in Props. 7, 8,
as it is these propositions which are actually cited later.
Prop. 7. If D : E is any ratio > BM : MO, it is required (Fig. 2)
to draw OP / F / meeting the circle in P' and AB produced in
F / so that
FT*: P'B = D : E.
Draw OT parallel to AB, and let the tangent to the circle at
B meet OT in T.
ON SPIRALS
67
Than D : E > BM : MO, by hypothesis,
> OB : BT, by similar triangles.
Take a straight line P'H' (less than BT) such that D : E
= OB : P'H', and place P'H' between the circle and OT
' verging towards B ' (construction assumed).
Then F'P' : P'B = OP' : P'H'
= OB : P'H'
= D:E.
Prop. 8. If D : E is any given ratio < BM: MO, it is required
to draw OFPG meeting AB in F, the circle in P, and the
tangent at B to the circle in G so that
FP : BG = D : E.
If OT is parallel to AB and meets the tangent at B in T,
BM: MO = OB : BT, by similar triangles,
whence D:E<OB: BT.
Produce TB to C, making BG of such length that
D:E =OB:BC,
ip that BG > BT.
Describe a circle through the three points 0, T, C and let OB
produced meet this circle in K.
' Then, since BC > BT, and OK is perpendicular to GT, it is
possible to place QG [between the circle TKG and BC] equal to
BK and verging towards ' (construction assumed).
F- 2
68 ARCHIMEDES
Let QGO meet the original circle in P and AB in F. Then
OFPG is the straight line required.
For CG.GT=OG.GQ = OG. BK.
But OF: OG = BT: GT, by parallels,
whence OF.GT=OG.BT.
Therefore CG . GT : OF . GT = OG . BK : OG . BT,
whence CG:OF=BK:BT
= BC:OB
= BC:OP.
Therefore OP : OF = BC : CG,
and hence PF: OP = BG : BC,
or PF: BG = OU : J56 Y = 1) : K.
Pappus objects to Archimedes's use of the i/C(r*y assumed in
Prop. 8, in those words :
'it seems to be a grave error into which geometers fall
whenever any one discovers the solution of a plane problem
by means of conies or linear (higher) curves, or generally
solves it by means of a foreign kind, as is the case e.g. (1) with
the problem in the fifth Book of the Conies of Apollonius
relating to the parabola, and (2) when Archimedes assumes in
his work on the spiral a i/sCcny of a " solid " character with
reference to a circle ; for it is possible without calling in the
aid of anything solid to find the proof of the theorem given by
Archimedes, that is, to prove that the circumference of the
circle arrived at in the first revolution is equal to the straight
line drawn at right angles to the initial line to meet the tangent
to the spiral (i.e. the subtangent)/
There is, however, this excuse for Archimedes, that he only
assumes that the problem can be solved and does not assume
the actual solution. Pappus l himself gives a solution of the
particular i/evcris by means of conies. Apollonius wrote two
Books of i>i5<m9, and it is quite possible that by Archimedes's
time there may already have been a collection of such problems
to which tacit reference was permissible.
Prop. 10 repeats the result of the Lemma to Prop. 2 of On
1 Pappus, iv, pp. 298-302. *
ON SPIRALS 69
Conoicls and Spheroids involving the summation of the series
! 2 + 2 2 -f 3 2 -f ... + n 2 . Prop 11 proves another proposition in
summation, namely that
> (na) 2 : { na . a 4- -J (na a) 2 }
The same proposition is also true if the terms of the series
are a 2 , (a + 6) 2 , (a + 2b) 2 ... (a + ?i l6) a , and it is assumed in
the more general form in Props. 25, 26.
Archimedes now introduces his Definitions, of the spiral
itself, the origin, the initial line, the first distance (= the
radius vector at the end of one revolution), the second distance
(= the equal length added to the radius vector during the
second complete revolution), and so on ; the first area (the area
bounded by the spiral described in the first revolution and
the ' first distance '), the second area (that bounded by the spiral
described in the second revolution and the ' second distance '),
and so on; the first circle (the circle witli the 'first distance'
as radius), the second circle (the circle with radius equal to the
sum of the 'first* and 'second distances', or twice the first
distance), and so on.
Props. 12, 14, 15 give the fundamental property of the
spiral connecting the length of the radius vector with the angle
through which the initial line has revolved from its original
position, and corresponding to the equation in polar coordinates
r = a 0. As Archimedes does not speak of angles greater
than TT, or 2 TT, he has, in the case of points on any turn after
the first, to use multiples of the circumference
of a circle as well as arcs of it. He uses the
'first circle* for this purpose. Thus, if P, Q
are two points on the first turn,
OP : OQ = (arc AKP') : (arc AK (/) ;
if P, Q are points on the nth turn of the
spiral, and c is the circumference of the first circle,
Prop. 13 proves that, if a straight line touches the spiral, it
70 ARCHIMEDES
touches it at one point only. For, if possible, let the tangent
at P touch the spiral at another point Q. Then, if we bisect
the angle POQ by OL meeting PQ, in L and the spiral in 11,
OP + OQ20R by the property of the spiral. But by
the property of the triangle (assumed, but easily proved)
OP + OQ> 20L, so that OL < OR, and some point of PQ
lies within the spiral. Hence PQ cuts the spiral, which is
contrary to the hypothesis.
Props. 16, 17 prove that the angle made by the tangent
at a point with the radius vector to that point is obtuse on the
' forward ' sMe, and acute on the * backward ' side, of the radius
vector.
Props. 18-20 give the fundamental proposition about Jhe
tangent, that is to say, they give the length of the subtanyent
at any point P (the distance between and the point of inter-
section of the tangent with the perpendicular from to OP).
Archimedes always deals first with the first turn and then
with any subsequent turn, and with each complete turn before
parts or points of any particular turn. Thus he deals with
tangents in this order, (1) the tangent at A the end of the first
turn, (2) the tangent at the end of the second and any subse-
quent turn, (3) the tangent at any intermediate point of the
first or any subsequent turn. We will take as illustrative
the case of the tangent at any intermediate point P of the first
turn (Prop. 20).
If A be the initial line, P any point on the first turn, PT
the tangent at P and OT perpendicular to OP, then it is to be
proved that, if ASP be the circle through P with centre 0,
meeting PT in $, then
(subtangent OT) = (arc ASP).
I. If possible, let OT be greater than the arc ASP.
Measure off OU such that OU > arc AS'P but < OT.
Then the ratio PO:OU is greater than the ratio PO : OT,
i.e. greater than the ratio of %PS to the perpendicular from
on PS.
Therefore (Prop. 7) we can draw a straight line OQF meeting
TP produced in F, and the circle in Q, such that
ON SPIRALS
Let OF meet the spiral in Q'.
Then we have, alternando, since PO = QO,
71
< (arc PQ) : (arc ASP), by hypothesis and a fortiori.
Componendo, FO:QO < (arc 4#Q) : (arc ASP)
<OQ':OP.
But QO = OP; therefore FO < OQ'; which is impossible.
Therefore OT is not greater than the arc ASP.
II. Next suppose, if possible, that OT < arc ASP.
Measure OF along OT such that pF is greater than OTbut
less than the arc A SI*.
Then the ratio PO : OF is less than the ratio PO : OT, i.e.
than the ratio of %PS to the perpendicular from on P/S';
therefore it is possible (Prop. 8) to draw a straight line OF'RG
meeting P$, the circle PSA, and the tangent to the circle at P
in F', R, G respectively, and such that
72 ARCHIMEDES
Let OF'G meet the spiral in R'.
Then, since PO = RO, we have, alter nando,
> (arc PR) : (arc ASP), a fortiori,
whence F'O : RO < (arc ASR) : (arc 4/SP)
< OR': OP,
so that jF'O < OJS'; wliich is impossible.
Therefore OT is not less than the arc ASP. And it was
proved not greater than the same arc. Therefore
As particular cases (separately proved by Archimedes), if
P be the extremity of the first turn and C T the circumference
of the first circle, the subtangent = q ; if P be the extremity
of the second turn and c tt the circumference of the 'second
t
circle', the subtangent = 2r 2 ; and generally, if c n be the
circumference of the nth circle (the circle with the radius
vector to the extremity of the nth turn as radius), the sub-
tangent to the tangent at the extremity of the nth turn = nc n .
If P is a point on the nth turn, not the extremity, and the
circle with as centre and OP as radius cuts the initial line
in K, while p is the circumference of the circle, the sub-
tangent to the tangent at P = (n l)p + arc KP (measured
c forward ').*
The remainder of the book (Props. 21-8) is devoted to
finding the areas of portions of the spiral and its several
turns cut off by the initial line or any two radii vectores.
We will illustrate by tlie general case (Prop. 26). Take
OB y 0(7, two bounding radii vectores, including an arc BG
of the spiral. With centre and radius 00 describe a circle.
Divide the angle BOG into any number of equal parts by
radii of this circle. The spiral meets these radii in points
P, Q ... F, Z such that the radii vectores OJB, OP, OQ ... OZ, 00
1 On the whole course of Archimedes's proof of the property of the
subtangent, see note in the Appendix.
ON SPIRALS
73
are in, arithmetical progression. Draw arcs of circle* with
radii OB, OP, OQ ... as shown; this produces a figure circum-
scribed to the spiral and consisting of the sum of small sectors
of circles, and an inscribed figure of the same kind. As the
first sector in the circumscribed figure is equal to the second
sector in the inscribed, it is easily seen that the areas of the
circumscribed and inscribed figures differ by the difference
between the sectors OzG and OBp' '; therefore, by increasing
the number of divisions of the angle BOG, we can make the
difference between the areas of the circumscribed and in-
scribed figures as small as we please ; we have, therefore, the
elements necessary for the application of the method of
exhaustion.
If there are n radii OB, OP ... 00, there are (n 1) parts of
the angle BOG. Since the angles of tall the small sectors are
equal, the sectors are as the square on their radii.
Thus (whole sector 0//C 1 ) : (circumscribed figure)
= (TI- l)OC* : (OP 2 + OQ 2 + ... + OC 2 ),
and (whole sector Ob'C) : (inscribed figure)
74 ARCHIMEDES
And OB, OP, OQ, . . . OZ, OG is an arithmetical progression
of n terms ; therefore (cf. Prop. 1 1 and Cor.),
( - 1) OC 2 : (OP 2 + OQ 2 + . . . + OC 2 )
Compressing the circumscribed and inscribed figures together
in the usual way, Archimedes proves by exhaustion that
(sector OVC) : (area of spiral OBC)
If 05 = 6, OC=c, and (c-b) = (n-l)h, Archimedes's
result is the equivalent of saying that, when h diminishes and
11 increases indefinitely, while c b remains constant,
limit of h {1
that is, with our notation,
!'
Jb
Jb
In particular, the area included by the first turn and the
initial line is bounded by the radii vectores and
the area, therefore, is to the circle with radius 2?ra as ^|
to (2?ra) 2 , that is to say, it is ^ of the circle or ^
This is separately proved in Prop. 24 by means of Prop. 10
and Corr. 1, 2.
The area of the ring added while the radius vector describes
the second turn is the area bounded by the radii vectores 2 no,
and 4?ra, and is to the circle with radius 4?ra in the ratio
the ratio is 7 : 12 (Prop. 25).
If jRj be the area of the first turn of the spiral bounded by
the initial line, JK 2 the area of the ring added by the second
complete turn, R. 6 that of the ring added by the third turn,
and so on, then (Prop. 27)
Also ^=
ON SPIRALS 75
Lastly, if E be the portion of the sector b'OC bounded by
b'B, the arc b'zC of the circle and the arc BG of the spiral, and
F the portion cut oft 1 between the arc BG of the spiral, the
radius OG and the arc intercepted between OB and OG of
the circle with centre and radius OJ5, it is proved that
E:F= {05 + (0(7- 05)} : (05 + 4(0(7-05)} (Prop. 28).
On Plane Equilibriums, I, II.
In this treatise we have the fundamental principles of
mechanics established by the methods of geometry in its
strictest sense. There were doubtless earlier treatises on
mechanics, but it may be assumed that none of them had
been worked out with such geometrical rigour. Archimedes
begins with seven Postulates including the following prin-
ciples. Equal weights at equal distances balance ; if unequal
weights operate at equal distances, the larger weighs down
the smaller. If when equal weights are in equilibrium some-
thing be added to, or subtracted from, one of them, equilibrium
is not maintained but the weight which is increased or is not
diminished prevails. When equal and similar plane figures
coincide if applied to one another, their centres of gravity
similarly coincide; and in figures which are unequal but
similar the centres of gravity will be 'similarly situated'.
In any figure the contour of which is concave in one and the
same direction the centre of gravity must be within the figure.
Simple propositions (15) follow, deduced by reductio ad
absurdum', these load to the fundamental theorem, proved
first for commensurable and then by reductio ad abswrdum
for incommensurable magnitudes, that Two magnitudes,
whether commensurable or incommensurable, balance at. dis-
tances reciprocally proportional to the magnitudes (Props.
6, 7). Prop. 8 shows how to find the centre of gravity of
a part of a magnitude when the centres of gravity of the
other part and of the whole magnitude are given. Archimedes
then addresses himself to the main problems of Book I, namely
to find the centres of gravity of (l) a parallelogram (Props.
9, 10), (2) a triangle (Props. 13, 14), and (3) a parallel-
trapezium (Prop. 15), and here we have an illustration of the
extraordinary rigour which he requires in his geometrical
76 ARCHIMEDES
proofs. We do not find him here assuming, as in The Method,
that, if all the lines that can be drawn in a figure parallel to
(and including) one side have their middle points in a straight
line, the centre of gravity must lie somewhere on that straight
line ; he is not content to regard the figure as made up of an
infinity of such parallel lines; pure geometry realizes that
the parallelogram is made up of elementary parallelograms,
indefinitely narrow if you please, but still parallelograms, and
the triangle of elementary trapezia, not straight lines, so
that to assume directly that the centre of gravity lies on the
straight line bisecting the parallelograms would really be
a petitio principii. Accordingly the result, no doubt dis-
covered in the informal way, is clinched by a proof by reductio
ad absurdum in each case. In the case of the parallelogram
ABCJJ (Prop. 9), if the centre of gravity is not on the straight
line EF bisecting two opposite sides, let it be at H. Draw
HK parallel to AD. Then it is possible by bisecting AE, ED,
then bisecting the halves, and so on, ultimately to reach
a length less than KIL Let this be done, and through the
points of division of AD draw parallels to AB or DO making
a number of equal and similar parallelograms as in the figure.
The centre of gravity of each of these parallelograms is
similarly situated with regard to it. Hence we have a number
of equal magnitudes with their centres of gravity at equal
distances along a straight line. Therefore the centre of
gravity of the whole is on the line joining the centres of gravity
of the two middle parallelograms (Prop. 5, Cor. 2). But this
is impossible, because // is outside those parallelograms.
Therefore the centre of gravity cannot but lie on EF.
Similarly the centre of gravity lies on the straight line
bisecting the other opposite sides AB, CD; therefore it lies at
the intersection of this line with EF, i.e. at the point of
intersection of the diagonals.
ON PLANE EQUILIBRIUMS, I 77
Tho proof in the case of the triangle is similar (Prop. 13).
Let AD be .the median through A. The centre of gravity
must lie on AD.
For, if not, let it be at //, and draw 111 parallel to BC.
Then, if we, bisect DC, then bisect the halves, and so on,
we shall arrive at a length DE less than IH. Divide BC into
lengths equal to DE, draw parallels to DA through the points
of division, and complete the small parallelograms as shown in
the figure.
The centres of gravity of the whole parallelograms 8N, Tl\
lie on AD (Prop. 9) ; therefore the centre of gravity of the
ligure formed by them all lies on AD\ let it bo (>. Join OH,
and produce it to meet in F the parallel through C to AD.
Now it is easy to see that, if n be the number of parts into
which DC, AC are divided respectively,
(sum of small teAMR, MLti ... ARN, NUP ...) : (AttC)
~ 1 : it, ;
whence
(sum of small As) : (sum of parallelograms) = 1 : (?i I).
Therefore the centre of gravity of the figure made up of all
the small triangles is at a point X on OH produced such that
But VII : 110 < CE : ED or (n - 1) : 1 ; therefore X H > VII.
It follows that the centre of gravity of all the small
triangles taken together lies at X notwithstanding that all
the triangles lie on one side of the parallel to A D drawn
through X : which is impossible.
78 ARCHIMEDES
Hence the centre of gravity of the whole triangle cannot
but lie on AD.
It lies, similarly, on either of the other two medians; so
that it is at the intersection of any two medians (Prop. 14).
Archimedes gives alternative proofs of a direct character,
both for the parallelogram and the triangle, depending on the
postulate that the centres of gravity of similar figures are
* similarly situated' in regard to them (Prop. 10 for the
parallelogram, Props. 11, 12 and part 2 of Prop. 13 for the
triangle).
The geometry of Prop. 15 deducing the centre of gravity of
a trapezium is also interesting. It is proved that, if AD, BO
are the parallel sides (AD being the smaller), and EF is the
straight line joining their middle points, the centre of gravity
is at a point (? on EF such that
GE:GF = (2BC + AD) : (2AD + BC).
Book II of the treatise is entirely devoted to finding tho
centres of gravity of a parabolic segment (Props. 1-8) and
of a portion of it cut off by a parallel to the base (Props. 9, 10).
Prop. 1 (really a particular case of I. 6, 7) proves that, if P, P'
be the areas of two parabolic segments and 7J, E their centres
of gravity, the centre of gravity of both taken together is
at a point on DE such that
P:P'=CE:CD.
ON PLANE EQUILIBRIUMS, I, II 79
This is % merely preliminary. Then begins the real argument,
the course of which is characteristic and deserves to be set out.
Archimedes uses a series of figures inscribed to the segment,
as he says, 'in the recognized manner' (yi/o>pf/bia>?). The rule
is as follows. Inscribe in the segment the triangle ABB' witli
the same base and height; the vertex A is then the point
of contact of the tangent parallel to BB'. Do the same with
the remaining segments cut off by AB, AB', then with the
segments remaining, and so on. If BRQl^AP'Q'R'B' is such
a figure, the diameters through Q, Q', P, P', 11, R' bisect the
straight lines AB y AB', AQ, AQ\ QB y $]? respectively, and
BB / is divided by the diameters into parts which are all
equal. It is easy to prove also that PP', QQ', RR' are all
parallel to BB' y and that AL:LM:MN:NO = 1:3:5:7, the
same relation holding if the number of sides of the polygon
is increased; i.e. the segments of AO arc always in the ratio
of the successive, odd numbers (Lemmas to Pi-op. 2). The
centre of gravity of the inscribed figure lies on AO (Prop. 2).
If there be two parabolic segments, and two figures inscribed
in them 'in the recognized manner' with an equal ndmber of
sides, the centres of gravity divide the respective axes in the
same proportion, for the ratio depends on the same ratio of odd
numbers 1:3:5:7... (Prop. 3). The centre of gravity of the
parabolic segment itself lies on the diameter AO (this is proved
in Prop. 4 by reductio ad absurdum in exactly the same way
as for the triangle in I. 13). It is next proved (Prop. 5) that
the centre of gravity of the segment is nearer to the vertex A
than the centre of gravity of the inscribed figure is; but that
it is possible to inscribe in the segment in the recognized
manner a figure such that the distance between the centres of
gravity of the segment and of the inscribed figure is less than
any assigned length, for we have only to increase the number
of sides sufficiently (Prop. 6). Incidentally, it is observed in
Prop. 4 that, if in any segment the triangle with the same
base and equal height is inscribed, the triangle is greater than
half the segment, whence it follows that, each time we increase
the number of sides in the inscribed figure, we take away
more than half of the segments remaining over ; and in Prop. 5
that corresponding segments on opposite sides of the axis, e. g.
QRB y Q'R'B' have their axes equal- and therefore are equal in
80
ARCHIMEDES
area. Lastly (Prop. 7), if there be two parabolic segments,
their centres of gravity divide their diameters in the same
ratio (Archimedes enunciates this of similar segments only,
but it is true of any two segments and is required of any two
segments in Prop. 8). Prop. 8 now finds the centre of gravity
of any segment by using the last proposition. It is the
geometrical equivalent of the solution of a simple equation in
the ratio (m, say) of A G to AO, where G is the centre of
gravity of the segment.
Since the segment = f (&ABB') t the sum of the two seg-
ments AQB, AQ'B' = %(&ABB').
Further, if QD, Q'D' are the diameters of these segments,
QD, Q'D' are equal, and, since the centres
of gravity //, H' of the segments divide
QD, Q'D' proportionally, HH' is parallel
to QQ', and the centre of gravity of the
two segments together is at K, the point
where HH' meets AO.
Now AO = 4^17 (Lemma 3 to Prop.
2), and QD = %AO-AV=AV. But
H divides QD in the same ratio as (7
divides AO (Prop. 7); therefore
VK = QH = m. QD = m.AV.
Taking moments about A of the segment, the triangle A KB'
and the sum of the small segments, we have (dividing out by
AV and A ABB')
or
and in = -|.
That is,
= 9,
or A (f : (]() = 3:2.
The final proposition (10) finds the centre of gravity of the
portion of a parabola cut oft* between two parallel chords PP' 9
BB'. If PP' is the shorter of the chords and the diameter
bisecting PP', BB' meets them in N, respectively, Archi-
medes proves that, if NO be divided into five equal parts of
which LM is the middle one (L being nearer to J\T than M is),
ON PLANE EQUILIBRIUMS, II 81
the centre of gravity G of the portion of the parabola between
PP' and BB' divides LM in such a way that
LG : GM = BO* . (2PN+ BO) : PN* . (2 BO + PN).
The geometrical proof is somewhat difficult, and uses a very
remarkable Lemma which forms Prop. 9. If a, 6, c, d, x, y are
straight lines satisfying the conditions
a b c ,
d _ x
a d ~~~ (&
, 2a + 4b + 6v + 3d
and - -
5a+106 + 10c + 5
then must x + y = fa.
The proof is entirely geometrical, but amounts of course to
the elimination of three quantities 6, c, (/ from the above four
equations.
The Sand-reckoner (Psammites or Arenarius).
I have already described in a previous chapter the remark-
able system, explained in this treatise and in a lost work,
'Apxai, Principles, addressed to Zeuxippus, for expressing very
large numbers which were beyond the range of the ordinary
Greek arithmetical notation. Archimedes showed that his
system would enable any number to be expressed up to that
which in our notation would require 80,000 million million
ciphers and then proceeded to prove that this system more
than sufficed to express the number of grains of sand which
it would take to fill the universe, on a reasonable view (as it
seemed to him) of the size to be attributed to the universe.
Interesting as the book is for the course of the argument by
which Archimedes establishes this, it is, in addition, a docu-
ment of the first importance historically. It is here that we
learn that Aristarchus put forward the Copernican theory of
the universe, with the *^un in the centre and the planets
including the earth revolving round it, and that Aristarchus
further discovered the angular diameter of the sun to be ^oth
of the circle of the zodiac or half a degree. Since Archimedes,
in order to calculate a safe figure (not too small) for the size
1533.2 ft
82 ARCHIMEDES
of the universe, has to make certain assumptions as to the
sizes and distances of the sun and moon and their relation
to the size of the universe, he takes the opportunity of
quoting earlier views. Some have tried, he says, to prove
that the perimeter of the earth is about 300,000 stades; in
order to be quite safe he will take it to be about ten times
this, or 3,000,000 stades, and not greater. The diameter of
the earth, like most earlier astronomers, he takes to be
greater than that of the moon but less than that of the sun.
Eudoxus, he says, declared the diameter of the sun to be nine
times that of the moon, Phidias, his own father, twelve times,
while Aristarchus tried to prove that it is greater than 1 8 but
less than 20 times the diameter of the moori; he will again be
on the safe side and take it to be 30 times, but not mom The
position is rather more difficult as regards the ratio of the
distance of the sun to the size of the universe. Here he seizes
upon a dictum of Aristarchus that the sphere of the fixed
stars is so great that the circle in which he supposes the earth
to revolve (round the sun) ' bears such a proportion to the
distance of the fixed stars as the centre of the sphere bears to
its surface '. If this is taken in a strictly mathematical sense,
it means that the sphere of the fixed stars is infinite in size,
which would not suit Archimcdes's purpose ; to get another
meaning out of it he presses the point that Aristarchus's
words cannot be taken quite literally because the centre, being
without magnitude, cannot be in any ratio to any other mag-
nitude ; hence he suggests that a reasonable interpretation of
the statement would be to suppose that, if we conceive a
sphere with radius equal to the distance between the centre
of the sun and the centre of the earth, then
(diam. of earth) : Cdiam. of said sphere)
= (diam. of said sphere) : (diam. of sphere of fixed stars).
This is, of course, an arbitrary interpretation ; Aristarchus
presumably meant no such thing, but merely that the size of
the earth is negligible in comparison with that of the sphere
of the fixed stars. However, the solution of Archimedes's
problem demands some assumption of the kind, and, in making
this assumption, he was no doubt awai;c that he was taking
a liberty with Aristarchus for the sake of giving his hypo-
thesis an air of authority.
THE SAND-RECKONER
83
Arahimedes has, lastly, to compare the diameter of the sun
with the circumference of the circle described by its centre.
Aristarchus had made the apparent diameter of the sun y^th
of the said circumference ; Archimedes will prove that the
said circumference cannot contain as many as 1,000 sun's
diameters, or that the diameter of the sun is greater than the
side of a regular chiliagon inscribed in the circle. First he
made an experiment of his own to determine the apparent
diameter of the sun. With a small cylinder or disc in a plane
at right angles to a long straight stick and movcablc along it,
he observed the sun at the moment when it cleared the
horizon in rising, moving the disc till it just covered and just
failed to cover the sun as he looked along the straight stick.
He thus found the angular diameter to lie between T f^JB and
^ J Li, where R is a right angle. But as, under his assump-
tions, the size of the earth is not negligible in comparison with
the sun's circle, he had to allow for parallax and find limits
for the angle subtended by the sun at the centre of the earth.
This hr does by a geometrical argument very much in the
manner of Aristarchus.
Let the circles with centres 0, C represent sections of the sun
and earth respectively, E the position of the observer observing
G2
84 ARCHIMEDES
the sun when it has just cleared the horizon. Draw from E
two tangents EP, EQ to the circle with centre 0, and from
C let CF, GG be drawn touching the same circle. With centre
C and radius CO describe a circle : this will represent the path
of the centre of the sun round the earth. Let this circle meet
the tangents from C in A, B, and join AB meeting CO in M.
Archimedes's observation has shown that
^R> Z.PEQ >jfaR;
and he proceeds to prove that AB is less than the side of a
regular polygon of 656 sides inscribed in the circle AOB,
but greater than the side of an inscribed regular polygon of
1,000 sides, in other words, that
The first relation is obvious, for, since CO > EO,
L PEQ > Z FCG.
Next, the perimeter of any polygon inscribed in the circle
AOB is less than ^ CO (i.e. - 2 T 2 - times the diameter) ;
Therefore AB < ^ -\ 4 - CO or T | CO,
and, a fortiori, AB < T ^ CO.
Now, the triangles CAM, COF being equal in all respects,
AM= OF, so that AB = 20F= (diameter of sun) > C//+ OK,
since the diameter of the sun is greater than that of the earth ;
therefore CH+OK < yfoCO, and HK > -f^CO.
And CO > CF, while HK < EQ, so that EQ > ft
We can now compare the angles OCF, OEQ ;
L OCF r tan OCF}
\C\\* I ^ I
/ f\ ~Hi~\ I i. f\ I/V1 I
EQ
> CF
> -loci a fortiori.
Doubling the angles, we have
THE SAND-RECKONER 85
Hence AB is greater than the side of a regular polygon of
812 sides, and a fortiori greater than the side of a regular
polygon of 1,000 sides, inscribed in the circle AOB.
The perimeter of the chiliagon, as of any regular polygon
with more sides than six, inscribed in the circle AOB is greater
than 3 times the diameter of the sun's orbit, but is less than
1,000 times the diameter of the sun, and a fortiori less than
30,000 times the diameter of the earth;
therefore (diameter of sun's orbit) < 10,000 (diam. of earth)
< 10,000,000,000 stades.
But (diam. of earth) : (diam. of sun's orbit)
= (diam. of sim's orbit) : (diam. of universe) ;
therefore the universe, or the sphere of the fixed stars, is less
than 10,000 3 times the sphere in which the sun's orbit is a
great circle.
Archimedes takes a quantity of sand not greater than
a poppy-seed and assumes that it contains not more than 1 0,000
grains ; the diameter of a poppy-seed he takes to be not less
than 4*Q-th of a finger-breadth ; thus a sphere of diameter
1 finger-breadth is not greater than 64,000 poppy-seeds and
therefore contains not more than 640,000,000 grains of sand
('6 units of second order + 40,000,000 units of first order')
and a fortiori not more than 1,000,000,000 ('10 units of
second order of numbers '). Gradually increasing the diameter
of the sphere by multiplying it each time by 100 (making the
sphere 1,000,000 times larger each time) and substituting for
10,000 finger-breadths a stadium (< 10,000 finger-breadths),
lie finds the number of grains of sand in a sphere of diameter
10,000,000,000 stadia to bo less than '1,000 units of seventh
order of numbers' or 10 fll , and the number in a sphere 10,000 3
times this size to be less than ' 10,000,000 units of the eighth
order of numbers' or 10 c:i .
The Quadrature of the Parabola.
In the preface, addressed to Dositheus after the death of
Coiion, Archimedes claims originality for the solution of the
problem of finding the area of a segment of a parabola cut off
by any chord, which he says he first discovered by means of
mechanics and then confirmed by means of geometry, using
the lemma that, if there are two unequal areas (or magnitudes
86 ARCHIMEDES
generally), then however small the excess of the greater over
the lesser, it can by being continually added to itself be made
to exceed the greater ; in other words, he confirmed the solution
by the method of exhaustion. One solution by means of
mechanics is, as we have seen, given in The Method; the
present treatise contains a solution by means of mechanics
confirmed by the method of exhaustion (Props. 1-17), and
then gives an entirely independent solution by means of pure
geometry, also confirmed by exhahstion (Props. 18-24).
I. The mechanical solution depends upon two properties of
the parabola proved in Props. 4, 5. If Qq be the base, and P
the vertex, of a parabolic segment, P is the point of contact
of the tangent parallel to Qq, the diameter PV through P
bisects Qq in V 9 and, if VP produced meets the tangent at Q
in T, then TP = PV. These properties, along with the funda-
mental property that QV' 2 varies as PV 9 Archimedes uses to
prove that, if EO be any parallel to TV meeting QT, QP
(produced, if necessary), the curve, and Qq in E, t\ 11,
respectively, then
QV: VO = OF:FR,
and QO : Oq = ER : RO. (Props. 4, 5.)
Now suppose a parabolic segment Ql^q so placed in relation
to a horizontal straight line Q A through Q that the diameter
bisecting Qq is at right angles to QA, i.e. vertical, and let the
tangent at Q meet the diameter qO through q in E. Produce
QO to A, making OA equal to OQ.
Divide Qq into any number of equal parts at O l , 2 . . . O n ,
and through these points draw parallels to OE, i. e. vertical
lines meeting OQ in H 19 H. 2 , ..., EQ in E 19 E^ ..., and the
THE QUADRATURE OF THE PARABOLA 87
curve in R ly R^ ... . Join QR l , and produce it to meet OE in
F, Q/? 2 meeting O l E l in t\, and so on.
O HI H 2 H 3
Now Archimedes has proved in a series of propositions
(6-13) that, if a trapezium such as 1 E 1 E^O% is suspended
from //^Y^and an area P suspended at A balances 1 /? 1 A ? 2 2
so suspended, it will take a greater area than P suspended at
A to balance the same trapezium suspended from J/ 2 and
a less area than P to balance the same trapezium suspended
from //, . A similar proposition holds with regard to a triangle
such as E n ll lt Q suspended where it is and suspended at Q and
ll n respectively.
Suppose (Props. 14, 15) the triangle QqE suspended where
it is from OQ, and suppose that the trapezium EO^ suspended
where it is, is balanced by an area l\ suspended at A, the
trapezium A T ,(A,, suspended where it is, is balanced by 7
suspended at A, and so on, and finally the triangle M n O n Q,
suspended where it is, is balanced by P n+l suspended at A ;
then P l + J^ + . . . -f jfj i+1 at A balances the whole triangle, so that
.since the whole triangle may be regarded as suspended from
the point on OQ vertically above its centre of gravity.
Now AO:OJI l = QO:OJl l
= #,0^0^, by Prop. 5,
= (trapezium EO^) : (trapezium
88 ARCHIMEDES
that is, it takes the trapezium F0 l suspended at A to balance
the trapezium E0 l suspended at H r And P l balances E0 l
where it is.
Therefore (FOJ > P r
Similarly (^1^2) > P 'ind so n -
Again AO'.OH^ = E, O l : 0^
= (trapezium E^O^ : (trapezium K^0^ 9
that is, (R>i0 2 ) at A will balance (# a 2 ) suspended at H 19
while P 2 at A balances (Efl^) suspended where it is
whence F 2 > U/^.
Therefore (^0 2 ) > ^2 > (#1^)*
(^2^3) > ^ > ^2^a> an( l so on ;
and finally, AA T w O n Q > 7^ 41 >
By addition,
therefore, a fortiori,
That is to say, we have an inscribed figure consisting of
trapezia and a triangle which is less, and a circumscribed
figure composed in the same way which is greater, than
.e.
It is therefore inferred, and proved by the method of ex-
haustion, that the segment itself is equal to %AJKqQ (Prop. 16).
In order to enable the method to be applied, it has only
to be proved that, by increasing the number of parts in Qq
sufficiently, the difference between the circumscribed and
inscribed figures can be made as small as we please. This
can be seen thus. We have first to show that all the parts, as
qF, into which qE is divided are equal.
We have E^iO^ = QO -.01^ = (n+1): 1,
or 0^ = ---- - . E l 19 whence also 2 $ = - - . O^E 2 .
THE QUADRATURE OF THE PARABOLA 89
And E 2 2 : 2 jK 2 = QO : 0# 2 = (71+ 1) : 2,
2
2 2 ,^ +1 "' 2 '*'
It follows that 2 # = /S'JB 2 , and so on.
Consequently 1 R IJ O^JR. 2J 3 7i a ... are divided into 1, 2, 3 ...
equal parts respectively by the lines from Q meeting qE.
It follows that the difference between the circumscribed and
inscribed figures is equal to the triangle FqQ, which can be
made as small as we please by increasing the number of
divisions in Qq y i.e. in qE.
Since the area of the segment is equal to %AEqQ, and it is
easily proved (Prop. 17) that AEqQ =: 4 (triangle with same
base and equal height with segment), it follows that the area
of the segment = -3 times the latter triangle.
It is easy to see that this solution is essentially the same as
that given in The Method (see pp. 29-30, above), only in a more
orthodox form (geometrically speaking). For there Archi-
medes took the sum of all the straight line*, as O l R l , 2 R 2 >
as making up the segment notwithstanding that there are an
infinite number of them and straight lines have no breadth.
Here he takes inscribed and circumscribed trapezia propor-
tional to the straight lines and having finite breadth, and then
compresses the figures together into the segment itself by
increasing indefinitely the number of trapezia in each figure,
i.e. diminishing their breadth indefinitely.
The procedure is equivalent to an integration, thus :
If X denote the area of the triangle FqQ, we have, if n be
the number of parts in (<)</,
(circumscribed figure)
= sum of AsQqF, QR Y F ly QU^, ...
= sum of AnQqF, QO^,, QO^S, ...
n 2
Similarly, we find that
(inscribed figure) = - 2 . X \X* + 2*X*+ ... + (n- 1) 2 .Y 2 }-
90
ARCHIMEDES
Taking the limit, we have, if A denote the area of the
triangle EqQ, so that A = nX,
1 C A
area of segment = -r^ J
" Jo
II. The purely geometrical method simply exhausts the
parabolic segment by inscribing successive figures ' in the
recognized manner' (see p. 79, above). For this purpose
it is necessary to find, in terms of the triangle with the same
base and height, the area added to the
inscribed figure by doubling the number of
sides other than the base of the segment.
Let QPq be the triangle inscribed c in the
recognized manner', P being the point of
contact of the tangent parallel to Qq, and
PV the diameter bisecting Qq. If QV, Vq
be bisected in M, m, and RM, rm be drawn
parallel to PV meeting the curve in R, r,
the latter points are vertices of the next
figure inscribed c in the recognized manner ',
for RY, ry are diameters bisecting PQ, Pq
respectively.
Now QV* = 4/ilK 2 , so that PV = 4PW, or RM = 3P\\ r .
But YM = APF = 2P H', so that YM = 2 RY.
Therefore A PRQ = | A PQ M = A PQ ] r .
Similarly
APr0 = JAPFg; whence (APRQ + &Prq)= %PQq. (Prop. 21.)
In like manner it can be proved that the next addition
to the inscribed figure adds % of the sum of AsPRQ, Prq,
and so on.
Therefore the area of the inscribed figure
= { 1+ i + (J) a +...} AP<&. (Prop. 22.)
Further, each addition to the inscribed figure is greater
than half the segments of the parabola left over before the
addition is made. For, if we draw the tangent at P and
complete the parallelogram EQqe with side EQ parallel to PV,
THE QUADRATURE OF THE PARABOLA 91
the triangle PQq is half of the parallelogram and therefore
more than half the segment. And so on (Prop. 20).
We now have to sum u terms of the above geometrical
series. Archimedes enunciates the problem in the form, Given
a series of areas A, B, C, D . . . %, of which A is the greatest, and
each is equal to four times the next in order, then (Prop. 23)
The algebraical equivalent of this is of course
To find the area of the segment, Archimedes, instead of
taking the limit, as we should, uses the method of reductio ad
abswrdwin.
Suppose K = - A7 J (^/.
(1) If possible, let the area of the segment be greater than K.
We then inscribe a figure ' in the recognized manner ' such
that the segment exceeds it by an area less than the excess of
the segment over K. Therefore the inscribed figure must be
greater than A r , which is impossible since
where A = A 7% (Prop. 23).
(2) If possible, let the area of the segment be less than K.
Ff then &PQq = A, K = %A t = J/J, and so on, until we
arrive at an area A" less than the excess of K over the area of
the segment, we have
A +]* + (! + ... +X + $X = $A = K.
Thus K exceeds A -f K + (J+ ... + X by an area less than A",
and exceeds the segment by an area greater than X.
It follows that A +H + G+ ... +X> (the segment) ; which
is impossible (Prop. 22).
Therefore the area of the segment, being neither greater nor
less than K, is equal to A' or %
On Floating Bodies, I, II.
In Book J of this treatise Archimedes lays down the funda-
mental principles of the science of hydrostatics. These are
92 ARCHIMEDES
deduced from Postulates which are only two in number. The
first which begins Book I is this :
' let it be assumed that a fluid is of such a nature that, of the
parts of it which lie evenly and are continuous, that which is
pressed the less is driven along by that which is pressed the
more; and each of its parts is pressed by the fluid whicli is
perpendicularly above it except when the fluid is shut up in
anything and pressed by something else ' ;
the second, placed after Prop. 7, says
c let it be assumed that, of bodies which are borne upwards in
a fluid, each is borne upwards along the perpendicular drawn
through its centre of gravity '.
Prop. 1 is a preliminary proposition about a sphere, and
then Archimedes plunges in med'Utu res with the theorem
(Prop. 2) that 'the surface of any fluid at rest is a sjthere the
centre of which is the same as that of the earth ', and in the
whole of Book I the surface of the fluid is always shown in
the diagrams as spherical. The method of proof is similar to
what we should expect in a modern elementary textbook, the
main propositions established being the following. A solid
which, size for size, is of equal weight with a fluid will, if let
down into the fluid, sink till it is just covered but not lower
(Prop. 3) ; a solid lighter than a fluid will, if let down into it,
be only partly immersed, in fact just so far that the weight
of the solid is equal to the weight of the fluid displaced
(Props. 4, 5), and, if it is forcibly immersed, it will be driven
upwards by a force equal to the difference between its weight
and the weight of the fluid displaced (Prop. 6).
The important proposition follows (Prop. 7) that a solid
heavier than a fluid will, if placed in it, sink to the bottom of
the fluid, arid the solid will, when weighed in the fluid, be
lighter than its true weight by the weight of the fluid
displaced.
The problem of the Crown.
This proposition gives a method of solving the famous
problem the discovery of which in his bath sent Archimedes
home naked crying evpijKa, tvprjKa, namely the problem of
ON FLOATING BODIES, I 93
determining the proportions of gold and silver in a certain
crown.
Let W be the weight of tho crown, w l and u\ 2 the weights of
the gold and silver in it respectively, so that W = w l + w 2 .
(1) Take a weight PFof pure gold and weigh it in the fluid.
The apparent loss of weight is then equal to the weight of the
fluid displaced ; this is ascertained by weighing. Let it be F r
It follows that the weight of the fluid displaced by a weight
M! of gold is ^ . F r
(2) Take a weight W of silver, and perform the same
operation. Let the weight of the fluid displaced be F 2 .
Then the weight of the fluid displaced by a weight iv 2 of
silver is ^ F 2 .
(3) Lastly weigh the crown itself in the fluid, and let F be
loss of weight or the weight of the fluid displaced.
We have then * . F l + TTJ? . F 2 = F,
that is, '</>! F l -f w 2 F^ = (w i + w>) F,
whence = J 2 , T
According to the author of the poem de ponderibus et men-
suriH (written 'probably about A.D. 500) Archimedes actually
used a muthod of this kind. We first take, says our authority,
two equal weights of gold and silver respectively and weigh
them against each other when both are immersed in water;
this gives the relation between their weights in water, and
therefore between their losses of weight in water. Next we
take the mixture of gold and silver and an equal weight of
silver, and weigh them against each other in water in the
same way.
Nevertheless I do not think it probable that this was the
way in which the solution of the problem was discovered. As
we are told that Archimedes discovered it in his bath, and
that he noticed that, if the bath was full when he entered it,
so much water overflowed as was displaced by his body, he is
more likely to have discovered the solution by the alternative
94 ARCHIMEDES
method attributed to him by Vitruvius, 1 namely by measuring
successively the volumes of fluid displaced by three equal
weights, (1) the crown, (2) an equal weight of gold, (3) an
equal weight of silver respectively. Suppose, as before, that
the weight of the 'crown is W and that it contains weights
w l and w.j, of gold and silver respectively. Then
(1) the crown displaces a certain volume of the fluid, V, say ;
(2) the weight W of gold displaces a volume V v say, of the
fluid;
7/'
therefore a weight ^v l of gold displaces a volume TJT V l of
the fluid ;
(3) the weight W of silver displaces F 2 , say, of the fluid;
therefore a weight m 2 of silver displaces -y^- F 2 .
i u f if
It follows that V = -yj F, + '.; - F,,,
W W
whence we derive (since W = iu l + w> 2 )
. _ T/ 2 ~ v
<" ~ V-Vi
the latter ratio being obviously equal to that obtained by the
other method.
The last propositions (8 and 9) of Book I deal with the case
of any segment of a sphere' lighter than a fluid and immersed
in it in such a way that either (1) the curved surface is down-
wards and the base is entirely outside the fluid, or (2) tho
curved surface is upwards and the base is entirely submerged,
and it is proved that in either case the segment is in stable
equilibrium when the axis is vertical. This is expressed hero
and in the corresponding propositions of Book II by saying
that, ' if the figure be forced into such a position that the base
of the segment touches the fluid (at one point), the figure will
not remain inclined but will return to the upright position'.
Book II, which investigates fully the conditions of stability
of a right segment of a paraboloid of revolution floating in
a fluid for different values of the specific gravity and different
ratios between the axis or height of the segment and the
1 De architechira, ix. 3.
ON FLOATING BODIES, I, II 95
principal parameter of the generating parabola, is a veritable
tour de force which must be read in full to be appreciated.
Prop. 1 is preliminary, to the effect that, if a solid lighter than
a fluid be at rest in it, the weight of the solid will be to that
of the same volume of the fluid as the immersed portion of
the solid is to the whole. The results of the propositions
about the segment of a paraboloid may be thus summarized.
Let h be the axis or height of the segment, p the principal
parameter of the generating parabola, s the ratio of the
specific gravity of the solid to that of the fluid (s always < 1).
The segment is supposed to be always placed so that its base
is either entirely above, or entirely below, the surface of the
fluid, and what Archimedes proves in each case is that, if
the segment is so placed with its axis inclined to the vertical
at any angle, it will not rest there but will return to the
position of stain lity.
I. If h is not greater than f /?, the position of stability is with
the axis vertical, whether the curved surface is downwards or
upwards (Props. 2, 3).
II. If h is greater than f p, then, in order that the position of
stability may be with the axis vertical, s must be not loss
than (// f-/>) 2 //<' 2 if tin; curved surface is downwards, and not
greater than {A 2 (A f j />) 2 }//^ 2 if the curved surface is
upwards (Props. 4, 5).
III. If 7i>f/>, but &/|^< 15/4, the segment, if placed with
one point of the base touching the surface, will never remain
there whether the curved surface be downwards or upwards
(Props. 6, 7). (The segment will move hi the direction of
bringing the axis nearer to the vertical position.)
IV. If /*>/?, but A/i/x 15/4, and if s is less than
(h %p) 2 /Jr in the case where the curved surface is down-
wards, but greater than {h 2 (h %p)*]/Jr in the case where
the curved surface is upwards, then the position of stability is
one in which the axis is not vertical but inclined to the surface
of the fluid at a certain angle (Props. 8, 9). (The angle is drawn
in an auxiliary figure. The construction for it in Prop. 8 is
equivalent to the solution of the following equation in 0,
96
ARCHIMEDES
where k is the axis of the segment of the paraboloid cut oft' by
the surface of the fluid.)
V. Prop. 10 investigates the positions of stability in the cases
where h/%p>15/4, the base is entirely above the surface, and
s has values lying between five pairs of ratios respectively.
Only in the case where s is not less than (h-^pY/h? is the
position of stability that in which the axis is vertical.
BA B l is a section of the paraboloid through the axis AM.
G is a point on AM such that AC = 2 CM, K is a point on OA
stoch that AM: OK = 15:4. CO is measured along CA such
that CO = %p, and E is a point on AM such that MR = f CO.
A 2 is the point in which the perpendicular to AM from K
meets AH, and A 3 is the middle point of AB. BA 2 B 2 , BA.^ M
are parabolic segments on A 2 M^ A^M.^ (parallel to AM) as axes
and similar to the original segment. (The parabola
is proved to pass through C by using the above relation
AM: CK =15:4 and applying Prop. 4 of the Quadrature of
the Parabola.) The perpendicular to AM from meets the
parabola BA 2 B 2 in two points P 2 , (<J 2 , and straight lines
through these points parallel to AM meet the other para-
bolas inPj, Q l and P 3 , Q :J respectively. P/l 1 and Q } U are
tangents to the original parabola meeting the axis MA pro-
duced in T, U. Then
(i) if * is not less than AR*:AM* or (A-fy) 2 :^ 2 , there is
stable equilibrium when AM is vertical ;
THE CATTLE-PROBLEM 97
(ii) if s< A R* : AM* but > Q, Q 3 2 : A M*, the solid will not rest
with its base touching the surface of the fluid in one point
only, but in a position with the base entirely out of the fluid
and the axis making with the surface an angle greater
than U ;
(iiia) if s = Q^^iAM 2 , there is stable equilibrium with one
point of the base touching the surface and AM inclined to it
at an angle equal to U\
(iiib) if s = P 1 P^ : AAf 2 , there is stable equilibrium with one
point of the base touching the surface and with AM inclined
to it at an angle equal to T ;
(iv) ittoP^iAM* but <Q 1 Q.*:AM 2 , there will be stable
equilibrium in a position in which the base is more submerged ;
(v) if 8<P 1 P.?:AM 2 , there will be stable equilibrium with
the base entirely out of the fluid and with the axis AM
inclined to the surface at an angle less than T.
It remains to mention the traditions regarding other in-
vestigations by Archimedes which have readied us in Greek
or through the Arabic.
(a) The
This is a difficult problem in indeterminate analysis. It is
required to find the number of bulls and cows of each of four
colours, or to find 8 unknown quantities. The first part of
the problem connects the unknowns by seven simple equations ;
and the second part adds two more conditions to which the
unknowns must be subject. If W, w be the numbers of white
bulls and cows respectively and (X, x)> (F, y), (Z, z) represent
the numbers of the other three colours, we have first the
following equations :
(I) W=& + $X + Y, (a)
, (/S)
, (y)
(II) w=(i + i)(* + s), (S)
, ()
98 ARCHIMEDES
Secondly, it is required that
W+X = a square, (6)
Y + Z = a triangular number. (t)
There is an ambiguity in the text which makes it just possible
that W+ X need only be the product of two whole numbers
instead of a square as in (0). Jul. Fr. Wurm solved the problem
in the simpler form to which this change reduces it. The
complete problem is discussed and partly solved by Amthor. 1
The general solution of the first seven equations is
TF= 2.3.7.53.466771= 10366482^,
Z= 2. 3 2 . 89. 4657?i = 7460514 n,,
F= 3 4 . 11. 465771 = 414938771,
Z 2 2 .5.79.4657n = 7358060??,
w = 2\3. 5. 7. 23. 37371 = 720636071,
X = 2. 3 2 . 17. 15991 n = 4893246/1,
y= 3 2 . 13.4648971 = 543921371,
2 =2 2 . 3. 5. 7. 11. 761 71 = 3515820n.
It is not difficult to find such a value of n that W+ X = a
square number; it is ri/ = 3 . 11 . 29 . 4657 2 = 4456 749 2 ,
where is any integer. We then have to make
a triangular number, i.e. a number of the form ^#(<7+
This reduces itself to the solution of the * Pellian ' equation
which leads to prodigious figures ; one of the eight unknown
quantities alone would have more than 206,500 digits !
()3) On semi-regular polyhedra.
In addition, Archimedes investigated polyhedra of a certain
type. This we learn from Pappus. 2 The polyhedra in question
are semi-regular, being contained by equilateral and equi-
1 Zeitschrift fur Math. u. Physik (Hist.-litt. Abt.) xxv. (1880), pp.
156 sqq.
2 Pappus, v, pp. 352-8.
ON SEMI-REGULAR POLYHEDRA 99
angular, but not similar, polygons; those discovered by
Archimedes were 13 in number. If we for convenience
designate a polyhedron contained by m regular polygons
of oc sides, n, regular polygons of ft sides, &c., by (?>i a , %...),
the thirteen Archimedean polyhedra, which we will denote by
1\, 1*, ... /| 3 , are as follows:
Figure with 8 faces: 1\ = (4,, 4 C ).
Figures with 14 faces: P 2 = (8 3 , 6 4 ), I\ = (6 4 , 8 G ),
I\ = (83, 8 8 ).
Figures with 2(> faces: P 5 = (8 3 , 18 4 ), 7 J 6 =(12 4 , 8 G , 6 8 ).
Figures with 32 faces: P 7 = (20 3 , 12 5 ), P 8 = (12 5 , 20 6 ),
P = (20 3 ,12 10 ).
Figure with 38 faces: P 10 ~ (32.,, 6 4 ).
Figures with 62 faces: P n = (20 3 , 30 4 , 12 5 ),
P ]2 EE(30 4 ,20 G ,12 ]0 ).
' Figure with 92 faces: P 1:J = (80 3 , 12 5 ).
Kepler 1 showed how these figures can be obtained. A
method of obtaining some of them is indicated in a fragment
of a scholium to the Vatican MS. of Pappus. If a solid
angle of one of the regular solids be cut oft* symmetrically by
a plane, i.e. in such a way that the plane cuts oft* the same
length from each of the edges meeting at the angle, the
section is a regular polygon which is a triangle, square or
pentagon according as the solid angle is formed of three, four,
or five plane angles. If certain equal portions be so cut off*
from all the solid angles respectively, they will leave regular
polygons inscribed in the faces of the solid ; this happens
(A) when the cutting planes bisect the sides of the faces and
so leave in each face a polygon of the same kind, and (B) when
the cutting planes cut off a smaller portion from each angle in
such a way that a regular polygon is left in each face which
has double the number of sides (as when we make, say, an
octagon out of a square by cutting off the necessary portions,
1 Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6.
H 2
100
ARCHIMEDES
symmetrically, from the corners). We have seen that, accord-
ing to Heron, two of the semi-regular solids had already been
discovered by Plato, and this would doubtless be his method.
The methods (A) and (B) applied to the five regular solids
give the following out of the 13 semi-regular solids. We
obtain (1) from the tetrahedron, P l by cutting off angles
so as to leave hexagons in the faces ; (2) from the cube, P 2 by
leaving squares, and P 4 by leaving octagons, in the faces ;
(3) from the octahedron, P 2 by leaving triangles, and P 3 by
leaving hexagons, in the faces ; (4) from the icosahedron,
Pj by leaving triangles, and J^ by leaving hexagons, in the
faces; (5) from the dodecahedron, P 7 by leaving pentagons,
and P 9 by leaving decagons in the faces.
Of the remaining six, four arc obtained by cutting off all
the edges symmetrically and eqiially by planes parallel to the
edges, and then cutting off angles. Take first the cube.
(1) Cut off from each four parallel edges portions which leave
an octagon as the section of the figure perpendicular to the
edges ; then cut off equilateral triangles from the corners
(see Fig. 1) ; this gives P 5 containing 8 equilateral triangles
and 18 squares. (P 5 is also obtained by bisecting all the
edges of J^ and cutting off corners.) (2) Cut off from the
edges of the cube a smaller portion so as to leave in each
face a square such that the octagon described in it has its
side equal to the breadth of the section in which each edge is
cut ; then cut off hexagons from each angle (see Fig. 2) ; this
FIG. 1.
FIG. 2.
gives 6 octagons in the faces, 12 squares under the edges and
8 hexagons at the corners; that is, we have P 6 . An exactly
ON SEMI-REGULAR POLYHEDRA
101
similar procedure with the icosahedron and dodecahedron
produces P n and P n (see Figs. 3, 4 for the case of the icosa-
hedron).
FIG. 3.
FIG. 4.
The two remaining solids JJ , P VA cannot be so simply pro-
duced. They are represented in Figs. 5, 6, which I have
FIG. 5.
FIG. G.
taken from Kepler. 1* is the snub cube in which each
solid angle is formed by the angles of four equilateral triangles
arid one square; P ri is the snub dodecahedron, each solid
angle of which is formed by the angles of four equilateral
triangles and one regular pentagon.
We are indebted to Arabian tradition for
(y) The Liber Asswnptorum.
Of the theorems contained in this collection many are
so elegant as to afford a presumption that they may really
be due to Archimedes. In three of them the figure appears
which was called ap/Si/Ao?, a shoemaker's knife, consisting of
three semicircles with a common diameter as shown in the
annexed figure. If N be the point at which the diameters
102
ARCHIMEDES
of the two smaller semicircles adjoin, and NP be drawn at
right angles to AS meeting the external semicircle in P, the
area of the apjSijAos (included between the three semicircular
arcs) is equal to the circle on PN as diameter (Prop. 4). In
Prop. 5 it is shown that, if a circle be described in the space
between the arcs AP> AN and the straight line PN touching
all three, and if a circle be similarly described in the space
between the arcs PB, NB and the straight line PN touching
all three, the two circles are equal. If one circle be described
in the #pj8r/Aoy touching all three semicircles, Prop. 6 shows
that, if the ratio of AN to NB be given, we can find the
relation between the diameter of the circle inscribed to the
apftrfXo? and the straight line AB] the proof is for the parti-
cular case AN=%BN 9 and shows that the diameter of the
inscribed circle = ^AB.
Prop. 8 is of interest in connexion with the problem of
trisecting any angle. If AB be any chord of a circle with
centre 0, and BG on AB produced be made equal to the radius,
draw CO meeting the circle in /), E ; then will the arc BD be
one-third of the arc AE (or BF, if EF be the chord through E
parallel to AB). The problem is by this theorem reduced to
(cf. vol. i, p. 241).
a
THE LIBER ASSUMPTORUM
103
Lastly, we may mention the elegant theorem about the
area of the Salinon (presumably 'salt-cellar') in Prop. 14.
ACB is a semicircle on AS as diameter, AD, EB are equal
lengths measured from A and B 011 AB. Semicircles are
drawn with AD, EB as diameters on the side towards (7, and
a semicircle with DE as diameter is drawn on the other side of
AB. CF is the perpendicular to AB through 0, the centre
of the semicircles ACB, DFE. Then is the area bounded by
all the semicircles (the tialiuon) equal to the circle on CF
as diameter.
The Arabians, through whom the ttook of Lemmas has
readied us, attributed to Archimedes other works (1) on the
Circle, (2) on the Heptagon in a Circle, (3) on Circles touch-
ing one another, (4) on Parallel Lines, (5) on Triangles, (6) on
the properties of right-angled triangles, (7) a book of Data,
(8) De clepsydris: statements which we are not in a position
to check. But the author of a book on the finding of chords
in a circle, 1 Abu'l Raihan Muh. al-Birunl, quotes some alterna-
tive proofs as coming from the first of these works.
(8) Formula for area of triangle.
More important, however, is the mention in this same work
of Archimedes as the discoverer of two propositions hitherto
attributed to Heron, the first being the problem of finding
the perpendiculars of a triangle when the sides are given, and
the second the famous formula for the area of a triangle in
terms of the sides,
V{s(s-a)(s-b)(s-c)}.
1 See Bibliotheca mathematica, xi s , pp. 11-78.
c
104 ERATOSTHENES
Long as the present chapter is, it is nevertheless the most
appropriate place for ERATOSTHENES of Cyrene. It was to him
that Archimedes dedicated The Method, and the Cattle-Problem
purports, by its heading, to have been sent through him to
the mathematicians of Alexandria. It is evident from the
preface to The Method that Archimedes thought highly of his
mathematical ability. He was, indeed, recognized by his con-
temporaries as a man of great distinction in all branches of
knowledge, though in each subject he just fell short of the
highest place. On the latter ground he was called Beta, and
another nickname applied to him, Pentathlos, has the same
implication, representing as it does an all-round athlete who
was not the first runner or wrestler but took the second prize
in these contests as well as in others. He was very little
younger than Archimedes ; the date of his birth was probably
284 B.C. or thereabouts. He was a pupil of the philosopher
Ariston of Chios, the grammarian Lysanias of Cyrene, and
the poet Callimachus ; he is said also to have been a pupil of
Zeno the Stoic, and he may have come under the influence of
Arcesilaus at Athens, where he spent a considerable time.
Invited, when about 40 years of age, by Ptolemy Euergetes
to be tutor to his son (Philopator), he became librarian at
Alexandria ; his obligation to Ptolemy he recognized by the
column which he erected with a graceful epigram inscribed on
it. This is the epigram, with which we are already acquainted
(vol. i, p. 260), relating to the solutions, discovered up to date,
of the problem of the duplication of the cube, and commend-
ing his own method by means of an appliance called /*eo-oAa/3oi>,
itself represented in bronze on the column.
Eratosthenes wrote a book with the title UAaroowKoy, and,
whether it was a sort of commentary on the Timaeus of
Plato, or a dialogue in which the principal part was played by
Plato, it evidently dealt with the fundamental notions of
mathematics in connexion with Plato's philosophy. It was
naturally one of the important sources of Theon of Smyrna's
work on the mathematical matters which it was necessary for
the student of Plato to know ; and Theon cites the work
twice by name. It seems to have begun with the famous
problem of Delos, telling the story quoted by Theon how the
god required, as a means of stopping a plague, that the altar
PLATONIGUS AND ON MEANS 105
there, "which was cubical in form, should be doubled in size.
The book evidently contained a disquisition on proportion
(dva\oyia), a quotation by Theon on this subject shows that
Eratosthenes incidentally dealt with the fundamental defini-
tions of geometry and arithmetic. The principles of music
were discussed in the same work.
We have already described Eratosthenes' s solution of the
problem of Delos, and his contribution to the theory of arith-
metic by means of his sieve (KQVKLVQV) for finding successive
prime numbers.
He wrote also an independent work On means. This was in
two Books, and was important enough to be mentioned by
Pappus along with works by Euclid, Aristaeus and Apol-
lonius as forming part of the Treasury of Analysis 1 ; this
proves that it was a systematic geometrical treatise. Another
passage of Pappus speaks of certain loci which Eratosthenes
called 'loci with reference to means' (TOTTOL Trpoy /xeo-orTjray) 2 ;
these were presumably discussed in the treatise in question.
What kind of loci these were is quite uncertain ; Pappus (if it
is not an interpolator who speaks) merely says that these loci
1 belong to the aforesaid classes of loci ', but as the classes are
numerous (including ' plane ', ' solid ', * linear ', ' loci on surfaces ',
&c.), we are none the wiser. Tannery conjectured that they
wore loci of points such that their distances from three fixed
straight lines furnished a 'imnlitftd', i.e. loci (straight lines
and conies) which wo should represent in trilinear coordinates
by such equations as 2y = x + 3, y/ 2 = o:s, y(x + z) = 2 ore,
x(x y) = z(y z), x(x y) = y(y z), the first three equations
representing the arithmetic, geometric and harmonic means,
while the last two represent the ' subcontraries ' to the
harmonic and geometric means respectively. Zeutheii has
a different conjecture? He points out that, if QQ' be the
polar of a given point C with reference to a conic, and CPOP'
be drawn through C meeting QQ' in and the conic in P, P',
then CO is the harmonic mean to CP, CP' ; the locus of for
all transversals CPP' is then the straight line QQ'. If A, tt
are points on PP f such that CA is the arithmetic, and Cti the
1 Pappus, vii, p. 636. 24. 2 lb. 9 p. 662. 15 sq.
8 Zeuthen, Die Lehre von den Keyelschnitten im Altertum, 1886, pp.
320, 321.
106
ERATOSTHENES
geometric mean between CP, OP', the loci of A, G respectively
are conies. Zeuthen therefore suggests that these loci and
the corresponding loci of the points on CPP* at a distance
from equal to the subcontraries of the geometric and
harmonic means between CP and GP' are the 'loci with
reference to means ' of Eratosthenes ; the latter two loci are
'linear', i.e. higher curves than conies. Needless to say, we
have no confirmation of this conjecture.
Eratosthenes s measurement of the Earth.
But the most famous scientific achievement of Eratosthenes
was his measurement of the earth. Archimedes mentions, as
we have seen, that some had tried to prove that the circum-
ference of the earth is about 300,000 stades. This was
evidently the measurement based on observations made at
Lysirnachia (on the Hellespont) and Syene. It was observed
that, while both these places were on one meridian, the head
of Draco was in the zenith at Lysimachia, and Cancer in the
zenith at Syeiie ; the arc of the meridian separating the two
in the heavens was taken to be I/ 15th of the complete circle.
The distance between the two towns
was estimated at 20,000 stades, and
accordingly the whole circumference of
the earth was reckoned at 300,000
stades. Eratosthenes improved on this.
He observed (l) that at Syene, at
noon, at the summer solstice, the
sun cast no shadow from an upright
gnomon (this was confirmed by the
observation that a well dug at the
same place was entirely lighted up at
the same time), while (2) at the same moment the gnomon fixed
upright at Alexandria (taken to be on the same meridian with
Syene) cast a shadow corresponding to an angle between the
gnomon and the sun's rays of I/ 50th of a complete circle or
four right angles. The sun's rays are of course assumed to be
parallel at the two places represented by S and A in the
annexed figure. If a be the angle made at A by the sun's rays
with the gnomon (0 A produced), the angle 80 A is also equal to
MEASUREMENT OF THE EARTH 107
a, or r/50th of four right angles. Now the distance from S
to A was known by measurement to be 5,000 stades; it
followed that the circumference of the earth was 250,000
stades. This is the figure given by Cleomedes, but Theon of
Smyrna and Strabo both give it as 252,000 stades. The
reason of the discrepancy is not known ; it is possible that
Eratosthenes corrected 250,000 to 252,000 for some reason,
perhaps in order to get a figure divisible by 60 and, inci-
dentally, a round number (700) of stades for one degree. If
Pliny is right in saying that Eratosthenes made 40 stades
equal to the Egyptian crxo/oy, then, taking the o^oo/cy at
12,000 Royal cubits of 0-525 metres, we get 300 such cubits,
or 157-5 metres, i.e. 516-73 feet, as the length of the stade.
On this basis 252,000 stades works out to 24,662 miles, and
the diameter of the earth to about 7,850 miles, only 50 miles
shorter than the true polar diameter, a surprisingly close
approximation, however much it owes to happy accidents
in the calculation.
We learn from Heron's Dioptra that the measurement of
the earth by Eratosthenes was givon in a separate work On
the Measurement of the Earth. According to Galen 1 this work
dealt generally with astronomical or mathematical geography,
treating of ' the size of the equator, the distance of the tropic
and polar circles, the extent of the polar zone, the size and
distance of the sun and moon, total and partial eclipses of
those heavenly bodies, changes in the length of the day
according to the different latitudes and seasons'. Several
details are preserved elsewhere of results obtained by
Eratosthenes, which were doubtless contained in this work.
He is supposed to have estimated the distance between the
tropic circles or twice the obliquity of the ecliptic at 1 l/83rds
of a complete circle or 47 42' 39"; but from Ptolemy's
language on this subject it is not clear that this estimate was
not Ptolemy's own. What Ptolemy says is that he himself
found the distance between the tropic circles to lie always
between 47 40' and 47 45', 'from which we obtain about
(vytMv) the same ratio as that of Eratosthenes, which
Hipparchus also used. For the distance between the tropics
becomes (or is found to be, yivtrai) very nearly 11 parts
Galen, Instit. Logica, 12 (p. 26 Kalbfleiscb).
108 ERATOSTHENES
out of 83 contained in the whole meridian circle'. 1 The
mean of Ptolemy's estimates, 47 42' 30", is of course nearly
ll/83rds of 360. It is consistent with Ptolemy's language
to suppose that Eratosthenes adhered to the value of the
obliquity of the ecliptic discovered before Euclid's time,
namely 24, and Hipparchus does, in his extant Commentary
on the Phaenomena of Aratus and Eudoxus, say that the
summer tropic is ' very nearly 24 north of the equator '. .
The Doxographi state that Eratosthenes estimated the
distance of the moon from the earth at 780,000 stades and
the distance of the sun from the earth at 804,000,000 stades
(the versions of Stobaeus and Joannes Lydus admit 4,080,000
as an alternative for the latter figure, but this obviously
cannot be right). Macrobius 2 says that Eratosthenes made
the 'measure' of the sun to be 27 times that of the earth.
It is not certain whether measure means ' solid content ' or
' diameter ' in this case ; the other figures on record make the
former more probable, in which case the diameter of the sun
would be three times that of the earth. Macrobius also tells
us that Eratosthenes's estimates of the distances of the sun
and moon were obtained by means of lunar eclipses.
Another observation by Eratosthenes, namely that at Syene
(which is under the summer tropic) and throughout a circle
round it with a radius of 300 stades the upright gnomon
throws no shadow at noon, was afterwards made use of by
Posidoiiius in his calculation of the size of the sun. Assuming
that the circle in which the sun apparently moves round the
earth is 1 0,000 times the size of a circular section of the earth
through its centre, and combining with this hypothesis the
datum just mentioned, Posidonius arrived at 3,000,000 stades
as the diameter of the sun.
Eratosthenes wrote a poem called Hermes containing a good
deal of descriptive astronomy; only fragments of this have
survived. The work Catasterismi (literally ' placings among
the stars ') which is extant can hardly be genuine in the form
in which it has reached us ; it goes back, however, to a genuine
work by Eratosthenes which apparently bore the same name ;
alternatively it is alluded to as KaraAoyoi or by the general
1 Ptolemy, Syntaxis, i. 12, pp. 67. 22-68. 6.
2 Macrobius, In Sown. Scip. i. 20. 9.
ASTRONOMY, ETC. 109
word 'Acrrpovojiia (Suidas), which latter word is perhaps a mis-
take for 'Aa-Tpodeo-ic, corresponding to the title 'Ao-rpoOeo-iai
g<p8ia)v found in the manuscripts. The work as we have it
contains the story, mythological and descriptive, of the con-
stellations, &c., under forty-four heads; there is little or
nothing belonging to astronomy proper.
Eratosthenes is also famous as the first to attempt a scientific
chronology beginning from the siege of Troy; this was the
subject of his Xpovoypafyiai, with which must be connected
the separate 'OXvpirtoftKai in several books. Clement of
Alexandria gives a short resumt of the main results of the
former work, and both works were largely used by Apollo-
dorus. Another lost work was on the Octaeteris (or eight-
yi'ars' period), which is twice mentioned, by Geminus and
Achilles; from the latter wo learn that Eratosthenes re-
garded tin* work on the same subject attributed to Eudoxus
as not genuine. His (ieograpliica in three books is mainly
known to us through Suidas's criticism of it. It began with
a history of geography down to his own time ; Eratosthenes
then proceeded to mathematical geography, the spherical form
of the earth, the negligibility in comparison with this of the
unevennesses caused by mountains and valleys, the changes of
features due to floods, earthquakes and the like. It would
appear from Theon of Smyrna's allusions that Eratosthenes
estimated the height of the highest mountain to be 10 stades
or about I/ 8000th part of the diameter of the earth.
XIV
CONIC SECTIONS. APOLLONIUS OF PERGA
A. HISTORY OF CONICS UP TO APOLLONIUS
Discovery of the conic sections by Menaechmus.
WE have seen that Menaechmus solved the problem of the
two mean proportionals (and therefore the duplication of
the cube) by means of conic sections, and that he is credited
with the discovery of the three curves ; for the epigram of
Eratosthenes speaks of ' the triads of Monaechmus ', whereas
of course only two conies, the parabola and the rectangular
hyperbola, actually appear in Menaechmus's solutions. The
question arises, how did Menaechmus come to think of obtain-
ing curves by cutting a cone ? On this we have no informa-
tion whatever. Democritus had indeed spoken of a. section of
a cone parallel and very near to the base, which of course
would be a circle, since the cone would certainly be the right
circular cone. But it is probable enough that the attention
of the Greeks, whose observation nothing escaped, would be
attracted to the shape of a section of a cone or a cylinder by
a plane obliquely inclined to the axis when it occurred, as it
often would, in real life; the case where the solid was cut
right through, which would show an ellipse, would presum-
ably be noticed first, and some attempt would be made to
investigate the nature and geometrical measure of the elonga-
tion of the figure in relation to the circular sections of the
same solid ; these would in the first instance be most easily
ascertained when the solid was a right cylinder; it would
then be a natural question to investigate whether the curve
arrived at by cutting the cone had the same property as that
obtained by cutting the cylinder. As we have seen, the
DISCOVERY OF THE CONIC SECTIONS 111
observation that an ellipse can be obtained from a cylinder
as well as a cone is actually made by Euclid in his Phaeno-
mena : ' if ', says Euclid, * a cone or a cylinder be cut by
a plane not parallel to the base, the resulting section is a
section of an acute-angled cone which is similar to a Ovptos
(shield)/ After this would doubtless follow the question
what sort of curves they are which are produced if we
cut a cone by a plane which does not cut through the
cone completely, but is either parallel or not parallel to
a generator of the cone, whether these curves have the
same property with the ellipse arid with one another, and,
if not, what exactly are their fundamental properties respec-
tively.
As it is, however, we are only told how the first writers on
conies obtained them in actual practice. We learn on the
authority of Geminus l that the ancients defined a cone as the
surface described by the revolution of a right-angled triangle
about one of the sides containing the right angle, and that
they knew no cones other than right cones. Of these they
distinguished throe kinds ; according as the vertical angle of
the cone was less than, equal to, or greater than a right angle,
they called the cone acute-angled, right-angled, or obtuse-
angled, and from each of these kinds of cone they produced
one and only one of the three sections, the section being
always made perpendicular to one of the generating lines of
the cone ; the curves were, on this basis, called ' section of an
acute-angled cone* (= an ellipse), 'section of a right-angled
cone' (= a parabola), and 'section of an obtuse-angled cone '
(= a hyperbola) respectively. These names were still used
by Euclid and Archimedes.
Menaechmuss probable procedure.
Menaechmus's constructions for his curves would presum-
ably be the simplest and the most direct that would show the
desired properties, and for the parabola nothing could be
simpler than a section of a right-angled cone by a plane at right
angles to one of its generators. Let OBG (Fig. 1) represent
1 Eutocius, Comm. on Conies of Apollonius.
112
CONIC SECTIONS
a section through the axis OL of a right-angled cone, and
conceive a section through AG (perpendicular to OA) and at
right angles to the plane of the paper.
FIG. 1.
If P is any point on the curve, and PN perpendicular to
AG, let J3(7be drawn through N perpendicular to the axis of
the cone. Then P is on the circular section of the cone al>out
BO as diameter.
Draw AD parallel to EG, and DF, CG parallel to OL meet-
ing AL produced in F, G. Then AD, AF are both bisected
by OL.
N = y, AN '=
Know
But jB, A , (7, G are concyclic, so that
BN.NC=AN.NG
Therefore
y* = AN. 2AL
. x,
and 2 A L is the parameter ' of the principal ordinates y.
In the case of the hyperbola Menaechmus had to obtain the
MENAECHMUS J S PROCEDURE
113
particular hyperbola which we call rectangular or equilateral,
and also to obtain its property with reference to its asymp-
totes, a considerable advance on what was necessary in the
case of the parabola. Two methods of obtaining the particular
hyperbola were possible, namely (1) to obtain the hyperbola
arising from the section of any obtuse-angled cone by a plane
at right angles to a generator, and then to show how a
rectangular hyperbola can be obtained as a particular case
by finding the vertical angle which the cone must have to
give a rectangular hyperbola when cut in the particular way,
or (2) to obtain the rectangular hyperbola direct by cutting
another kind of cone by a section not necessarily perpen-
dicular to a generator.
(1) Taking the first method, we draw (Fig. 2) a cone with its
vertical angle BO (1 obtuse. Imagine a section perpendicular
to the plane of the paper and passing through AG which is
perpendicular to OB. Let GA produced meet CO produced in
A* ', and complete the same construction as in the case of
the parabola.
FIG. 2.
In this case we have
PN* = BN.
= AN.NG.
114 CONIC SECTIONS
But, by similar triangles,
NO:AF=NC:AD
= A'N:AA'.
A F
Hence P^V 2 = A Jf . A'N . ~,
AA
which is the property of the hyperbola, AA' being what we
call the transverse axis, and 2 AL the parameter of the principal
ordinates.
Now, in order that the hyperbola may be rectangular, we
must have 2 AL: A A' equal to 1. The problem therefore now
is< given a straight line A A', and AL along A A produced
equal to ^AA \ to find a cone such that L is on its axis and
the section through AL perpendicular to the generator through
A is a rectangular hyperbola with A f A as transverseaxis. In
other words, we have to find a point on the straight line
through A perpendicular to AA r such that OL bisects the
angle which is the supplement of the angle A'OA.
This is the case if A'Q : OA = A'L : LA = 3:1 ;
therefore is on the circle which is the locus of all points
such that their distances from the two fixed points A', A
are in the ratio 3:1. This circle is the circle on KL as
diameter, where A'K : KA = A'L : LA = 3:1. Draw this
circle, and is then determined as the point in which AO
drawn perpendicular to AA intersects the circle.
It is to be observed, however, that this deduction of a
particular from a more general case is not usual in early
Greek mathematics ; on the contrary, the particular usually
led to the more general. Notwithstanding, therefore, that the
orthodox method of producing conic sections is said to have
been by cutting the generator of each cone perpendicularly,
I am inclined to think that Menaechmus would get his rect-
angular hyperbola directly, and in an easier way, by means of
a different cone differently cut. Taking the right-angled cone,
already used for obtaining a parabola, we have only to make
a section parallel to the axis (instead of perpendicular to a
generator) to get a rectangular hyperbola.
MENAECHMUS'S PROCEDURE
115
For, let the right-angled cone HOK (Fig. 3) be cut by a
plane through A' AN parallel
to the axis M and cutting the
sides of the axial triangle HOK
in A', A, N respectively. Let
P be tho point on the curve
for which PN is the principal
ordinate. Draw 00 parallel
to HK. We have at once
= CN*-CA*, since MK = OM, and MN= OC=CA.
This is the property of the rectangular hyperbola having A' A
as axis. To obtain a particular rectangular hyperbola with
axis of given length we have only to choose the cutting plane
so that the intercept A' A may have the given length.
But Menaechmus had to prove the asymptote-property of
his rectangular hyperbola. As he can hardly be supposed to
have got as far as Apollonius in investigating the relations of
the hyperbola to its asymptotes, it is probably safe to assume
that he obtained the particular property in the simplest way,
i. e. directly from the property of the curve in relation, to
its axes.
R
FIG. 4.
If (Fig. 4) OR, OR' be the asymptotes (which are therefore
116 CONIC SECTIONS
at right angles) and A' A the axis of a rectangular hyperbola,
P any point on the curve, PN the principal ordinate, draw
PK, PIC perpendicular to the asymptotes respectively. Let
PN produced meet the asymptotes in U, R'.
Now, by the axial property,
= 2PK.PK', since /.PRK is half a right angle ;
therefore PK . PK' =
Works by Aristaeus and Euclid.
If Menaechmus was really the discoverer of the three conic
sections at a date which we must put at about 360 or 350 B.C.,
the subject must have been developed very rapidly, for by the
end of the century there were two considerable works on
conies in existence, works which, as we learn from Pappus,
were considered worthy of a place, alongside the (Ionics of
Apollonius, in the Treasury of Analysis. Euclid flourished
about 300 B.C., or perhaps 10 or 20 years earlier; but his
Conies in four books was preceded by a work of Aristaeus
which was still extant in the time of Pappus, who describes it
as * five books of tiolid Loci connected (or continuous, crvvt^)
with the conies'. Speaking of the relation of Euclid's Conies
in four books to this work, Pappus says (if the passage is
genuine) that Euclid gave credit to Aristaeus for his dis-
coveries in conies and did not attempt to anticipate him or
wish to construct anew the same system. In particular,
Euclid, when dealing with what Apollonius calls the three-
and four-line locus, ' wrote so much about the locus as was
possible by means of the conies of Aristaeus, without claiming
completeness for his demonstrations '.* We gather from these
remarks that Euclid's Conies was a compilation and rearrange-
ment of the geometry of the conies so far as known in his
1 Pappus, vii, p. 678. 4.
WORKS BY ARISTAEUS AND EUCLID 117
time; whereas the work of Aristaeus was more specialized and
more original.
' Solid loci 9 and 'solid problems'.
' Solid loci ' are of course simply conies, but the use of the
title ' Solid loci ' instead of ' conies ' seems to indicate that
the work was in the main devoted to conies regarded as loci.
As we have seen, ' solid loci ' which are conies are distinguished
from ' plane loci ', on the one hand, which are straight lines
and circles, and from ' linear loci ' on the other, which are
curves higher than conies. There is some doubt as to the
real reason why the term ' solid loci ' was applied to the conic
sections. We are told that ' plane ' loci are so called because
they are generated in a plane (but so are some of the higher
curves, such as the quadratrijc and the spiral of Archimedes),
ft-nd that 'solid loci' derived their name from the fact that
they arise as sections of solid figures (but so do some higher
curves, e.g. the spiric curves which are sections of the onreipa
or tore). But some light is thrown on the subject by the corre-
sponding distinction which Pappus draws between 'plane',
' solid ' and ' linear ' problems.
'Those problems', he says, 'which can be solved by means
of a straight line and a circumference of a circle may pro-
perly be called />lanr ; for the lines by means of which such
problems are solved have their origin in a plane. Those,
however, which are solved by using for their discovery one or
more of the sections of the cone have been called solid', for
their construction requires the use of surfaces of solid figures,
namely those of cones. There remains a third kind of pro-
blem, that which is called linear ; for other lines (curves)
besides those mentioned are assumed for the construction, the
origin of which is more complicated and less natural, as they
are generated from more irregular surfaces and intricate
movements.' 1
The true significance of the word ' plane ' as applied to
problems is evidently, not that straight lines and circles have
their origin in a plane, but that the problems in question can
be solved by the ordinary plane methods of transformation of
1 Pappus, iv, p. 270. 5-17.
118 CONIC SECTIONS
areas, manipulation of simple equations between areas and, in
particular, the application of areas ; in other words, plane
problems were those which, if expressed algebraically, depend
on equations of a degree not higher than the second.
Problems, however, soon arose which did not yield to ' plane '
methods. One of the first was that of the duplication of the
cube, which was a problem of geometry in three dimensions or
solid geometry. Consequently, when it was found that this
problem could be solved by means of conies, and that no
higher curves were necessary, it would be natural to speak of
them as * solid ' loci, especially as they were in fact produced
from sections of a solid figure, the cone. The propriety of the
term would be only confirmed when it was found that, just as
the duplication of the cube depended on the solution of a pure
cubic equation, other problems such as the trisection of an
angle, or the cutting of a sphere into two segments bearing
a given ratio to one another, led to an equation between
volumes in one form or another, i.e. a mixed cubic equation,
and that this equation, which was also a solid problem, could
likewise be solved by means of conies.
Aristaeus's Solid Loci
The Solid Loci of Aristaeus, then, presumably dealt with
loci which proved to be conic sections. In particular, he must
have discussed, however imperfectly, the locus with respect to
three or four lines the synthesis of which Apollonius says that
he found inadequately worked out in Euclid's (Ionics. The
theorems relating to this locus are enunciated by Pappus in
this way :
' If three straight lines be given in position and from one and
the same point straight lines be drawn to meet the three
straight lines at given angles, and if the ratio of the rectangle
contained by two of the straight lines so drawn to the square
on the remaining one be given, then the point will lie on a
solid locus given in position, that is, on one of the three conic
sections. And if straight lines be so drawn to meet, at given
angles, four straight lines given in position, and the ratio of
the rectangle contained by two of the lines so drawn to the
rectangle contained by the remaining two be given, then in
ARISTAEUS'S SOLID LOCI 119
the same way the point will lie on a conic section given in
position/ l
The reason why Apollonius referred in this connexion to
Euclid and not to Aristaeus was probably that it was Euclid's
work that was on the same lines as his own.
A very large proportion of the standard properties of conies
admit of being stated in the form of locus-theorems; if a
certain property holds with regard to a certain point, then
that point lies on a conic section. But it may be assumed
that Aristaeus's work was not merely a collection of the
ordinary propositions transformed in this way ; it would deal
with new locus-theorems not implied in the fundamental
definitions and properties of the conies, such as those just
mentioned, the theorems of the three- and four-line locus.
But one (to us) ordinary property, the focus-directrix property,
was, as it seems to me, in all probability included.
Focus-directrix property known to Euclid.
It is remarkable that the directrix does not appear at all in
Apollonius's great treatise 1 on conies. The focal properties of
the central conies are given by Apollonius, but the foci are
obtained in a different way, without any reference to the
directrix; the focus of the, parabola does not appear at all.
We may perhaps conclude, that neither did Euclid's Conies
contain the focus-directrix property; for, according to Pappus,
Apollonius based his first four books on Euclid's four books,
while filling them out and adding to them. Yet Pappus gives
the proposition as a lemma to Euclid's Harfuce-Lwi, from
which we cannot but infer that it was assumed in that
treatise without proof. If, then, Euclid did not take it from
his own ( 1 onic$, what more likely than that it was contained
in Aristaeus's Solid Loei ?
Pappus's enunciation of the theorem is to the etfect that the
locus of a point such that its distance, from a given point is in
a given ratio to its distance from a fixed straight line is a conic
section, and is an ellipse, a parabola, or a hyperbola according
as the given ratio is less than, equal to, or greater than unity.
1 Pappus, vii, p. 678. 15-24.
120 CONIC SECTIONS
Proof from Pappus.
The proof in the case where the given ratio is different from
unity is shortly as follows.
Let S be the fixed point, SX the perpendicular from 8 on
the fixed line. Let P be any point on the locus and PN
p
Jv
K AN SK' A 1
A'
A N
perpendicular to SX, so that HP is to NX in the given
ratio (e);
thus e 2
Take K on SX such that
then, if K' be another point on SN, produced it' necessary,
such that NK = NK',
: NK 2
= PN*:XK.XK'.
The positions of N, K, K' change with the position of P.
If A, A' be the points on which N falls when K, K' coincide
with X respectively, we have
'= SA' : A'X.
Therefore 8X:BA = SK :SN=(l+e):e,
whence (1 +e):e = (SX-SK)-.(SA-SN)
= XK:AN.
FOCUS-DIRECTRIX PROPERTY 121
Similarly it can be shown that
(1 ~e):e = XK':A'N.
By multiplication, XK . XK': AN. A'N = (1 - e 2 ) : e 2 ;
and it follows from above, ex aequuli, that
PN*: AN. A'N = (I <*#):!,
which is the property of a central conic.
When e < 1, A and A' lie on the same side of X, while
X lies on A A', and the conic is an ellipse ; when e > 1, A and
A' lie on opposite sides of X, while N lies on A'-A produced,
and the conic is a hyperbola.
The case where e, = 1 and the curve is a parabola is easy
and need not be reproduced here.
The treatise would doubtless contain other loci of types
similar to that which, as Pappus says, was used for the
trisection of an angle : I refer to the proposition already
quoted (vol. i, p. 243) that, if A, B are the base angles of
a triangle with vertex P, and L 11 = 2 /.A, the locus of P
is a hyperbola with eccentricity 2.
Propositions included in Euclid's Conies.
That Euclid's Conic* covered much of the same ground as
the first three Books of Apollonius is clear from the language
of Apollonius himself. Confirmation is forthcoming in the
quotations by Archimedes of propositions (1) 'proved in
the elements of conies ', or (2) assumed without remark as
already known. The former class include the fundamental
ordinate properties of the conies in the following forms :
(1) for the ellipse,
PN* : AN. A'N = P'N'* : AN'. A'N' = BU* : AC* ;
(2) for the hyperbola,
PN* : AN. A'N = P'.V" 2 : AN' . A'N' ;
(3) for the parabola, PN* = p a .AN',
the principal tangent properties of the parabola ;
the property that, if there are two tangents drawn from one
point to any conic section whatever, and two intersecting
122 CONIC SECTIONS
chords drawn parallel to the tangents respectively, the rect-
angles contained by the segments of the chords respectively
are to one another as the squares of the parallel tangents ;
the by no means easy proposition that, if in a parabola the
diameter through P bisects the chord QQ' in F, and QD is
drawn perpendicular to PF, then
where p a is the parameter of the principal ordinatcs and p is
the parameter of the ordinates to the diameter P V.
Conic sections in Archimedes.
But we must equally regard Euclid's Conies as the source
from which Archimedes took most of the other ordinary
properties of conies which he assumes without proof. Before
summarizing these it will be convenient to refer to Archi-
medes's terminology. We have seen that the axes of an
ellipse are not called axes but diameters, greater and lesser ;
the axis of a parabola is likewise its diameter and the other
diameters are 'lines parallel to the diameter', although in
a segment of a parabola the diameter bisecting the base is
the ' diameter ' of the segment. * The two ' diameters ' (axes)
of an ellipse are conjugate. In the case of the hyperbola the
1 diameter' (axis) is the portion of it within the (single-branch)
hyperbola ; the centre is not culled the ' centre ', but the point
in which the ' nearest lines to the section of an obtuse-angled
cone' (the asymptotes) meet; the half of the axis (OA) is
1 the line adjacent to the axis ' (of the hyperboloid of revolution
obtained by making the hyperbola revolve about its ' diameter '),
and A' A is double of this line. Similarly CP is the line
' adjacent to the axis J of a segment of the hyperboloid, and
P'P double of this line. It is clear that Archimedes did not
yet treat the two branches of a hyperbola as forming one
curve ; this was reserved for Apollonius.
The main properties of conies assumed by Archimedes in
addition to those above mentioned may be summarized thus.
Central Conies.
1. The property of the ordinates to any diameter PP',
CONIC SECTIONS IN ARCHIMEDES 123
In the case of the hyperbola Archimedes does not give
any expression for the constant ratios PN*:AN.A'N and
QV*:PV.P'V respectively, whence we conclude that he had
no conception of diameters or radii of a hyperbola not meeting
the curve.
2. The straight line drawn from the centre of an ellipse, or
the point of intersection of the asymptotes of a hyperbola,
through the point of contact of any tangent, bisects all chords
parallel to the tangent.
3. In the ellipse the tangents at the extremities of either of two
conjugate diameters are both parallel to the other diameter.
4. If in a hyperbola the tangent at P meets the transverse
axis in r i\ and PN is the principal ordinate, AN > AT. (It
is not easy to see how this could be proved except by means
of the general property that, if PP' be any diameter of
a hyperbola, QV the ordinate to it from Q, and QT the tangent
at Q meeting P*P in 7 7 , then TP : TP' = PV: 1>'V.)
5. If a cone, right or oblique, be cut by a plane meeting all
the generators, the section is either It circle or an ellipse.
<>. If a line between the asymptotes meets a hyperbola and
is bisected at the point of concourse, it will touch the
hyperbola.
7. If x, y are straight lines drawn, in fixed directions respec-
tively, from a point on a hyperbola to meet the asymptotes,
the rectangle xy is constant.
8. If 7 J aV be the principal ordinate of P, a point on an ellipse,
and if iYP be produced to meet the auxiliary circle in p, the
ratio pN:l*N is constant.
9. The criteria of similarity of conies and segments of
conies are assumed in practically the same form as Apollonius
gives them.
The Parabola.
1. The fundamental properties appear in the alternative forms
PN? : /"JV= AN: AN', or P.V 2 = p a . AN,
QV*:$V'*=PV:PV', or QV*=p.PV.
Archimedes applies the term parameter (a wap 9 $LV Svvavrou
at airo ray ro/zay) to the parameter of the principal ordinates
124 CONIC SECTIONS
only : p is simply the line to which the rectangle equal to QF 2
and of width equal to PV is applied.
2. Parallel chords are bisected by one straight line parallel to
the axis, which passes through the point of contact of the
tangent parallel to the chords.
3. If the tangent at Q meet the diameter P V in T, and QV be
the ordinate to the diameter, P V = PT.
By the aid of this proposition a tangent to the parabola can
be drawn (a) at a point on it, (6) parallel to a given chord.
4. Another proposition assumed is equivalent to the property
of the subnormal, NG = -|/> tt .
5. If QQ' be a chord of a parabola perpendicular to the axis
and meeting the axis in M y while QVq another chord parallel
to the tangent at P meets the diameter through P in V, and
RIIK is the principal ordinate of any point R on the curve
meeting PV in // and the axis in K, then PViPJI > or
= MK:KA ; 'for this is proved 1 (0/6 Floating Bodies, II. 6).
Where it was proved we do not know; the proof is not
altogether easy. 1
6. All parabolas are similar.
As we have seen, Archimedes had to specialize in the
parabola for the purpose of his treatises on the Quadrature
of the Parabola, Conoids and tipherouls, Floating Bodies,
Book II, and Plane Equilibriums, Book II ; consequently he
had to prove for himself a number of special propositions, which
have already been given in their proper places. A few others
are assumed without proof, doubtless as being easy deductions
from the prgpositions which he does prove. They refer mainly
to similar parabolic segments so placed that their buses are in
one straight line and have one common extremity.
1. If any three similar and similarly situated parabolic
segments JBQ 19 Q 2 , BQ.^ lying along the same straight line
as bases (BQ l < BQ 2 < BQ 3 ), and if E be any point on the
tangent at B to one of the segments, and EO a straight line
through E parallel to the axis of one of the segments and
meeting the segments in jR 3 , JJ 2 , R l respectively and BQ Z
in 0, then
R A R 2 : M.K, = (Q,Q 3 : BQ 3 ) . (BQ l : Q, Q 2 ).
1 See Apollonius ofPerga, ed. Heath, p, liv.
CONIC SECTIONS IN ARCHIMEDES 125
2. If two similar parabolic segments with bases BQ lt BQ 2 be
placed as in the last proposition, and if BR 1 R 2 be any straight
line through B meeting the segments in R l , /J 2 respectively,
These propositions are easily deduced from the theorem
proved in the Quadrature of the Parabola, that, if through E y
a point on the tangent at B, a straight line ERO be drawn
parallel to the axis and meeting the curve in R and any chord
BQ through B in 0, then
3. On the strength of these propositions Archimedes assumes
the solution of the problem of placing, between two parabolic
segments similar to one, another and placed as in the above
propositions, a straight lino of a given length and in a direction
parallel to the diameters of either parabola.
Euclid and Archimedes no doubt adhered to the old method
of regarding the three conies as arising from sections of three
kinds of right circular cones (right-angled, obtuse-angled arid
acute-angled) by planes drawn in each case at right angles to
a generator of the cone. Yet neither Euclid nor Archimedes
was unaware that the 'section of an acute-angled cone', or
ellipse, could bo otherwise produced. Euclid actually says in
his Pkaenomena that 'if a cone or cylinder (presumably right)
be cut by a plane not parallel to the base, the resulting section
is a section of an acute-angled cone which is similar to
a dvpeos (shield) '. Archimedes know that the non-circular
sections even of an oblique circular cono made by planes
cutting all the generators are ellipses ; for he shows us how,
given an ellipse, to draw a cone (in general oblique) of which
it is a section and which has its vertex outside the plane
of the ellipse on any straight line through the centre of the
ellipse in a plane at right angles to the ellipse and passing
through one of its axes, whether the straight line is itself
perpendicular or not perpendicular to the plane of the ellipse ;
drawing a cone in this case of course means finding the circular
sections of the surface generated by a straight line always
passing through the given vertex and all the several points of
the given ellipse. The method of proof would equally serve
126 APOLLONIUS OF PERGA
for the other two conies, the hyperbola and parabola, and we
can scarcely avoid the inference that Archimedes was equally
aware that the parabola and the hyperbola could be found
otherwise than by the old method.
The first, however, to base the theory of conies on the
production of all three in the most general way from any
kind of circular cone, right or oblique, was Apollonius, to
whose Work we now come.
B. APOLLONIUS OF PERGA
Hardty anything is known of the life of Apollonius except
that he was born at Perga, in Pamphylia, that he went
when quite young to Alexandria, where ho studied with the
successors of Euclid and remained a long time, and that
he flourished (yeyoi/e) in the reign of Ptolemy Eucrgetes
(247-222 B.C.). Ptolemaeus Chennus mentions an astronomer
of the same name, who was famous during the reign of
Ptolemy Philopator (222 205 B.C.), and it is clear that our
Apollonius is meant. As Apollonius dedicated the fourth and
following Books of his Comes to King Attains I (241-197 B.C.)
we have a confirmation of his approximate date. He was
probably born about 262 B.C., or 25 years after Archimedes.
We hear of a visit to Pergamum, where he made the acquain-
tance of Eudemus of Pergamum, to whom he dedicated the
first two Books of the Conies in the form in which they have
come down to us ; they were the first two instalments of a
second edition of the work.
The text of the Comes.
The Conies of Apollonius was at once recognized as the
authoritative treatise on the subject, and later writers regu-
larly cited it when quoting propositions in conies. Pappus
wrote a number of lemmas to it ; Serenus wrote a commen-
tary, as also, according to Suidas, did Hypatia. Eutocius
(fl. A.D. 500) prepared an edition of the first four Books and
wrote a commentary on them ; it is evident that he had before
him slightly differing versions of the completed work, and he
may also have had the first unrevised edition which had got
into premature circulation, as Apollonius himself complains in
the Preface to Book I.
THE TEXT OF THE CONICS 127
The edition of Eutocius suffered interpolations which were
probably made in the ninth century when, under the auspices
of Leon, mathematical studies were revived at Constantinople ;
for it was at that date that the* uncial manuscripts were
written, from which our best manuscripts, V (= Cod. Vat. gr.
206 of the twelfth to thirteenth century) for the Conies, and
W (= Cod. Vat. gr. 204 of the tenth century) for Eutocius,
were copied.
Only the first four Books survive in Greek; the eighth
Book is altogether lost, but the three Books V-VII exist in
Arabic. It was Ahmad and al-Hasan, two sons of Muh. b.
Musa b. Shakir, who first contemplated translating the Conies
into Arabic. They were at first deterred by the bad state of
their manuscripts; but afterwards Ahmad obtained in Syria
a copy of Kutocius's edition of Books 1-IV and had them
translated by Hilal b. Abl Ililal al-Himsi (died 883/4).
Books V-V1I were translated, also for Ahmad, by Thabit
b. Qurra( 826 901) from another manuscript. Nasiraddin's
recension of this translation of the seven Books, made in 1248,
is represented by two copies in the Bodleian, one of the year
1301 (No. 943) and the other of 1626 containing Books V-VII
only (No. 885).
A Latin translation of Books I-IV was published by
Johannes Baptista Memus at Venice in 1537 ; but the first
important edition was the translation by Commandinus
(Bologna, 1566), which included the lemmas of Pappus and
the commentary of Eutocius, and was the first attempt to
make the book intelligible by means of explanatory notes.
For the Greek text Commandinus used Cod. Marcianus 518
and perhaps also Vat. gr. 205, both of which were copies of V,
but not V itself.
The first published version of Books V-VII was a Latin
translation by Abraham Echellensis and Giacomo Alfonso
Borelli (Florence, 1661) of a reproduction of the Books written
in 983 by Abu '1 Fath al-Isfahiim.
The editio .princeps of the Greek text is the monumental
work of Halley (Oxford, 1710). The original intention was
that Gregory should edit the four Books extant in Greek, with
Eutocius's commentary and a Latin translation, and that
Halley should translate Books V-VI1 from the Arabic into
128 APOLLONIUS OF PERGA
Latin. Gregory, however, died while the work was proceeding,
and Halley then undertook responsibility for the whole. The
Greek manuscripts used were two, one belonging to Savile
and the other lent by D. Baynard ; their whereabouts cannot
apparently now be traced, but they were both copies of Paris,
gr. 2356, which was copied in the sixteenth century from Paris,
gr. 2357 of the sixteenth century, itself a copy of V. For the
three Books in Arabic Halley used the Bodleian MS. 885, but
also consulted (a) a compendium of the three Books by 'Abdel-
melik al-Shlrazi (twelfth century), also in the Bodleian (913),
(b) Borelli's edition, and (c) Bodl. 943 above mentioned, by means
of which he revised and corrected his translation when com-
pleted. Halley's edition is still, so far as I know, the only
available source for Books V-VII, except for the beginning of
Book V (up to Prop. 7) which was edited by L. Nix (Leipzig,
1889).
The Greek text of Books I-IV is now available, with the
commentaries of Eutocius, the fragments of Apollonius, &c.,
in the definitive edition of Heiberg (Teubner, 1891-3).
Apollonius's own account of the Conies.
A general account of the contents of the great work which,
according to Geminus, earned for him the title of the ' great
geometer' cannot be better given than in the words of the
writer himself. The prefaces to the several Books contain
interesting historical details, and, like the prefaces of Archi-
medes, state quite plainly and simply in what way the
treatise differs from those of his predecessors, and how much
in it is claimed as original. The strictures of Pappus (or
more probably his interpolator), who accuses him of being a
braggart and unfair towards his predecessors, are evidently
unfounded. The prefaces are quoted by v. Wilamowitz-
Moellendorff as specimens of admirable Greek, showing how
perfect the style of the ^ great mathematicians could be
when they were free from the trammels of mathematical
terminology.
Book I. General Preface.
Apollonius to Eudemus, greeting.
If you are in good health and things are in other respects
as you wish, it is well ; with rne too things are moderately
THE CONIG8 129
well. During the time I spent with you at Pergamum
I observed your eagerness to become acquainted with my
work in conies; I am therefore sending you the first book,
which I have corrected, and I will forward the remaining
books when I have finished them to my satisfaction. I dare
say you have not forgotten my telling you that I undertook
the investigation of this subject at the request of Naucrates
the geometer, at the time when he came to Alexandria and
stayed with me, and, when I had worked it out in eight
books, I gave them to him at once, too hurriedly, because he
was on the point of sailing; they had therefore not been
thoroughly revised, indeed I had put down everything just as
it occurred to me, postponing revision till the end. Accord-
ingly I now publish, as opportunities serve from time to time,
instalments of the work as they are corrected. In the mean-
time it has happened that some other persons also, among
those whom I have met, have got the first and second books
before they were corrected ; do not be surprised therefore if
you come across them in a different shape.
Now of the eight books the first four form an elementary
introduction. The first contains the modes of producing the
three sections and the opposite branches (of the hyperbola),
arid the fundamental properties subsisting in them, worked
out more fully and generally than in th writings of others.
The second book contains the properties of the diameters and
tho axes of the sections as well as the asymptotes, with other
things generally and necessarily used for determining limits
of possibility (& op * 07/01') ; and what I mean by diameters
and axes respectively you will learn from this book. The
third book contains many remarkable theorems useful for
tho syntheses of solid loci and for dioriumi ; the most and
prettiest of those theorems are new, and it was their discovery
which made me aware that Euclid did not work out the
synthesis of the locus with rospect to throe and four lines, but
only a chance portion of it, and that not successfully ; for it
was not possible for the said synthesis to be completed without
the aid of the additional theorems discovered by me: The
fourth book shows in how many ways the sections of cones
can meet one another and the circumference of a circle ; it
contains other things in addition,- none of which have been
discussed by earlier writers, namely the questions in how
many points a section of a cone or a circumference of a circle
can meet [a double-branch hyperbola, or two double-branch
hyperbolas can meet one another].
The rest of the books are more by way of surplusage
(7Tpiov(ria<rTiK<oTpa) : one of them deals somewhat fully with
130 APOLLONIUS OF PERGA
minima and maxima, another with equal and similar sections
of cones, another with theorems of the nature of determina-
tions of limits, and the last with determinate conic problems.
But of course, when all of them are published, it will be open*
to all who read them to form their own judgement about them,
according to their own individual tastes. Farewell.
The preface to Book II merely says that Apollonius is
sending the second Book to Eudemus by his son Apollomus,
and begs Eudemus to communicate it to earnest students of the
subject, and in particular to Philonides the geometer whom
Apollonius had introduced to Eudemus at Ephesus. There is
no preface to Book III as we have it, although the preface to
Book IV records that it also was sent to Eudemus.
Preface to Book IV.
Apollonius to Attains, greeting.
Some time ago I expounded and sent to Eudemus of Per-
gamum the first three books of my conies which I have
compiled in eight books, but, as he has passed away, I have
resolved to dedicate the remaining books to you because of
your earnest desire to possess my works. I am sending you
on this occasion the fourth book. It contains a discussion of
the question, in how many points at most it is possible for
sections of cones to meet one another and the circumference
of a circle, on the assumption that they do not coincide
throughout, and further in how many points at most a
section of a cone or the circumference of a circle can meet the
hyperbola with two branches, [or two double-branch hyper-
bolas can meet one another]; and, besides these questions,
the book considers a number of others of a similar kind.
Now the first question Conon expounded to Thrasydaeus, with-
out, however, showing proper mastery of the proofs, and on
this ground Nicoteles of Gyrene, not without reason, fell foul
of him. The second matter has merely been mentioned by
Nicoteles, in connexion with his controversy with Conon,
as one capable of demonstration; but I have not found it
demonstrated either by Nicoteles himself or by any one else.
The third question and the others akin to it I have not found
so much as noticed by any one. All the matters referred to,
which I have not found anywhere, required for their solution
many and various novel theorems, most of which I have,
as a matter of fact, set out in the first three books, while the
rest are contained in the present book. These theorems are
of considerable use both for the syntheses of problems and for
THE CONWtt 131
diorismi* Nicoteles indeed, on account of his controversy
with Conon, will not have it that any use can be made of the
discoveries of Corion for the purpose of diorismi', he is,
however, mistaken in this opinion, for, even if it is possible,
without using them at all, to arrive at results in regard to
limits of possibility, yet they at all events afford a readier
means of observing some things, e.g. that several or so many
solutions am possible, or again that no solution is possible;
and such foreknowledge secures a satisfactory basis for in-
vestigations, while the theorems in question are again useful
for the analyses of dioriami. And, even apart from such
usefulness, they will be found worthy of acceptance for the
sake of the demonstrations themselves, just as we accept
many other things in mathematics for this reason and for
no other.
The prefaces to Books V Vll now to be given are repro-
duced for Book V from the translation of L. Nix and for
Books VI, VII from that of Halley.
Preface to Book V.
Apollonius to Attalus, greeting.
In this fifth book I have laid down propositions relating to
maximum arid minimum straight lines. You must know
that my predecessors and contemporaries have only super-
ficially touched upon the investigation of the shortest lines,
and have only proved what straight lines touch the sections
and. conversely, what properties they have in virtue of which
they are tangents. For my part, 1 have proved these pro-
perties in the first book (without however making any use, in
the proofs, of the doctrine of the shortest lines), inasmuch as
1 wished to place them in close connexion with that part
of the subject in which 1 treat of the production of the three
conic sections, in order to show at the same time that in each
of the three sections countless properties and necessary results
appear, as they do with reference to the original (transverse)
diameter. The propositions in which I discuss the shortest
lines 1 have separated into classes, and I have dealt with each
individual case by careful demonstration ; I have also con-
nected the investigation of them with the investigation of
the greatest lines above mentioned, because 1 considered that
those who cultivate this science need them for obtaining
a knowledge of the analysis, and determination of limits of
possibility, of problems as well as for their synthesis: in
addition to which, the subject is one of those which seem
worthy of study for their own sake. Farewell.
132 APOLLONIUS OF PERGA
Preface to Book VI.
Apollonius to Attalus, greeting.
I send you the sixth book of the conies, which embraces
propositions about conic sections and segments of conies equal
and unequal, similar and dissimilar, besides some other matters
left out by those who have preceded me. In particular, you
will find in this book how, in a given right cone, a section can
be cut which is equal to a given section, and how a right cone
can be described similar to a given cone but such as to contain
a given conic section. And these matters in truth I have
treated somewhat more fully and clearly than those who wrote
before my time on these subjects. Farewell.
Preface to Book VII.
Apollonius to Attalus, greeting.
I send to you with this letter the seventh book on conic
sections. In it are contained a large number of new proposi-
tions concerning diameters of sections and the figures described
upon them ; and all these, propositions have their uses in many
kinds of problems, especially in the determination of the
limits of their possibility. Several examples of those occur
in the determinate conic problems solved and demonstrated
by me in the eighth book, which is by way of an appendix,
and which I will make a point of sending to you as soon
as possible. Farewell.
Extent of claim to originality.
We gather from these prefaces a very good idea of the
plan followed by Apollonius in the arrangement of the sub-
ject and of the extent to which he claims originality. The
first four Books form, as he says, an elementary introduction,
by which he means an exposition of the elements of conies,
that is, the definitions and the fundamental propositions
which are of the most general use and application ; the term
' elements ' is in fact used with reference to conies in exactly
the same sense as Euclid uses it to describe his great work.
The remaining Books beginning with Book V are devoted to
more specialized investigation of particular parts of the sub-
ject. It is only for a very small portion of the contend of the
treatise that Apollonius claims originality ; in the first three
Books the claim is confined to certain propositions bearing on
the ' locus with respect to three or four lines ' ; and in the
fourth Book (on the number of points at which two conies
THE CONICS 133
may intersect, touch, or both) the part which is claimed
as new is the extension to the intersections of the parabola,
ellipse, and circle with the double-branch hyperbola, and of
two double-branch hyperbolas with one another, of the in-
vestigations which had theretofore only taken account of the
single-branch hyperbola. Even in Book V, the most remark-
able of all, Apolloiiius does not say that normals as ' the shortest
lines ' had not been considered before, but only that they had
been superficially touched upon, doubtless in connexion with
propositions dealing with the tangent properties. He explains
that lie found it convenient to treat of the tangent properties,
without any reference to normals, in the first Book in order
to connect them with the chord properties. It is clear, there-
fore, that in treating normals as maxima and minima, and by
themselves, without any reference to tangents, as he does in
Book V, he was making an innovation ; and, in view of the
extent to which the theory of normals as maxima and minima
is developed by him (in 77 propositions), there is 110 wonder
that he should devote a whole Book to the subject. Apart
from the developments in Books III, IV, V, just mentioned,
and the numerous new propositions in Book VII with the
problems thereon which formed the lost Book VI11, Apollonius
only claims to have treated the whole subject more fully and
generally than his predecessors.
Great generality of treatment from the beginning.
So far from being a braggart and taking undue credit to
himself for the improvements which he made upon his prede-
cessors, Apollonius is, if anything, too modest in his descrip-
tion of his personal contributions to the theory of conic
sections. For the ' more fully and generally ' of his first
preface scarcely conveys an idea of the extreme generality
with which the whole subject is worked out. This character-
istic generality appears at the very outset.
Analysis of the Conies.
Book 1.
Apollonius begins by describing a double oblique circular
cone in the most general way. Given a circle and any point
outside the plane of the circle and in general not lying on the
134 APOLLONIUS OP PERGA
straight line through the centre of the circle perpendicular to
its plane, a straight line passing through the point and pro-
duced indefinitely in both directions is made to move, while
always passing through the fixed point, so as to pass succes-
sively through all the points of the circle ; the straight line
thus describes a double cone which is in general oblique or, as
Apollonius calls it, scalene. Then, before proceeding to the
geometry of a cone, Apollonius gives a number of definitions
which, though of course only required for conies, are stated us
applicable to any curve.
1 In any curve,' says Apollonius, ' I give the name diameter to
any straight line which, drawn from the curve, bisects all thu
straight lines drawn in the curve (chords) parallel to any
straight line, and I call the extremity of the straight line
(i.e. the diameter) which is at the curve a vertex of the curve
and each of the parallel straight lines (chords) an ordinate
(lit. drawn ordinate- wise, Tray/xej>a>y Kar^\6ai) to the
diameter/
He then extends these terms to a pair of curves (the primary
reference being to the double-branch hyperbola), giving the
name transverse diameter to any straight line bisecting all the
chords in both curves which are parallel to a given straight
line (this gives two vertices where the diameter meets the
curves respectively), and the name erect dmmeter (opOia) to
any straight line which bisects all straight lines drawn
between one curve and the other which are parallel to any
straight line; the ordinates to any diameter are again the
parallel straight lines bisected by it. Conjugate diameters in
any curve or pair of curves are straight lines each of which
bisects chords parallel to the other. Axes are the particular
diameters which cut at right angles the parallel chords which
they bisect ; and conjugate axes are related in the same way
as conjugate diameters. Here we have practically our modern
definitions, and there is a great advance on Archimedes's
terminology.
The conies obtained in the most general way from an
oblique cone.
Having described a cone (in general oblique), Apollonius
defines the axis as the straight line drawn from the vertex to
THE CONICti, BOOK I 135
the centre of the circular base. After proving that all
sections parallel to the base are also circles, and that there
is another set of circular sections subcontrary to these, he
proceeds to consider sections of the cone drawn in any
manner. Taking any triangle through the axis (the base of
the triangle being consequently a diameter of the circle which
is the base of the cone), he is careful to make his section cut
the base in a straight line perpendicular to the particular
diameter which is the base of the axial triangle. (There is
110 loss of generality in this, for, if any section is taken,
without reference to any axial triangle, we have only to
select the particular axial triangle the base of which is that
diameter of the circular base which is
at right angles to the straight line in
which the section of the cone cuts the
base.) Let ABC be any axial triangle,
and let any section whatever cut the
base in a straight line ])E at right
angles to BC\ if then PM be the in-
tersection of the cutting plane and the
axial triangle, and if QQ' be any chord
in the section parallel to DE, Apollonius
proves that QQ' is bisected by PM. In
other words, PM is a diameter of the section. Apollonius is
careful to explain that,
' if the cone is a right cone, the straight line in the base (DK)
will be at right angles to the common section (PM) of the
cutting plane, and the triangle through the axis, but, if the
cone is scalene, it will not in general be at right angles to PM,
but will be at right angles to it only when the plane through
the axis (i.e. the axial triangle) is at right angles to the base
of the cone ' (I. 7).
That is to say, Apollonius works out the properties of the
conies in the most general way with reference to a diameter
which is not one of the principal diameters or axes, but in
general has its ordinates obliquely inclined to it. The axes do
not appear in his exposition till much later, after it has been
shown that each conic has the same property with reference
to any diameter as it has with reference to the original
diameter arising out of the construction ; the axes then appear
136
APOLLONIUS OF PERGA
as particular cases of the new diameter of reference, xne
three sections, the parabola, hyperbola, and ellipse are made
in the manner shown in the figures. In each case they pass
through a straight line DE in the plane of the base which
is at right angles to BO, the base of the axial triangle, or
to EG produced. The diameter PM is in the case of the
THE CONICS, BOOK I 137
parabola parallel to AC] in the case of the hyperbola it meets
the other half of the double cone in P' ; and in the case of the
ellipse it meets the cone itself again in P'. We draw, in
K
the cases of the hyperbola and ellipse, AF parallel to PM
to meet BC or ]$C produced in F.
Apollonius expresses the properties of the three curves by
means of a certain straight line PL drawn at right angles
to PM in the plane of the section.
In the case of the parabola, PL is taken such that
PL: PA = BW-.BA.AC;
and in the case of the hyperbola and ellipse such that
In the latter two cases we join P'L, and then dr
parallel to PL to meet P'L, produced if necessary, in h.
If UK be drawn through V parallel to J1C and meeting
AB, AC in //, K respectively, IIK is the diameter of the circular
section of the cone made by a plane parallel to the base.
Therefore Q V 2 = 11 V . VK.
Then (1) for the parabola we have, by parallels and similar
triangles,
11V:PV=BC:CA,
and VK:PA = B
138 APOLLONIUS OF PERGA
Therefore QV* : P V . PA = H V. VK : PV. PA
= BC* : . BA . AC
= PL: PA, by hypothesis,
= PL.PV:PV.PA,
whence QV* = PL . PV.
(2) In the case of the hyperbola and ellipse,
UV:PV=BF:FA,
VK:P'V=FC:AF.
Therefore QV' 2 : PV. P'V = HV . VK : PV. P'V
= BF.FO:AF*
= PL : PP', by hypothesis,
= RV:P'V
= PV.VR:PV.P'V,
whence QV* = PV . VR.
New names, 'parabola', 'ellipse 9 , 'hyperbola'.
Accordingly, in the case of the parabola, the square of the
ordinate (QV 2 ) is equal to the rectangle applied to PL and
with width equal to the abscissa (PV) ;
in the case of the hyperbola the rectangle applied to PL
which is equal to QV 2 and has its width equal to the abscissa
PV overlaps or exceeds (irrrepfidXXti) by the small rectangle LR
which is similar and similarly situated to the rectangle con-
tained by PL, PP' ;
in the case of the ellipse the corresponding rectangle falls
short (eXXeiTTti) by a rectangle similar and similarly situated
to the rectangle contained by PL, PP'.
Here then we have the properties of the three curves
expressed in the precise language of the Pythagorean applica-
tion of areas, and the curves are named accordingly : ^>ara6o/ct
(irapa/3o\rj) where the rectangle is exactly applied, hyperbola
(irTrepfJoXri) where it exceeds, and ellipse (eAAet^/rts') where it
falls short.
THE CONICS, BOOK I 139
PL is called the latus rectum (opOia) or the parameter of
the orditiates (irap 9 $v Svvavrai al Karayofjitvat reray/zli/a)?) in
each case. In the case of the central conies, the diameter PP'
is the transverse (17 irXayia] or transverse diameter ; while,
even more commonly, Apollonius speaks of the diameter and
the corresponding parameter together, calling the latter the
latus rectum or erect side (opOia nXtvpd) and the former
the transverse side of tiiefigurr (elSos) on, or applied to, the
diameter.
Fundamental properties equivalent to Cartesian equations.
If p is the parameter, and d the corresponding diameter,
the properties of the curves are the equivalent of the Cartesian
equations, referred to the diameter and the tangent at its
extremity as axes (in general oblique),
y 2 - = px (the parabola),
y 2 = px -f 1 x~ (the hyperbola and ellipse respectively).
a
Thus Apollonius expresses the fundamental property of the
central conies, like that of the parabola, as an equation
between areas, whereas in Archimedes it appears as a
proportion
which, however, is equivalent to the Cartesian equation
referred to axes with the centre as origin. The latter pro-
perty with reference to the original diameter is separately
proved in I. 21, to the effect that QV 2 varies as PF.P'F, as
is really evident from the fact that QV* : PV.P'V = PL : PP',
seeing that PL : PP' is constant for any fixed diameter PP'.
Apollonius has a separate proposition (1. 14) to prove that
the opposite branches of a hyperbola have the same diameter
and equal latera recta corresponding thereto. As he was the
first to treat the double-branch hyperbola fully, he generally
discusses the hyperbola (i.e. the single branch) along with
the ellipse, and the opposites, as he calls the double-branch
hyperbola, separately. The properties of the single-branch
hyperbola are, where possible, included in one enunciation
with those of the ellipse and circle, the enunciation beginning,
140 APOLLONIUS OP PERGA
c If in a hyperbola, an- ellipse, or the circumference of a circle ' ;
sometimes, however, the double-branch hyperbola and the
ellipse come in one proposition, e.g. in I. 30: 'If in an ellipse
or the opposites (i. e. the double hyperbola) a straight line be
drawn through the centre meeting the curve on both sides of
the centre, it will be bisected at the centre/ The property of
conjugate diameter 9 in an ellipse is proved in relation to
the original diameter of reference and its conjugate in I. 15,
where it is shown that, if DD' is the diameter conjugate to
PP' (i.e. the diameter drawn ordinate-wise to PP'), just as
PP' bisects all chords parallel to /)/)', so DD' bisects all chords
parallel to PP' ; also, if DL' be drawn at right angles to DD'
and such that DJJ '. DD' = PP /2 (or DL' is a third proportional
to DD', PP 7 ), then the ellipse has the same property in rela-
tion to DD' as diameter and DL' as parameter that it has in
relation to PP' as diameter and PL as the corresponding para-
meter. Incidentally it appears that PL . PP' = DD'*, or PL is
a third proportional to PP',/)//, as indeed, is obvious from the
property of the curve Q V* : PV. P7' = PL: PP' = DD'* : PP'*.
The next proposition, I. 16, introduces the secondai^y diameter
of the double-branch hyperbola (i.e. the diameter conjugate to
the transverse diameter of reference), which does not meet the
curve; this diameter is defined as that straight line drawn
through the centre parallel to the ordinates of the transverse
diameter which is bisected at the centre and is of length equal
to the mean proportional between the ' sides of the figure ',
i.e. the transverse diameter PP' and the corresponding para-
meter PL. The centre is defined as the middle point of the
diameter of reference, and it is proved that all other diameters
are bisected at it (1. 30).
Props. 17-19, 22-9, 31-40 are propositions leading up to
and containing the tangent properties. On lines exactly like
those of Eucl. III. 16 for the circle, Apollonius proves that, if
a straight line is drawn through the vertex (i. e. the extremity
of the diameter of reference) parallel to the ordjnates to the
diameter, it will fall outside the conic, and no other straight
line can fall between the said straight line and the conic ;
therefore the said straight line touches the conic (1.17, 32).
Props. I. 33, 35 contain the property of the tangent at any
point on the parabola, and Props. I. 34, 36 the property of
THE CONICS, BOOK I 141
the tangent at any point of a central conic, in relation
to the original diameter of reference ; if Q is the point of
contact, QV the ordinate to the diameter 'through P, and
if QT, the tangent at Q, meets the diameter produced in T y
then (1) for the parabola PV FT, and (2) for the central
conic TP:TP' = PV: VP'. The method of proof is to take a
point T on the diameter produced satisfying the respective
relations, and to prove that, if TQ be joined and produced,
any point on TQ on either side of Q is outside the curve : the
form of proof is by reductio ad absnrdum, and in each
case it is again proved that no other straight line can fall
between TQ and the curve. The fundamental property
TP:TP'= PV\VP' for the central conic is then used to
prove that UV. <!T = (!P 2 and QV* : (<V . VT = p:PP' (or
(! I)' 2 : (!P 2 ) and the corresponding properties with reference to
tho diameter l)D' conjugate to PP' and r, t, the points where
/)/)' is met by the ordinate to it from Q and by the tangent
at Q respectively (Props. T. 37-40).
Tr<insiti<ni in \\c,m diameter <i,nd tanyent tit Its extremity.
An important section of the Book follows (1. 41-50), con-
sisting of propositions leading up to what amounts to a trans-
formation of coordinates from the original diameter and the
tangent at its extremity to any diameter and the tangent at
its extremity; what Apollonius proves is of course that, if
any other diameter bo taken, the ordinate-property of the
ctmic with reference to that diameter is of the same form as it
is with reference to the original diameter. It is evident that
this is vital to the exposition. The propositions leading up to
the result in I. 50 are not usually given in our text-books of
geometrical conies, but aro useful and interesting.
Suppose that the tangent at any point Q meets the diameter
of reference PV in T, and that the tangent at P meets the
diameter through Q in K. Let R be any third point on
the curve; lot the ordinate RW to PV moot the diameter
through Q in F, and let liU parallel to the tangent at Q meet
PV in U. Then *
(1) in tho parabola, the triangle KUW = tho parallelogram
KW, and
142
APOLLONIUS OF PERGA
e F
T U P\W
w'
THE CONICS, BOOK I 143
(2) in the hyperbola or ellipse?, A RUW = the difference
between the triangles GFW and CPE.
(1) In the parabola &RUW: &QTV = RW* : QV*
= PW:PV
But, since TV=2PV, AQTV =
therefore ARUW = C3EW.
(2) The proof of the proposition with reference to the
central conic depends on a Lemma, proved in I. 41, to the effect
that, if PX 9 VY be similar parallelograms on Cl\ C'Fas bases,
and if VX be an equiangular parallelogram on QFas base and
such that, if the ratio of (JP to the other side of PX is m, the
ratio of QV to the other side of VZ is m.p/Pl*, then VZ is
equal to the difference between FFand PX. The proof of the
Lernma by Apollonius is difficult, but the truth of it can bo
easily seen thus.
By the property of the curve, QV' 2 : ( !V* - (T 2 = y> : PP' ;
therefore (! V* ^ ( ?P* = PP . O V*.
P
Now C3PX= p. (!!**/ m, where p is a constant depending
on the angle of the parallelogram.
Similarly
a VY - n . CT'Ym, and Q VZ = p . -^ QV 2 /m.
It follows that D VY * DAY = D VZ.
Taking now the triangles OF\\ r 9 CPE and RVW in the
(ellipse or hyperbola, we see that CF]\ r , CPE are similar, and
RUW has one angle (at W) e([tial or supplementary to the
angles at P and V in the other two triangles, while we have
whence QV: VT =r (p : PP') . (CV: QV),
and, by parallels,
RW: WU = (p : PP') . (OP : PE).
144 APOLLONIUS OF PERGA
Therefore RUW, CPE, CFWsuce the halves of parallelograms
related as in the lemma ;
therefore A R UW = A GFW - A CPE.
The same property with reference to the diameter secondary
to <7PFis proved in I. 45.
It is interesting to note the exact significance of the property
thus proved for the central conic. The proposition, which is
the foundation of Apollonius's method of transformation of
coordinates, amounts to this. If CI\ CQ are fixed semi-
diameters and Ji a variable point, the area of the quadrilateral
CFRU is constant for all positions of R on the conic. Suppose
now that CP, CQ are taken as axes of x and y respectively.
If we draw RX parallel to CQ to meet CP and RY parallel to
CP to meet CQ, the proposition asserts that (subject to the
proper convention as to sign)
&RYF+CUCXRY+&RXU = (const).
But since RX, RY, RF, RU are in fixed directions,
ARYF varies as RY* or x\ &CXRY as RX . RY or xij,
and &RXU as RX* or y*.
Hence, if x, y arc the coordinates of R,
otxt + pxy + yy* = A,
which is the Cartesian equation of the conic referred to the
centre as origin and any two diameters as axes.
The properties so obtained are next used to prove that,
if UR meets the curve again in R' and the diameter through
Q in JI, then RR' is bisected at M. (I. 46-8).
Taking (1) the case of the parabola, we have,
and &li'UW'=C3KW'.
By subtraction, (RWW'R') = 1=1 1" W,
whence &RFM = bR'F'M,
and, since the triangles are similar, RM = R'M.
The same result is easily obtained for the central conic.
It follows that EQ produced in the case of the parabola,
THE CON1CS, BOOK I 145
or CQ in the case of the central coriic, bisects all chords as
RR' parallel to the tangent at Q. Consequently EQ and CQ
are diameters of the respective conies.
In order to refer the conic to the new diameter and the
corresponding ordinates, we have only to determine the para-
meter of these ordinates and to show that the property of the
conic with reference to the new parameter and diameter is in
the same form as that originally found.
The propositions I. 49, 50 do this, and show that the new
parameter is in all the cases p', where (if is the point of
intersection of the tangents at I* and Q)
OQ:QE = p':2QT.
(1) In the case of the parabola, we have TP = PF= EQ,
whence A EOQ = A POT.
Add to each the figure POQF'W;
therefore QTW'F' = a EW = bR'UW,
whence, subtracting AlUW'F' from both, we have
Therefore KM . MF' = 2 QT . QM.
But KM : MF' = OQ : QE = p' : 2 QT, by hypothesis ;
theref cms KM* : li'M . MF' = p . QM : 2 QT . QM.
And KM . MF' = 2QT. QM, from above ;
therefore KW^tf.QM,
which is the desired property. 1
1 The proposition that, in the case of the parabola, if p be the para-
meter of the ordinates to the diameter through Q, then (see the first figure
on p. 142)
has an interesting application ; for it enables us to prove the proposition,
assumed without proof by Archimedes (but not easy to prove otherwise),
that, if in a parabola the diameter through P bisects the chord QQ' in K,
and QD is drawn perpendicular to PF, then
146
APOLLONIUS OF PERGA
(2) In the case of the central conic, we have
AR'UW = ACF'W ~ ACPE.
(Apollonius here assumes what he does not prove till III. 1,
namely that ACPE = AGQT. This is proved thus.
We have CV: CT = GV* : CP Z ; (I. 37, 39.)
therefore , ACQV: ACQT = ACQV: ACPE,
so that ACQT = ACPE.)
Therefore A E'UW = ACF'W - ACQT,
and it is easy to prove that in all cases
AR'MF'=QTUM.
Therefore R'M . MF' = QM(QT +MU).
Let QL be drawn at right angles to tJQ and equal to p'.
Join Q'L and draw MK parallel to QL to meet Q'L in K.
Draw CH parallel to Q'L to meet QL in 11 and MK in N.
Now R'M: MF' = OQ : QE
= QL : 2QT, by hypothesis,
= Q1I : QT.
But QT : MU = CQ : CM = QII : MN,
so that (Q H + MN) : (QT + MU) = QIt:QT
= R f M:MF', from above.
where p a is the parameter of the principal ordinates and p the para-
meter of the ordinates to the diameter
FV. ,
If the tangent at the vertex A meets
VP produced in E, and PT, the tangent
at P, in 0, the proposition of Apollonius
proves that
OP:PE=p:2PT.
But OP=%PT;
therefore PT" = p.PE
Thus
QV* : QD* = PT 2 : PN*, by similar triangles,
= p.AN:p a . AN
THE CONICS, BOOK I 147
It follows that
QM(QH+ MN) : QM(QT+ MU) = KM* : R'M . MF' ;
but, from above, QM(QT + Mlf) = R'M . MF' ;
therefore li'M* = QM(QH+ MN)
= QM. MK,
wliicli is the desired property.
In the case of the hyperbola, the same property is true for
the opposite branch.
These important propositions show that the ordinate property
of the three conies is of the same form whatever diameter is
taken as the diameter of reference. It is therefore a matter
of indifference to which particular diameter and ordinates the
* .
conic is referred. Tins is stated by Apollonius in a summary
which follows I. 50.
First apiwinmcc of ^r^je^wi <t.res.
The axes appear for the first time in the propositions next
following (I. 52-8), where Apollonius shows how to construct
each of the conies, given in each case (1) a diameter, (2) the
length of the corresponding parameter, and (3) the inclination
of the ordinates to the diameter. In each case Apollonius
first assumes the angle between the ordinates and the diameter
to be a right angle ; then he reduces the case where the angle
is oblique to the case where it is right by his method of trans-
formation of coordinates; i.e. from the given diameter and
parameter hefimts the axis of the conic and the length of the
corresponding parameter, and he then constructs the conic as
in the first case where the ordinates are at right angles to the
diameter. Here then we have a case of the proof of existence
by means of construction. The conic is in each case con-
structed by finding the cone of which the given conic is a
section. The problem of finding the axis of a parabola and
the centre and the axes of a central conic when the conic (and
not merely the {elements, as here) is given comes later (in II.
44-7), where it is also proved (II. 48) that no central conic
can have more than two axes.
148 APOLLONIUS OF PERGA
It has been my object, by means of the above detailed
account of Book I, to show not merely what results are
obtained by Apollonius, but the way in which he went to
work ; and it will have been realized how entirely scientific
and general the method is. When the foundation is thus laid,
and the fundamental properties established, Apollonius is able
to develop the rest of the subject on lines more similar to
those followed in our text-books. My description of the rest
of the work can therefore for the most part be confined to a
summary of the contents.
Book II begins with a section devoted to the properties of
the asymptotes. They are constructed in II. 1 in this way.
Beginning, as usual, with any diameter of reference and the
corresponding parameter and inclination of ordimites, Apol-
lonius draws at P the vertex (the extremity of the diameter)
a tangent to the hyperbola and sets off along it lengths PL, PL'
on either side of P such that PL*=PL'*= t ) . PP' [ = M> a l>
where p is the parameter. He then proves that (!L, (Hf pro-
duced will not meet the curve in any finite point and are there-
fore asymptotes. II. 2 proves further that no straight line
through G within the angle between the asymptotes can itself
be an asymptote. II. 3 proves that the intercept made by the
asymptotes on the tangent at any point P is bisected at P, and
that the square on each half of the intercept is equal to one-
fourth of the ' figure ' corresponding to the diameter through
P (i.e. one-fourth of the rectangle contained by the 'erect'
side, the latus rectum or parameter corresponding to the
diameter, and the diameter itself) ; this property is used as a
means of drawing a hyperbola when the asymptotes and one
point on the curve are given (II. 4). II. 5-7 are propositions
about a tangent at the extremity of a diameter being parallel
to the chords bisected by it. Apollonius returns to the
asymptotes in II. 8, and II. 8-14 give the other ordinary
properties with reference to the asymptotes (II. 9 is a con-
verse of II. 3), the equality of the intercepts between the
asymptotes and the curve of any chord (II. 8), the equality of
the rectangle contained by the distances between either point
in which the chord meets the curve and the points of inter-
section with the asymptotes to the square on the parallel
semi-diameter (II. 10), the latter property with reference to
THE CONICS, BOOK II 149
the portions of the asymptotes which include between them
a branch of the conjugate hyperbola (II. 11), the constancy of
the rectangle contained by the straight lines drawn from any '
point of the curve in fixed directions to meet the asymptotes
(equivalent to the Cartesian equation with reference to the
asymptotes, xy = const.) (II. 12), and the fact that the curve
and the asymptotes proceed to infinity and approach con-
tinually nearer to one another, so that the distance separating
them can be made smaller than any given length (II. 14). II. 15
proves that the two opposite branches of a hyperbola have the
same asymptotes and II. 16 proves for the chord connecting
points on two branches the property of II. 8. II. 1 7 shows that
' conjugate opposites ' (two conjugate double-branch hyper-
bolas) have the same asymptotes. Propositions follow about
coftjugate hyperbolas; any tangent to the conjugate hyper-
bola will meet both branches of the original hyperbola
and will be bisected at the point of contact (II. 19); if Q be
any point on a hyperbola, and (IE parallel to the tangent
at Q meets the conjugate hyperbola in E, the tangent at
E will be parallel to CQ and CQ, CE will be conjugate
diameters (II. 20), while the tangents at Q, K will meet on one
of the asymptotes (II. 21) ; if a chord Qq in one branch of
a hyperbola meet the asymptotes in R, r and the conjugate
hyperbola in Q', <f, then Q'Q.Qq' = 2V IP (IT. 23). Of the
rest of the propositions in this part of the Book the following
may be mentioned : if TQ, TQ' are two tangents to a conic
and V is the middle point of QQ', TV is a diameter (II. 29,
30, 38) ; if tQ, tQ' be tangents to opposite branches of a hyper-
bola, RR' the chord through t parallel to QQ', v the middle
point of QQ', then v;Z?, vli' are tangents to the hyperbola
(II. 40) ; in a conic, or a circle, or in conjugate hyperbolas, if
two chords not passing through the centre intersect, they do not
bisect each other (II. 26, 41, 42). II. 44-7 show how to find
a diameter of a conic and the centre of a central conic, the
axis of a parabola and the axes of a central conic. The Book
concludes with problems of drawing tangents to conies in
certain ways, through any point on or outside the curve
(II. 49), making with the axis an angle equal to a given acute
angle (II. 50), making a given angle with the diameter through
the point of contact (II. 51, 53) ; II. 52 contains a Siopurpos for
150 APOLLONIUS OF PERGA
the last problem, proving that, if the tangent to an ellipse at
any point P meets the major axis in T, the angle OPT is not
greater than the angle ABA', where JB is one extremity of the
minor axis.
Book III begins with a series of propositions about the
equality of certain areas, propositions of the same kind as, and
easily derived from, the propositions (I. 41-50) by means of
which, as already shown, the transformation of coordinates is
effected. We have first the proposition that, if the tangents
at any points P, Q of a conic meet in 0, and if they meet
the diameters through Q, P respectively in E, T, then
A OPT = AOQE (III. ] , 4) ; and, if P, Q be points on adjacent
branches of conjugate hyperbolas, AOP7^ T = &CQT (III. 13).
With the same notation, if R be any other point on the conic,
and if we draw RU parallel to the tangent at Q meeting the
diameter through P in 'U and the diameter ^hrough Q in M,
and RW parallel to the tangent at P meeting QT in II and
the diameters through Q, P in F, W, then AHQF = quadri-
lateral HTUR (III. 2. 6) ; this is proved at once from the fact
that &RMF= quadrilateral QTUM (see I. 49, 50, or pp. 145-6
above) by subtracting or adding the area HRMQ on each
side. Next take any other point R', and draw R'U', F'lI'R'W
in the same way as before ; it is then proved that, if RU, R'W
meet in / and R'U', R W in J, the quadrilaterals F'IRF, I U U'lt'
are equal, and also the quadrilaterals FJR'F', JU'UH (III. 3,
7, 9, 10). The proof varies according to the actual positions
of the points in the figures.
In Figs. 1, 2 AHFQ = quadrilateral UTUR,
By subtraction, FHH'F'= WU'R' + (IU)\
whence, if IH be added or subtracted, F'IRF= IVU'R',
and again, if IJ be added to both, FJR'F' = JU'UR.
In Fig. 3 bR'U'W = &CF'W'-&CQT,
so that ACQT = CU'R'F'.
THE CONICS, BOOK III 151
E F 1 F
Fia. 1.
FIG. 2.
Fm. 3.
152 APOLLONIUS OF PERGA
Adding the quadrilateral CF'H'T, we have
AH'F'Q = H'TU'tt,
and similarly AHFQ = HTUR.
By subtraction, F'H'HF= H'TU'R'-IITUR.
Adding H'IRH to each side, we have
If each of these quadrilaterals is subtracted from /,/",
FJR'F' = ./I
The corresponding results are proved in III. 5, 11, 12, 14
for the case where the ordinates through RR' are drawn to
a secondary diameter, and in III. 15 for the case where P, Q
are on the original hyperbola and R, R' on the conjugate
hyperbola.
The importance of these propositions lies in the fact that
they are immediately used to prove the well-known theorems
about the rectangles contained by the segments of intersecting
chords and the harmonic properties of the pole and polar.
The former question is dealt with in III. 16-23, which give
a great variety of particular cases. We will give the proof
of one case, to the effect that, if OP, ()Q be two tangents
to any conic and Rr, R'r' be any two chords parallel to
them respectively and intersecting in J y an internal or external
point,
then RJ . Jr : R'J . Jr' = OP 2 : ()Q 2 = (const.).
We have
RJ.Jr = RW**JW*> and RW 2 : JW* = &RUW: AJ^IF;
therefore
RJ . Jr : RW 2 = (RW 2 - JW*) : RW* = JU'VR : &RUW.
But RW 2 : OP 2 = ARUW : A'OPT ;
therefore, ex aequali, RJ . Jr : OP 2 = JU'UR : A OPT.
THE CONICS, BOOK III 153
Similarly KM'* : JM' 2 = kll'F'M' : &JFM',
whence R'J . Jr' : R'M' 2 = FJR'F' : A R'F'M'.
But R'M'* : OQ 2 =-- A R'F'M' : A OQE ;
therefore, ex aeqwili, R'J . Jr' : OQ 2 = FJR'F' : A OQfl.
It follows, since FJR'F' = JU'UR, and A07T = AOQA T ,
that III . Jr : OP' 2 = #',/. ,/r' : Of/ 2 ,
or RJ . Jr : R'J . Jr' = OP 2 : OQ*.
If we had taken chords /h'j, /i'r/ parallel respectively to
OQ, OP and intersecting in /, an internal or external point,
we should have in like manner
ft/. /> H'T. Trf = OQ^: OP 2 .
As a particular case, if PP' IH a diameter, and /i? 1 , R'r' he
chords parallel respectively to the tangent at P and the
diameter PP' and intersecting in 7, then (as is separately
proved)
RI.Ir:R'l.Tr' = p:PP'.
The corresponding results are proved in the cases where certain
of the points lie on the conjugate hyperbola.
The six following propositions about the segments of inter-
secting chords (III. 24- 9) refer to two chords in conjugate
hyperbolas or in an ellipse drawn parallel respectively to two
conjugate diameters PP\ /)//, and the results in modern form
are perhaps worth quoting. If lii\ R'r' be two chords so
drawn and intersecting in 0, then
(<i) in the conjugate hyperbolas
RO . Or R
~
and , (RO* + Or 2 ) : ( R'O* + Or' 2 ) = CP 2 : ( /7) 2 ;
(/>) in the ellipse
_
CD* ~~
154
APOLLONIUS OF PERGA
The general propositions containing the harmonic properties
of the pole and polar of a conic are III. 37-40, which prove
that in any conic, if TQ, Tq be tangents, and if Qq the chord
of contact be bisected in F, then
(1) if any straight line through T meet the conic in R', R and
Qq in /, then (Fig. 1) RT : TR' = RI : IR' ;
(2) if any straight line through Fmeet the conic in R, R'
the parallel through T to Qq in 0, then (Fig. 2)
FIG. 2.
The above figures represent theorem (1) for the parabola and
theorem (2) for the ellipse.
THE CONICS, BOOK III 155
To prove (1) we have
R'L* : lR* = H'(f : QH*= bll'F'Q : AHFQ = H'TU'R' : HTUR
(III. 2, 3, &c.).
Also XfT* : TR* = R'U'* : UR> = kR'U'W : ARUW,
and , R'T* : TR- = TW* : TW* = AT7/'1F : A THW,
so that R'T-:TR~ = bTH'W - AR'U'W: A.THW ~ &RUW
= U'TU'R'-.HTUR
= R'l* : IR\ from above.
To prove (2) we have
RV-.VR'- = RU*: R'U'* = &RUW : AR'U'}\",
and also
= HQ Z : Qll'* = AHFQ : AJI'F'Q = HTUR*: H'TU'R',
KO that
^ : VR' 2 = UTllR + A RUW-Jl'TU'li' + bR'U'W
= HO* : OH' 2 .
Props, 111. 30-6 deal separately with the particular cases
in which (a) the transversal is parallel to an asymptote of the
hyperbola or (b) the chord of contact is parallel to an asymp-
tote, i.e. where one of the tangents is an asymptote, which is
the tangent at infinity.
Next we have propositions about intercepts made by two
tangents on a third: If the tangents at three points of a
parabola form a triangle, all three tangents will be cut by the
points of contact in the same proportion (III. 41); if the tan-
gents at the extremities of a diameter PP' of a central conic
are cut in r, r' by any other tangent, Pr . PV = CD 2 (III. 42) ;
if the tangents at P, Q to a hyperbola meet the asymptotes in
* Where a quadrilateral, as HTUR in the figure, is a cross-quadri-
lateral, the area is of course the difference between the two triangles
which it forms, as HTW ^ RUW.
156 APOLLONIUS OF PERGA
Ly IS and My M' respectively, then L'M, ZJlf'are both parallel
to PQ (III. 44).
The first of these propositions asserts that, if the tangents at
three points P, Q, R of a parabola form a triangle pqr, then
Pr : rq = rQ : Qp = qp : pR.
From this property it is easy to deduce the Cartesian
equation of a parabola referred to two fixed tangents as
coordinate axes. Taking qR, qP as fixed coordinate axes, we
find the locus of Q thus. Let x, y be the coordinates of Q.
Then, if qp = x lt qr = y l , qR = h, qP = k, we have
From these equations we derive
i #1 2/i t . *
also, since = t/1 3 we have h - - = 1.
2/1 -y ^i 2/1
By substituting for o^, 2/1 the values V(ltx), V(ky) we
obtain
'+ (!)'= '
The focal properties of central conies are proved in
III. 45-52 without any reference to the directrix ; there is
no mention of the focus of a parabola. The foci aro called
' the points arising out of the application ' (ra 4/c rr/y Trapa-
floXfjs yivonsva crrjfjLtTa), the meaning being that >V, A S y/ are taken
on the axis AA' such that A8.SA' = Atf .tfA' = \l\ L .AA'
or 6 r 2 , that is, in the phraseology of application of areas,
a rectangle is applied to A A' as base equal to one-fourth
part of the 'figure', and in the case of the hyperbola ex-
ceeding, but in the case of the ellipse falling short, by a
square figure. The foci being thus found, it is proved that,
if the tangents Ar, A'r' at the extremities of the axis are met
by the tangent at any point P in r, ?' respectively, rr' subtends
a right angle at $, &', and the angles rrOS f , A'r'S' are equal, as
also are the angles r'r&', ArS (III. 45, 46)." It is next shown
that, if be the intersection of r>S y/ , r x >S f , then OP is perpen-
dicular to the tangent at P (III. 47). These propositions are
THE CONICS, BOOK III 157
used to prove that the focal distances of P make equal angles
with the tangent at P (III. 48). In III. 49-52 follow the
other ordinary properties, that, if SY be perpendicular to
the tangent at P, the locus of F is the circle on A A' as
diameter, that the lines from C drawn parallel to the focal
distances to meet the tangent at P are equal to CA, and that
the sum or difference of the focal distances of any point is
equal to A A'.
The last propositions of Book III are of use 'with reference
to the locus with respect to three or four lines. They are as
follows.
1. If PP / be a diameter of a central conic, and if PQ, P'Q
drawn to any other point Q of the conic meet the tangents at
1", P in JK', li respectively, then PR.P'R' = 4r'D 2 (III. 53).
2. If TQ, TQ' be two tangents to a conic, V the middle point
of QQ', P the point of contact of the tangent parallel to QQ',
and li any other point on the conic, let Qr parallel to TQ'
meet (// in r, and Q'/ parallel to TQ meet QR in / ; then
Qr . CJV : QQ'* = (PV* : PT*) . (TQ . TQ': QV*). (III. 54, 56.)
3. If the tangents are tangents to opposite branches of a
hyperbola and meet in /, and if R, /*, / are taken as before,
while t([ is half the chord through t parallel to QQ', then
Qr . Q'r' : QQ? = tQ . tQ' : tq*. (III. 55.)
The second of these propositions leads at once to the three-
line locus, and from this we easily obtain the Cartesian
equation to a conic with reference to two fixed tangents as
axes, where the lengths of the tangents are h, , viz.
Book IV is on the whole dull, and need not be noticed at
length. Props. 1-23 prove the converse of the propositions in
Book III about the harmonic properties of the pole and polar
for a large number of particular cafees. One of the proposi-
tions (IV. 9) gives a method of drawing two tangents to
a conic from an external point T. Draw any two straight
lines through T cutting the conic in Q, Q' and in R, R' respec-
158 APOLLONIUS OP PERGfA
tively. Take on QQ' and 0' on RR' so that TQ', TR' are
harmonically divided. The intersections of 00' produced with
the conic give the two points of contact required.
The remainder of the Book (IV. 24-57) deals with intersecting
conies, and the number of points in which, in particular cases,
they can intersect or touch. IV. 24 proves that no two conies
can meet in such a way that part of one of them is common
to both, while the rest is not. The rest of the propositions
can be" divided into five groups, three of which can be brought
under one general enunciation. Group I consists of particular
cases depending on the more elementary considerations affect-
ing conies: e.g. two conies having their concavities in oppo-
site directions will not meet in more than two points (IV. 35);
if a conic meet one branch of a hyperbola, it will not meet
the other branch in more points than two (IV. 37); a conic
touching one branch of a hyperbola with its concave side
will not meet the opposite branch (IV. 39). IV. 36, 41, 42, 45,
54 belong to this group. Group II contains propositions
(IV. 25, 38, 43, 44, 46, 55) showing that no two conies
(including in the term the double-branch hyperbola) can
intersect iiimore than four points. Group III (IV. 26, 47 )t 48,
49, 50, 56) are particular cases of the proposition that two
conies which touch at one point cannot intersect at move than
two other points. Group IV (IV. 27, 28, 29, 40, 51, 52, 53, 57)
are cases of the proposition that no two conies which touch
each other at two points can intersect at any other point.
Group V consists of propositions about double contact. A
parabola cannot touch another parabola in more points than
one (IV. 30); this follows from the property TP = PV. A
parabola, if it fall outside a hyperbola, cannot have double
contact with it (IV. 31); it is shown that for the hyperbola
PV>PT, while for the parabola P'V = PT; therefore the
hyperbola would fall outside the parabola, which is impossible.
A parabola cannot have internal double contact with an ellipse
or circle (IV. 32). A hyperbola cannot have double contact
with another hyperbola having the same centre (IV. 33) ;
proved by means of GV. GT = GP 2 . If an ellipse have double
contact with an ellipse or a circle, the chord of contact will
pass through the centre (IV. 34).
Book V is of an entirely different order, indeed it is the
THE CONICS, BOOKS IV-V 159
most remarkable of the extant Books. It deals with normals
to conies regarded as maximum and minimum straight lines
drawn from particular points to the curve. Included in it are
a series of propositions which, though worked out by the
purest geometrical methods, actually lead immediately to the
determination of the evolute of each of the three conies ; that
is to say, the Cartesian equations to the eyolutes can be easily
deduced from the results obtained by Apollonius. There can
be no doubt that the Book is almost wholly original, and it is
a veritable geometrical tour deforce.
Apollonius in this Book considers various points and classes
of points with reference to the maximum or minimum straight
lines which it is possible to draw from them to the conies,
i.e. as the feet of normals to the curve. He begins naturally
with points on the axis, and he takes first the point E where
AE measured along the axis from the vertex A is ^>, p being
the principal parameter. The first throe propositions prove
generally and for certain particular cases that, if in an ellipse
or a hyperbola AM be drawn at right angles to A A' and equal
to I y>, and if CM/ moot the ordinate PN of any point P of the
curve in 7/, then PjV 2 = 2 (quadrilateral MANIl) ; this is a
lemma used in the proofs of later propositions, V. 5, 6, &c.
Next, in V. 4, 5, 6, he proves that, if AE = |^, then AE is the
minimum straight line from E to the curve, and if P be any
other point on it, PE increases as P moves farther away from
A on either side ; he proves in fact that, if PN be the ordinate
from 1\
(1) in the case of the parabola PE* = AE*
(2) in the case of the hyperbola or ellipse
PE* = AN 2 + AN*
where of course p = BB' 2 /AA' y and therefore (AA' p)/AA'
is equivalent to what we call e 2 , the square of the eccentricity.
It is also proved that KA' is the maxivmim ^straight line from
E to the curve. It is next proved that, if be any point on
the axis between A and E, OA is the minimum straight line
from to the curve and, if P is any other point on the curve,
OP increases as P moves farther from A (V, 7).
160
APOLLONIUS OF PERGA
Next Apollonius takes points G on the axis at a distance
from A greater than ^p, and he proves that the minimum
straight line from G to the curve (i.e. the normal) is Gl\
where P is such a point that
(1) in the case of the parabola NG = \p ;
(2) in the case of the central conic NG : ON = y> : A A' ;
and, if P' is any other point on the conic, P'G increases as P'
moves away from P on either side ; this is proved by show-
ing that
(1) for the parabola P'G 2 = PGP + NN'* ;
(2) for the central conic P'G* = PG a + JViV /a . ~-
v AA
As these propositions contain the fundamental properties of
the subnormals, it is worth while to reproduce Apollonius' B
proofs.
(1) In the parabola, if G be any point on the axis such that
AG > %p, measure GN towards A equal to -Jy ^ e ^ 1*^ ^ >e
the ordinate through N t P' any other point on the curve.
Then shall PG be the minimum line from G to the curve, &c.
THE CONICS, ' BOOK V 161
We have P'iV' 2 = p . AN' = 2NG. AN' ;
and N'G* = NN' 2 + NG 2 2NG. NN',
according to the position of N'.
Therefore P'G 2 = 2NG.AN+ NG 2 + NN'*
and the proposition is proved.
(2) In the case of the central conic, take G on the axis such
that AG > %p, and measure GN towards A such that
NG:CN = p:AA'.
Draw the ordinate PN through N, and also the ordinate P'N'
from any other point l v .
We have first to prove the lemma (V. 1, 2, 3) that, if AM be
drawn perpendicular to A A' and equal to %p, and if CM,
produced if necessary, meet P^Vin //, then
PiV 2 = 2 (quadrilateral MANll).
This is easy, for, if AL(= 2AM) be the parameter, and A'L
meet PN in R, then, by the property of the curve,
= AN (Nil + AM)
= 2 (quadrilateral MANH).
Let GU, produced if necessary, meet P'N' in 11'. From H
draw 111 perpendicular to P'H'.
Now, since, by hypothesis, NG : (>N = /> : A A'
= AM:AC
= I1N:NC,
NH = NG, whence also H'N' = N'G.
Therefore NG* = 2 AUNG, N'G' = 2AI1'N'G.
And PN* = 2(MANH);
therefore PG* = NG 2 + PiV a = 2 (AMHG).
162 APOLLONIUS OF PEEGA
Similarly, if CM meets P'N' in K,
= 2 AH'N'G + 2(AMKN')
Therefore, by subtraction,
P'G*-PG 2 =
= HI.(H'IIK)
= HI.(HT1K)
CA
which proves the proposition.
If be any point on PG, OP is the minimum straight line
from to the curve, and OP' increases as P' moves away from
P on either side; this is proved in V. 12. (Since P'G > 'PG,
L GPP' > L GP'P ; therefore, a fortiori, L OPP' > L OP'P,
and OP' > OP.)
Apolloiiius next proves the corresponding propositions with
reference to points on the minor axis of an ellipse. If p' be
the parameter of the ordinates to the minor axis, 2>'=AA ' 2 /BB',
or ij/= CA Z /CB. If now E' be so taken that BE'=\p',
then BE' is the maximum straight line from E' to the curve
and, if P be any other point on it, E'P diminishes as P moves
farther from B on either side, and E'B' is the minimum
straight line from E' to the curve. It is, in fact, proved that
where Bti is the abscissa of P
(V. 16-18). If be any point on the minor axis such that
BO > BE', then OB is the maximum straight line from to
the curve, &c. (V. 19).
If g be a point on the minor axis such that Bg > BG, but
Bg < % p', and if Gn be measured towards B so that
then n is the foot of the ordinates of two points P such that
PC/ is the maximum straight line from g to the curve. Also,
THE CONICS, BOOK V 163
if P' be any other point on it, P'g diminishes as P' moves
farther from P on either side to B or B', and
P> > '2
-P = nu 2 .
or nn
If be any point on Pg produced beyond the minor axis, PO
is the maximum straight line from to the same part of the
ellipse, for which *Pgr is a maximum, i.e. the semi-ellipse BPB',
&c. (V. 20-2).
In V. 23 it is proved that, if (j is on the minor axis, and gP
a maximum straight line to the curve, and if Pg meets AA'
in G y then GP is the minimum straight line from G to the
curve ; this is proved by similar triangles. Only one normal
can be drawn from any one point on a conic (V. 24-6). The
normal at any point P of a conic, whether regarded as a
minimum straight line from G on the major axis or (in the
case of the ellipse) as a maximum straight line from g on the
minor axis, is perpendicular to the tangent at P (V. 27-30);
in general (1) if () be any point within a conic, and OP be
a maximum or a minimum straight line from to the conic,
the straight line through P perpendicular to PO touches the
conic, and (2) if (Y be any point on OP produced outside the
conic, O'P is the minimum straight line from 0' to the conic,
&c. (V. 31-4).
Number of normals from a point.
We now come to propositions about two or more normals
meeting at a point. If the normal at P meet the axis of
a parabola or the axis A A' of a hyperbola or ellipse in G, the
angle PGA increases as P or (i moves farther away from A,
but in the case of the hyperbola the angle will always be less
than the complement of half the angle between the asymptotes.
Two normals at points on the same side of A A' will meet on
the opposite side of that axis ; and two normals at points on
the same quadrant of an ellipse as AB will meet at a point
within the angle ACB' (V. 35-40). In a parabola or an
ellipse any normal PG will meet the curve again; in the
hyperbola, (1) if AA' be not greater than p, no normal can
meet the curve at a vsecond point on the same branch, but
164 APOLLONIUS OF PERGA
(2) if A A' > p } some normals will meet the same branch again
and others not (V. 41-3).
If P 1 G V P 2 r 2 b normals at points on one side of the axis of
a conic meeting in 0, and if be joined to any other point P
on the conic (it being further supposed in the case of the
ellipse that all three lines OP lf OP 2 , OP cut the same half of
the axis), then
(1) OP cannot be a normal to the curve ;
(2) if OP meet the axis in K, and PG be the normal at P, AG
is less or greater than AK according as P does or does not lie
between P l and P 2 .
From this proposition it is proved that (1) three normals at
points on one quadrant of an ellipse cannot meet at one point,
and (2) four normals at points on one semi-ellipse bounded by
the major axis cannot meet at one point (V. 44-8).
In any conic, if M be any point on the axis such that AM
is not greater than \<p, and if be any point on the double
ordinate through M, then no straight line drawn to any point
on the curve on the other side of the axis from and meeting
the axis between A and M can be a normal (V. 49, 50).
Propositions leadiny immediately to the determination
of tJte e volute of a conic.
These great propositions are V. 51, 52, to the following
effect :
If AM measured along the axis be greater than %p (but in
the case of the ellipse less than AG), and if MO be drawn per-
pendicular to the axis, then a certain length (;y, say) can be
assigned such that
(a) if M > y, no normal can be drawn through which cuts
the axis ; but, if OP be any straight line drawn to the curve
cutting the axis in K, NK<NG, where PN is the ordinate
and PG the normal at P ;
(6) if OM = y y only one normal can be so drawn through 0,
and, if OP be any other straight line drawn to the curve and
cutting the axis in K, NK < NG, as before ;
(c) if OM<y, two normals can be so drawn through 0, and, if
OP be any other straight line drawn to the curve, NK is
THE CONICS, BOOK V 165
greater or less than NG according as OP is or is not inter-
mediate between the two normals (V. 51, 52).
The proofs are of course long and complicated. The length
y is determined in this way :
(1) In the case of the parabola, measure MH towards the
vertex equal to /;, and divided// at-A^ so that HN l = 2N^A.
The length y is then taken such that
where P l N l is the ordinate passing through N^ ;
(2) In the case of the hyperbola and ellipse, we have
AM>%p, so that OA \AAI<AA'\p\ therefore, if 11 be taken
on AM such that ClliHM= AA':p, H will fall between A
and M.
Take two mean proportionals G'JV,, CI between CA and (JH,
and let P l N' l be the ordinate through JV^.
The length y is then taken such that
y : P^ = (CM : Mil) . (HN, : N,C).
In the case (ft), where OM = y, is the point of intersection
of consecutive normals, i.e. is the centre of curvature at the
point P\ and, by considering the coordinates of with reference
to two coordinate axes, we can derive the Cartesian equations
of the evolutes. E.g. (1) in the case of the parabola let the
coordinate axes be the axis and the tangent at the vertex.
Then AM = x, OM = y. Let p = 4 a ; then
and AN=x
But 7/ 2 : P^ = iV/ 2 : H Jlf 2 , by hypothesis,
or
therefore ay- =
or
(2) In the case of the hyperbola or ellipse we naturally take
CM, CB as axes of a? and y. The work is here rather more
complicated, but there is no difficulty in obtaining, as the
locus of 0, the curve
166 APOLLONIUS OF PERGA
The propositions V. 53, 54 are particular cases of* the pre-
ceding propositions.
Construction of normals.
The next section of the Book (V. 55-63) relates to the con-
struction of normals through various points according to their
position within or without the conic and in relation to the
axes. It is proved that one normal can be drawn through any
internal point and through any external point which is not
on the axis through the vertex A . In particular, if is any
point below the axis A A' of an ellipse, and OM is perpen-
dicular to A A', then, if AM>AC y one normal can always be
drawn through cutting the axis between A and (7, but never
more than one such normal (V. 55-7). The points on the
curve at which the straight lines through are normals are
determined as the intersections of the conic with a certain
rectangular hyperbola. The procedure
of Apollonius is equivalent to the fol-
lowing analytical method. Let AM be
the axis of a conic, PGO one of the
normals which passes through the given
point 0, PN the ordinate at P ; and let
OM be drawn perpendicular to the axis.
Take as axes of coordinates the axes in the central conic and,
in the case of the parabola, the axis and the tangent at the
vertex.
If then (x, y) be the coordinates of P and (x lt y v ) those of
we have y NO
2/1 ~~ x^x NG '
Therefore (1) for the parabola
-J .
or ^-(ai-*#)y--yi.*2> = ; (1)
(2) in the ellipse or hyperbola
Xy T a*) ~ l2/ 51 ' y & = ' * < 2 >
The intersections of these rectangular hyperbolas respec-
THE CONIC3, BOOKS V, VI 167
lively with the conies give the points at which the normals
passing through are normals.
Pappus criticizes the use of the rectangular hyperbola in
the case of the parabola as an unnecessary resort to a ' solid
locus'; the meaning evidently is that the same points of
intersection can be got by means of a certain circle taking
the place of the rectangular hyperbola. We can, in fact, from
the equation (1) above combined with y 2 = px, obtain the
circle
= o.
The Book concludes with other propositions about maxima
and minima. In particular V. 68-71 compare the lengths of
tangents TQ, TQ', where Q is nearer to the axis than Q'.
V. 72, 74 compare the lengths of two normals from a point
from which only two can be drawn and the lengths of other
straight lines from to the curve ; V. 75-7 compare the
lengths of throe normals to an ellipse drawn from a point
below the major axis, in relation to the lengths of other
straight lines from to the curve.
Book "VI is of much less interest. The first part (VI. 1-27)
relates to equal (i.e. congruent) or similar conies and segments
of conies ; it is naturally preceded by some definitions includ-
ing those of ' equal ' and ' similar ' as applied to conies and
segments of conies. Conies are said to be similar if, the same
number of ordinates being drawn to the axis at proportional
distances from the vertices, all the ordinates are respectively
proportional to the corresponding abscissae. The definition of
similar segments is the same with diameter substituted for
axis, and with the additional condition that the angles
between the base and diameter in each are equal. Two
parabolas are equal if the ordinates to a diameter in each are
inclined to the respective diameters at equal angles and the
corresponding parameters are equal; two ellipses or hyper-
bolas are equal if the ordinates to a diameter in each are
equally inclined to the respective diameters and the diameters
as well as the corresponding parameters are equal (VI. 1. 2).
Hyperbolas or ellipses are similar when the 'figure' on a
diameter of one is similar (instead of equal) to the c figure ' on
a diameter of the other, and the ordinates to the diameters in
168 APOLLONIUS OF PERGA
each make equal angles with them ; all parabolas are similar
(VI. 11, 12, 13). No conic of one of the three kinds (para-
bolas, hyperbolas or ellipses) can be equal or similar to a conic
of either of the other two kinds (VI. 3, 14, 15). Let QPQ',
qpq' be two segments of similar conies in which QQ', q<f are
the bases and PV,pv are the diameters bisecting them ; then,
if PT, pt be the tangents at P, p and meet the axes at T, t at
equal singles, and if PV : PT = pv : pt, the segments are similar
and similarly situated, and conversely (VI. 17, 18). If two
ordinates be drawn to the axes of two parabolas, or the major or
conjugate axes of two similar central conies, as PN, P'N' and
pn, p'n' respectively, such that the ratios AN: an and AN': an'
are each equal to the ratio of the respective latera recta, the
segments PP', pp' will be similar ; also PP* will not be similar
to any segment in the other conic cut off by two ordinates
other than pn, p'n', and conversely (VI. 21, 22). If any cone
be cut by two parallel planes making hyperbolic or elliptic
sections, the sections will be similar but not equal (VI. 26, 27).
The remainder of the Book consists of problems of con-
struction; we are shown how in a given right cone to find
a parabolic, hyperbolic or elliptic section equal to a given
parabola, hyperbola or ellipse, subject in the case of the
hyperbola to a certain Siopiarfios or condition of possibility
(VI. 28-30); also how to find a right cone similar to a given
cone and containing a given parabola, hyperbola or ellipse as
a section of it, subject again in the case of the hyperbola to
a certain Siopta-pos (VI. 31-3). These problems recall the
somewhat similar problems in I. 51-9.
Book VII begins with three propositions giving expressions
for AP 2 ( = AN* + PN 2 ) in the same form as those for PN 2 in
the statement of the ordinary property. In the parabola All
is measured along the axis produced (i. e. in the opposite direc-
tion to AN) and of length equal to the latus rectum, and it is
proved that, for any point P, AP* = AN.NH (VII. 1). In
the case of the central conies A A' is divided at H, internally
for the hyperbola and externally for the ellipse (AH being the
segment adjacent to A) so that AH: A'H ~ p:AA', where p
is the parameter corresponding to A A', or p = BB"* /AA', and
it is proved that
AP*:AN.NH=AA':A'H.
THE CONWS, BOOKS VI, VII
169
The same is true if A A' is the minor axis of an ellipse and p
the corresponding parameter (VII. 2, 3).
If AA' be divided at //' as well as H (internally for the
hyperbola and externally for the ellipse) so that H is adjacent
to A and //' to A', and if A' II : All = AH': A'H' = A A' :p,
the lines AH, A' 11' (corresponding to p in the proportion) are
called by Apollonius komoloyues, and lie makes considerable
use of the auxiliary points //, //' in later propositions from
VII. G onwards. Meantime he proves two more propositions,
which, like VII. 1 3, are by way of lemmas. First, if CD be
the semi-diameter parallel to the tangent at P to a central
conic, and if the tangent meet the axis AA' in T, then
PT* : CD* = NT : CX. (VII. 4.)
Draw AE, TF at right angles to CA to meet CP, and let AE
meet PT in 0. Then, if p' be the parameter of the ordinates
to GP, we have
$p':PT=OP:PE (1.49,50.)
or
Therefore
PT* : CD* = |?/. PF: $p' . CP
= PF: CP
170 APOLLONIUS OF PERGA
Secondly, Apollonius proves that, if PN be a principal
ordinate in a parabola, p the principal parameter, p' the
parameter of the ordinates to the diameter through P, then
p'=p + AN (VII. 5); this is proved by means of the same
property as VII. 4, namely %p' : PT = OP : PE.
Much use is made in the remainder of the Book of two
points Q and M, where AQ is drawn parallel to the conjugate
diameter CD to meet the curve in Q, and M is the foot of
the principal ordinate at Q; since the diameter OP bisects
both AA' and QA, it follows that A'Q is parallel to OP.
Many ratios between functions of PP', DD' are expressed in
terms of AM, A'M, MH, MH' , AH, A'H,&c. The first pro-
positions of the Book proper (VII. 6, 7) prove, for instance,
that PP' 2 : DD"* = MH': MH.
For PT 2 : CD 3 = NT: ON = AM: A'M, by similar triangles.
Also CP i : PT 2 = A'Q* : A Q*.
Therefore,, ex aequaU,
CP 2 : CD* = (AM : A'M) x (A'Q 2 : AQ 2 )
= (AM: A'M) x (A'Q 2 : A'M. MH')
x (A'M. MH': AM. MH) x (AM.MH : AQ 2 )
= (AM : A'M) x (AA': All') x (A'M: AM)
x (MH': MH) x (A'H : A A'), by aid of VII. 2, 3.
Therefore PP' 2 : DD' 2 = M H ' : MH.
Next (VII. 8, 9, 10, 11) the following relations are proved,
namely
(1) AA f2 :(PP' + DDJ=A f II.MH': {MH' + V(MH.MH')} 2 ,
(2) AA' 2 : PP'JID' = A'H : V(MH. MH')~
(3) A A'* : (PP 1 * + DD' 2 ) = A'H : MH+ MH'.
The steps by which these results are obtained are as follows.
First, A A' 2 : PP' 2 =A'H: MH' (a)
= A'H.MH':MH\
(Tliis is proved thus :
AA' 2 :PP'*=CA 2 :CP 2
= CN.CT:CP Z
= A'M. A' A : A'Q 2 .
THE VONICS, BOOK VII 171
But A'Q*:A'M.MH'=AA':AH' (VII. 2, 3)
= AA':A'H
= A'M. AA': A'M. A'H,
so that, alternately,
A'M. A A': A'ip = A'M. A'H : A'M . ME'
= A' 11 -.Mil'.)
Next, PP'- : DD" = MH ' : Mil, as above, (/3)
= MH'*'.MH.MU' I
whence PP': DD' = Mil': V(MH . MH'), (y)
and PI "* : (PP' + DD') = Mil'* : { Mil ' + V(MH . Mil') } * ;
(1) above follows from this relation and (ex) ex aequali;
(2) follows from (a) nnd (y) ex aequali, and (3) from (a)
and (/?).
We now obtain immediately the important proposition that
PP"* + DD' 2 is constant, whatever be the position of P on an
ellipse or hyperbola (the upper sijfn referring to the ellipse),
and is equal to AA U + J3B'' Z (VII. 12, 13, 29, 30).
For AA* : BB'* = AA':>p = A'H :All = A'H : A'H',
by construction :
therefore A A" 2 : A A'* + BB' Z = A'H : HH' ;
also, from (a) above,
and, by means of (/3),
' + Mil
Ex aequali, from the last two relations, we have
A A' 2 : (PP'* + DD" 2 ) = A'H : HH'
= AA'*:AA'*BB"\ from above,
whence PP>* DD'* = A A' 2 BB' 2 .
172 APOLLONIUS OF PERGA
A number of other ratios are expressed in terms of the
straight lines terminating at A, A', H, H', M, M' as follows
(VII. 14-20).
In the ellipse A A'* : PP'* * DD' 2 = A'H:2 CM,
and in the hyperbola or ellipse (if p be the parameter of the
ordinates to PP')
AA'*:p> = A'H.MH':MH*,
A A' 2 : (PP' + pf = A' 11 . Mir : (MHMH')*>
A A'* : PP' . p = A'H : MH,
and A A'* : (PP' 2 + F 2 ) = A' II . MH': (Mil'* + MW).
Apollonius is now in a position, by means of all these
relations, resting on the use of the auxiliary points H, H', M,
to compare different functions of any conjugate diameters
with the same functions of the axes, and to show how the
former vary (by way of increase or diminution) as P moves
away from A. The following is a list of the functions com-
pared, where for brevity I shall use a, b to represent AA' y BE' \
a', I/ to represent PP', DD' ; and p, p' to represent the para-
meters of the ordinates to AA', PP / respectively.
In a hyperbola, according as a > or < 6, of > or < &', and the
ratio a':// decreases or increases as P moves from A on
either side; also, if a = b, a'=b' (VII. 21-3); in an ellipse
a:b>af:b', and the latter ratio diminishes as P moves from
A to B (VII. 24).
In a hyperbola or ellipse a + b<a' + b', and a' + Z/in the
hyperbola increases continually as P moves farther from A,
but in the ellipse increases till a', b' take the position of the
equal conjugate diameters when it is a maximum (VII.
25, 26). .
In a hyperbola in which a y b are unequal, or in an ellipse,
a ^6 >a'^ 6', and a'^b' diminishes as P moves away from A,
in the hyperbola continually, and in the ellipse till a', // are
the equal conjugate diameters (VII. 27).
ab < a'6', and a'6' increases as P moves away from A, in the
hyperbola continually, and in the ellipse till a', b' coincide with
the equal conjugate diameters (VII. 28).
. VII. 31 is the important proposition that, if PP', DD' are
THE CONICS, BOOK VII
173
conjugate diameters in an ellipse or conjugate hyperbolas, and
if the tangents at thoir extremities form the parallelogram
LL'MM', then
the parallelogram LL'MM' = rcct. A A'. BB'.
The proof is interesting. Let the tangents at P, D respec-
tively meet the major or transverse axis in T, T',
Now (by VII. 4) PT* : VIP = NT : ON ;
therefore 2 A CPT :2&T'DC = NT : CN.
But 2 A CPT : (GL) = PT : CD,
= CP : DT', by similar triangles,
That is, (CL) is a mean proportional between 2 &CPT and
' DC.
Therefore, since '/(NT. CN) is a mean proportional between
NT and CN,
174 APOLLONIUS OF PERGA
2 ACPT: (GL) = V(GN. NT) : GN
(1.37,39)
therefore (GL) = CA .
The remaining propositions of the Book trace the variations
of different functions of the conjugate diameters, distinguishing
the maximum values, &c. The functions treated are the
following :
p', the parameter of the ordinates to PP' in the hyperbola,
according as A A' is (1) not less than p, the parameter corre-
sponding to A A* ', (2) less than p but not less than \ p, (3) less
than |p (VII. 33-5).
PP'<+>p' 9 as compared with AA'^p in the hyperbola (VII. 36)
or the ellipse (VII. 37).
PP'+p' AA'+p in the hyperbola (VII.
38-40) or the ellipse (VII. 41).
PP'./ AA'.p in the hyperbola (VII. 42)
or the ellipse (VII. 43).
., AA"~+p 2 in the hyperbola, accord-
ing as (1) A A' is not less than
p, or (2) AA'< p, but A A'* not
less than ^(AA^pY 9 or (3)
PP' 2 +// 2 AA'^+yP in the ellipse, according
as AA' 2 is not greater, or is
greater, than (AA' + p) 2 (VII.
47, 48).
Pl**fP A A'* ~p* in the hyperbola, accord-
ing as AA f > or <p (VII.
49, 50).
PP' 2 - 2>* A A'* - p* or BB^+p!* in the ellipse,
according as PP' > or < p'
(VII. 51).
THE CONICU, BOOK VII 175
As we have said, Book VIII is lost. The nature of its
contents can only be conjectured from Apollonius's own
remark that it contained determinate conic problems for
which Book VII waS useful, particularly in determining
limits of possibility. Unfortunately, the lemmas of Pappus
do not enable us to form any clearer idea. But it is probable
enough that the Book contained a number of problems having
for their object the finding of conjugate diameters in a given
conic such that certain functions of their lengths have given
values. It was on this assumption that Halley attempted
a restoration of the Book.
If it be thought that the above account of the Conies is
disproportionately long for a work of this kind, it must be
remembered that the treatise is a great classic which deserves
to be more known than it is. What militates against its
being read in its original form is the great extent of the
exposition (it contains 387 separate propositions), due partly
to the Greek habit of proving particular cases of a general
proposition separately from the proposition itself, but more to
the cumbrousness of the enunciations of complicated proposi-
tions in general terms (without the help of letters to denote
particular points) and to the elaborateness of the Euclidean
form, to which Apollonius adheres throughout.
Other works by Apollonius.
Pappus mentions and gives a short indication of the con-
tents of six other works of Apollonius which formed part of the
Treasury of A tialysis. 1 Three of these should be mentioned
in close connexion with the (Joules.
(a) Om the Cutting-off of <t Ratio (\6yov anoTOfiri),
two Books.
This work alone of the six mentioned has survived, and
that only in the Arabic ; it was published in a Latin trans-
lation by Edmund Halley in 1706. It deals with the general
problem, ' Given two straight lines, parallel to one another or
intersecting, and a fixed point on each line, to draw through
1 Pappus, vii, pp. 640-8, 660-72.
176 APOLLONIUS OF PERGA
a given point a straight line which shall cut off segments from
each line (measured from the fixed points) bearing a given
ratio to one another. 9 Thus, let A, B be fixed points on the
two given straight lines AC, BK, arid let be the given
point. It is required to draw through a straight line
cutting the given straight lines in points M, N respectively
such that AM is to UN in a given ratio. The two Books of
the treatise discussed the various possible cases of this pro-
blem which arise according to the relative positions of the
given straight lines and points, and also the necessary condi-
tions and limits of possibility in cases where a solution is not
always possible. The first Book begins by supposing the
given lines to be parallel, and discusses the different cases
which arise ; Apollonius then passes to the cases in which the
straight lines intersect, but one of the given points, A or B, is
at the intersection of the two lines. Book II proceeds to the
general case shown in the above figure, and first proves that
the general case can be reduced to the case in Book I where
one of the given points, A or J5, is at the intersection of the
two lines. The reduction is easy. For join OB meeting A(J
in J3', and draw ffN' parallel to BN to meet OM in N'. Then
the ratio ffN' : BN, being equal to the ratio OB' : OB, is con-
stant. Since, therefore, BN: A M is a given ratio, the ratio
11' N' : AM is also given.
Apollonius proceeds in all cases by the orthodox method of
analysis and synthesis. Suppose the problem solved and
OMN drawn through in such a way that B'N'iAM is a
given ratio = A, say.
O.V THE CUTTJNG-OFF OF A RATIO 177
Draw 0(7 parallel to ZLV or WN' to meet AM in V. Take
D on AM such that OC : AD = X = B'N' : 4 J/.
Then AM:AD = ffN':OU
therefore MD :AD = B'C : CM,
or CM . MD = AD. B'C, a given rectangle.
.Henco tlie problem is reduced to one of applying to CD a
rectcnu/le (CM . MD) equal to <t given rectangle (A D . B'C) but
falliiifj dtort ly ti square fiywe. In the case as drawn, what-
ever be the value of X, the solution is always 'possible because
the given rectangle AD .CB' is always less than CA . AD, and
therefore always loss than -Jf'/) 2 ; one of the positions of
M falls between A and D because CM.MD<(!A . AD.
The proposition TIL 41 of the Conies aboift the intercepts
made on two tangents to a parabola by a third tangent
(pp. 155-6 above) suggests an obvious application of our pro-
blem. We had, with the notation of that proposition,
Prirq = rQ:Q;> = <]/):pR.
Suppose that the two tangents c/I\ qR are given as iixocl
tangents with their points of contact P, R. f rhen we can
draw another tangent if we can draw a straight line
intersecting qP,qli in such a way that Pr:rq=z<ip:pR or
P<j : c/r = (fit :pU, i. v. (jr : pR = P</ : qR (a constant ratio) ;
i.e. we have to draw a straight line such that the intercept by
it on qP measured from q has a given ratio to the intercept
by it on qR measured from A*. This is a particular case of
our problem to which, as a matter of fact, Apollonius devotes
special attention. In the annexed figure the letters have the
B 7 C M D A
same meaning as before, and N'M has to be drawn through
such that B N' : AM = A. In this case there are limits to
178 APOLLONIUS OF PERGA
the value of X in order that the solution may be possible.
Apollonius begins by stating the limiting case, saying that we
obtain a solution in a special manner in the case where M is
the middle point of CD, so that the rectangle CM . MI) or
CB' . AD has its maximum value.
The corresponding limiting value of \ is determined by
finding the corresponding position of D or M.
We have B'C : MD = CM: AD, as before,
= B'M: MA',
whence, since MD = CM,
B'C : B'M = CM: MA
= B'M: B'A,
so that B'M* = B'C. B'A.
Thus M is found and therefore D also.
According, therefore, as X is less or greater than the par-
ticular value of OG-.AD thus determined, Apollonius finds no
solution or two solutions.
Further, we have >
AD = B'A + B'C- (B'D + B'C)
= B'A + B'C-2B'M
= B'A + B'C- 2 VB'A . B'C.
If then we refer the various points to a system of co
ordinates in which B'A, B'N' are the axes of x and y, and ii
we denote by (x, y) and the length B'A by h,
X = OC/AD = y/(h + x-2<Shx).
If we suppose Apollonius to have used these results for tin
parabola, he cannot have failed to observe that the limitinj
case described is that in which is on the parabola, whil
N'OM is the tangent at ; for, as above,
B'M : B'A = B'C : B'M = N'O : N'M, by parallels,
so that B'A, N'M are divided at M, respectively in the sam
proportion.
O.Y THE CUTTING-OFF OF A RATIO 179
Further, it' we put for A the ratio between the lengths of the
two fixed tangents, then if h, k be those lengths,
k - y
__
which can easily be reduced to
the equation of the parabola referred to the two fixed tangents
as axes.
(ft) On the cuttiny-off of an area (\<*>piov dnoro^),
two Books.
This work, also in two Books, dealt with a similar problem,
with the difference that the intercepts on the given straight
lines measured from the given points are required, not to
have a given ratio, but to contain a given rectangle. Halley
included an attempted restoration of this work in his edition
of the De sectione rationis.
The general case can here again be reduced to the more
special one in which one of the fixed points is at the inter-
section of the two given straight lines. Using the same
figure as before, but with D taking the position shown by (D)
iu the figure, we take that point such that
0(J .A I) = the given rectangle.
We have then to draw ON'M through such that
B'N' .AM=OC.AD,
or &N'\()G=AD:AM.
But, by parallels, B'N' : OG = B'M: CM',
therefore AM : CM = AD: B'M
so that ' B f M .MD = AD. B'C.
Hence, as before, the problem is reduced to an application
of a rectangle in the well-known manner. The complete
180 APOLLONIUS OF PERGA
treatment of tins problem in all its particular cases with their
8iopt<r/Aoi could present no difficulty to Apollonius.
If the two straight lines are parallel, the solution of the
problem gives a means of drawing any number of tangents
to an ellipse when two parallel tangents, their points of con-
tact, and the length of the parallel semi-diameter are given
(see Conies, III. 42). In the case of the hyperbola (III. 43)
the intercepts made by any tangent on the asymptotes contain
a constant rectangle. Accordingly the drawing of tangents
(Jepends upon the particular cane of our problem in which both
fixed points are the intersection of the two fixed lines.
(y) On determinate xeetioii, (8i<opicrfjLvri Tofirj), two Books.
The general problem here is, Given four points ,4, B, (!, D on
a straight line, to determine another point P on the same
straight line such that the ratio AP.CP \BP.DP has a
given value. It is clear from Pappus's account 1 of the contents
of this work, and from his extensive collection. of lemmas to
the different propositions in it, that the question was very
exhaustively discussed. To determine P by means of the
equation
where A, .B, (7, /), A are given, is in itself an easy matter since
the problem can at once be put into the form of a quadratic
equation, and the Greeks would have no difficulty in reducing
it to the usual application of areas. If, however (as we may
fairly suppose), it was intended for application in further
investigations, the complete discussion of it would naturally
include not only the finding of a solution, but also the deter-
mination of the limits of possibility and the number of possible
solutions for different positions of the point-pairs A, (! and
B, D, for the cases in which the points in either pair coincide,
or in which one of the points is infinitely distant, and so on.
This agrees with what we find in Pappus, who makes it clear
that, though we do not meet with any express mention of
series of point-pairs determined by the equation for different
values of A, yet the treatise contained what amounts to a coiu-
1 Pappus, vii, pp. 642-4.
ON DETERMINATE SECTION 181
pletc Theory of Involution. Pappus says that the separate
cases were dealt with in which the given ratio was that of
either (1) the square of one abscissa measured from the
required point or (2) the rectangle contained by two. such
abscissae to any one of the following: (1) the square of one
abscissa, (2) the rectangle contained by one abscissa and
another separate line of given length independent of the
position of the required point, (3) the rectangle contained by
two abscissae. We learn also that maxima and minima were
investigated. From the lemmas, too, we may draw other
conclusions, e. g.
(1) that, in the case where X = 1, or AP.VP =
Apollonius used the relation III* : DP = AK . Il( ' : A D . DC,
(2) that Apollonius probably obtained a double point E of the
involution determined by the point-pairs A<(* and B y 1) by
means of the relation
AB .UC: A 1) . DC = BE* : DW.
A possible application of the 1 problem was the determination
of the points of intersection of the given straight line with a
conic determined as a four-line locus, since A, B, C, D are in
fact the points of intersection of the given straight line with
the four lines to which the locus has reference.
(8) ()iu Contacts or Ta agencies (trrafyai), two Books.
Pappus again comprehends in one enunciation the varieties
of problems dealt with in the treatise, which we may repro-
duce as follows: Given three things, each of which may be
either a ^oinl, a straight line or a circle, to draw a circle
which shall jtass through each of the giceu, points (so far as it
is poiids that are given) ami touch the straight lines or
circles. 1 The possibilities as regards the different data are
ten. We may have any one of the following: (1) three
points, (2) three straight lines, (3) two points and a straight
line, (4) two straight lines and a point, (5) two points and
a circle, (6) two circles and a point, (7) two straight lines and
1 Pappus, vii, p, 644, 25-8.
182 APOLLONIUS OF PERGA
a circle, (8) two circles and a straight line, (9) a point, a circle
and a straight line, (10) three circles. Of these varieties the
first two are treated in Eucl. IV ; Book I of Apollonius's
treatise treated of (3), (4), (5), (6), (8), (9), while (7), the case of
two straight lines and a circle, and (10), that of the three
circles, occupied the whole of Book II.
The last problem (10), where the data are three circles,
has exercised the ingenuity of many distinguished geometers,
including Vieta and Newton. Vieta (1540-1603) set the pro-
blem to Adrianus llomanus (van Roomeu, 1561-1615) who
solved it by means of a hyperbola. Vieta was not satisfied
with this, and rejoined with his Apollonius Galhis (1600) in
which he solved the problem by plane methods. A solution
of the same kind is given by Newton in his Arithmetica
Universalis (Prob. xlvii), while an equivalent problem is
solved by means of two hyperbolas in the Principia, Lemma
xvi. The problem is quite capable of a ' plane ' solution, and,
as a matter of fact, it is not difficult to restore the actual
solution of Apollonius (which of course used the ' plane' method
depending on the straight line and circle only), by means of
the lemmas given by Pappus. Three things are necessary to
the solution. (1) A proposition, used by Pappus elsewhere 1
and easily proved, that, if two circles touch internally or
externally, any straight lino through the point of contact
divides the circles into segments respectively similar. (2) The
proposition that, given three circles, their six centres of simili-
tude (external and internal) lie three by three on four straight
lines. This proposition, though not proved in Pappus, was
certainly known to the ancient geometers ; it is even possible
that Pappus omitted to prove it because it was actually proved
by Apollonius in his treatise. (3) An auxiliary problem solved
by Pappus and enunciated by him as follows. 2 Given a circle
ABC, and given three points D, E, F in a straight line, to
inflect (the broken line) DAE (to the circle) so as to make BG
in a straight line with CF\ in other words, to inscribe in the
circle a triangle the sides of which, when produced, pass
respectively through three given points lying in a straight
line. This problem is interesting as a typical example of the
ancient analysis followed by synthesis. Suppose the problem
1 Pappus, iv, pp. 194-6. 2 lb. vii, p. 848.
CONTACTS OR TANGENCIES 183
solved, i.e. suppose DA, EA drawn to the circle cutting it in
points B, C such that BC produced passes through F.
Draw BG parallel to DF; join GC
and produce it to meet DE in H.
Then
LBAQ=LRGC
= supplement of Z CH D ;
therefore A, D y //, C lie on a circle, and
DE.EH=AE.EC. o H K e F
Now AE.E(J is given, being equal to the square on the
tangent from E to the circle ; and DE is given ; therefore HE
is given, and therefore the point //.
But F is also given ; therefore the problem is reduced to
drawing HC, FC to meet the circle in such a way that, if
HC, FC produced meet the circle again in G, B, the straight
line BG is parallel to II F: a problem which Pappus has
previously solved. 1
Suppose this done, and draw BK the tangent at B meeting
HF'mK. Then
Z K BC = Z BGC, in the alternate segment,
Also the angle (JFK is common to the two triangles KBF>
CHF\ therefore the triangles are similar, and
or
Now BF .FC is given, and so is HF\
therefore FK is given, and therefore K is given.
The synthesis is as follows. Take a point H on DE such
that DE . EH is equal to the square on the tangent from E to
the circle.
Next take K on II F such that HF.FK = the square on the
tangent from F to the circle.
Draw the tangent to the Circle from K, and let B be the
point of contact. Join BF meeting the circle in (7, and join
1 Pappus, vii, pp. 830-2.
184
APOLLONIUS OF PERGA
HO meeting the circle again in (?. It is then easy to prove
that BG is parallel to DF.
Now join EC, and produce it to meet the circle again at A ;
join AB.
We have only to prove that A B, BD are in one straight line.
Since DE.EH = AE.EC, the points A, D, II, are con-
cyclic.
Now the angle CHF, which is the supplement of the angle
CHD, is equal to the angle JiGC, and therefore to tlie
angle BAG.
Therefore the angle BAC is equal to the supplement of
angle DEC, so that the angle BAG is equal to the angle DAG,
and AB, BD are in a straight line.
The problem of Apollonius is now easy. We will take the
case in which the required circle touches all the three given
circles externally as shown in the figure. Let the radii of the
OX CONTACTS OR TANGENOIEH 185
given circles be a, ft, c and their centres A, B, C. Let D, #, F
be the external centres of similitude so that BD: 7)C'=&:c, &c.
Suppose the problem solved, and let P, Q, R be the points
of contact. Let PQ produced meet the circles with centres
A y B again in K, L. Then, by the proposition (1) above, the
segments KGP, QHL are both similar to the segment PYQ ;
therefore they are similar to one another. It follows that PQ
produced beyond L passes through F. Similarly QR, PR
produced pass respectively through />, E.
Let PE, QD meet the circle with centre (! again in M y N.
Then, the segments PQR, RXM being similar, the angles
PQR, RNM are equal, and therefore MN is parallel to PQ.
Produce XM to meet EF in V.
Then E V : EF = EM: EP = EC : EA = c : a ;
therefore the point V is given.
Accordingly the problem reduces itself to this: Given three
points J r , E, D in a straight line, it is required to draw DR, ER
to a point R on the circle with centre (! so that, if DR, ER meet
the circle again in iV, M, XM produced shall pass through V.
This is the problem of Pappus just solved.
Thus R is found, and DR, ER produced meet the circles
with centres B and A in the other required points Q, P
respectively.
(e) Plane loci, two Books.
Pappus gives a pretty full account of the contents of this
work, which has sufficed to enable restorations of it to
be made by three distinguished geometers, Fermat, van
Schooten, and (most completely) by Robert Simson. Pappus
prefaces his account by a classification of loci on two
different plans. Under the first classification loci are of three
kinds: (1) tfaKTiKOL, lioldin<j-i u or Jived ; in this case the
locus of a point is a point, of a line a line, and of a solid
a solid, where presumably the line or solid can only move on
itself so that it does not change its position : (2) Sitgo-
SLKOL, pasxiny-aloiuj : this is the ordinary sense of a locus,
where the locus of a point is a line, and of a liile a solid :
(3) dva(TTpo<f>iKot, woviiuj backwards and forwards, as it were,
in which sense a plane may be the locus of a point and a solid
186 APOLLON1US OF PERGA
of a line. 1 The second classification is the familiar division into
plane, solid, and linear loci, plane loci being straight lines
and circles only, solid loci conic sections only, and linear loci
those which are not straight lines nor circles nor any of the
conic sections. The loci dealt with in our treatise are accord-
ingly all straight lines or circles. The proof of the pro-
positions is of course enormously facilitated by the use of
Cartesian coordinates, and many of the loci are really the
geometrical equivalent of fundamental theorems in analytical
or algebraical geometry. Pappus begins with a composite
enunciation, including a number of propositions, in these
terms, which, though apparently confused, are not difficult
to follow out:
' If two straight lines be drawn, from one given point or from
two, which are (a) in a straight line or (/>) parallel or
(c) include a given angle, and either (a) bear a given ratio to
one another or (]8) contain a given rectangle, then, if the locus
of the extremity of one of the lines is a plane locus given in
position, the locus of the extremity of the other will also be a
plane locus given in position, which will sometimes be of the
same kind as the former, sometimes of the other kind, and
will sometimes be similarly situated with reference to the
straight line, and sometimes contrarily, according to the
particular differences in the suppositions/ 2
(The words ' with reference to the straight line ' are obscure, but
the straight line is presumably some obvious straight line in
each figure, e.g., when there are two given points, the straight
line joining them.) After quoting thrue obvious loci ' added
by Charmaridrus ', Pappus gives three loci which, though con-
taining an unnecessary restriction in the third case, amount
to the statement that any equation of the first degree between
coordinates inclined at fixed angles to (a) two axes perpen-
dicular or oblique, (/>) to any number of axes, represents a
straight line. The enunciations (5-7) are as follows/'
5. ' If, when a straight line is given in magnitude and is
Inoved so as always to be parallel to a certain straight line
given in position, one of the extremities (of the moving
straight line), lies on a straight line given in position, the
1 Pappus, vii, pp. 660. 18-662. 5. 2 Ib. vii, pp. 662. 25-664. 7.
3 Ib., pp. 664. 20-666. 6.
PLANE LOCI 187
other extremity will also lie on a straight line given in
position/
(That is, x = a or y = 6 in Cartesian coordinates represents a
straight line.)
6. 'If from any point straight lines be drawn to meet at given
angles two straight lines either parallel or intersecting, and if
the straight lines so drawn have a given ratio to one another
or if the sum of one of them and a line to which the other has
a given ratio be given (in length), then the point will lie on a
straight line given in position/
(This includes the equivalent of saying that, if x, y be the
coordinates of the point, each of the equations x = my,
= c represents a straight line.)
7. 'If any number of straight lines be given in position, and
straight lines be drawn from a point to meet them at given
angles, and if the straight lines so drawn be such that the
rectangle contained by one of them and a given straight line
added to the rectangle contained by another of them and
(another) given straight line is equal to the rectangle con-
tained by a third and a (third) giveil straight line, and simi-
larly with the others, the point will lie on a straight line given
in position/
(Here we have trilinear or multilinear coordinates propor-
tional to the distances of the variable point from each of the
three or more fixed lines. When there are three fixed lines,
the statement is that ax + by = cz represents a straight line.
The precise meaning of the words 'and similarly with the
the others' or 'of the others 1 KOI r>v \onrSw 6/io/coy is
uncertain; the words seem to imply that, when there were
more than three rectangles a,i\ lnj.cz ..., two of them were
taken to be equal to the sum of all the others ; but it is quite
possible that Pappus meant that any linear equation between
these rectangles represented a straight line. Precisely how
far Apollonius went in generality we are not in a position to
judge.)
The last enunciation (8) of Pappus referring to Book I
states that,
' If from any point (two) straight lines be drawn to meet (two)
parallel straight lines given in position at given angles, and
188 APOLLONIUS OF PERGA
cut off from the parallels straight lines measured from given
points on them such that (a) they have a given ratio or
(6) they contain a given rectangle or (c) the sum or difference
of figures of given species described on them respectively is
equal to a given "area, the point will lie on a straight line
given in position/ 1
The contents of Book II are equally interesting. Some of
the enunciations shall for brevity be given by means of letters
instead of in general terms. If from two given points A, B
two straight lines be ' inflected ' (K\aorQSxnv) to a point P, then
(1), if AP* * BP* is given, the locus of P is a straight line ;
(2) if AP, BP are in a given ratio, the locus is a straight line
or a circle [this is the proposition quoted by Eutocius in his
commentary on the Conies, but already known to Aristotle] ;
(4) if AP 2 is 'greater by a given area than in a given ratio '
to J9P 2 , i.e. if AP* = d z + m . BP' 1 , the locus is a circle given in
position. An interesting proposition is (5) that, 'If from any
number of given points whatever straight lines be inflected to
one point, and the figures (given in species) described on all of
them be together equal to a given area, the point will lie on
a circumference (circle) given in position ' ; that is to say, if
a.AP* + p. BP* + y.G'P 2 +... = a given area (where a,/3, y ...
are constants), the locus of P is a circle. (3) states that, if
AN be a fixed straight line and A a fixed point on it, and it'
AP be any straight line drawn to a point P such that, if PN
is perpendicular to AN, AP 2 = a . AN or a . BN, where a is a
given length and B is another fixed point on AN, then the
locus of P is a circle given in position ; this is equivalent
to the fact that, if A be the origin, AN the axis of x, and
x = A N, y = PN bo the coordinates of P, the locus ,/: 2 + y' 1 = ax
or <B 2 -f-2/ 2 = a (x b) is a circle. (6) is somewhat obscurely
enunciated : ' If from two given points straight lines be in-
flected (to a point), and from the point (of concourse) a straight
line be drawn parallel to a straight line given in position and
cutting off from another straight line given in position an
intercept measured from a given point on it, and if the sum of
figures (given in species) described on the two inflected lines
be equal to the rectangle contained by a given straight line
and the intercept, the point at which the straight lines are
1 Pappus, vii, p. 666. 7-13.
PLANE LOCI 189
inflected lies on a .circle given in position/ The meaning
seems to be this : Given two fixed points A, J3, a length a,
a straight line OX with a point fixed upon it, and a direc-
tion represented, say, l>y any straight line OZ through 0, then,
if AP, BP be drawn to P, and PM parallel to OZ meets OX
in M, the locus of P will b(3 a circle given in position if
whore a, f$ are constants. The last two loci are again
obscurely expressed, but the sense is this : (7) If PQ be any
chord of a circle passing through a fixed internal point 0, and
If be an external point on PQ produced such that either
(a) OE 2 = PR . JtQ or (/>) Ofi* + PO . OQ= PR . RQ y the locus
of Ji is a straight line given in position. (8) is the reciprocal
of this: Given tho fixed point (), the straight line which is
the locus of R, and also the; relation (a) or (6), the locus of
P, Q is a circle.
() Neva-eis (Verylnys or Inclinations), two Books.
As we have seen, the problem in a i/cvcrt? is to place
between two straight lines, a straight line and a curve, or
two curves, a straight line of given length in such a way
that it rertfe* towards a fixed point, i.e. it will, if pro-
duced, pass through a fixed point. Pappus observes that,
when we conic* to particular cases, the problem will be
'plane', * solid' or 'linear', according to the nature of the
particular hypotheses; but a selection had been made from
the class which could be solved by plane methods, i.e. by
means of the straight line*, and circle, the object being to give
those which wore more generally useful in geometry. The
following were the cases thus selected and proved. 1
I. Given (a) a semicircle and u straight line at right angles
to the base, or (/;) two semicircles with their bases in a straight
line, to insert a straight line of given length verging to an
angle of the semicircle [or of one of the semicircles].
II. Given a rhombus with one side produced, to insert
a straight line of given length in the external angle so that it
verges to the opposite angle.
1 Pappus, vii, pp. 670-2.
190 APOLLONIUS OF PERGA
III. Given a circle, to insert a chord of given length verging
to a given point.
In Book I of Apollonius's work there were four cases of
I (a), two cases of III, and two of II ; the second Book con-
tained ten cases of I (b).
Restorations were attempted by Marino Ghetaldi (Apollonius
redivivus, Venice, 1607, and Apollonius redivivus . . . Liber
secundus, Venice, 1613), Alexander Anderson (in a tiupple-
meutum Apollo mi redivivi, 1612), and Samuel Horsley
(Oxford, 1770); the last is much the most complete.
In the case of the rhombus (II) the construction of Apollonius
can be restored with certainty. It depends on a lemma given
by Pappus, which is as follows: Given a rhombus AD with
diagonal BC produced to E, if F be taken on EC such that EF
is a mean proportional between BE and EC, and if a circle be
described with E as centre and EF as radius cutting CD
in K and AC produced in //, then shall J8, K y H be in one
straight line. 1
Let the circle cut AC in L, join LK meeting BC in M, and
join HE, LE, KE.
Since now CL, CK are equally inclined to the diameter of
the circle, CL = CK. Also EL = EK y and it follows that the
triangles ECK, ECL are equal in all respects, so that
LCKE = LCLE = LCIIE.
By hypothesis, EB:EF=EF: EC,
or EB:EK = EK:EC.
1 Pappus, vii, pp. 778-80.
NETSEIS (VERGING^ OR INCLINATIONS) 191
Therefore the triangles BEK, KEC, which have the angle
BEK common, are similar, and
Z CBK = Z GK K = Z G ## (from above).
But Z 7/C'# = Z AGE = Z
Therefore in the triangles G'-fi/f, C7/JF two angles are
respectively equal, so that Z (JElf = Z (7/i 7i also.
But since LGKK = LCHK (from above), K, G f , E, II are
concyclic.
Hence Z <7AY/ + Z (/#// = (two right angles) ;
therefore, since Z C KH = Z GVi.fi,
LVKB + LVKH = (two riglit angles),
and BKJI is a straight line.
It is certain, from the nature of this lemma, that Apollonius
made his construction by drawing the circle shown in the
figure.
He would no doubt arrive at it by analysis somewhat as
follows.
Suppose the problem solved, and HK inserted as re-
quired ( = &).
Bisect HK in JV, and draw NE at right angles to KH
meeting S(- produced in E. Draw KM perpendicular to BC\
and produce it to meet AC in L. Then, by the property of
the rhombus, LM = MI\, and, since KX = Nil also, MX is
parallel to LH.
Now, since the angles at M, N are right, M, K, N, E are
concyclic.
Therefore VEK = LMNK = LG11K, so that (7, 7v, H, E
are concyclic.
Therefore Z J3CD = supplement of KOE = Z A^A r =
and the triangles EKH, DOB are similar.
Lastly,
therefore the triangles EBK, EKC are similar, and
or
192 APOLLONIUS OF PERGA
But, by similar triangles EKH, DOB,
EK:KH=J)G:GB,
and, since the ratio DG:GB, as well as KH, is given, EK
is given.
The construction then is as follows.
If k be the given length, take a straight line /; such that
apply to BG a rectangle BE . EG equal to p 2 and exceeding by
a square ; then with E as centre and radius equal to /> describe a
circle cutting A(! produced in II and (H) in K. UK is then
equal to k and, by Pappus's lemma, verges towards B.
Pappus adds an interesting solution of the same problem
with reference to a square instead of a rhombus ; the solution
is by one Heraclitus and depends on a lemma which Pappus
also gives. 1
We hear of yet other lost works by Apollonius.
(77) A Comparison of the dodecahedron with the icosahedron.
This is mentioned by Hypsicles in the preface to the so-called
Book XIV of Euclid. Like the Conies, it appeared in two
editions, the second of which contained the proposition that,
if there be a dodecahedron and an icosahedron inscribed in
one and the same sphere, the surfaces of the solids are in the
same ratio as their volumes ; this was established by showing
'that the perpendiculars from the centre of the sphere to
a pentagonal face of the dodecahedron and to a triangular
face of the icosahedron are equal.
(&} Marinus on Euclid's Data speaks of a General Treatise
(KaOoXov TTpay/jiaTeia) in which Apollonius used the word
assigned (rtrayfjitvov) as a comprehensive term to describe the
datum in general. It would appear that this work must
have dealt with the fundamental principles of mathematics,
definitions, axioms, &c., and that to it must be referred the
various remarks on such subjects attributed to Apollonius by
Proclus, the elucidation of the notion of a line, the definition
1 Pappus, vii, pp. 780-4.
OTHER LOST WORKS 193
of plane and solid angles, and his attempts to prove the axioms ;
it must also have included the three definitions (13-15) in
Euclid's Data which, according to a scholium, were due to
Apollonius and must therefore have been interpolated (they
are definitions of Karrj-yfievr], avr\y /jL^vq , and the elliptical
phrase wapa 0cre*, which means ' parallel to a straight line
given in position '). Probably the same work also contained
Apollonius' s alternative constructions for the problems of
Eucl. I. 10, 11 and 23 given by Proclus. Pappus speaks
of a mention by Apollonius 'before his own elements' of the
class of locus called e^/crj/coy, and it may be that the treatise
now in question is referred to rather than the Plane Loci
itself.
(i) The work On the Cochliau was on the cylindrical helix.
It included the theoretical generation of the curve on the
surface of the cylinder, and the proof that the curve is
homoeomeric or uniform, i.e. such that any part will fit upon
or coincide with any other.
(K) A work on Unordered Irrationals is mentioned by
Proclus, and a scholium on Eucl. X. 1 extracted from Pappus's
commentary remarks that c Euclid did not deal with all
rationals and irrationals, but only with the simplest kinds by
the combination of which an infinite- number of irrationals
are formed, of which latter Apollonius also gave some'.
To a like effect is a passage of the fragment of Pappus's
commentary on Eucl. X discovered in an Arabic translation
by Woepcke: 'it was Apollonius who, besides the ordered
irrational magnitudes, showed the existence of the unordered,
and by accurate methods set forth a great number of them'.
The hints given by the author of the commentary seem to imply
that Apollonius's extensions of the theory of irrationals took
two directions, (1) generalizing the medial straight line of
Euclid, 011 the basis that, between two lines commensurable in
square (only), we may take not only one sole medial line but
three or four, and so on ad infinitwin, since we can take,
between any two given straight lines, as many lines as
we please in continued proportion, (2) forming compound
irrationals by the addition and subtraction of more than two
terms of the sort composing the binomials, (tpotomes, &c.
1523.2
194 APOLLONIUS OF PERGA
(A) On the burning-miwor (rrpi rov irvptov) is the title of
another work of Apollonius mentioned by the author of the
Fragmentum mathematicum Boliense, which is attributed by
Heiberg to Anthemius but is more likely (judging by its sur-
vivals of antiquated terminology) to belong to a much earlier
date. The fragment shows that Apollonius discussed the
spherical form of mirror among others. Moreover, the extant
fragment by Anthemius himself (on burning mirrors) proves the
property of mirrors of parabolic section, using the properties of
the parabola (a) that the tangent at any point makes equal
angles with the axis and with the focal distance of the point,
and (&) that the distance of any point on the curve from the
focus is equal to its distance from a certain straight line
(our ' directrix ') ; and we can well believe that the parabolic
form of mirror was also considered in Apollonius's work, and
that he was fully aware of the focal properties of the parabola,
notwithstanding the omission from the Conies of all mention
of the focus of a parabola.
(/*) In a work called &KVTOKLOV ( quick-delivery ') ApolloniuH
is said to have found an approximation to the value of TT ' by
a different calculation (from that of Archimedes), bringing it
within closer limits '} Whatever these closer limits may have
been, they were considered to be less suitable for practical use
than those of Archimedes.
It is a moot question whether Apollonius's system of arith-
metical notation (by tetrads) for expressing large numbers
and performing the usual arithmetical operations with them,
as described by Pappus, was included in this same work.
Heiberg thinks it probable, but there does not seem to be any
necessary reason why the notation for large numbers, classify-
ing them into myriads, double myriads, triple myriads, &c.,
i.e. according to powers of 10,000, need have been connected
with the calculation of the value of TT, unless indeed the num-
bers used in the calculation were so large as to require the
tetradic system for the handling of them.
We have seen that Apollonius is credited with a solu-
tion of the problem of the two mean proportionals (vol. i,
pp. 262-3).
1 y. Eutocius on Archimedes, Measurement of a Circle,
OTHER LOST WORKS 195
Astronomy.
We are told by Ptolemaeus Chennus l that Apollouiua was
famed for his astronomy, and was called (Epsilon) because
the form of that letter is associated with that of the moon, to
which his accurate researches principally related. Hippolytus
says he made the distance of the moon's circle from the sur-
face of the, earth to be 500 myriads of stades. 2 This figure'
can hardly be right, for, the diameter of the earth being,
according to Eratosthenes's evaluation, about eight myriads of
stades, this would make the distance of the moon from the
earth about 125 times the earth's radius. This is an unlikely
figure, seeing that Aristarchus had given limits for the ratios
between the distance of the moon and its diameter, and
between the diameters of the moon and the earth, which lead
to about 19 as the ratio of the moon's distance to the earth's
radius. Tannery suggests that perhaps Hippolytus made a
mistake in copying from his source and took the figure of
5,000,000 stades to be the length of the radius instead of the
diameter of the moon's orbit.
But we have better evidence of the achievements of Apol-
lonius in astronomy. In Ptolemy's tiynt(u:is :} he appears as
an authority upon the hypotheses of epicycles and eccentrics
designed to account for the apparent motions of the planets.
The propositions of Apollonius quoted by Ptolemy contain
exact statements of the alternative hypotheses, and from this
fact it was at one time concluded that Apollonius invented
the two hypotheses. This, however, is not the case. The
hypothesis of epicycles was already involved, though with
restricted application, in the theory of Heraclides of Pontus
that the two inferior planets, Mercury and Venus, revolve in
circles like satellites round the sun, while the sun itself
revolves in a circle round the earth ; that is, the two planets
describe epicycles about the material suji as moving centre.
In order to explain the motions of the superior planets by
means of epicycles it was necessary to conceive of an epicycle
about a point as moving centre which is not a material but
a mathematical point. It was some time before this extension
of the theory of epicycles took place, and in the meantime
1 apud Phottunt, Cod. cxc, p. 151 b 18, ed. Bekker.
2 Hippol. Be fitt. iv. 8, p. 66. ed, Duncker. 3 Ptolemv. Suntaxis. xii. 1.
196 APOLLONIUS OF PERGA
another hypothesis, that of eccentrics, was invented to account
for the movements of the superior planets only. We are at this
stage when we come to Apollonius. His enunciations show
that he understood the tlieory of epicycles in all its generality,
but he states specifically that the theory of eccentrics can only
be applied to the three planets which can be at any distance
from the sun. The reason why he says that the eccentric
hypothesis will not serve for the inferior planets is that, in
order to make it serve, we should have to suppose the circle
described by the centre of the eccentric circle to be greater
than the eccentric circle itself. (Even this generalization was
made later, at or before the time of Hipparchus.) Apollonius
further says in his enunciation about the eccentric that 'the
centre of the eccentric circle moves about the centre of the
zodiac in the direct order of the signs and at a speed equal to
that of the sun, while the star moves on the eccentric about
its centre in the inverse order of the signs and at a speed
equal to the anomaly '. It is clear from this that the theory
of eccentrics was invented for the specific purpose of explain-
ing the movements of Mars, Jupiter, and Saturn about the
sun and for that purpose alone. This explanation, combined
with the use of epicycles about the sun as centre to account
for the motions of Venus and Mercury, amounted to the
system of Tycho Brahe ; that system was therefore anticipated
by some one intermediate in date between Heraclides and
Apollonius and probably nearer to the latter, or it may
have been Apollonius himself who took this important step.
If it was, then Apollonius, coining after Aristarchus of
Samos, would be exactly the Tycho lirahe of the Copernicus
of antiquity. The actual propositions quoted by Ptolemy as
proved by Apollonius among others show mathematically at
what points, under each of the two -hypotheses, the apparent
forward motion changes into apparent retrogradation and
vice versa, or the planet appears to be stationfiw/.
XV
THE SUCCESSORS OF THE GREAT GEOMETERS
WITH Archimedes and Apollonius Greek geometry reached
its culminating point. There remained details to be filled
in, and no doubt in a work such as, for instance, the Cuiiics
geometers of the requisite calibre could have found proposi-
tions containing the germ of theories which were capable of
independent development. But, speaking generally, the fur-
ther progress of geometry on general lines was practically
barred by the restrictions of method and form which were
inseparable from the classical Greek geometry. True, it was
opeit to geometers to discover and investigate curves of a
higher order than conies, such as spirals, conchoids, and the
like. Bat the Greeks could not get very far even on these
lines in the absence of some system of coordinates and without
freer means of manipulation such as are afforded by modern
algebra, in contrast to the geometrical algebra, which could
only deal with equations connecting lines, areas, and volumes,
but involving no higher dimensions than three, except in so
far as the use of proportions allowed a very partial exemp-
tion from this limitation. The theoretical methods available
enabled quadratic, cubic and bi-quadratic equations or their
equivalents to be solved. But all the solutions were (jeometri-
eal ; in other words, quantities could only be represented by
lines, areas and volumes, or ratios between them. There was
nothing corresponding to operations with general algebraical
quantities irrespective of what they represented. There were
no symbols for such quantities. In particular, the irrational
was discovered in the form of incommensurable lines ; hence
irrationals came to be represented by straight lines as they
are in Euclid, Book X, and the Greeks had no other way of
representing them. It followed that a product of two irra-
tionals could only be represented by a rectangle, and so on.
Even when Diophantus came to use a symbol for an unkngwii
198 SUCCESSORS OF THE GREAT GEOMETERS
quantity, it was only an abbreviation for the word
with the meaning of ' an undetermined multitude of units ',
not a general quantity. The restriction then of the algebra
employed by geometers to the geometrical form of algebra
operated as an insuperable obstacle to any really new depar-
ture in theoretical geometry.
It might be thought that there was room for further exten-
sions in the region of solid geometry. But the fundamental
principles of solid geometry had also been laid down in Euclid,
Books XI XIII ; the theoretical geometry of the sphere had
been fully treated in the ancient sphaeric ; and any further
application of solid geometry, or of loci in three dimensions,
was hampered by the same restrictions of method which
hindered the further progress of plane geometry.
Theoretical geometry being thus practically at the end of
its resources, it was natural that mathematicians, seeking for
an opening, should turn to the applications of geometry. One
obvious branch remaining to be worked out wavS the geometry
of measurement, or mensuration, in its widest sense, which of
course had to wait on pure theory and to be based on its
results. One species of mensuration was immediately required
for astronomy, namely the measurement of triangles, especially
spherical triangles; in other words, trigonometry plane and
spherical. Another species of mensuration was that in which
an example had already been set by Archimedes, namely the
measurement of areas and volumes of different shapes, and
arithmetical approximations to their true values in cases
where they involved surds or the ratio (IT) between the
circumference of a circle and its diameter ; the object of such
mensuration was largely practical. 0f these two kinds of
mensuration, the first (trigonometry) is represented by Hip-
parchus, Menelaus and Ptolemy ; the second by Heron of
Alexandria. These mathematicians will be dealt with in later
chapters ; this chapter will be devoted to the successors of the
great geometers who worked on the same lines as the latter.
Unfortunately we have only very meagre information as to
what these geometers actually accomplished in keeping up the
tradition. No geometrical works by them have come down
to us in their entirety, and we are dependent on isolated
extracts or scraps of information furnished by commen-
NICOMEDES 199
tators, and especially by Pappus and Eutocius. Some of
these are very interesting, and it is evident from the
extracts from the works of such writers as Diocles and
Dionysodorus that, for some time after Archimedes and
Apollonius, mathematicians had a thorough grasp of the
contents of the works of the great geometers, and were able
to use the principles and methods laid down therein with
ease and skill.
Two geometers properly belonging to this chapter have
already been dealt with. The first is NICOMEDES, the inventor
of the conchoid, who was about intermediate in date between
Eratosthenes and Apollonius. The conchoid lias already been
described above (vol. i, pp. 238-40). It gave a general method
of solving any vtv<Ti$ where ono of the lines which cut off' an
intercept of given length on the line verging to a given point
is a straight line ; and it was used both for the finding of two
mean proportionals and for the trisection of any angle, these
problems being alike reducible to a reCcny of this kind. How
far Nicomedcs discussed the properties of the curve in itself
is uncertain ; we only know from Pappus that ho proved two
properties, (1) that the so-called ' ruler' in the instrument for
constructing the curve is an asymptote, (2) that any straight
lino drawn in the, space between the ' ruler ' or asymptote and
the conchoid must, if produced, bo cut by the conchoid. 1 The
equation of the curve referred to polar coordinates is, as wo
have soon, r = a + b sec 0. According to Eutocius, Nicomedos
prided himself inordinately on his discovery of this curve,
contrasting it with Eratosthenes's mechanism for finding any
number of mean proportionals, to which ho objected formally
and at length on the ground that it was impracticable and
entirely outside the spirit of geometry. 2
^Ni comedos is associated by Pappus with Dinostratus, the
brother of Menaechmus, and others as having applied to the
squaring of the circle the curve invented by Hippias and
known as the qwtdratrir* which was originally intended for
the purpose of trisecting any angle. These facts are all that
we know of Nicornedes's achievements.
1 Pappus, iv, p. 244. 21-8.
2 Kutoc. on Archimedes, On the Spliere and Cylinder, Archimedes,
vol. iii, p. 98.
8 Pappus, iv, pp. 250. 33-252. 4. Cf, vol. i, p. 225 sq.
200 SUCCESSORS OF THE GREAT GEOMETERS
The second name is that of DIOCLES. We have already
(vol. i, pp. 264-6) seen him as the discoverer of the curve
known as the cissoid, which he used to solve the problem
of the two mean proportionals, and also (pp. 47-9 above)
as the author of a method of solving the equivalent of
a certain cubic equation by means of the intersection
of an ellipse and a hyperbola. We are indebted for our
information on both these subjects to Eutocius, 1 who tells
us that the fragments which he quotes came From Diocles's
work ?T/oi wvpefcw, On burning-mirrors. The connexion of
the two things with the subject of this treatise is not obvious,
and we may perhaps infer that it was a work of considerable
scope. What exactly were the forms of the burning-mirrors
discussed in the treatise it is not possible* to say, but it is
probably safe to assume that among them were concave
mirrors in the forms (1) of a sphere, (2) of a paraboloid, and
(3) of the surface described by the revolution of an ellipse
about its major axis. The author of the Frayment'iim nuithe-
maticum Boliense says that Apollonius in his book On lite
burniny-mirror disclosed the case of the concave spherical
mirror, showing about what point ignition would take place ;
and it is certain that Apollonius was aware that an ellipse has
the property of reflecting all rays through one focus to the
other focus. Nor is it likely that the corresponding property
of a parabola with reference to rays parallel to the axis was
unknown to Apollonius. Diodes therefore, writing a century
or more later than Apollonius, could hardly have failed to
deal with all three cases. True, Anthemius (died about
A.D. 534) in his fragment on burning-mirrors says that the
ancients, while mentioning the usual burning-mirrors and
saying that such figures are conic sections, omitted to specify
which conic sections, and how produced, and gave no geo-
metrical proofs of their properties. But if the properties
were commonly known and quoted, it is obvious that they
must have been proved by the ancients, and the explanation
of Anthemius's remark is presumably that the original works
in which they were proved (e.g. those of Apollonius and
Diodes) were already lost when he wrote. There appears to
be no trace of Diocles's work left either in Greek or Arabic,
1 Eutocius. loc. cit.. n. 6G. 8 so., n. 160. 3 so.
DIOCLES 201
unless we have a fragment from it in the
iMithematicum Jiobiense. But Moslem writers regarded Diocles
as the discoverer of the parabolic burning-mirror; 'the ancients',
says al Singrirl (Sachawl, Ansari), * made mirrors of plane
surfaces. Some made them concave (i.e. spherical) until
Diocles (Diuklis) showed and proved that, if the surface of
these mirrors has its curvature in the form of a parabola, they
then have the greatest power and burn most strongly. There
is a work on this subject composed by Ibn al-Haitham/ This
work survives in Arabic and in Latin translations, and is
reproduced by Heiberg and Wiedemann 1 : it does not, how-
ever, mention the name of Diocles, but only those of Archi-
medes and Aiithemius. Urn al-Haitham says that famous
men like Archimedes and Anthemius had used mirrors made
up of a number of spherical rings; afterwards, he adds, they
considered the form of curves which would reflect rays to one
point, and found that the concave surface of a paraboloid of
revolution has this property. It is curious to find Ibn al-
Haitham saying that the ancients had not set out the proofs
sufficiently, nor the method by which they discovered them,
words which almost exactly recall those of Antheniius himself.
Nevertheless the whole course of Ibn al-Haitham's proofs is
on the CJreek model, Apollonius being actually quoted by name
in tin* proof of the main property of the parabola required,
namely that the tangent at any point of the curve makes
equal angles with the focal distance of the, point and the
straight line, drawn through it parallel to the axis. A proof
of the property actually survives in the (ireek Fratjinentiifii
mtithmnitwimi Hohicn^e, which evidently came from some
treatise on the parabolic burning-mirror; but Ibn al-Haitham
does not seem to have had even this fragment at his disposal,
since his proof takes a different course, distinguishing three
different cases, reducing the property by analysis to the
known property A X = A 2\ and then working out the syn-
thesis. Tim proof in the Fnuj'nwntutH is worth giving. It is
substantially as follows, beginning with the preliminary lemma
that, if PT } the tangent at any point P, meets the axis at r l\
and if AM be measured along the axis from the vertex A
equal to %AL, where A L is the parameter, then &P = ST.
1 Bibliotheca mathematics, x 3 , 1910, pp. 201-37,
202 SUCCESSORS OF THE GREAT GEOMETERS
Let PN be the ordinate from P; draw AY at right angles
to the axis meeting PT in Y, and join HY.
Now PN*
= 4 AH. AN
= 4 AH. AT (since AN =
But PN = ZAY (since 4JV = AT) ;
therefore 4 F 2 = TA . AH,
and the angle TYH is right.
The triangles 8YT, HYP being right-angled, and 2T being
equal to YP, it follows that SP = AT.
With the same figure, let #7' be a ray parallel to AX
impinging on the curve at P. It is required to prove that
the angles of incidence and reflection (to H) are equal.
We have HP = &1\ so that < the angles at the points 7 f , P
are equal. So', says the author, 'arc the angles TPA, KPli
[the angles between the tangent and the curve, on each side, of
the point of contact]. Let the difference between the angles
be taken; therefore the angles HP A, RPIt which remain
[again * mixed* angles] are equal. Similarly we shall show
that all the lines drawn parallel to AH will be reflected at
equal angles to the point H.'
The author then proceeds: 'Thus burning-mirrors con-
structed with the surface of impact (in the form) of the
section of a right-angled cone may easily, in the manner
DIOCLES. PERSEUS 203
above shown, be proved to bring about ignition at the point
indicated/
Heiberg held that the style of this fragment is By/antine
and that it is probably by Anthemius. Cantor conjectured
that here we might, after all, have an extract from Diocles's
work. Heiberg's supposition seems to me untenable because
of the author's use (1) of the ancient terms 'section of
a right-angled cone' for parabola and 'diameter' for axis
(to say nothing of tin* use of the parameter, of which there is
no word in the genuine fragment of Anthemius), and (2) of
the mixed 'angles of contact*. Nor does it seem likely that
even Diodes, living a century after Apollonius, would have
spoken of the 'section of a right-angled cone' instead of a
parabola, or used the 4 mixed ' angle of which there is only the
merest survival in Euclid. The assumption of the equality
of the two angles made by the curve with the tangent on
both sides of the point of contact reminds us of Aristotle's
assumption of the equality of the angles * nf a segment 1 of
a eirde as prior to the truth proved in Eucl. I. 5. I am
inclined, therefore, to date the fragment much earlier oven
than Diodes. Zeuthen suggested that the property of the
paraholoidal mirror may him? been discovered by Archimedes,
who, according to a Greek tradition, wrote (\ito^tncn. This,
however, does not receive any confirmation in Ibn al-Haitham
or in Anthemius, and we can only say that the fragment at
least goes back to an original which was probably not later
than 'Apollonius.
PKUSKI'S is only known, from allusions to him in Proclus, 1
as the. discoverer mid investigator of the >/>/?'/< wet in tit*. They
are classed by Prod us among curves obtained by cutting
solids, and in this respect they are associated with the conic
sections. We may safely infer that they were discovered
after the conic sections, and only after the theory of conies
had been considerably developed. This was already the case
in Euclid's time, and it is probable, therefore, that Perseus was
not earlier than Euclid. On the other hand, by that time
the investigation of conies had brought the exponents of the
subject such fame that it would be natural for mathematicians
to see whether there was not an opportunity for winning a
1 Proclus on Eucl. I, pp. 111. 23-112. 8, 356. 12. Cf. vol. i, p. 226.
204 SUCCESSORS OF THE GREAT GEOMETERS
like renown as discoverers of other curves to be obtained by
cutting well-known solid figures other than the cone and
cylinder. A particular case of one such solid figure, the
orrerpa, had already been employed by Archytas, and the more
general form of it would not unnaturally be thought of as
likely to give sections worthy of investigation. Since Geininus
is Proclus's authority, Perseus may have lived at any date from
Euclid's time to (say) 75 B.C., but the most probable supposi-
tion seems to be that he came before Apollonius and near to
Euclid in date.
The spire in one of its forms is what we call a tore, or an
anchor-ring. It is generated by the revolution of a circle
about a straight line in its plane in such a w$ty that the plane
of the circle always passes through the axis of revolution. It
takes three forms according as the axis of revolution is
(a) altogether outside the circle, when the .spire is open
(8i\r)s), (b) a tangent to the circle, when the .surface, is con-
tinuous ((rvv*\ri$), or (c) a chord of the circle, when it is inter-
laced (c/ZTreTrAcy/zli/Ty), or crossing- itself (cTraAAarrovo-a) ; an
alternative name for the surface was *pt'/coy, a riiuj. Perseus
celebrated his discovery in an epigram to the effect that
c Perseus on his discovery of three lines (curves) upon five
sections gave thanks to the gods therefor'. 1 There is omo
doubt about the meaning of * three JinevS u/xm five sections'
(rpcf? ypa/jipas TTI rrli/re Topa^s). We gather from Proclus's
account of three sections distinguished by Perseus that the
plane of section was always parallel to the axis of revolution
or perpendicular to the plane which cuts the tore symmetri-
cally like the division in a split-ring. It is difficult to inter-
pret the phrase if it means three curves made by five different
sections. Proclus indeed implies that the three curves were
sections of the three kinds of tore respectively (the open, the
closed, and the interlaced), but this is evidently a slip.
Tannery interprets the phrase as meaning 'three curves in,
addition to five sections '. a Of these the five sections belong
to the open tore, in which the distance (c) of the centre of the
generating circle from the axis of revolution is greater than
the radius (a) of the generating circle. If d be the perpen-
1 Proclus on Eucl. I, p. 112. 2.
2 See Tannery, Mtmoirett scientijiques, II, pp. 24-8.
PERSEUS 205
dicular distance of the plane of section from the axi*s of rota-
tion, we can distinguish the following cases :
(1) c + a>d>c. Here the curve is an oval.
(2) d = c: transition from case (1) to the next case.
(3) c>J>c a. The curve is now a closed curve narrowest
in the middle.
(4) eZ = r ft. In this case the curve is the hipi>opede
(horse-fetter), a curve in the shape of the figure, of 8. The
lemniscate of Bernoulli is a particular case of this curve, that
namely in which c = 2a.
(5) r a>iZ>0. In this case the section consists of two
ovals symmetrical with one another.
The three curves specified l>y Prcx-lus are those correspond-
ing to (1), (3) and (4).
When the ton* is 'continuous 7 or closed, c = , and we have
sections corresponding to (1), (2) and (3) only; (4) reduces to
two cireles touching one another.
But Tannery finds in the third, the interlaced, form of tore
three new sections corresponding to(l) (2) (3), each with an
oval in the middle. This would make three curves in addi-
tion to the five sections, or eight curves in all. We cannot he
certain that this is the true explanation of the phrase in the
epigram: but it seems to l>e the l>est suggestion that has
been made.
According to Proelus, Perseus worked out the property of
his curves, as Nicomedes did that of the conchoid, Hippias
that of the ij-mulrutrix, and Apollonius those of the three
conic sections. That is, Perseus must have given, in some
form, the equivalent of the Cartesian equation by which we
can represent the different curves in question. If we refer the
tore to three axes of coordinates at right angles to one another
with the, centre of the tore as origin, the axis of y being taken
I/O be the axis of revolution, and those of z, sc being perpen-
dicular to it in the plane bisecting the tore (making it a split-
ring), the equation of the tore is
206 SUCCESSORS OF THE GREAT GEOMETERS
where c, a have the same meaning as above.. The different
sections parallel to the axis of revolution are obtained by
giving (say) z any value between and c + a. For the value
z = a the curve is the oval of Cassini which has the property
that, if r, r' be the distances of any point on the curve from
two fixed points as poles, rr'= const. For, if z' = a, the equa-
tion becomes
(#* + y* + c 2 )- = 4 c a a 2 + 4 c 2 a 2 ,
or Jc a"
and this is equivalent to rr'= + 2ra if ./,?/ are the coordinates
of any point on the curve referred to Ox, Oy as axes, whew
is the middle point of the line (2v in length) joining the two
poles, and Ox lies along that line in either direction, while Oy
is perpendicular to it. Whether Perseus discussed this case
and arrived at the property in relation to the two poles is of
course quite uncertain.
Isoperirnetric figures.
The subject of isoperimetric figures, that is to say, the com-
parison of the areas of figures having different shapes but the
same perimeter, was one which would naturally appeal to the
early Greek mathematicians. We gather from Proclus's notes
on Eucl. I. 36, 37 that those theorems, proving that parallelo-
grams or triangles on the same or equal bases and between
the same parallels are equal in area, appeared to the ordinary
person paradoxical because they meant that, by moving the
side opposite to the base in the parallelogram, or the vertex
of the triangle, to the right or left as far as we please, we may
increase the perimeter of the figure to any extent while keep-
ing the same area. Thus the perimeter in parallelograms or
triangles is in itself no criterion as to their area. Misconcep-
tion on this subject was rife among non-mathematicians.
Proclus tells us of describers of countries who inferred
the size of cities from their perimeters; he mentions also
certain members of communistic societies in his own time who
cheated their fellow-members by giving them land of greater
perimeter but less area than the plots which they took
ISOPERIMETRIC FIGURES. ZENODORUS 207
themselves, so that, while they got a reputation for greater
honesty, they in fact took more than their share of the
produce. 1 Several remarks by ancient authors show the
prevalence of the same misconception, Thucydides estimates
the size of Sicily according to the time required for circum-
navigating it. 2 About 130 B.C. Poly bias observed that there
were people who could not understand that camps of the same
periphery might have different capacities. 3 Quintilian has a
similar remark, and Cantor thinks he may have had in his
mind the calculations of Pliny, who compares the size of
different parts of the earth by adding their lengths to their
breadths. 4
/ENonours wrote, at some date lx*tween (Ray) 200 B.C. and
A.I). 90, a treatise ncpi iero/*rpan> o-x^ara)!/, On isometric
jiyure*. A number of propositions from it are preserved in
the commentary of Theon of Alexandria on Book I of
Ptolemy's tfyutaxix; and they are reproduced in Latin in the
third volume of Hultsch's edition of Pappus, for the purpose
of comparison with Pappus's own exposition of the same
propositions at the beginning of his B(x>k V, where he appears
to have followed Zenodorus pretty elosely while making some
changes in detail. 5 From the closeness with which the style
of Zenodorus follows that of Euclid and Archimedes we may
judge that his date was not much later than that of Archi-
medes, whom he mentions as the author of the proposition
(Afeamrenient <tf (Urcle, Prop. 1) that the area of a circle i*
half that of the rectangle contained by the perimeter of the
circle and its radius. The important propositions proved by
Zenodorus and Pappus include the following: (1) Of all
reyiUar jMilytjon* of equal jierimcter, that is the greatest in
area wlwh /m,x the tno^t unifies. (2) A circle is greater tlian
any regular 'polyyini of equal contour. (3) Of all polygons of
the same number of sides and equal perimeter tlw equilateral
and eqummjular jxtfygon is the greatest hi area. Pappus
added the further proposition that Of all seynwttis of a circle
having the same circumference the semicircle is the greatest in
1 Proclus on Eucl. I, p. 403. Tisq. 2 Thuc. vi. 1.
3 Polybius, ix. 21. 4 Pliny, Hist. vat. vi. 208.
5 Pappus, v, p. 308 sq.
208 SUCCESSORS OF THE GREAT GEOMETERS
area. Zenodorus's treatise was not confined to propositions
about plane figures, but gave also the theorem that Of all
solid figures the surfaces of which are equal, the sphere is the
greatest in solid content.
We will briefly indicate Zenodorus's method of proof. To
begin with (1) ; let ABC, DBF be equilateral and equiangular
polygons of the same perimeter, DEF having more angles
than ABC. Let G, H be the centres of the circumscribing
circles, OK, HL the perpendiculars from <z f // to the sides
AB, DE y so that A', L bisect those sides.
AM
K
Since the perimeters are equal, AB > l)E t and AK > DL.
Make KM equal to J)L and join GM.
Since AB is the same fraction of the perimeter that the
angle AGB is of four right angles, and 1)E is the same fraction
of the same perimeter that the angle DUE is of four right
Angles, it follows that
that is, AK : MK= LAGK-.L DHL.
But AK:MK > LAGK\LMGK
(this is easily proved in a lemma following by the usual
method of drawing an arc of a circle with G as centre and GM
as radius cutting GA and GK produced. The proposition is of
course equivalent to tail a/ tan |9 > <x/j8, where \TT > a > ft).
Therefore Z MGK > Z DHL,
and -consequently Z GM K < Z HDL.
Make the angle NMK equal to the angle HDL, so that MN
meets KG produced in N.
ZENODORUS 209
The triangles NMK, HDL are now equal in all respects, and
NK is equal to IIL, so that OK < HL.
But the area of the polygon ABC is half the rectangle
contained by GK and the perimeter, while the area of the
polygon DEF is half the rectangle contained by HL and
the same perimeter. Therefore the area of the polygon DEF
is the greater.
(2) The proof that a circle is greater than any regular
polygon with the same perimeter is deduced immediately from
Archimedes'** proposition that the area of a circle is equal
to the right-angled triangle with perpendicular side equal to
the radius and base equal to the perimeter of the circle;
Zenodorus inserts a proof in exteiiso of Archimedes's pro-
position, with preliminary lemma. The perpendicular from
the centre of the circle circumscribing the polygon is easily
proved to IK; less than the radius of the given circle with
perimeter equal to that of the polygon : whence the proposition
follows.
(3) The proof of this proposition depends on some pre-
liminary lemmas. The first proves that, if there be two
triangles on the same base and with the
puinc perimeter, one being isosceles and
the other scalene, the isosceles triangle
has the greater area. (Given the scalene
triangle BDC on the base B(\ it is easy to
draw on BC as base the isosceles triangle
having the same perimeter. We have
only to take BU equal to (/M)+/>f f ).
bisect li(! at fa\ and erect at K the per-
pendicular AK such that AK* = Bll*-BK\)
Produce BA to Fso that BA = AF, and join AD, DF.
Then Jil) + DF> BF< i.e. BA +A( 1 , i.e. BD + DC, by hypo-
thesis; therefore DF > DC, whence in the triangles FAD,
CAD the angle FA I) > the angle CAD.
Therefore L FA 1) > \ L FA ( '
> LBV A.
Make the angle FAG equal to the angle BCA or ABC, so
that AG is parallel tojBC; let BD produced meet AG in G,
and join GC.
1BSS.S P
210 SUCCESSORS OF THE GREAT GEOMETERS
Then
> &DBC.
The second lemrna is to the effect that, given two isosceles
triangles not similar to one another, if we construct on the
same bases two triangles similar to one another such that the
sum of their perimeters is equal to the sum of the perimeters
of the first two triangles, then the sum of the areas of the
similar triangles is greater than the sum of the areas of
the non-similar triangles. (The easy construction of the
similar triangles is given in a separate lemma.)
Let the bases of the isosceles triangles, KB, BC, bo placet I in
one straight line, BC being greater than KB.
Let ABC, DKB be the similar isosceles triangles, and FttC,
GKB the non-similar, the triangles being such that
BA+AC + ED + DB = BF+ FC+EG + GB.
Produce AF, GD to meet the bases in K, L. Then clearly
AK, GL bisect BC, EB at right angles at K, L.
Produce GL to //, making LH equal to GL.
Join HB and produce it to N\ join I IF.
Now, since the triangles ABC, DEB are similar, the angle
ABC is equal to the angle DEB or DBE.
Therefore Z NBC ( = L HBE = Z GBE) > Z DBE or Z A BC ;
therefore the angle ABH, and a fortiori the angle FBH, is
less than two right angles, and HF meets B K in some point M.
ZENODORUS 211
Now, by hypothesis, DB + BA = GB + BF:
therefore DB + BA = H B + BF> HF.
By an easy lemma, since the triangles DEB, A BC are similar,
(DB + BA) 2 = (DL + AK)* + (BL + BK)*
Therefore (DL + AK)* + LK* :> HF*
whence DL + A K > GL + FK,
and it follows that AF > GD.
But BK > BL] therefore AF.BK > GD.BL.
Hence the * hollow-angled (figure)' (KoiXoywvtov) ABFC is
greater than the hollow-angled (figure) (fEDB.
Adding A DEB + & fi/*V to each, we have
A DKH + & A B( 1 > A HEB+A FBC.
The above is the only case taken by Zenodorus. The proof
still holds if Kit = B(\ so that BK = BL. But it fails in the
case in which EB > IK 1 and the vertex G of the triangle EB
belonging to the non-similar pair is still above /) and not
below it (as F is below A in the preceding case). This was
no doubt the reason why Pappus gave a pnx)f intended to
apply to all the cases without distinction. This proof is the
same as the above proof by Zenodorus up to the point where
it is proved that
I>L + AK > GL + FK,
but then* diverges. Unfortunately the text is bad, and gives
no sufKcient indication of the course of the proof; but it would
seem that Pappus used the relations
DL : GL = A DEB : A GEB,
AK : FK = AABC:AFB(\
and Al\*:DL t2 =AAB(':&DEB,
combined of course with the fact that GB+ BF = DB + BA,
in order to prove the proposition that,
according as DL + AK > or < GL + FK,
A DEB + & ABC > or < A GEB+& FBC.
212 SUCCESSORS OF THE GREAT GEOMETERS
The proof of his proposition, whatever it was, Pappus
indicates that he will give later ; but in the text as we have it
the promise is not fulfilled.
Then follows the proof that the maximum polygon of given
perimeter is both equilateral and
equiangular.
(1) It is equilateral.
For, if not, let two sides of the
maximum polygon, as AB, lid, be
unequal. Join A(\ and on AC as
base draw the isosceles triangle AFC
such that AF+ FC= AB + BC. The
area of the triangle AFC is then
greater than the area of the triangle ARC, and the area of
the whole polygon has been increased by the construc-
tion: which is impossible, as by hypothesis the area is a
maximum.
Similarly it can be proved that no other side is unequal
to any other.
(2) It is also equiangular.
For, if possible, let the maximum polygon ABCJ)E (which
we have proved to be equilateral)
have the angle at B greater than
the angle at I). Then BAC, DECmv
non-similar isosceles triangles. On
AC, CE as bases describe the two
isosceles triangles FAG, GKC similar
to one another which have the sum
of their perimeters equal to the
sum of the perimeters of BA(\
DEC. Then the sum of the areas of the two similar isosceles
triangles is greater than the sum of the areas of the triangles
BAC y DEC] the area of the polygon is therefore increased,
which is contrary to the hypothesis.
Hence no two angles of the polygon can be unequal.
The maximum polygon of given perimeter is therefore both
equilateral and equiangular.
Dealing with the sphere in relation to other solids having
ZENODORUS. HYPSICLES 213
their surfaces equal to that of the sphere, Zenodorus confined
himself to proving (1) that the sphere is greater if the other
solid with surface equal to that of the sphere is a solid formed
by the . revolution of a regular polygon about a diameter
bisecting it as in Archimedes, On the Sphere and (lylinder,
Book I, and (2) that the sphere is greater than any of
the regular solids having its surface equal to that of the
sphere.
Pappus's treatment of the subject is more complete in that
he proves that the sphere is greater than the cone or cylinder
the surface of which is equal to that of the sphere, and further
that of the five regular solids which have the same surface
that which has more faces is the greater. 1
HYPSICLKS (second half of second century B.C.) has already
been mentioned (vol. i, pp. 419 20) as the author of the con-
tinuation of the JJlententa known as 15ook XIV. Me is quoted
by Diophantus as having given a definition of a polygonal
number as follows:
^
4 If there are as many numbers as we please beginning from
1 and increasing by the same common difference, then, when
the common difference is 1, the sum of all the numbers in
a triangular number ; when 2, a square : when 3, a pentagonal
number [and so on]. And the number of angles is called
after the number which exceeds the common difference by 2,
and the side after the number of terms including 1.'
This definition amounts to saying that the /<th a-gonal num-
ber (1 counting as the first) is -|/i ; 2 + (u-- 1 ) (a 2) \. If, as is
probable, Hypsicles wrote a treatise on polygonal numbers, it
has not survived. On the other hand, the 'AvafyopiKos (Ascen-
xiones) known by his name has survived in Greek as well as in
Arabic, and has been edited with translation. 2 True, the
treatise (if it really be by Hypsicles, and not a clumsy effort
by a beginner working from an original by Hypsicles)
does no credit to its author; but it is in some respects
interesting, and in particular because it is the first Greek
1 Pappus, v, Props. 19, 38-56.
2 Manitius, /Ms Ht/psikli's tfchrift Anaphorikos, Dresden, Lehmannsche
Buchdruekerei, 1888.
214 SUCCESSORS OF THE GREAT GEOMETERS
work in which we find the division of the ecliptic circle into
360 ' parts ' or degrees. The author says, after the preliminary
propositions,
* The circle of the zodiac having been divided into 360 equal
circumferences (arcs), let each of the latter be called a degree
in space (polpa TOTTIKTJ, ' local ' or ' spatial part '). And simi-
larly, supposing that the time in which the zodiac circle
returns to any position it has left is divided into 360 equal
times, let each of these be called a degree in time (fioipa
From the word KaXeiada) (' let it be called ') we may perhaps
infer that the terms were new in Greece. This brings us to
the question of the origin of the division (1) of the circle of
the zodiac, (2) of the circle in general, into 360 parts. On this
question innumerable suggestions have been made. With
reference to (1) it was suggested as long ago as 1788 (by For-
maleoiii) that the division was meant to correspond to the
number of days in the year. Another suggestion is that it
would early be discovered that, in the case of any circle the
inscribed hexagon dividing the circumference into six parts
has each of its sides equal to the radius, and that this would
naturally lead to the circle being regularly divided into six
parts ; after this, the very ancient sexagesimal system would
naturally come into operation and each of the parts would be
divided into 60 subdivisions, giving 360 of these for the whole
circle. Again, there is an explanation which is not even
geometrical, namely that in the Babylonian numeral system,
which combined the use of 6 and 10 as base**, the numbers 6,
60, 360, 3600 were fundamental round numbers, and these
numbers were transferred from arithmetic to the heavens.
The obvious objection to the first of these explanations
(referring the 360 to the number of days in the solar year) is
that the Babylonians were well acquainted, as far back as the
monuments go, with 365-2 as the number of days in the year.
A variant of the hexagon-theory is the suggestion .that a
natural angle to be discovered, and to serve as a measure of
others, is the angle of an equilateral triangle, found by draw-
ing a star * like a six-spoked wheel without any circle. If
the base of a sundial was so divided into six angles, it would be
HYPSICLES 215
natural to divide each of the sixth parts into either 10 or 60
parts; the former division would account for the attested
division of the day into 60 hours, while the latter division on
the sexagesimal system would give the 360 time-degrees (each
of 4 minutes) making up the day of 24 hours. The purely
arithmetical explanation is defective in that the series of
numbers for which the Babylonians had special names is not
60, 360, 3600 but 60 (Soss), COO (Ner), and 3600 or 60 2 (Sar).
On the whole, after all that has been said, I know of no
better suggestion than that of Tannery. 1 It is certain that
both the division of the ecliptic into 360 degrees and that of
the vv\Qrinepov into 360 time-degrees were adopted by the
Greeks from Babylon. Now the earliest division of the
ecliptic was into 12 parts, the signs, and the question is, how
were the signs subdivided? Tannery observes that, accord-
ing to tho cuneiform inscriptions, as well as the testimony of
Greek authors, the sign was divided into parts one of which
(tlarfjatu) was double of the other (wwnm), the former being
l/30th, the other (called stadium by Manilius) l/60th, of the
sign ; the former division would give 360 parts, the latter 720
parts for the whole circle. The latter division was more
natural, in view of the long-established system of sexagesimal
fractions; it also had the advantage of corresponding toler-
ably closely to the apparent diameter of the sun in comparison
with the circumference of the sun's apparent circle. But, on
the other hand, the double fraction, the l/30th, was contained
in the circle of the zodiac approximately the same number of
times as there are days in the year, and consequently corre-
sponded nearly to the distance described by the sun along the
xodiac in one day. It would seem that this advantage was
sufficient to turn the scale in favour of dividing each sign of
the zodiac into 30 parts, giving 360 parts for the whole
circle. While the Chaldaeans thus divided the ecliptic into
360 parts, it does not appear that they applied the same divi-
sion to the equator or any other circle. They measured angles
in general by dls y an ell representing 2, so that the complete
circle contained 180, not 360, parts, which they called ells.
The explanation may perhaps be that the Chald&eans divided
1 Tannery. ' La coudee aatronomique et les anciennes divisions du
cercle ' (IKmoircs scieHtifiquea, ii, pp. 256-68).
216 SUCCESSORS OF THE GREAT GEOMETERS
the diameter of the circle into 60 ells in accordance with their
usual sexagesimal division, and then came to divide the cir-
cumference into 180 such ells on the ground that the circum-
ference is roughly three times the diameter. The measure-
ment in ells and dactyli (of which there were 24 to the ell)
survives in Hipparchus (On the Phaeiiometia ofEudoxus and
Aratus), and some measurements in terms of the same units
are given by Ptolemy. It was Hipparchus who first divided
the circle in general into 360 parts or degrees, and the
introduction of this division coincides with his invention of
trigonometry.
The contents of Hypsicles's tract need not detain us long.
The problem is : If we know the ratio which the length of the
longest day bears to the length of the shortest day at any
given place, to find how many time-degrees it takes any given
sign to rise ; and, after this has been found, the author tinds
what length of time it takes any given degree in any sign to
rise, i.e. the interval between the rising of one degree-point on
the ecliptic and that of the next following. It is explained
that the longest clay is the time during which one half of the
zodiac (Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius) rises,
and the shortest day the time during which the other half
(Capricornus, Aquarius, Pisces, Aries, Taurus, Gemini) rises.
Now at Alexandria the longest day is to the shortest as 7
to 6; the longest therefore contains 210 ' time-degrees', the
shortest 150. The two quadrants Cancer- Virgo and Libra-
Sagittarius take the same time to rise, namely 105 time-
degrees, and the two quadrants Capricornus -Pisees and Aries-
Gemini each take the same time, namely 75 time-degrees.
It is further assumed that the times taken by Virgo, Leo,
Cancer, Gemini, Taurus, Aries are in descending arithmetical
progression, while the times taken by Libra, Scorpio, Sagit-
tarius, Capricornus, Aquarius, Pisces continue the same de-
scending arithmetical series. The following lemmas are used
and proved :
I. If a ls a 2 ...a n , a w + 1 , a n+2 ... a 2n is a descending arithmeti-
cal progression of 2n terms with 8 (= a^a^ = a 2 -a 3 = ...)
as common difference,
HYPSICLES 217
II. If MJ, rt 2 '-- a n-*- a 2n-i * s a Descending arithmetical pro-
gression of 2 /i-- 1 terms with 5 as common difference and a n
is the middle term, then
i + 2 + ... + 2n -i = (2u l)a n .
III. If a T , tt a ...a n , ^ n+1 ...a 2n is a descending arithmetical
progression of 2 jt, terms, then
Now let ;1, 5, (/ ho the descending series the sum of which
is 105, and /), K, Fthe next three terms in the same series
the sum of which is 75, the common difference being <S; we
then have, by (1),
-iD + K+F) = 9(5, or 30 = 95,
and 5 = 3$ .
Next, by (II), ^l + 5+f f =35. or 35 = 105, and 5 = 35:
therefore A, B, (', I), K, /'are equal to 38$, 35, 31|, 28$, 25,
21-f time-degrees respectively, which tlie author of the tract
expresses in time-degrees and minutes as 38' 20', 35', 31' 40',
28' 20', 25', 21' -10'. We have now to carry through the same
procedure for each degree in each sign. If the difference
between the times taken to rise by one sign and the next
is 3' 20', what is the difference for each of the 30 degrees in
the sign? We have here 30 terms followed by 30 other terms
of the same descending arithmetical progression, and the
formula (I) gives 3' . 20' = (30) a </, where d is the common
difference ; therefore d = J x 3' . 20'= 0' 0' 1 3" 20'". Lastly,
take the sign corresponding to 21' 40'. This is the sum of
a descending arithmetical progression of 30 terms a lt 2 ... a. w
with common difference 0' 0' 13" 20'". Therefore, by (III),
21' 40'= IS^ + ajJ, whence a^a^^ l' 26' 40". Now,
since there are 30 terms a p a, 2 ... a 30 , we have
04-0^= 29(2 = 0' 6' 26" 40'".
It follows that rtgo = 0' 40' 6" 40'" and a t = 0' 46' 33" 20"',
218 SUCCESSORS OF THE GREAT GEOMETERS
and' from these and the common difference 0< 0'13"20'" all
the times corresponding to all the degrees in the circle can be
found,
The procedure was probably, as Tannery thinks, taken
direct from the Babylonians, who would no doubt use it for
the purpose of enabling the time to be determined at any
hour of the night. Another view is that the object was
astrological rather than astronomical (Manitius). In either
case the method was exceedingly rough, and the assumed
increases and decreases in the times of the risings of the signs
in arithmetical progression are not in accordance with the
facts. The book could only have been written before the in-
vention of trigonometry by Hipparchus, for the problem of
finding the times of rising of the signs is really one of
spherical trigonometry, and these times were actually cal-
culated by Hipparchus and Ptolemy by means of tables of
chords.
DIONYSODORUS is known in the first place as the author of
a solution of the cubic equation subsidiary to the problem of
Archimedes, On the Spliere and Cylinder, II. 4, To cut a given
sphere by a plane so that the volumes of the segments have to
one another a given ratio (see above, p. 46). Up to recently
this Dionysodorus was supposed to be Dionysodorus of Amisene
in Pontus, whom Suidas describes as ' a mathematician worthy
of mention in the field of education'. But we now learn from
a fragment of the Herculaneum Roll, No. 1044, that ' Philonides
was a pupil, first of Eudeinus, and afterwards of Dionysodorus,
the son of Dionysodorus the Caunian \ Now Eudeiuus is
evidently Eudemus of Pergamum to whom Apolloriiu.s dedi-
cated the first two Books of his Co ivies, and Apollonius actually
asks him to show Book II to Philonides. In another frag-
ment Philonides is said to have published some lectures by
Dionysodorus. Hence our Dionysodonis may be Dionysodorus
of Caunus and a contemporary of Apollonius, or very little
later. 1 A Dionysodorus is also mentioned by Heron 2 as the
author of a tract On the Spire (or tore) in which he proved
that, if d be the diameter of the revolving circle
1 W. Schmidt in BiUiotheca mathematics iv s , pp. 321-5.
8 Heron, Metrica, ii. 13, p. 128. 3.
DIONYSODORUS 219
generates the tore, and c the distance of its centre from the
axis of revolution,
(volume of tore) : nc 2 . d = %ird* : ^ccZ,
that is, (volume of tore) = %n 2 . cd*,
which is of course the product of the area of the generating
circle and the length o the path of its centre of gravity. The
form in which the result is stated, namely that the tore is to
the cylinder with height (/ and radius c as the generating
circle of the tore is to half the parallelogram cd t indicates
quite clearly that Dionysodorus proved his result by the same
procedure as that employed by Archimedes in the Method and
in the book On Conoids and Spheroids \ and indeed the proof
on Archimedean lines is not difficult.
Before passing to the mathematicians who are identified
with the discovery and development of trigonometry, it will
be convenient, I think, to dispose of two more mathematicians
belonging to the last century B.C., although this involves
a slight departure. from chronological order ; I mean Posidonius
and Geminus.
POSIDONIUS, a Stoic, the teacher of Cicero, is known as
Posidonius of Apamea (where, he was born) or of Rhodes
(where he taught) ; his date may be taken as approximately
135-51 iu_'. In pure mathematics he is mainly quoted as the
author of certain definitions, or for views on technical terms,
e.g. 'theorem' and 'problem', and subjects belonging to ele-
mentary geometry. More important were his contributions
to mathematical geography and astronomy. He gave his
great work on geography the title On the Ocean, using the
word which had always had such a fascination for the Greeks ;
its contents are, known to us through the copious quotations
from it in Strabo; it dealt with physical as well as mathe-
matical geography, the zones, the tides and their connexion
with the moon, ethnography and all sorts of observations made
during extensive travels. His astronomical book bore the
title Meteoroloyica or irepl /Lierea>pa>i>, -and, while Geminus
wrote a commentary on or exposition of this work, we may
assign to it a number of views quoted from Posidonius in
220 SUCCESSORS OF THE GREAT GEOMETERS
Cleomedes's work De motu circulars corporum caelestium.
Posidonius also wrote a separate tract on the size of the sun.
The two things which are sufficiently important to deserve
mention here are (1) Posidonius's measurement of the circum-
ference of the earth, (2) his hypothesis as to the distance and
size of the sun.
(1) He estimated the circumference of the earth in this
way. He assumed (according to Cleomedes ] ) that, whereas
the star Canopus, invisible in Greece, was just seen to graze the
horizon at Rhodes, rising and setting again immediately, the
meridian altitude of the same star at Alexandria was * a fourth
part of a sign, that is, one forty-eighth part of the zodiac
circle' ( = 7^); and he observed that the distance between
the two places (supposed to lie on the same meridian) * was
considered to be 5,000 stades'. The circumference of the
earth was thus made out to be 240,000 stades. Unfortunately
the estimate of the difference of latitude, 7^, was very far
from correct, the true difference being 5^ only ; moreover
the estimate of 5,000 stades for the distance was incorrect,
being only the maximum estimate put upon it by mariners,
while some put it at 4.000 and Eratosthenes, by observations
of the shadows of gnomons, found it to be 3,750 studes only.
Strabo, on the other hand, says that Posidonius favoured ' the
latest of the measurements which gave the smallest dimen-
sions to the earth, namely about 180,000 stades 1 . 2 This is
evidently 48 times 3,750, so that Posidonius combined Erato-
sthenes's figure of 3,750 stades with the incorrect estimate
of 7| for the difference of latitude, although Eratosthenes
presumably obtained the figure of 3,750 stades from his own
estimate (250,000 or 252,000) of the circumference of the earth
combined with an estimate of the difference of latitude which
was about 5f and therefore near the truth.
(2) Cleomedes rj tells us that Posidonius supposed the circle
in which the sun apparently moves round the earth to be
10,000 times the size of a circular section of the earth through
its centre, and that with this assumption he combined the
1 Cleomedes, De motu circular}, i. 10, pp. 92-4.
2 Strabo, ii. c. 95.
8 Cleomedes, op. cit. ii. 1, pp. 144-6, p. 98. 1-5.
POSIDONIUS 221
statement of Eratosthenes (based apparently upon hearsay)
that at Syene, which is under the summer tropic, and
throughout a circle round it of 300 stades in diameter, the
upright gnomon throws no shadow at noon. It follows from
this that the diameter of the sun occupies a portion of the
sun's circle 3,000,000 stades in length ; in other words, the
diameter of the sun is 3,000,000 stades. The assumption that
the sun's circle is 10,000 times as large as a great circle of the
earth was presumably taken from Archimedes, who had proved
in the tfaitd-reckoiwr that the diameter of the sun's orbit is
le#8 than 10,000 times that of the earth; Posidonius in fact
took the maximum value to be the true value ; but his esti-
mate of the sun's size is far nearer the truth than the estimates
of Aristarchus, Hipparchus, and Ptolemy. Expressed in terms
of the mean diameter of the earth, the estimates of these
astronomers give for the diameter of the sun the figures 6|,
12^, and 5J respectively; Posidonius's estimate gives 39J, the
true figure being 108-9.
In elementary geometry Posidonius is credited by Proclus
with certain definitions. He defined 'figure' as 'confining
limit' (ntpas <rvyK\ioi>) } mid 'parallels' as 'those lines which,
being in one plane, neither converge nor diverge, but have all
the perpendiculars equal which are drawn from the points of
one line to the other'.- (Both these definitions are included
in the Dejinitio'HN of Heron.) He also distinguished seven
species of quadrilaterals, and had views on the distinction
between theorem and problem. Another indication of his
interest in the fundamentals of elementary geometry is the
fact 3 that he wrote a separate work in refutation of the
Epicurean Zeno of Sidon, who had objected to the very begin-
nings of the Elements on the ground that they contained un-
proved assumptions. Thus, said Zeno, even Eucl.1. 1 requires it
to be admitted that f two straight lines cannot have a common
segment ' ; and, as regards the ' proof ' of this fact deduced
from the bisection of a circle by its diameter, he would object
that it has to be assumed that two arcs of circles cannot have
a common part. Zeno argued generally that, even if we
admit the fundamental principles of geometry, the deductions
1 Proclus on Eucl. I, p. 143. 8. 2 /&., p. 176. 6-10.
3 76., pp. 199. 14-200. 3.
222 SUCCESSORS OF THE GREAT GEOMETERS
from them cannot be proved without the admission of some-
thing else as well which has not been included in the said
principles, and he intended by means of these criticisms to
destroy the whole of geometry. 1 We can understand, there-
fore, that the tract of Posidonius was a serious work.
A definition of the centre of gravity by one ' Posidonius a
Stoic ' is quoted in Heron's Mechanics, but, as the writer goes
on to say that Archimedes introduced a further distinction, we
may fairly assume that the Posidonius in question is not
Posidonius of Rhodes, but another, perhaps Posidonius of
Alexandria, a pupil of Zeno of Cittium in the third cen-
tury B.C.
We now come to GEMINUS, a very important authority on
many questions belonging to the history of mathematics, as is
shown by the numerous quotations from him in Proclus's
Commentai-y on Euclid, Book I. His date and birthplace are
uncertain, and the discussions on the subject now form a whole
literature for which reference must be made to Manitius's
edition of the so-called Gemini elementa astronomiae (Teubner,
1898) and the article 'Geminus' in Pauly-Wissowa's Real-
Encyclopddie. The doubts begin with his name. Petau, who
included the treatise mentioned in his Umnologion (Paris,
1630), took it to be the Latin Gemlnus. Manitius, the latest
editor, satisfied himself that it was Gernlnus, a Greek name,
judging from the fa^t that it consistently appears with the
properispomenon accent in Greek (Ptfjiivos), while it is also
found in inscriptions with the spelling re/*/oy; Manitius
suggests the derivation from yc//, as '/jy^oy from spy, and
MXe^o/oy from d\$; he compares also the unmistakably
Greek names 'Ifcro/oy, KpaTivos. Now, however, we are told
(by Tittel) that the name is, after all, the Latin GAnlnus,
that T*IJUVO$ came to be so written through false analogy
with i4A6|u/oy, &c., and that re[/i]ea/oy, if the reading is
correct, is also wrongly formed on the model of 'Avrwcivos,
'Aypnrirttva. The occurrence of a Latin name in a centre
of Greek culture need not surprise us, since Romans settled in
such centres in large numbers during the last century B.C.
Geminus, however, in spite of his name, was thoroughly Greek.
1 Proclus on Eucl. I, pp. 214. 18-215. 13, p. 216. 10-19, p. 217. 10-23.
GEMINUS 223
An upper limit for his date is furnished by the fact that he
wrote a corhmentary on or exposition of Posidonius's work
TTtpl /zerccopcoi/ ; on the other hand, Alexander Aphrodisiensis
(about A.D. 210) quotes an important passage from an 'epitome*
of this 6^177170-19 by Ocminus. The view most generally
accepted is that he was a Stoic philosopher, born probably
in the island of Rhodes, and a pupil of Posidonius, and that
he wrote about 73-67 B.C.
Of Geminus's works that which has most interest for us
is a comprehensive work on mathematics. Proclus, though
he makes great use of it, does not mention its title, unless
indeed, in the passage where, after quoting from Geminus
a classification of lines which never meet, he says ' these
remarks I have selected from the <f>i\oKa\ta of Geminus', 1
the word <f>iXoKa\ia is a title or an alternative title. Tappus,
however, quotes a work of Geminus ' on the classification of
the mathematics' (tv rS> irepl TTJS rS>v fjia()r)fjLaTa>i' ra^6o>y),
while Eutocius quotes from ' the sixth book of the doctrine of
the mathematics ' (iv r ZKTW rfjs rS>v /jLaOrjfJidTwv 0ci>pias).
The former title? corresponds well enough to the long extract
on the division of the mathematical sciences into arithmetic,
geometry, mechanics, astronomy, optics, geodesy, canonic
(musical harmony) and logistic which Proclus gives in his
first prologue, and also to the fragments contained in the
Anonym i variae coUectioues published by Hultsch in his
edition of Heron; but it does not suit most of the other
passages borrowed by Proclus. The correct title was most
probably that given by Eutocius, The Doctrhw, or Tlieory,
of the Mathematics', and Pappus probably refers to one
particular section of the work, say the first Book. If the
sixth Book treated of conies, as we may conclude from
Eutocius's reference, there must have been more Books to
follow; for Proclus has preserved us details about higher
curves, which must have come later. If again Geminus
finished his work and wrote with the same fullness about the
other branches of mathematics as he did about geometry,
there must have been a considerable number of Books
altogether. It seems to have been designed to give a com-
plete view of the whole science of mathematics, and in fact
1 Proclus on Eucl. I, p. 177. 24.
224 SUCCESSORS OF THE GREAT GEOMETERS
to have been a sort of encyclopaedia of the subject. The
quotations of Proclus from Geminus's work do not stand
alone; we have other collections of extracts, some more and
some less extensive, and showing varieties of tradition accord-
ing to the channel through which they came down. The
scholia to Euclid's Elements, Book I, contain a considerable
part of the commentary on the Definitions of Book I, and are
valuable in that they give Geminus pure and simple, whereas
Proclus includes extracts from other authors. Extracts from
Geminus of considerable length are included in the Arabic
commentary by an-Nairizi (about A.D. 900) who got them
through the medium of Greek commentaries on Euclid,
especially that of Simplicius. It does not appear to be
doubted any longer that 'Aganis' in an-NahizI is really
Geminus ; this is inferred from the close agreement between
an-NairizI's quotations from c Aganis' and the correspond-
ing passages in Proclus; the difficulty caused by the fact
that Simplicius calls Aganis 'socius nostcr' is met by the
suggestion that the particular word socius is either the
result of the double translation from the Greek or means
nothing more, in the mouth of Simplicius, than ' colleague '
in the sense of a worker in the same field, or ' authority '.
A few extracts again are included in the Aitonymi variae
collectiones in Hultsch's Heron. Nos. 5-14 give definitions of
geometry, logistic, geodesy and their subject-matter, remarks
on bodies as continuous magnitudes, the three dimensions as
* principles ' of geometry, the purpose of geometry, and lastly
on optics, with its subdivisions, optics proper, Catoptriea and
<r/t?7i>oypa0i/c?j, scene-painting (a sort of perspective), with some
fundamental principles of optics, e.g. that all light travels
along straight lines (which are broken in the cases of reflection
and refraction), and the division between optics and natural
philosophy (the theory of light), it being the province of the
latter to investigate (what is a matter of indifference to optics)
whether (1) visual rays issue from the eye, (2) images proceed
from the object and impinge on the eye, or (3) the intervening
air is aligned or compacted with the beam-like breath or
emanation from the eye.
Nos. 80-6 again in the same collection give the Peripatetic
explanation of the name mathematics, adding that the term
GEMINUS 225
was applied by the early Pythagoreans more particularly
to geometry and arithmetic, sciences which deal with the pure,
the eternal and the unchangeable, but was extended by later
writers to cover what we call ' mixed ' or applied mathematics,
which, though theoretical, has to do with sensible objects, e.g.
astronomy and optics. Other extracts from Geminus are found
in extant manuscripts in connexion with Damianus's treatise
on optics (published by R. Schone, Berlin, 1897). The defini-
tions of logistic and geometry also appear, but with decided
differences, in the scholia to Plato's (-harmtdes 165 K. Lastly,
isolated extracts appear in Eutocius, (1) a remark reproduced
in the commentary on Archimedes's Plane Equilibriums to
the effect that Archimedes in that work gave the name of
postulates to what are really axioms, (2) the statement that
before Apollonius's time the conies were produced by cutting
different cones (right-angled, acute-angled, and obtuse-angled)
by sections perpendicular in each case to a generator. 1
The object of Geminus's work was evidently the examina-
tion of the first principles, the logical building up of mathe-
matics on the basis of those admitted principles, and the
defence of the whole structure against the criticisms of
the enemies of the science?, the Epicureans and Sceptics, some
of whom questioned the unproved principles, and others the
logical validity of the deductions from them. Thus in
geometry Geminus jlealt first with the principles or hypotheses
(dpxai, viro0t(ri$) and then with the logical deductions, the
theorems and problems (ra fiera ray dp\d$). The distinction
is between the things which must be taken for granted but
arc incapable of proof and the things which must not be
assumed but are matter for demonstration. The principles
consisting of definitions, postulates, and axioms, Geminus
subjected them severally to a critical examination from this
point of view, distinguishing carefully between postulates and
axioms, and discussing the legitimacy or otherwise of those
formulated by Euclid in each class. In his notes on the defini-
tions Geminus treated them historically, giving .the various
alternative definitions which had been suggested for each
fundamental concept such as ' line ', ' surface ', ' figure '/body',
* angle ', &c., and frequently adding instructive classifications
1 Eutocius, COMM. on ApoUoniuis Conies, ad itnf.
228 SUCCESSORS OF THE GREAT GEOMETERS
of the different species of the thing defined. Thus in the
case of 'lines' (which include curves) he distinguishes, first,
the composite (e.g. a broken line forming an angle) and the
incomposite. The incomposite are subdivided into those
* forming a figure ' (crx^^^oTrotova-ai) or determinate (e.g.
circle, ellipse, cissoid) and those not forming a figure, inde-
terminate and extending without limit (e.g. straight lino,
parabola, hyperbola, conchoid). In a second classification
incomposite lines are divided into (1) ' simple ', namely the circle
and straight line, the one ' making a figure ', the other extend-
ing without limit, and (2) ' mixed '. ' Mixed ' lines again are
divided into (a) ' lines in planes ', one kind being a line meet-
ing itself (e.g. the cissoid) and another a line extending
without limit, and (6) 'lines on solids', subdivided into lines
formed by sections (e.g. conic sections, spiric curves) and
'lines round solids' (e.g. a helix round a cylinder, sphere, or
cone, the first of which is uniform, homoeomeric, alike in all
its parts, while the others are non-uniform). Geminus gave
a corresponding division of surfaces into simple and mixed,
the former being plane surfaces and spheres, while examples
of the latter are the tore or anchor-ring (though formed by
the revolution of a circle about an axis) and the conicoids of
revolution (the right-angled conoid, the obtuse-angled conoid,
and the two spheroids, formed by the revolution of a para-
bola, a hyperbola, and an ellipse respectively about their
axes). He observes that, while there are three homoeomeric
or uniform 'lines' (the straight line, the circle, and the
cylindrical helix), there are only two homoeomeric surfaces,
the plane and the sphere. Other classifications are those of
' angles ' (according to the nature of the two lines or curves
which form them) and of figures and plane figures.
When Proclus gives definitions, &c., by Posidonius, it is
evident that he obtained them from Gerninus's work. Such
are Posidonius's definitions of ' figure ' and ' parallels ', and his
division of quadrilaterals into seven kinds. We may assume
further that, even where Geminus did not mention the name
of Posidonius, he was, at all events so far as the philosophy of
mathematics was concerned, expressing views which were
mainly those of his master.
GEMINUS 227
Attempt to prove the Parallel-Postulate.
Geminus devoted much attention to the distinction between
postulates and axioms, giving the views of earlier philoso-
phers and mathematicians (Aristotle, Archimedes, Euclid,
Apollonius, the Stoics) on the subject as well as his own. It
was important in view of the attacks of the Epicureans and
Sceptics on mathematics, for (as Geminus says) it is as futile
to attempt to prove the indemonstrable (as Apollonius did
when he tried to prove the axioms) as it is incorrect to assume
what really requires proof, ' as Euclid did in the fourth postu-
late [that all right angles are equal] and in the fifth postulate
[the parallel-postulate] V
The fifth postulate was the special stumbling-block.
Geminus observed that the converse is actually proved by
Euclid in I. 17; also that it is conclusively proved that an
angle equal to a right angle is not necessarily itself a right
angle (e.g. the ' angle ' between the, circumferences of two semi-
circles on two equal straight lines with a common extremity
and at right angles to one another) ; we cannot therefore admit
that the converses are incapable of demonstration. 2 And
' we have learned from the very pioneers of this science not to
have, regard to mere plausible imaginings when it is a ques-
tion of the reasonings to be included inour geometrical
doctrine. As Aristotle says, it is as justifiable to^isk scien-
tific proofs from a rhetorician as to accept mere plausibilities
from a geometer. . . So in this case (that of the parallel-
postulate) the fact that, when the right angles are lessened, the
straight lines converge is true and necessary ; but the state-
ment that, since they converge more and more as they are
produced, they will sometime meet is plausible but not neces-
sary, in the absence of some argument showing that this is
true in the case of straight lines. For the fact that some lines
exist which approach indefinitely but yet remain non-secant
(acrtf/zTTTooro*), although it seems improbable and paradoxical,
is nevertheless true and fully ascertained with reference to
other species of lines [the hyperbola and its asymptote and
the conchoid and its asymptote, as Geminus says elsewhere].
May not then the same thing be possible in the case of
1 Proclus on Eucl. I, pp. 178-82. 4> 183. 14-184. 10.
2 /&., pp. 183. 26-184. 5.
Q 2
228 SUCCESSORS OF THE GREAT GEOMETERS
straight lines which happens in the case of the lines referred
to ? Indeed, until the statement in the postulate is clinched
by proof, the facts shown in the case of the other lines may
direct our imagination the opposite way. And, though the
controversial arguments against the meeting of the straight
lines should contain much that is surprising, is there not all
the more reason why we should expel from our body of
doctrine this merely plausible and unreasoned (hypothesis) ?
It is clear from this' that we must seek a proof of the present
theorem, and that it is alien to the special character of
postulates/ l
Much of this might have been written by a modern
geometer. Geminus's attempted remedy was to substitute
a definition of parallels like that of Posidonius, based on the
notion of equidistanee. An-Naiiizi gives the definition as
follows: 'Parallel straight lines are straight lines situated in
the same plane and such that the distance between them, if
they are produced without limit in both directions at the same
time, is everywhere the same', to which Geminus adds the
statement that the said distance is the shortest straight line
that can be drawn between them. Starting from this,
Geminus proved to his own satisfaction the propositions of
Euclid regarding parallels and finally the parallel-postulate.
He first gave the propositions (1) that the 'distance ' between
the two lines as Defined is perpendicular to both, and (2) that,
if a straight line is perpendicular to each of two straight lines
and meets both, the two straight lines are parallel, and the
'distance' is the intercept on the perpendicular (proved by
reductio ad absurdum). Next comp (3) Euclid's propositions
I. 27, 28 that, if two lines are parallel, the alternate angles
made by any transversal are equal, &c. (easily proved by
drawing the two equal 'distances' through the points of
intersection with the transversal), and (4) Eucl. I. 29, the con-
verse of I. 28, which is proved by reductio (id absurdum, by
means of (2) and (3). Geminus still needs Eucl. I. 30, 31
(about parallels) and I. 33, 34 (the first two propositions
relating to parallelograms) for his final proof of the postulate,
which is to the following effect.
Let AS, CD be two straight lines met by the straight line
1 Proclus on Eucl. I, pp. 192. 5-193. 3.
GEMINUS 229
EF, and let the interior angles BEF, EFD be together less
than two right angles.
Take any point 11 on FD and draw HK parallel to AB
meeting EF in K. Then, if we bisect EF at L, LFnt, M, MF
at X, and so on, we shall at last have a length, as FX, less
U
than FK. Draw FG, XOP parallel to AB. Produce FO to Q,
and let /< T y be thr, same multiple, of jFO that FE is of #A r ;
then shall AB, (7) meet in (<>.
Lrt >S f be the middle point of FQ and Ji the middle point of
Fti. Draw through /f, #, Q respectively the* straight lines
7UW, jSTZ7, QK parallel to EF. Join J/7?, Z6 f and produce
them to 7 T , K. Produce FG to T.
Then, in the triangles JWiY, jffOP, two angles are equal
respectively, the vertically opposite angles FOX, HOP and
tlu> alternate angles XFO, PRO ; and ^ T = OR ; therefore
7w j = #y.
And AW, 7^G in the parallelogram FXPG are equal ; there-
fore KG = 2 A T .Y = / T J/ (whence J//i is parallel to FG or AB)
Similarly we prove that SU=>2FM = FL, and L8 is
parallel to FG or 4U.
Lastly, by the triangles FL8, QVti, in which the sides FM,
ti(J are equal and two angles are respectively equal, Q V =
Since then EL, QV are equal and parallel, so are EQ t LV,
and (says Gominus) it follows that AB passes through Q.
230 SUCCESSORS OF THE GREAT GEOMETERS
What follows is actually that both EQ and AB (or EB)
are parallel to LV, and Geminus assumes that EQ, AB
are coincident (in other words, that through a given point
only one parallel can be drawn to a given straight line, an
assumption known as Playfair's Axiom, though it is actually
stated in Proclus on Eucl. I. 31).
The proof therefore, apparently ingenious as it is, breaks
down. Indeed the method is unsound from the beginning,
since (as Saccheri pointed out), before even the definition of
parallels by Geminus can be used, it has to be proved that
' the geometrical locus of points equidistant from a straight
line is a straight line ', and this cannot be proved without a
postulate. But the attempt is interesting as the first which
has come down to us, although there must have been many
others by geometers earlier than Geminus.
Coming now to the things which follow from the principles
(ra perk ray />X^ y )> we gather from Proclus that Geminus
carefully discussed such generalities as the nature of elements,
the different views which had been' held of the distinction
between theorems and problems, the nature and significance
of StopiarfjioL (conditions and limits of possibility), the meaning
of * porism ' in the sense in which Euclid used the word in his
Porisms as distinct from its other meaning of c corollary ', the
different sorts of problems and theorems, the two varieties of
converses (complete and partial), topical or focus-theorems,
with the classification of loci. He discussed also philosophical
questions, e.g. the question whether a line is made up of
indivisible parts (e a/ze/D<j/), which came up in connexion
with Eucl. I. 10 (the bisection of a straight line).
The book was evidently not less exhaustive as regards
higher geometry. Not only did Gerninus mention the &piric
curves, conchoids and cissoids in his classification of curves ;
he showed how they were obtained, and gave proofs, presum-
ably of their principal properties. Similarly he gave the
proof that there are three homoeomeric or uniform lines or
curves, the straight line, the circle and the cylindrical helix.
The proof of f uniformity ' (the property that any portion of
the line or. curve will coincide with any other portion of the
same length) was preceded by a proof that, if two straight
lines be drawn from any point to meet a uniform line or curve
GEMINUS 231
and make equal angles with it, the straight lines are equal. 1
As Apollonius wrote on the cylindrical helix and proved the
fact of its uniformity, we may fairly assume that Geminus
was here drawing upon Apollonius.
Enough has been said to show how invaluable a source of
information Geminus's work furnished to Proclus and all
writers on the history of mathematics who had access to it.
In astronomy we know that Geminus wrote an 7777/0-* 9 of
Posidonius's work, the Meteorologica or ?rep2 /ireo>pa>j>. This
is the source of the famous extract made from Geminus by
Alexander Aphrodisiensis, and reproduced by Simplicius in
his commentary on the Physics of Aristotle, 2 on which Schia-
parelli relied in his attempt to show that it was Heraclides of
Pontus, not Aristarchus of Sanaos, who first put forward the
heliocentric hypothesis. The extract is on the distinction
between physical and astronomical inquiry as applied to the
heavens. It is the business of the physicist to consider the
substance of the heaven and stars, their force and quality,
their coming into being and decay, and lie is in a position to
prove the facts about their size, shape, and arrangement;
astronomy, on the other hand, ignores the physical side,
proving the arrangement of the heavenly bodies by considera-
tions based on the view that the heaven is a real 007*09, and,
when it tells us of the shapes, sizes and distances of the earth,
sun and moon, of eclipses and conjunctions, and of the quality
and extent of the movements of the heavenly bodies, it is
connected with the mathematical investigation of quantity,
size, form, or shape, and uses arithmetic and geometry to
prove its conclusions. Astronomy deals, jiot with causes, but
with facts; hence it often proceeds by hypotheses, stating
certain expedients by which the phenomena may be saved.
For example, why do the sun, the moon and the planets
appear to move irregularly ? To explain the observed facts
we may assume, for instance, that the orbits are eccentric
circles or that the stars describe epicycles on a carrying
circle; and then we have to go farther and examine other
ways in which it is possible for the phenomena to be brought
about. ' Hence we actually find a certain person [Heraclides
1 Proclus on Eucl. I, pp. 112. 22-113. 3, p. 251. 3-11.
2 Simpl. in Phy*., pp. 291-2, ed. Diels.
282 SUCCESSOES OF THE GREAT GEOMETERS
of Pontus] coming forward and saying that, even on the
assumption that the earth moves in a certain way, while
the sun is in a certain way at red, the apparent irregularity
with reference to the sun may be saved! Philological con-
siderations as well as the other notices which we possess
about Heraclides make it practically certain that ' Heraclides
of Pontus' is an interpolation and that Geminus said m
simply, 'a certain person', without any name, though he
doubtless meant Aristarchus of Samos. 1
Simplicius says that Alexander quoted this extract from
the epitome of the grjyrj<ri$ by Geminus. As the original
work was apparently made the subject of an abridgement, we
gather that it must have been of considerable scope. It is
a question whether egrjyrjoris means 'commentary' or Ex-
position ' ; I am inclined to think that the latter interpretation
is the correct one, and that Geminus reproduced Posidonius's
work in its entirety with elucidations and comments; this
seems to me to be suggested by the words added by Simplicius
immediately after the extract c this is the account given by
Geminus, or Posidouius in Geminus, of the difference between
physics and astronomy ', which seems to imply that Geminus
in our passage reproduced Posidonius textually.
'Introduction to the PJiaenomena* attributed to Geminus.
There remains the treatise, purporting to be l>y Geminus,
which has come down to us under the title Tepivov e/crayooyr;
e/s ra $aiv6p.tva? What, if any, is the relation of this work
to the exposition of Posidonius's Meteorologies or the epitome
of it just mentioned ? One view is that the original Isayoye
of Geminus and the 17777079 of Posidonius were one and the
same work, though the Isagoye as we have it is not by
Geminus, but by an unknown compiler. The objections to
this are, first, that it does not contain the extract given by
Simplicius, which would have come in usefully at the begin-
ning of an Introduction to Astronomy, nor the other extract
given by Alexander from Geminus and relating to the rainbow
which seems likewise to have come from the
1 Of. Aristarchus of Samos, pp. 275-83.
2 Edited by Manitius (Teubner, 1898).
3 Alex. Aphr. on Aristotle's Meteorologica, iii. 4, 9 (Ideler, ii, p. 128;
p. 152. 10, Hayduck).
GEMINUS 233
secondly, that it docs not anywhere mention the name of
Posidonius (not, perhaps, an insuperable objection) ; and,
thirdly, that there are views expressed in it which are not
those held by Posidonius but contrary to them. Again, the
writer knows how to give a sound judgement as between
divergent views, writes in good style on the whole, and can
hardly have been the mere compiler of extracts from Posi-
donius which the view in question assumes him to be. It
seems in any case safer to assume that the Isagoge and the
grjyrj(ri$ were separate works. At the same time, the Isagoge,
as we have it, contains errors which we cannot attribute to
Gemirms. The choice, therefore, seems to lie between two
alternatives : either the book is by Geminus in the main, but
has in the course of centuries suffered deterioration by inter-
polations, mistakes of copyists, and so on, or it is a compilation
of extracts from an original Isagoge by Geminus with foreign
and inferior elements introduced either by the compiler him-
self or by other prentice hands. The result is a tolerable ele-
mentary treatise ilhitable for teaching purposes and containing
the most important doctrines of Greek astronomy represented
from the standpoint of Hipparchus. Chapter 1 treats of the
zodiac, the solar year, the irregularity of the sun's motion,
which is explained by the eccentric position* of the sun's orbit
relatively to tho zodiac, the order and the periods of revolution
of the planets and the moon. In 23 we are told that all
the fixed stars do not lie on one spherical surface, but some
are farther away than others a doctrine due to the Stoics.
Chapter 2, again, treats of the twelve signs of the zodiac,
chapter 3 of the constellations, chapter 4 of the axis of
the universe and the poles, chapter 5 of the circles on the
sphere (the equator and the parallel circles, arctic, summer-
tropical, winter-tropical, antarctic, the colure-circles, the zodiac
or ecliptic, the horizon, the meridian, and the Milky Way),
chapter 6 of Day and Night, their relative lengths in different
latitudes, their lengthening and shortening, chapter 7 of
the times which the twelve signs take to rise. Chapter 8
is a clear, interesting and valuable chapter on the calendar,
the length of months and years and the various cycles, the
octaeteris, the 16-years and 160-years cycles, the 19-years
cycle of Euctemon (and Meton), and the cycle of Callippus
234 SUCCESSORS OF THE GREAT GEOMETERS
(76 years). Chapter 9 deals with the moon's phases, chapters
10, 11 with eclipses of the sun and moon, chapter 12 with the
problem of accounting for the motions of the sun, moon and
planets, chapter 13 with Risings and Settings and the various
technical terms connected therewith, chapter 14 with the
circles described by the fixed stars, chapters 15 and 16 with
mathematical and physical geography, the zones, &c. (Geminus
follows Eratosthenes's evaluation of the circumference of the
earth, not that of Posidonius). Chapter 17, on weather indica-
tions, denies the popular theory that changes of atmospheric
conditions depend on the rising and setting of certain stars,
and states that the predictions of weather (eTncrrj/jaoYa/.) in
calendars (rrapaTrTyy/zara) are only derived from experience
and observation, and have no scientific value. Chapter 18 is
on the eAiy//6y, the shortest period which contains an integral
number of synodic months, of days, and of anomalistic revolu-
tions of the moon ; this period is three times the Chaldaean
period of 223 lunations used for predicting eclipses. The end
of the chapter deals with the maximum, mean, and minimum
daily motion of the moon. The chapter as a whole does not
correspond to the rest of the book ; it deals with more difficult
matters, and is thought by Manitius to be a fragment only of
a discussion to which the compiler did not feel himself equal.
At the end of the work is a calendar (Parupeyma) giving the
number of days taken by the sun to traverse each sign of
the zodiac, the risings and settings of various stars and the
weather indications noted by various astronomers, Democritus,
Eudoxus, Dositheus, Euctemoii, Meton, Callippus ; this calendar
is unconnected with the rest of the book and the contents
are in several respects inconsistent with it, especially the
division of the year into quarters which follows Callippus
rather than Hipparchus. Hence it has been, since Boeckh's
time, generally considered not to be the work of Geminus.
Tittel, however, suggests that it is not impossible that Geminus
may have reproduced an older Parapegma of Callippus.
XVI
SOME HANDBOOKS
THE description- of the handbook on the elements of
astronomy entitled the Introduction to the Phaenomenct and
attributed to Geminus might properly have been reserved
for this chapter. It was, however, convenient to deal with
Geminus in close connexion with Posidonius; for Geminus
wrote an exposition of Posidonius's Meteorologica related to the
original work in such a way that Simplicius, in quoting a long
passage from an epitome of this work, could attribute the
passage to either Geminus or ' Posidonius in Geminus ' ; and it
is evident that, in other subjects too, Geminus drew from, and
was influenced by, Posidonius.
The small work De motu circular i corpomm caelestium by
CLEOMEDES (KXco/u^oyy KVK\iKtj Ot&pia) in two Books is the
production of a much less competent person, but is much more
largely based on Posidonius. This is proved by several refer-
ences to Posidonius by name, but it is specially true of the
very long first chapter of Book II (nearly half of the Book)
which seems for the most part to be copied bodily from
Posidonius, in accordance with the author's remark at the
end of Book I that, in giving the refutation of the Epicurean
assertion that the sun is just as large as it looks, namely one
foot in diameter, he will give so much as suffices for such an
introduction of the particular arguments used by 'certain
authors who have written whole treatises on this one topic
(i.e. the size of the sun), among whom is Posidonius'. The
interest of the book then lies mainly in what is quoted from
Posidonius ; its mathematical interest is almost nil.
The date of Cleomedes is not certainly ascertained, but, as
he mentions no author later than Posidonius, it is permissible
to suppose, with Hultsch, that he wrote about the middle of
236 SOME HANDBOOKS
the first century B. c. As he seems to know nothing of the
works of Ptolemy, he can hardly, in any case, have lived
later than the beginning of the second century A. D.
Book I begins with a chapter the object of which is to
prove that' the universe, which has the shape of a sphere,
is limited and surrounded by void extending without limit in
all directions, and to refute objections to this view. Then
follow chapters on the five parallel circles in the heaven and
the zones, habitable and uninhabitable (chap. 2) ; on the
motion of the fixed stars and the independent (rrpoaiptTiKaL)
movements of the planets including the sun and moon
(chap. 3); on the zodiac and the effect of the sun's motion in
it (chap. 4) ; on the inclination of the axis of the universe and
its effects on the lengths of days and nights at different places
(chap. 5); on the inequality in the rate of increase in the
lengths of the days and nights according to the time of year,
the different lengths of the seasons due to the motion of the
sun in an eccentric circle, the difference between a day-and-
night and an exact revolution of the universe owing to the
separate motion of the sun (chap. 6) ; on the habitable regions
of the globe including Britain and the ' island of Thulo ', said
to have been visited by Pytheas, where, when the sun is in
Cancer and visible, the day is a month long ; and so on (chap. 7).
Chap. 8 purports to prove that the universe is a sphere by
proving first that the earth is a sphere, and then that the air
about it, and the ether about that, must necessarily make up
larger spheres. The earth is proved to be a vsphere by the
method of exclusion ; it is assumed that the only possibilities
are that it is (a) flat and plane, or (6) hollow and deep, or
(c) square, or (d) pyramidal, or (e) spherical, and, the first four
hypotheses being successively disposed of, only the fifth
remains. Chap. 9 maintains that the earth is in the centre of
the universe ; chap. 10, on the size of the earth, contains the
interesting reproduction of the details of the measurements of
the earth by Posidonius and Eratosthenes respectively which
have been given above in their proper places (p. 220, pp. 1 06-7) ;
chap. 1 1 argues that the earth is in the relation of a point to,
i. e. is negligible in size in comparison with, the universe or
even the sun's circle, but not the moon's circle (cf. p. 3 above).
Book II, chap. 1, is evidently the piece de resistance, con-
CLEOMEDES 237
sisting of an elaborate refutation of Epicurus and his followers,
who held that the sun is just as large as it looks, and further
asserted (according to Cleomedes) that the stars are lit up as
they rise and extinguished as they set. The chapter seems to
bo almost wholly taken from Posidonius; it ends* with some
pages of merely vulgar abuse, comparing Epicurus with Ther-
sites, with more of the same sort. The value of the chapter
lies in certain historical traditions mentioned in it, and in the
account of Posidonius's speculation as to the size and distance
of the sun, which does, as a matter of fact, give results much
nearer the truth than those obtained by Aristarchus, Hippar-
chus, and Ptolemy. Cleomedes observes (1) that by means of
water-clocks it is found that the apparent diameter of the sun
is l/750tli of the sun's circle, and that this method of
measuring it is said to have been first invented by the
Egyptians; (2) that Hipparchus is said to have found that
the sun is 1,050 times the size of the earth, though, as regards
this, we have the better authority of Adrastus (in Theon of
Smyrna) and of Chalcidius, according to whom Hipparchus
made the sun nearly 1,880 times the size of the earth (both
figures refer of course to the solid content). We have already
described Posidonius's method of arriving at the size and
distance of the sun (pp. 220-1). After he has given this, Cleo-
medes, apparently deserting his guide, adds a calculation of
his own relating to the sizes and distances of the moon and
the sun which shows how little he was capable of any scien-
tific inquiry. 1 Chap. 2 purports to prove that the sun is
1 He says (pp. 146. 17-148. 27) that in an eclipse the breadth of the
earth's shadow is stated to be two moon-breadths ; hence, he says, it
seems credible (mQavov) that the earth is twice the size of the moon (this
practically assumes that the breadth of the earth's shadow is equal to
the diameter of the earth, or that the cone of the earth's shadow is
a cylinder!). Since then the circumference of the earth, according to
Kratostherios, is 250,000 stades, and its diameter therefore ' more than
80,000 ' (he evidently takes TT = 8), the diameter of the moon will be
40,000 stades. Now, the moon's circle being 750 times the moon's
diameter, the radius of the moon's circle, i.e. the distance of the moon
from the earth, will be Jtli of this (i.e. TT = 3) or 125 moon-diameters;
therefore the moon's distance is 5,000,000 stades (which is much too
great). Again, since the moon traverses its orbit 13 times to the sun's
once, he assumes that the sun's orbit is 13 times as large as the moon's,
and consequently that the diameter of the sun is 13 times that of the
moon, or 520,000 stades and its distance 13 times 5,000,000 or 65,000,000
stades !
238 SOME HANDBOOKS
larger than the earth ; and the remaining chapters deal with
the size of the moon and the stars (chap. 3), the illuminatipn
of the moon by the sun (chap. 4), the phases of the moon and
its conjunctions with the sun (chap. 5), the eclipses of the
moon (chap. 6), the maximum deviation in latitude of the five
planets (given as 5 for Venus, 4 for Mercury, 2| for Mars
and Jupiter, 1 for Saturn), the maximum elongations of
Mercury and Venus from the sun (20 and 50 respectively),
and the synodic periods of the planets (Mercury 116 days,
Venus 584 days, Mars 780 days, Jupiter 398 days, Saturn
378 days) (chap. 7).
There is only one other item of sufficient interest to be
mentioned here. In Book II, chap. 6, Cleomedes mentions
that there were stories of extraordinary eclipses which ' the
more ancient of the mathematicians had vainly tried to ex-
plain'; the supposed c paradoxical' case was that in which,
while the sun seems to be still above the horizon, the eclipsed
moon rises in the east. The passage has been cited above
(vol. i, pp. 6-7), where I have also shown that Cleomedes him-
self gives the true explanation of .the phenomenon, namely
that it is due to atmospheric refraction.
The first and second centurfes of the Christian era saw
a continuation of the work of writing manuals or introduc-
tions to the different mathematical subjects. About A. D. 100
came NICOMACHUS, who wrote an Introduction to Arithmetic
and an Introduction to Harmony] if we may judge by a
remark of his own, 1 he would appear to have written an intro-
duction to geometry also. The Arithmetical Introduction has
been sufficiently described above (vol. i, pp. 97-112).
There is yet another handbook which needs to be mentioned
separately, although we have had occasion to quote from it
several times already* This is the book by THEON OF SMYRNA
which goes by the title Expositio rerum mathematicarum ad
legendum Platonem utilium. There are two main divisions
of this work, contained in two Venice manuscripts respec-
tively. The first was edited by Bullialdus (Paris, 1644), the
second by .T. H. Martin (Paris, 1849); the whole has been
1 Nicom. Arith. ii. 6. 1.
THEON OP SMYRNA 239
edited by E. Killer (Teubner, 1878) and finally, with a French
translation, by J. Dupuis (Paris, 1892).
Theon's date is approximately fixed by two considerations.
He is clearly the person whom Theon of Alexandria called
'the old Theoii', rov -rraXaiov Ion/a, 1 and there is no reason
to doubt that he is the ' Theoii the mathematician ' (6 fiaOrj-
fiaTtKos) who is credited by Ptolemy witfi four observations
of the planets Mercury and Venus made in A.D. 127, 129, 130
and 132. 2 The latest writers whom Theoii himself mentions
are Thrasyllus, who lived in the reign of Tiberius, and
Adrastus the Peripatetic, who belongs to the middle of the
second century A.D. Thecm's work itself is a curious medley,
valuable, not intrinsically, but for the numerous historical
notices which it contains. The title, which claims that the
book contains things useful for the study of Plato, must not
be taken too seriously. It was no doubt an elementary
introduction or vade-mecum for students of philosophy, but
there is little in it which has special reference to the mathe-
matical questions raised in Plato. The connexion consists
mostly in the long proem quoting the views of Plato on the
paramount importance of mathematics in the training of
the philosopher, and the mutual relation of the five different
branches, arithmetic, geometry, stereometry, astronomy and
music. The want of care shown by Theon in the quotations
from particular dialogues of Plato prepares us for the patch-
work character of the whole book.
In the first chapter he promises to give the mathematical
theorems most necessary for the student of Plato to know,
in arithmetic, music, and geometry, with its application to
stereometry and astronomy/* But the promise is by no means
kept as regards geometry and stereometry : indeed, in a
later passage Theoii seems to excuse himself from including
theoretical geometry in his plan, on the ground that all those
who are likely to read his work or the writings of Plato may
be assumed to have gone through an elementary course of
theoretical geometry. 4 But he writes at length on figured
1 Theon of Alexandria, Comm. on Ptolemy's Syntaxis, Basel edition,
pp. 390, 395, 396.
2 Ptolemy, Syntaxis, ix. 9, x. 1, 2.
8 Theon of Smyrna, ed. Hiller, p. 1. 10-17.
< #., p. 16. 17-20-
240 SOME HANDBOOKS
numbers, plane and solid, which are of course analogous to
the corresponding geometrical figures, and he may have con-
sidered that he was in this way sufficiently fulfilling his
promise with regard to geometry and stereometry. Certain
geometrical definitions, of point, line, straight line, the three
dimensions, rectilinear plane and solid figures, especially
parallelograms and parallelepipedal figures including cubes,
pi intitules (square bricks) and SoKities (beams), and scalene
figures with sides unequal every way ( = P&pfoKoi in the
classification of solid numbers), are dragged in later (chaps.
53-5 of the section on music) 1 in the middle of the discussion
of proportions and means; if this passage is not an inter-
polation, it confirms the supposition that Theon included in
his work only this limited amount of geometry and stereo-
metry.
Section I is on Arithmetic in the same sense as Nicomachus's
Introduction. At the beginning Theon observes that arith-
metic will be followed by music. Of music in its three
aspects, music in instruments (*v opydvois), music in numbers,
i.e. musical intervals expressed in numbers or pure theoretical
music, and the music or harmony in the universe, the first
kind (instrumental music) is not exactly essential, but the other
two must be discussed immediately after arithmetic. 2 The con-
tents of the arithmetical section have been sufficiently indicated
in the chapter on Pythagorean arithmetic (vol. i, pp. 112-13) ;
it deals with the classification of numbers, odd, even, and
their subdivisions, prime numbers, composite numbers with
equal or unequal factors, plane numbers subdivided into
square, oblong, triangular and polygonal numbers, with their
respective '.gnomons' and their properties as the sum of
successive terms of arithmetical progressions beginning with
1 as the first term, circular and spherical numbers, solid num-
bers with three factors, pyramidal numbers and truncated
pyramidal numbers, perfect numbers with their correlatives,
the over-perfect and the deficient; this is practically what
we find in Nicomachus. But the special value of Theoii's
exposition lies in the fact that it contains an account of the
famous ' side- ' and c diameter- ' numbers of the Pythagoreans. 3
1 Theon of Smyrna, ed. Hiller, pp. 111-13. 2 /&., pp. 16. 24-17. 11.
3 /&., pp. 42. 10-45. 9. Of. vol. i, pp. 91-3.
THEON OF SMYRNA 241
In the Section on Music Theon says he will first speak of
the two kinds of music, the audible or instrumental, and the
intelligible or theoretical subsisting in numbers, after which
he promises to deal lastly with ratio as predicable of mathe-
matical entities in general and the ratio constituting the
harmony in the universe, ' not scrupling to set out once again
the things discovered by our predecessors, just as we have
given the things handed down in former times by the Pytha-
goreans, with a view to making them better known, without
ourselves claiming to have discovered any of them'. 1 Then
follows a discussion of audible music, the intervals which
give harmonies, &c., including substantial quotations from
Thrasyllus and Adrastus, and references to views of Aris-
toxenus, Hippasus, Archytas, Eudoxus and Plato. With
chap. 17 (p. 72) begins the account of the 'harmony in
numbers', which turns into a general discussion of ratios,
proportions and means, with more quotations from Plato,
Eratosthenes and Thrasyllus, followed by Thrasyllus's divisio
canonis, chaps. 35, 36 (pp. 87-93). After a promise to apply
the latter division to the sphere of the universe, Theon
purports to return to the subject of proportion and means.
This, however, does not occur till chap. 50 (p. 106), the
intervening chapters being taken up with a discussion of
the 5e*ay and rerpaKri/y (with eleven applications of the
latter) and the mystic or curious properties of the numbers
from 2 to 1 ; here we have a part of the theoloyumeiia of
arithmetic. The discussion of proportions and the different
kinds of means after Eratosthenes and Adrastus is again
interrupted by the insertion of the geometrical definitions
already referred to (chaps. 53-5, pp. 111-13), after which
Theon resumes the question of means for * more precise '
treatment.
The Section on Astronomy begins on p. 120 of Killer's
edition. Here again Theon is mainly dependent uon
Adrastus, from whom he makes long quotations. Thus, on
the sphericity of the earth, he says that for the neces-
sary conspectus of the arguments it will be sufficient to
refer to the grounds stated summarily by Adrastus. In
explaining (p. 124) that the unevennesses in the surface of
1 Theon of Smyrna, ed. Hillcr, pp. 46. 20-47. 14.
1621.2 R
242 SOME HANDBOOKS
the earth, represented e.g. by mou'ntains, are negligible in
comparison with the size of the whole, he quotes Eratosthenes
and Dicaearchus as claiming to have discovered that the
perpendicular height of the highest mountain above the normal
level of the land is no more than 10 stades ; and to obtain the
diameter of the earth he uses Eratosthenes's figure of approxi-
mately 252,000 stades for the circumference of the earth,
which, with the Archimedean value of ^- for TT, gives a
diameter of about 80,182 stades. The principal astronomical
circles in the heaven are next described (chaps. 5-12, pp.
129-35) ; then (chap. 12) the assumed maximum deviations in
latitude are given, that of the sun being put at 1, that of the
moon and Venus at 12, and those of the planets Mercury,
Mars, Jupiter and Saturn at 8, 5, 5 and 3 respectively; the
obliquity of the ecliptic is given as the side of a regular polygon
of 15 sides described in a circle, i.e. as 24 (chap. 23, p. 151).
Next the order of the orbits of the sun, moon and planets is ex-
plained (the system is of course geocentric) ; we are told (p. 13 8)
that ' some of the Pythagoreans ' made the order (reckoning
outwards from the earth) to be moon, Mercury, Venus, sun,
Mars, Jupiter, Saturn, whereas (p. 142) Eratosthenes put the
sun next to the moon, and the mathematicians, agreeing with
Eratosthenes in this, differed only in the order in which they
placed Venus and Mercury after the sun, some putting Mercury
next and some Venus (p. 143). The order adopted by 4 some
of the Pythagoreans ' is the Chaldacan order, which was not
followed by any Greek before Diogenes of Babylon (second
century B.C.); 'some of the Pythagoreans' are therefore the
later Pythagoreans (of whom Nicomachus was one) ; the other
order, moon, sun, Venus, Mercury, Mars, Jupiter, Saturn, was
that of Plato and the early Pythagoreans. In chap. 15
(p. 138sq.) Theon quotes verses of Alexander 'the Aetoliun'
(not really the ' Aetoliaii ', but Alexander of Ephesus, a con-
temporary of Cicero, or possibly Alexander of Miletus, as
Chalcidius calls him) assigning to each of the planets (includ-
ing the earth, though stationary) with the sun and moon and
the sphere of the fixed stars one note, the intervals between
the notes being so arranged as to bring the nine into an
octave, whereas with Eratosthenes and Plato the earth was
excluded, and the eight notes of the octachord were assigned
THEON OF SMYRNA 243
to the seven heavenly bodies and the sphere of the fixed stars.
The whole of this passage (chaps. 15 to 16, pp. 138-47) is no
doubt intended as the promised account of the ' harmony in
the universe ', although at the very end of the work Theon
implies that this has still to be explained on the basis of
Thrasyllus's exposition combined with what he has already
given himself.
The next chapters deal with the forward movements, the
stationary points, and the retrogradations, as they respectively
appear to us, of the five planets, and the ' saving of the pheno-
mena ' by the alternative hypotheses of eccentric circles and
epicycles (chaps. 17-30, pp. 147-78). These hypotheses are
explained, and the identity of the motion produced by the
two is shown by Adrastus in the case of the sun (chaps. 26, 27,
pp. 166-72). The proof is introduced with the interesting
remark that * Hipparchus says it is worthy of investigation
by mathematicians why, on two hypotheses so different from
one another, that of eccentric circles and that of concentric
circles with epicycles, the same results appear to follow '. It
is not to be supposed that the proof of the identity could be
other than easy to a mathematician like Hipparchus; the
remark perhaps merely suggests that the two hypotheses were
discovered quite independently, and it was not till later that
the effect was discovered to be the same, when of course the
fact would seem to be curious and a mathematical proof would
immediately be sought. Another passage (p. 188) says that
Hipparchus preferred the hypothesis of the epicycle, as being
his own. If this means that Hipparchus claimed to have
discovered the epicycle-hypothesis, it must be a misapprehen-
sion; for Apollonius already understood the theory of epi-
cycles in all its generality. According to Theon, the epicycle-
hypothesis is more ' according to nature ' ; but it was presum-
ably preferred because it was applicable to all the planets,
whereas the eccentric-hypothesis, when originally suggested,
applied only to the three superior planets ; in order to make
it apply to the inferior planets it is necessary to suppose the
circle described by the centre of the eccentric to be greater
than the eccentric circle itself, which extension of the hypo-
thesis, though known to Hipparchus, does not seem to have
occurred to Apollonius.
R 2
244 SOME HANDBOOKS
We next have (chap. 31, p. 178) an allusion to the systems
of Eudoxus, Callippus and Aristotle, and a description
(p. 180 sq.) of a system in which the 'carrying' spheres
(called ' hollow ') have between them ' solid spheres which by
their own motion will roll (dve\igov<n) the carrying spheres in
the opposite direction, being in contact with them '. These
'solid' spheres (which carry the planet fixed at a point on
their surface) act in practically the same way as epicycles.
In connexion with this description Theon (i.e. Adrastus)
speaks (chap. 33, pp. 186-7) of two alternative hypotheses in
which, by comparison with Chalcidius, 1 we recognize (after
eliminating epicycles erroneously imported into both systems)
the hypotheses of Plato and Heraclides respectively. It is
this passage which enables us to conclude for certain that
Heraclides made Venus and Mercury revolve in circles about
the sun, like satellites, while the sun in its turn revolves in
a circle about the earth as centre. Theon (p. 187) gives the
maximum arcs separating Mercury and Venus respectively
from the sun as 20 and 50, these figures being the same as
those given by Cleomedes.
The last chapters (chaps. 37-40), quoted from Adrastus, deal
with conjunctions, transits, occultations and eclipses. The
book concludes with a considerable extract from Dercyllides,
a Platonist with Pythagorean leanings, who wrote (before the
time of Tiberius and perhaps even before Varro) a book on
Plato's philosophy. It is here (p. 198. 14) that we have the
passage so often quoted from Eudernus :
' Eudemus relates in his Astronomy that it was Oenopides
who first discovered the girdling of the zodiac and the revolu-
tion (or cycle) of the Great Year, that Thales was the first to
discover the eclipse of the sun and the fact that the sun's
period with respect to the solstices is not always the same,
that Anaximander discovered that the earth is (suspended) on
high and lies (substituting KeTrat for the reading of the manu-
scripts, KivtiTai> moves) about the centre of the universe, and
that Anaximenes said that the moon has its light from the
sun and (explained) how its eclipses come about' (Anaxi-
menes is here apparently a mistake for Anaxagoras).
1 Chalcidius, Comm. on Timaeus, c. 110. Cf. Aristarcltus of Samos,
pp. 256-8.
XVII
TRIGONOMETRY: HIPPARCHUS, MENELAUS,
PTOLEMY
WE have seen that S^haeric, the geometry of the sphere,
was very early studied, because it was required so soon as
astronomy became mathematical ; with the Pythagoreans the
word fyrfiaeric, applied to one of the subjects of the quadrivium,
actually meant astronomy. The subject was so far advanced
before Euclid's time that there was in existence a regular
textbook containing the principal propositions about great
and small circles on the sphere, from which both Autolycus
and Euclid quoted the propositions as generally known.
These propositions, with others of purely astronomical in-
terest, were collected afterwards in a work entitled tiirtiaerica,
in three Books, by THEODOSIUS.
Suidas has a notice, 8. v. 0eo86<no$, which evidently con-
fuses the author of the tiphaerica, with another Theodosius,
a Sceptic philosopher, since it calls him ' Theodosius, a philoso-
pher', and attributes to him, besides the mathematical works,
* Sceptic chapters ' and a commentary 011 the chapters of
Theudas. Now the commentator on Theudas must have
belonged, at the earliest, to the second half of the second
century A.D., whereas our Theodosius was earlier than Meiie-
laus (fl. about A.D. 100), who quotes him by name. The next
notice by Suidas is of yet another Theodosius, a poet, who
came from Tripolis. Hence it was at one time supposed that
our Theodosius was of Tripolis. But Vitruvius x mentions a
Theodosius who invented a sundial 'for any climate'; and
Strabo, in speaking of certain Bithynians distinguished in
their particular sciences, refers to ' Hipparchus, Theodosius
and his sons, mathematicians ' 2 . We conclude that our Theo-
1 De architecture ix. 9. 2 Strabo, xii. 4, 9, p. 566.
246 TRIGONOMETRY
dosius was of Bithynia and not later in date than Vitruvius
(say 20 B.C.); but the order in which Strabo gives the
names makes it not unlikely that he was contemporary with
Hipparchus, while the character of his Sphaerica suggests a
date even earlier rather than later.
Works by Theodosius.
Two other works of Theodosius besides the Sphaerica,
namely On habitations and On Days and Nights, seem to
have been included in the ' Little Astronomy ' (fiiKpos dcrrpo-
vopovfjisvoS) sc. TOKOS). These two treatises need not detain us
long. They are extant in Greek (in the great MS. Vaticanus
Graecus 204 and others), but the Greek text has not appar-
ently yet been published. In the first, Oti habitations, in 12
propositions, Theodosius explains the different phenomena due
to the daily rotation of the earth, and the particular portions
of the whole system which are visible to inhabitants of the
different zones. In the second, On Days and Nights, contain-
ing 13 and 19 propositions in the two Books respectively,
Theodosius considers the arc of the ecliptic described by the
sun each day, with a view to determining the conditions to be
satisfied in order that the solstice may occur in the meridian
at a given place, and in order that the day and the night may
really be equal at the equinoxes; he shows also that the
variations in the day and night must recur exactly after
a certain time, if the length of the solar year is commen-
surable with that of the day, while on the contrary assump-
tion they will not recur so exactly.
In addition to the works bearing on astronomy, Theodosius
is said l to have written a commentary, now lost, on the tyoSiov
or Method of Archimedes (see above, pp. 27-34).
Contents of the Sphaerica.
We come now to the Sphaerica, which deserves a short
description from the point of view of this chapter. A text-
book on the geometry o the sphere was wanted as a supple-
ment to the Elements of Euclid. In the Elements themselves
1 Suidas, loc. cit.
THEODOSIUS'S SPHAERICA 247
(Books XII and XIII) Euclid included no general properties
of the sphere except the theorem proved in XII. 16-18, that
the volumes of two spheres are in the triplicate ratio of their
diameters ; apart from this, the sphere is only introduced in
the propositions about the regular solids, where it is proved
that they are severally inscribable in a sphere, and it was doubt-
less with a view to his proofs of this property in each case that
he gave a new definition of a sphere as the figure described by
the revolution of a semicircle about its diameter, instead of
the more usual definition (after the manner of the definition
of a circle) as the locus of all points (in space instead of in
a plane) which are equidistant from a fixed point (the centre).
No doubt the exclusion of the geometry of the sphere from
the Elements was due to the fact that it was regarded as
belonging to astronomy rather than pure geometry.
Theodosius defines the sphere as ' a solid figure contained
by one surface such that all the straight lines falling upon it
from one point among those lying within the figure are equal
to one another ', which is exactly Euclid's definition of a circle
with * solid ' inserted before * figure ' and ' surface ' substituted
for c line '. The early part of the work is then generally
developed on the lines of Euclid's Book III on the circle.
Any plane section of a sphere is a circle (Prop. 1). The
straight line from the centre of the sphere to the centre of
a circular section is perpendicular to the plane of that section
(1, For. 2 ; cf. 7, 23); thus a plane section serves for finding
the centre of the sphere just as a chord does for finding that
of a circle (Prop. 2). The propositions about tangent planes
(3-5) and the relation between the sizes of circular sections
and their distances from the centre (5, 6) correspond to
Euclid III. 16-19 and 15; as the small circle corresponds to
any chord, the great Circle (' greatest circle ' in Greek) corre-
sponds to the diameter. The poles of a circular section
correspond to the extremities of the diameter bisecting
a chord of a circle at right angles (Props. 8-10). Great
circles bisecting one another (Props. 11-12) correspond to
chords which bisect one another (diameters), and great circles
bisecting small circles at right angles and passing through
their poles (Props. 13-15) correspond to diameters bisecting
chords at right angles. The distance of any point of a great
248 TRIGONOMETRY
circle from its pole is equal to the side of a square inscribed
in the great circle and conversely (Props. 16, 17). Next come
certain problems : To find a straight line equal to the diameter
of any circular section or of the sphere itself (Props. 18, 19) ;
to draw the great circle through any two given points on
the surface (Prop. 20); to find the pole of any given circu-
lar section (Prop. 21). Prop. 22 applies Eucl. III. 3 to the
sphere.
Book II begins with a definition of circles on a sphere
which touch one another ; this happens * when the common
section of the planes (of the circles) touches both circles '.
Another series of propositions follows, corresponding again
to propositions in Eucl., Book III, for the circle. Parallel
circular sections have the same poles, and conversely (Props.
1, 2). Props. 3-5 relate to circles on the sphere touching
one another and therefore having their poles on a great
circle which also passes through the point of contact (cf.
Eucl. III. 11, [12] about circles touching one another). If
a great circle touches a small circle, it also touches another
small circle equal and parallel to it (Props. 6, 7), and if a
great circle be obliquely inclined to another circular section,
it touches each of two equal circles parallel to that section
(Prop. 8). If two circles on a sphere cut one another, the
great circle drawn through their poles bisects the intercepted
segments of the circles (Prop. 9). If there are any number of
parallel circles on a sphere, and any number of great circles
drawn through their poles, the arcs of the parallel circles
intercepted between any two of the great circles are similar,
and the arcs of the great circles intercepted between any two
of the parallel circles are -equal (Prop. 10).
The last proposition forms a sort of transition to the portion
of the treatise (II. 11-23 and Book III) which contains pro-
positions of purely astronomical interest, though expressed as
propositions in pure geometry without any specific reference
to the various circles in the heavenly sphere. The proposi-
tions are long and complicated, and it would neither be easj
nor worth while to attempt an enumeration. They deal with
circles or parts of circles (arcs intercepted on one circle by
series of other circles and the like). We have no difficulty ir
recognizing particular circles which come into many proposi
THEODOSIUS'S SPHAERICA 249
tions. A particular small circle is the circle which is the
limit of the stars which do not set, as seen by an observer at
a particular place on the earth's surface ; the pole of this
circle is the pole in the heaven. A great circle which touches
this circle and is obliquely inclined to the ' parallel circles ' is the
circle of the horizon ; the parallel circles of course represent
-the apparent motion of the fixed stars in the diurnal rotation,
and have the pole of the heaven as pole. A second great
circle obliquely inclined to the parallel circles is of course the
circle of the zodiac or ecliptic. The greatest of the ' parallel
circles ' is naturally the equator. All that need be said of the
various propositions (except two which will be mentioned
separately) is that the sort of result proved is like that of
Props. 12 and 13 of Euclid's Phaenomena to the effect that in
the half of the zodiac circle beginning with Cancer (or Capri-
cornus) equal arcs set (or rise) in unequal times ; those which
are nearer the tropic circle take a longer time, those further
from it a shorter; those which take the shortest time are
those adjacent to the equinoctial points ; those which are equi-
distant from the equator rise and set in equal times. In like
manner Theodosius (III. 8) in effect takes equal and con-
tiguous arcs of the ecliptic all on one side of the equator,
draws through their extremities great circles touching the
circumpolar c parallel ' circle, and proves that the correspond-
ing arcs of the equator intercepted between the latter great
circles are unequal and that, of the said arcs, that correspond-
ing to the arc of the ecliptic which is nearer the tropic circle
is the greater. The successive great circles touching the
cireumpolar circle are of course successive positions of the
horizon as the earth revolves about its axis, that is to say,
the same length of arc on the ecliptic takes a longer or shorter
time to rise according as it is nearer to or farther from the
tropic, in other words, farther from or nearer to the equinoctial
points.
'It is, however, obvious that investigations of this kind,
which only prove that certain arcs 'are greater than others,
and do not give the actual numerical ratios between them, are
useless for any practical purpose such as that of telling the
hour of the night by the stars, which was one of the funda-
mental uroblems in Greek astronomv; and in order to find
250
TRIGONOMETRY
the required numerical ratios a new method had to be invented,
namely trigonometry.
No actual trigonometry in Theodosius.
It is perhaps hardly correct to say that spherical triangles
are nowhere referred to in Theodosius, for in III. 3 the con-
gruence-theorem for spherical triangles corresponding to Eucl.
I. 4 is practically proved; but there is nothing in the book
that can be called trigonometrical. The nearest approach is
in III. 11, 12, where ratios between certain straight lines are
compared with ratios between arcs. ACc (Prop. 11) is a great
circle through the poles A, A' \ CDc, C'D are two other great
circles, both of which are at right angles to the plane of ACfc,
but CDc is perpendicular to AA', while C' D is inclined to it at
an acute angle. Let any other great circle AB'BA' through
AA' cut CD in any point B between C and /), and C'D in B'.
Let the ' parallel ' circle EB'e be drawn through B', and let
C'c' be the diameter of the ' parallel ' circle touching the great
circle C'D. Let L, K be the centres of the ' parallel ' circles,
and let R, p be the radii of the ' parallel ' circles CDc, Cfcf
respectively. It is required to prove that
2R : 2p > (arc CB) : (arc C'B').
Let (7'0, Ee meet in JV, and join NB'.
Then B'N, being the intersection of two planes perpendicu-
lar to the plane of AC'CA ', is perpendicular to that plane and
therefore to both Ee and C'U.
THEODOSIUS'S SPHAERICA 251
Now, the triangle NLO being right-angled* at L, NO > NL.
Measure NT along NO equal to NL, and join TB'.
Then in the triangles B'NT, B f NL two sides B'N, NT are
equal to two sides B'N, NL y and the included angles (both
being right) are equal ; therefore the triangles are equal in all
respects, and LNLB'= LNTB'.
Now 2R:2 = OC':C'K
= ON:NT
[= tan jmj': tan .V
> LCOB-./.NOB'
> (arc 6'): (are fi'C' ').
If a', //, c' are the sides of the spherical triangle AB'C', this
result is equivalent (since the angle COB subtended by the arc
GB is equal to A) to
1 : sin b' = tan A : tan of
> a : a',
where a = BG, the side opposite A in the triangle ABC.
The proof is based on the fact (proved in Euclid's
and assumed as known by Aristarchus of Samoa and Archi-
medes) that, if a, )8 are angles such that n > Oi > /?,
tan OL/ tan ft > a//3.
While, therefore, Theodosius proves the equivalent of the
formula, applicable in the solution of a spherical triangle
right-angled at (7, that tana = nin 6 tan A, he is unable, for
want of trigonometry, to find the actual value of a/a 7 , and
can only find a limit for it. He is exactly in the same position
as Aristarchus, who can only approximate to the values of the
trigonometrical ratios which he needs, e.g. sin 1, cos 1, sin 3,
by bringing them within upper and lower limits with the aid
of the inequalities
tana <x sin a
tan J8 sin^S'
where 4 TT > a > fl.
252 TRIGONOMETRY
We may contrast with this proposition of Theodosius the
corresponding proposition in Menelaus's Sphaerica (III. 15)
dealing with the more general case in which (?', instead of
being the tropical point on the ecliptic, is, like B', any point
between the tropical point and D. If R, p have the same
meaning as above and r lt r. z are the radii of the parallel circles
through J3' and the new C", Menelaus proves that
sin a
sin a 7
which, of course, with the aid of Tables, gives the means
of finding the actual values of a or a' when the other elements
are given.
The proposition III. 12 of Theodosius proves a result similar
to that of III. 11 for the case where the great circles AB'B,
AC'C, instead of being great circles through the poles, arc
great circles touching ' the circle of the always- visible stars ',
i. e. different positions of the horizon, and the points C', B' are
any points on the arc of the oblique circle between the tropical
and the equinoctial points ; in this case, with the same notation,
4 R : 2 p > (arc BG) : (arc 5'C")-
It is evident that Theodosius was simply a laborious com-
piler, and that there was practically nothing original in his
work. It has been proved, by .means of propositions quoted
verbatim or assumed as known by Autolycus in his Moving
Sphere and by Euclid in his Phaenomendt, that the following
propositions in Theodosius are pre-Euclidean, I. 1, 6 a, 7, 8, 11,
12, 13, 15, 20 ; II. 1, 2, 3, 5, 8, 9, 10 a, 13, 15, 17, 18, 19, 20, 22 ;
III. Ib, 2, 3, 7, 8, those shown in thick type being quoted
word for word.
The beginnings of trigonometry.
But this is not all. In Menelaus's Spliaerica,, III. 15, there
is a reference to the proposition (III. 11) of Theodosius proved
above, and in Gherard of Cremona's translation from the
Arabic, as well as in Halley's translation from the Hebrew
of Jacob b. Machir, there is an addition to the effect that this
proposition was used by Apollonius in a book the title of
which is given in the two translations in the alternative
BEGINNINGS OF TRIGONOMETRY 253
forms ' liber aygregativus ' and ' liber de principiis univorsa-
libus'. Each of these expressions may well mean the work
of Apollonius which Marinus refers to as the 'General
Treatise* (fi KaOoXov Tr/oay/iarem). There is no apparent
reason to doubt that the remark in question was really
contained in Menelaus's original work ; and, even if it is an
Arabian interpolation, it is not likely to have been made
without some definite authority. If then Apollonius was the
discoverer of the proposition, the fact affords some ground for
thinking that the beginnings of trigonometry go as far back,
at least, as Apollonius. Tannery 1 indeed suggested that not
only Apollonius but Archimedes before him may have com-
piled a ' table of chords ', or at least shown the way to such
a compilation, Archimedes in the work of which we possess
only a fragment in the Measurement of a Circle, and Apollonius
in the VKVTOKIOV, where he gave an approximation to the value
of TT closer than that obtained by Archimedes; Tannery
compares the Indian Table of Sines in the Su**ya-Siddhanta,
where the angles go by 24ths of a right angle (l/24th=3 45',
2/24ths=7 30', &c.), as possibly showing Greek influence.
This is, however, in the region of conjecture ; the first person
to make systematic use of trigonometry is, so far as we know,
Hipparchus.
HIPPAKCHUS, the greatest astronomer of antiquity, was
born at Nicaea in Bithynia. The period of his activity is
indicated by references in Ptolemy to observations made by
him the limits of which are from 161 B.C. to 126 B.C. Ptolemy
further says that from Hipparchus's time to the beginning of
the reign of Antoninus Pius (A.D. 138) was 265 years. 2 The
best and most important observations made by Hipparchus
were made at Rhodes, though an observation of the vernal
equinox at Alexandria on March 24, 146 B.C., recorded by him
may have been his own. His main contributions to theoretical
and practical astronomy can here only be indicated in the
briefest manner.
1 Tannery, Recherches sur Fhist. de Vastronomie ancienne, p. 64.
* 2 Ptolemy, Syntaxis^ vii. 2 (vol. ii, p. 15).
254 TRIGONOMETRY
The work of Hipparchus.
Discovery of precession.
1. The greatest is perhaps his discovery of the precession
of the equinoxes. Hipparchus found that the bright star
Spica was, at the time of his observation of it, 6 distant
from the autumnal equinoctial point, whereas he deduced from
observations recorded by Timocharis that Timocharis had
made the distance 8. Consequently the motion had amounted
to 2 in the period between Timocharis's observations, made in
283 or 295 B.C., and 129/8 B.C., a period, that is, of 154 or
166 years; this gives about 46-8" or 43-4" a year, as compared
with the true value of 50 -3 75 7".
Calculation of mean lunar month.
2. The same discovery is presupposed in his work On the
length of the Year, in which, by comparing an observation
of the summer solstice by Aristarchus in 281/0 B.C. with his
own in 136/5 B.C., he found that after 145 years (the interval
between the two dates) the summer solstice occurred half
a day-and-night earlier than it should on the assumption of
exactly 365 days to the year ; hence he concluded that the
tropical year contained about ^th of a day-and-night less
than 365 J days. This agrees very nearly with Censorinus's
statement that Hipparchus's cycle was 304 years, four times
the 76 years of Callippus, but with 111,035 days in it
instead of 111,036 ( = 27,759 x 4). Counting in the 304 years
12x304 + 112 (intercalary) months, or 3,760 months in all,
Hipparchus made the mean lunar month 29 days 12" hrs.
44 min. 2^ sec., which is less than a second out in comparison
with the present accepted .figure of 29-53059 days!
3. Hipparchus attempted a new determination of the sun's
motion by means of exact equinoctial and solstitial obser-
vations; he reckoned the eccentricity of the sun's course
and fixed the apogee at the point 5 30' of Gemini. More
remarkable still was his investigation of the moon's
course. He determined the eccentricity and the inclination
of the orbit to the ecliptic, and by means of records of
observations of eclipses determined the moon's period with
extraordinary accuracy (as remarked above). We now learn
HIPPARCHUS 255
that the lengths of the mean synodic, the sidereal, the
anomalistic and the draconitic month obtained by Hipparchus
agree exactly with Babylonian cuneiform tables of date not
later than Hipparchus, and it is clear that Hipparchus was
in full possession of all the results established by Babylonian
astronomy.
Im]yroved estimates of sizes awl distances of sun
and moo/6.
4. Hipparchus improved on Aristarchus's calculations of the
sizes and distances of the sun and moon, determining the
apparent diameters more exactly and noting the changes in
them ; he made the mean distance of the sun 1,245D, the mean
distance of the moon 33 7), the diameters of the sun and
moon 12$ D and J D respectively, where D is the mean
diameter of the earth.
Epicycles and eccentrics.
5. Hipparchus, in investigating the motions of the sun, moon
and planets, proceeded on the alternative hypotheses of epi-
cycles and eccentrics ; he did not invent these hypotheses,
which were already fully understood and discussed by
Apollonius. While the motions of the sun and moon could
with difficulty be accounted for by the simple epicycle and
eccentric hypotheses, Hipparchus found that for the planets it
was necessary to combine the two, i.e. to superadd epicycles to
motion in eccentric circles.
dat<tlogue of nturs.
6. He compiled c*i catalogue of fixed stars including 850 or
more such stars; apparently he was the first to state their
positions in terms of coordinates in relation to the ecliptic
(latitude and longitude), and his table distinguished the
apparent sizes of the stars. His work was continued by
Ptolemy, who produced a catalogue of 1,022 stars which,
owing to an error in his solar tables affecting all his longi-
tudes, has by many erroneously been supposed to be a mere
reproduction of Hipparchus's catalogue. That Ptolemy took
many observations himself seems certain. 1
1 See two papers by Dr. J. L. E. Dreyer in^the Monthly Notices of the
Royal Astronomical Society, 1917, pp. 528-39, and 1918, pp. 343-9.
356 TRIGONOMETRY
Improved Instruments.
7. He made great improvements in the instruments used for
observations. Among those which he used were an improved
dioptra, a ' meridian-instrument ' designed for observations in
the meridian only, and a universal instrument (aorpoAajSoi/
opyavov) for more general use. He also made a globe on
which he showed the positions of the fixed stars as determined
by him ; it appears that he showed a larger number of stars
on his globe than in his catalogue.
Geography.
In geography Hipparchus wrote a criticism of Eratosthenes,
in great part unfair. He checked Eratosthenes's data by
means of a sort of triangulation ; he insisted on the necessity
of applying astronomy to geography, of fixing the position of
places by latitude and longitude, and of determining longitudes
by observations of lunar eclipses.
Outside the domain of astronomy and geography, Hipparchus
wrote a book On, things borne down ly their weight from
which Siinplicius (on Aristotle's De caelo, p. 264 sq.) quotes
two propositions. It is possible, however, that even in this
work Hipparchus may have applied his doctrine to the case of
the heavenly bodies.
In pure mathematics he is said to have considered a problem
in permutations and combinations, the problem of finding the
number of different possible combinations of 10 axioms or
assumptions, which he made to be 103,049 (v. I. 101,049)
or 310,952 according as the axioms were affirmed or denied 1 :
it seems impossible to make anything of these figures. When
the Fihrist attributes to him works c On the art of algebra,
known by the title of the Rules ' and ' On the division of num-
bers ', we have no confirmation : Suter suspects some confusion,
in view of the fact that the article immediately following in
the Fihrist is on Diophantus, who also ' wrote on the art of
algebra 1 .
1 Plutarch, Quaest. Conviv. viii. 9. 3, 732 r, De Stoicorum repugn. 29.
1047 D.
HIPPARCHUS 257
First systematic use of Trigonometry.
We come now to what is the most important from the
point of view of this work, Hipparchus's share in the develop-
ment of trigonometry. Even if he did not invent it,
Hipparchus is the first person of whose systematic use of
trigonometry we have documentary evidence. (1) Theon
of 'Alexandria says on the Syntaxis of Ptolemy, & propos of
Ptolemy's Table of Chords in a circle (equivalent to sines),
that Hipparchus, too, wrote a treatise in twelve books on
straight lines (i.e. chords) in a circle, while another in six
books was written by Menelaus. 1 In the tiyntaxis I. 10
Ptolemy gives the necessary explanations as to the notation
used in his Table. The circumference of the circle is divided
into 360 parts or degrees; the diameter is also divided into
120 parts, and one of such parts is the unit of length in terms
of which the length of each chord is expressed; each part,
whether of the circumference or diameter, is divided into 60
parts, each of these again into 60, and so on, according to the
system of sexagesimal fractions. Ptolemy then sets out the
minimum number of propositions in plane geometry upon
which the calculation of the chords in the Table is based (Sia
rrjy K rail/ ypa/i/io>i> /ieflo&Kfjs 1 OLVT&V (ruorao'eooy). The pro-
positions are famous, and it cannot be doubted that Hippar-
chus used a set of propositions of the same kind, though his
exposition probably ran to much greater length. As Ptolemy
definitely set himself to give the necessary propositions in the
shortest form possible, it will be better to give them under
Ptolemy rather than here. (2) Pappus, in speaking of Euclid's
propositions about the inequality of the times which equal arcs
of the zodiac take to rise, observes that ' Hipparchus in his book
On the rising of the twelve signs of the zodiac shows by means
of numerical calculations (Si api6p.S>v} that equal arcs of the
semicircle beginning with Cancer which set in times having
a certain relation to one another do not everywhere show the
same relation between the times in which they rise ', 2 and so
on. We have seen that Euclid, Autolycus, and even Theo-
dosius could only prove that the said times are greater or less
1 Theon, Comm. on Syntax? s, p. 110, ed. Halma.
2 Pappus, vi, p. 600. 9-13.
258 TRIGONOMETRY
in relation to one another ; they could not calculate the actual
times. As Hipparchus proved corresponding propositions by
means of numbers, we can only conclude that he used proposi-
tions in spherical trigonometry, calculating arcs from others
which are given, by means of tables. (3) In the only work
of his which survives, the Commentary on the Phaenomena
of Eudoxus and Aratus (an early work anterior to the
discovery of the precession of the equinoxes), Hipparchus
states that (presumably in the latitude of Rhodes) a star which
lies 27^ north of the equator describes above the horizon an
arc containing 3 minutes less than 15/24ths of the whole
circle 1 ; then, after some more inferences, he says, 'For each
of the aforesaid facts is proved by means of lines (Sia rS>v
ypafifji&y) in the general treatises on these matters compiled
by me '. In other places 2 of the Commentary he alludes to
a work On simultaneous risings (ra IT* pi T>V (rvvava,To\S>v),
and in II. 4. 2 he says he will state summarily, about each of
the fixed stars, along with what sign of the zodiac it rises and
sets and from which degree to which degree of each sign it
rises or sets in the regions about Greece or wherever the
longest day is 1 4| equinoctial hours, adding that lie has given
special proofs in another work designed so that it is possible
in practically every place in the inhabited earth to follow
the differences between the concurrent risings and settings/ 5
Where Hipparchus speaks of proofs ' by means of lines ', he
does not mean a merely graphical method, by construction
only, but theoretical determination by geometry, followed by
calculation, just as Ptolemy uses the expression /c rS>v ypap-
p.S>v of his calculation of chords and the expressions <r(f>aipiKal
fleets and ypa/ifJiiKal Se^eiy of the fundamental proposition
in spherical trigonometry (Menelaus's theorem applied to the
sphere) and its various applications to particular cases. It
is significant that in the Syntaxis VIII. 5, where Ptolemy
applies the proposition to the very problem of finding the
times of concurrent rising, culmination and setting of the
fixed stars, he says that the times can be obtained ' by lines
only ' (8ia p.6va>v r&v ypa/jLfjLOH/)* Hence we may be certain
that, in the other books of his own to which Hipparchus refers
1 Ed. Manitius, pp. 148-50. 2 16., pp. 128. 5, 148. 20.
3 lb., pp. 182. 19-184. 5. 4 Syntaxis, vol. ii, p. 193.
HIPPARCHUS 259
in his Comment<M*y, he used the formulae of spherical trigono-
metry to get his results. In the particular case where it is
required to find the time in which a star of 27^ northern
declination describes, in the latitude of Rhodes, the portion of
its arc above the horizon, Hipparchus must have used the
equivalent of the formula in the solution of a right-angled
spherical triangle, tan b = cos A tan c, where (J is the right
angle. Whether, like Ptolemy, Hipparchus obtained the
formulae, such as this one, which he used from different
applications of the one general theorem (Menelaus's theorem)
it is not possible to say. There was of course no difficulty
in calculating the tangent or other trigonometrical function
of an angle if only a table of sines was given ; for Hippar-
chus and Ptolemy were both aware of the fact expressed by
sin 2 (X + cos 2 a = 1 or, as they would have written it,
(crd. 2<x) 2 + {crd. (180-2a)} 2 = 4r 2 ,
where (crd. 2 a) means the chord subtending an arc 2 a, and r
is the radius, of the circle of reference.
Table of Chords.
We have no details of Hipparchus's Table of Chords suffi-
cient to enable us to compare it with Ptolemy's, which goes
by half-degrees, beginning with angles of ^, 1, 1^, and so
on. But Heron 1 in his Metrica says that 'it is proved in the
books about chords in a circle ' that, if ci 9 and a u are the sides
of a regular enneagon (9-sided figure) and hendecagon (l 1 -sided
figure) inscribed in a circle of diameter d, then (1) a g = d 9
(2) a n = y-yd very nearly, which means that sin 20 was
taken as equal to 0.3333 ... (Ptolemy's table makes it
(20 + - + | ), so that the first approximation is -|), and
bO\ 60 60 /
sin T X T . 180 or sin 16 21' 49" was made equal to 0-28 (this cor-
responds to the chord subtending an angle of 32 43' 38",nearly
half-way between 32^ and 33, and the mean between the two
chords subtending the latter angles gives --Y + -+ . -) as
the required sine, while ^o (*<>&) ^ vf> which only differs
1 Heron, Metrica, I 22, 24, pp. 58. 19 and 62. 17.
S2
260 TRIGONOMETRY
by ^^ from $$$ or 5 7 T , Heron's figure). There is little doubt
that it is to Hipparchus's work that Heron refers, though the
author is not mentioned.
While for our knowledge of Hipparchus's trigonometry we
have to rely for the most part upon what we can infer from
Ptolemy, we fortunately possess an original source of infor-
mation about Greek trigonometry in its highest development
in the Hphaerica of Menelaus.
The. date of MENELAUS of Alexandria is roughly indi-
cated by the fact that Ptolemy quotes an observation of
his made in the first year of Trajan's reign (A.D. 98). He
was therefore a contemporary of Plutarch, who in fact
represents him as being present at the dialogue De facie in
orbe lunae, where (chap. 17) Lucius apologizes to Menelaus 'the
mathematician' for questioning the fundamental proposition
in optics that the angles of incidence and reflection are equal.
He wrote a variety of treatises other than the Sphaerica.
We have seen that Theon mentions his work on Chords in a
Circle in six Books. Pappus says that he wrote a treatise
(IT pay pare fa) on the setting (or perhaps only rising) of
different arcs of the zodiac. 1 Proclus quotes an alternative
proof by him of Eucl. I. 25, which is direct instead of by
reductio ad absurdum? and he would seem to have avoided
the latter kind of proof throughout. Again, Pappus, speaking
of the many complicated curves ' discovered by Demetrius of
Alexandria (in his "Linear considerations") and by Philon
of Tyana as the result of interweaving plectoids and other
surfaces of all kinds ', says that one curve in particular was
investigated by Menelaus and called by him c paradoxical '
(7ra/>a5ooy) 3 ; the nature of this curve can only be conjectured
(see below).
But Arabian tradition refers to other works by Menelaus,
(1) Elements of Geometry, edited by Thabit b. Qurra, in three
Books, (2) a Book on triangles, and (3) a work the title of
which is translated by Wenrich de cognitione quantitatis
discretae corporum perrnixtorum. Light is thrown on this
last title by one al-Chazim who (about A,D. 1121) wrote a
1 Pappus, vi, pp. 600-2.
2 Proclus ori Eucl. I, pp.
8 Pappus, iv, p. 270. 25.
2 Proclus ori Eucl. I, pp. 345. 14-346. 11.
MENELAUS OF ALEXANDRIA 261
treatise about the hydrostatic balance, i.e. about the deter-
mination of the specific gravity of homogeneous or mixed
bodies, in the course of which he mentions Archimedes and
Menelaus (among others) as authorities on the subject ; hence
the treatise (3) must have been a book on hydrostatics dis-
cussing such problems as that of the crown solved by Archi-
medes. The alternative proof of Eucl. I. 25 quoted by
Proclus might have come either from the Elements of Geometry
or the Book on triangles. With regard to the geometry, the
* liber trium f ratrum ' (written by three sons of Musa b. Shakir
in the ninth century) says that it contained a solution of the
duplication of the cube, which is none other than that of
Archytas. The solution of Archytas having employed the
intersection of a tore and a cylinder (with a cone as well),
there would, on the assumption that Menelaus reproduced the
solution, be a certain appropriateness in the suggestion of
Tannery 1 that the curve which Menelaus called the napdSogos
ypa/*/z?7 was in reality the curve of double curvature, known
by the name of Viviani, which is the intersection of a sphere
with a cylinder touching it internally and having for its
diameter the radius of the sphere. This curve is a particular
case of Eudoxus's hippopede, and it has the property that the
portion left outside the curve of the surface of the hemisphere
on which it lies is equal to the square on the diameter of the
sphere; the fact of the said area being squareable would
justify the application of the word napdSogos to the curve,
and the quadrature itself would not probably be beyond the
powers of the Greek mathematicians, as witness Pappus's
determination of the area cut off between a complete turn of
a certain spiral on a sphere and the great circle touching it at
the origin. 2
The Sphaerica of Menelaus.
This treatise in three Books is fortunately preserved in
the Arabic, and although the extant versions differ con-
siderably in form, the substance is beyond doubt genuine;
the original translator was apparently Ishaq b. Hunain
(died A.D. 910). There have been two editions, (1) a Latin
1 Tannery, MJmoires scientifiqueS) ii, p. 17. 2 Pappus, iv, pp. 264-8.
262 TRIGONOMETRY
translation by Maurolycus (Messina, 1558) and (2) Halley's
edition (Oxford, 1758). The former is unserviceable because
Maurolycus's manuscript was very imperfect, and, besides
trying to correct and restore the propositions, he added
several of his own. Halley seems to have made a free
translation of the Hebrew version of the work by Jacob b.
Machir (about 1273), although he consulted Arabic manuscripts
to some extent, following them, e.g., in dividing the work into
three Books instead of two. But an earlier version direct
from the Arabic is available in manuscripts of the thirteenth
to fifteenth centuries at Paris and elsewhere ; this version is
without doubt that made by the famous translator Gherard
of Cremona (1114-87). With the help of Halley's edition,
Gherard's translation, and a Ley den manuscript (930) of
the redaction of the work by Abu-Nasr-Mansur made in
A.D. 1007-8, Bjornbo has succeeded in presenting an adequate
reproduction of the contents of the Sphaerica^
Book I.
In this Book for the first time we have the conception and
definition of a spherical triangle. Menelaus does not trouble
to give the usual definitions of points and circles related to
the sphere, e.g. pole, great circle, small circle, but begins with
that of a spherical triangle as ' the area included by arcs of
great circles on the surface of a sphere ', subject to the restric-
tion (Def . 2) that each of the sides or legs of the triangle is an
arc less than a semicircle. The angles of the triangle are the
angles contained by the arcs of great circles on the sphere
(Def. 3), and one such angle is equal to or greater than another
according as the planes containing the arcs forming the first
angle are inclined at the same angle as, or a greater angle
than, the planes of the arcs forming the other (Dcfs. 4, 5).
The angle is a right angle if the planes of the arcs are at i^ight
angles (Def. 6). Pappus tells us that Menelaus in his Sphaerica
calls the figure in question (the spherical triangle) a ' three-
side ' (rp/TrAeirpoj/) 2 ; the word triangle (rpiytovov) was of course
1 BjOrnbo, Studien fiber Menelaos* Sphttrik (Abhandlungen zur Gesch. d.
math. Wissenschaften, Heft xiv. 1902).
2 Pappus, vi, p. 476. 16.
MENELAUS'S SPHAERICA 263
already appropriated for the plane triangle. We should gather
from this, as well as from the restriction of the definitions to
the spherical triangle and its parts, that the discussion of the
spherical triangle as such was probably new ; and if the pre-
face in the Arabic version addressed to a prince and beginning
with the words, ' O prince ! I have discovered an excellent
method of proof . . . ' is genuine, we have confirmatory evidence
in the writer's own claim.
Mcnelaus's object, so far as Book I is concerned, seems to
have been to give the main propositions about spherical
triangles corresponding to Euclid's propositions about plane
triangles. At the same time he does not restrict himself to
Euclid's methods of proof even where they could be adapted
to the case of the sphere; he avoids the form of proof by
reductio ad absurdum, but, subject to this, he prefers the
easiest proofs. In some respects his treatment is more com-
plete than Euclid's treatment of the analogous plane cases.
In the congruence-theorems, for example, we have I. 4 a
corresponding to Eucl. I. 4, I. 4b to Eucl. I. 8, I. 14, 16 to
Eucl. I. 26 a, b; but Menelaus includes (I. 13) what we know
as the ' ambiguous case ', which is enunciated on the lines of
Eucl. VI. 7. I. 12 is a particular case of I. 16. Menelaus
includes also the further case which has no analogue in plane
triangles, that in which the three angles of one triangle are
severally equal to the three angles of the other (1.17). He
makes, moreover, no distinction between the congruent and
the symmetrical, regarding both as covered by congruent. 1. 1
is a, problem, to construct a spherical angle equal to a given
spherical angle, introduced only as a lemma because required
in later propositions. I. 2, 3 are the propositions about
isosceles triangles corresponding to Eucl. I. 5, 6 ; Eucl. 1. 18, 19
(greater side opposite greater angle and vice versa) have their
analogues in I. 7, 9, and Eucl. I. 24, 25 (two sides respectively
equal and included angle, or third side, in one triangle greater
than included angle, or third side, in the other) in I. 8. I. 5
(two sides of a triangle together greater than the third) corre-
sponds to Eucl. I. 20. There is yet a further group of proposi-
tions comparing parts of spherical triangles, I. 6, 18, 19, where
I. 6 (corresponding to Eucl. I. 21) is deduced from I. 5, just as
the first part of Eucl. I. 21 is deduced from Eucl. I. 20.
264 TRIGONOMETRY
Eucl. I. 16, 32 are not true of spherical triangles, and
Menelaus has therefore the corresponding but different pro-
positions. L 10 proves that, with the usual notation a, 6, c,
A, B 9 (7, for the sides and opposite angles of a spherical
triangle, the exterior angle at C, or 180 (7, < = or >A
according as c + a > = or < 180, and vice versa. The proof
of this and the next proposition shall be given as specimens.
In the triangle ABC suppose that c + a > = or < 180 ; let
D be the pole opposite to A.
Then, according as c + a > = or < 180, BC> = or < BD
(since AD = 180),
and therefore LD > = or < /.BCD (= 180-G'), [I. 9]
i.e. (since LD = /.A) 180-(7< = or >A.
Menelaus takes the converse for granted.
As a consequence of this, I. 11 proves that A + B + G> 180.
Take the same triangle ABC, with the pole D opposite
to A, and from B draw the great circle BE such that
LDBE = LBDE.
Then CE+EB = CD < 180, so that, by the preceding
proposition, the exterior angle ACB to the triangle BCE is
greater than LCBE,
i.e. C > LCBE.
Add A or D (= LEBD) to the unequals ;
therefore C + A > Z.CBD,
whence A + B + C > LCBD + B or 180.
After two lemmas I. 21, 22 we have some propositions intro-
ducing M, N, P the middle points of a, 6, c respectively. I. 23
proves, e.g., that the arc MJf of a great circle >c, and I. 20
that AM < = or >|a according as A > = or < (B + C). .The
last group of propositions, 26-35, relate to the figure formed
MENELAUS'S SPHAERICA 265
by the triangle ABC witt great circles drawn through B to
meet AC (between A and 0) in D, E respectively, and the
case where D and E coincide, and they prove different results
arising from different relations between a and c (a>c), com-
bined with the equality of AD and EC (or DC), of the angles
ABD and EEC (or DBG), or of a + c and BD + BE (or 2BD)
respectively, according as a + c< = or > 180.
Book II has practically no interest for us. The object of it
is to establish certain propositions, of astronomical interest
only, which are nothing more than generalizations or exten-
sions of propositions in Theodosius's Mphaerica, Book III.
Thus Theodosius III. 5, 6, 9 are included in Menelaus II. 10,
Theodosius III. 7-8 in Menelaus II. 12, while Menelaus II. 11
is an extension of Theodosius III. 13. The proofs are quite
different from those of Theodosius, which are generally very
long-winded.
Book III. Trigonometry.
It will have been noticed that, while Book I of Menelaus
gives the geometry of the spherical triangle, neither Book I
nor Book II contains any trigonometry. This is reserved for
Book III. As I shall throughout express the various results
obtained in terms of the trigonometrical ratios, sine, cosine,
tangent, it is necessary to explain once for all that the Greeks
did not use this terminology, but, instead of sines, they used
the chords subtended by arcs of a
circle. In the accompanying figure
let the arc AD of a circle subtend an
angle a at the centre 0. Draw AM
perpendicular to OD y and produce it
to meet the circle again in A'. Then
sin a = AM/AO, and AM is \AA'
or half the chord subtended by an
angle 2 a at the centre, which may
shortly be denoted by |(crd. 2 a).
Since Ptolemy expresses the chords as so many 120th parts of
the diameter of the circle, while AM / AO = AA'/2AO, it
follows that sin a and (crd. 2 a) are equivalent. Cos a is
of course sin (90 a) and is therefore equivalent to crd.
(180-2a).
266 TRIGONOMETRY
(a) ' Menelaus' s theorem' for the sphere.
The first proposition of Book III is the famous ' Menelaus's
theorem ' with reference to a spherical triangle and any trans-
versal (great circle) cutting the sides of a triangle, produced
if necessary. Menelaus does not, however, use a spherical
triangle in his enunciation, but enunciates the proposition in
terms of intersecting great circles. ' Between two arcs ADB,
AEG of great circles are two other arcs of great circles DFC
and BFE which intersect them and also intersect each other
in F. All the arcs are less than a semicircle. It is required
to prove that
sin EA ~" sin FD sin BA
It appears that Menelaus gave three or four cases, sufficient
to prove the theorem completely. The proof depends on two
simple propositions which Menelaus assumes without proof;
the proof of them is given by Ptolemy.
(1) In the figure on the last page, if OD be a radius cutting
a chord AB in 0, then
For draw AM, BN perpendicular to OD. Then
= (crd. 2 AD): l(<srd. 2DB)
= sin AD: sin DB.
(2) If AB meet the radius OC produced in T, then
MENELAUS'S SPHAERTCA 267
For, if AM, jBJVare perpendicular to 0(7, we have, as before,
. 2BC)
Now let the arcs of great circles ADB, A EC be cut by the
arcs of great circles DFC, BFK which themselves meet in F.
Let G be the centre of the sphere and join GB, GF, GE, AD.
Then the straight lines AD, GB, being in one plane, are
either parallel or not parallel. If they are not parallel, they
will meet either in the direction of D, B or of A, G.
Let AD, GB meet in T.
Draw the straight lines ARC, DLC meeting GE, GFin K, L
respectively.
Then K, L, T must lie on a straight line, namely the straight
line which is the section of the planes determined by the arc
EFB and by the triangle ACD. 1
Thus we have two straight lines AC, AT cut by the two
straight lines (77), TK which themselves intersect in L.
Therefore, by Menelaus's proposition in plane geometry,
CK _ CL DT
KA~ LD'TA
1 So Ptolemy. In other words, since the straight lines GB, GE, GF,
which are in one plane, respectively intersect the straight lines AD, AC,
CD which are also in one plane, the points of intersection T, K, L are in
both planes, and therefore lie on the straight line in which the planes
intersect.
268 TRIGONOMETRY
But, by the propositions proved above,
CK sin GE CL sin OF 1)T sin PR
therefore, by substitution, we have
sin GE _ sinCF sin DB ^
sin EA ~~ sin jPD * sin BA *
Menelaus apparently also gave the proof for the cases in
which AD, GB meet towards A, G, and in which AD, GB are
parallel respectively, and also proved that in like manner, in
the above figure,
sin O A _ sin CD sin FB
sin AE~~ s5TS^*siirgA T
(the triangle cut by the transversal being here CFE instead of
ADC). Ptolemy 1 gives the proof of the above case only, and
dismisses the last-mentioned result with a ' similarly '.
()8) Deductions from Menelaus' s Theorem.
III. 2 proves, by means of I. 14, 10 and III. 1, that, if ABC,
A'B'C' be two spherical triangles in which A = A', and C, C/
are either equal or supplementary, sin c/sin a = siii-c'/sin a'
and conversely. The particular case in which C, C' are right
angles gives what was afterwards known as the ' regula
quattuor quantitatum' and was fundamental in Arabian
trigonometry. 2 A similar association attaches to the result of
III. 3, which is the so-called c tangent ' or c shadow-rule ' of the
Arabs. If ABC, A'B'C' be triangles right-angled at A, A', and
(7, C' are equal and both either > or < 90, and if P, P' be
the poles of AC, A'C', then
sinAB _ sinJ/jy sin BP
sin AC ~ sin A'C' ' sin B'P' "
Apply the triangles so that (7 falls on C 9 C'B' on GB as GE,
and G A' on GA as CD ; then the result follows directly from
III. 1. Since sin BP = cos AB, and sin B'P' = cos A'B', the
result becomes
sin CM tan.A.B
which is the c tangent-rule ' of the Arabs. 3
1 Ptolemy, Syntaxis, i. 13, vol. i, p. 76.
2 See Braunmiihl, Gesch. der Trig, i, pp. 17, 47, 58-60, 127-9,
8 Cf. Braunmtihl. 00. cit. i, DD. 17-18. 58. 67-9. &c.
MENELAUS'S SPH'AERIGA
269
It follows at once (Prop. 4) that, if AM, A'M' are great
circles drawn perpendicular to the bases BO, B'C' of two
spherical triangles ABC, A'B'W in which B = K,C=C',
sin BM sin MC / . . . . . , tan AM \
~~ * ir^f ( since both are equal to - J/T/"/ r
sin M 'C' \ ^ tan AM /
sin K'M'
III. 5 proves that, if there are two spherical triangles ABC,
p'
/
D(A')
'C 1
A'KC' right-angled at A, A' and such that (7=C", while 6
and b' are less than 90,
sin (a + b) sin (a' -f //)
sin (a //) sin (a' b')
from which we may deduce 1 the formula
sin (a -f b) _ 1 -f cos C
sin (a />) ~~ 1 cos C '
which is equivalent to tan b = tan a cos C.
(y) Anharmonic property of four great circles through
otie point.
But more important than the above result is the fact that
the proof assumes as known the anhar-
monic property of four great circles
drawn from a point on a sphere in rela-
tion to any great circle intersecting them
all, viz. that, if ABCD, A'R'Wl)' be two
transversals,
sin AD sin BC __ sin A'D'
" " i "\/^ A Tl """"" " f\t/^t * A / li/
sin DO sin A B sin D C sin A B
1 Braunmiihl, op. cit. i, p. 18; BjOrnbo, p. 96.
270 TRIGONOMETRY
It follows that this proposition was known before Mene-
laus's time. It is most easily proved by means of ' Menelaus's
Theorem', III. 1, or alternatively it may be deduced for the
sphere from the corresponding proposition in plane geometry,
just as Menelaus's theorem is transferred by him from the
plane to the sphere in III. 1. We may therefore fairly con-
clude that both the* anharmonic property and Menelaus's
theorem with reference to the sphere were already included
in some earlier text-book ; and, as Ptolemy, who built so much
upon Hipparchus, deduces many of the trigonometrical
formulae which he uses from the one theorem (III. 1) of
Menelaus, it seems probable enough that both theorems were
known to Hipparchus. The corresponding plane theorems
appear in Pappus among his lemmas to Euclid's Porisms? and
there is therefore every probability that they were assumed
by Euclid as known.
(S) Propositions analogous to End. VI. 3.
Two theorems following, III. 6, 8, have their analogy in
Eucl. VI. 3. In III. 6 the vertical angle A of a spherical
triangle is bisected by an arc of a great circle meeting BG in
D, and it is proved that sin BD/sin DC = sin JiA/sin AC\
in III. 8 we have the vertical angle bisected both internally
and externally by arcs of great circles meeting BC in D and
E 9 and the proposition proves the harmonic property
sin BE _ sin 7)
smEC ~~ sin DC '
III. 7 is to the effect that, if arcs of great circles be drawn
through B to meet the opposite side AC of a spherical triangle
in D, E so that ZABD = / EBC, then
sin EA . sin AD sin 2 AB
As this is analogous to plane propositions given by Pappus as
lemmas to different works included in the Treasury of
Analysis, it is clear that these works were familiar to
Menelaus.
1 Pappus, vii, pp. 870-2, 874.
MENELAUS'S SPHAERICA 271
III. 9 and III. 10 show, for a spherical triangle, that (1) the
great circles bisecting the three angles, (2) the great circles
through the angular points meeting the opposite sides at
right angles meet in a point.
The remaining propositions, III. 11-15, return to the same
sort of astronomical problem as those dealt with in Euclid's
Phaeiiomena, Theodosius's tiphaeriea and Book II of Mene-
laus's own work. Props. 11-14 amount to theorems in
spherical trigonometry such as the following.
Given arcs a lf 2 , or.., 4 , ft l9 2 , 3 , /? 4 , such that
90
and also a x > @ 19 a 2 > a , 3 > 3I 4 > /3 4 ,
(1) If sin a t : sin a 2 : sin 3 : sin a 4 = sin )3j : sin/J 2 : sin /3 3 : sin 4 ,
then .=^>i=,.
If sin( I +/g 1 ) _ sin ( 2 + /3 2 ) _ sMa :) L + j8 3
sin(a l -/3 1 ) sin (a 2 -/? 2 ) sin(a 3 -^ a
,1
then
(S\ If sin fai-ot,) sn(-/ 2 )
sin(a 3 4 ) sin (/3., - /3 4 )
Again, given three series of three arcs such that
! > 2 > a 3 , t > 2 > |8 3I 90 > y l > y 2 > y 3 ,
nd sin (o^ y a ) : sin (a 2 y 2 ) : sin (a 3 y :j )
= sin (j8 1 - y t ) : sin (0 2 - y 2 ) : sin (/3 3 - y 3 )
= siny^sinyg-.sinyg
TRIGONOMETRY
i- 2 ,
> o o 5 an(i
272
(1) K ,>/,
then
(2) If ft l < !
then
III. 15, the last proposition, is in four parts. The first part
is the proposition corresponding to Theodosius III. 11 above
alluded to. Let BA, BC be two quadrants of great circles
(in which we easily recognize the equator and the ecliptic),
P the pole of the former, PA 19 PA% quadrants of great circles
meeting the other quadrants in A 19 A% and O l9 0% respectively.
Let R be the radius of the sphere, r, r, , r :j the radii of the
'parallel circles' (with pole P) through 0, C 19 C :J respectively.
Then shall
sn
Rr
sn
In the triangles PCG^ BA^C. A the angles at C,
and the angles at (7 3 equal ; therefore (III. 2)
sin PG
are right,
sn
sin PC*
MENELAUS'S SPHAERICA 273
But, by III. 1 applied to the triangle BC^ cut by the
transversal
sn sn C sin
sin BA% sin #C 3 sin PC\
sin A^AS _ sin PJ. 1 sin BA. } __ sin P^ T sin PC
sin (7, 6 Y ., "" sin PC. sin J5(7o " " sin PC* sin PC,
I t> I O 1 .)
from above,
Part 2 of the proposition proves that, if PC Z A 2 be drawn
such that sin 2 P6 f 2 = sin PA 2 . sin PC, or ?' 2 2 = jRr (where r 2 is
the radius of the parallel circle through (7 2 ), BC 2 BA 2 is a
maximum, while Parts 3, 4 discuss the limits to the value of
the ratio between the arcs A^A^ and C\C^
Nothing is known of the life of CLAUDIUS PTOLEMY except
that he was of Alexandria, made observations between the
years 
