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Full text of "A History Of Greek Mathematics Vol II"

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OSMANIA UNIVERSITY LIBRARY 
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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 A.D. 125 and 141 or perhaps 151, and therefore presum- 
ably wrote his great work about the middle of the reign of 
Antoninus Pius (A.I). 138-61). A tradition handed down by 
the Byzantine scholar Theodoras Meliteniota (about 1361) 
states that he was born, not at Alexandria, but at Ptolemais 
fl 'EpfjLtiov. Arabian traditions, going back probably to 
Hunain b. Ishaq, say that he lived to the age of 78, and give 
a number of personal details to which too much weight must 
not be attached. 



The MadrjfjLartKr) crvvragi* (Arab. AlmcKjesf). 

Ptolemy's great work, the definitive achievement of Greek 
astronomy, bore the title MadijfjiaTiKfjs Swrdgecos fttftXfa iy, 
the Mathematical Collection in thirteen Books. By the time 
of the commentators who distinguished the lesser treatises on 
astronomy forming an introduction to Ptolemy's work as 
fUKpb? acrrpoi>o//oi$/<tJ'oy (TOTTO?), the 'Little Astronomy', the 
book came to be called the ' Great Collection ', fitydXr) <TI;J>- 
raty. Later still the Arabs, combining the article Al with 



274 TRIGONOMETRY 

the superlative pfyurros, made up a word Al-majisti, which 
became Almagest ; and it has been known by this name ever 
since. The complicated character of the system expounded 
by Ptolemy is no doubt responsible for the fact that it 
speedily became the subject of elaborate commentaries. 

Commentaries on the Syntaxis. 

Pappus 1 cites a passage from his own commentary on* 
Book I of the Mathematics, which evidently means Ptolemy's 
work. Part of Pappus's commentary on Book V, as well as 
his commentary on Book VI, are actually extant in the 
original. Theon of Alexandria, who wrote a commentary on 
the Syntaxis in eleven Books, incorporated as much as was 
available of Pappus's commentary on Book V with full 
acknowledgement, though not in Pappus's exact words. In 
his commentary on Book VI Theon made much more partial 
quotations from Pappus ; indeed the greater part of the com- 
mentary on this Book is Theon's own or taken from other 
sources. Pappus's commentaries are called scholia, Theon's 
vTTOfjLvriijiaTa. Passages in Pappus's commentary on Book V 
allude to ' the scholia preceding this one ' (in the plural), and 
in particular to the scholium on Book IV. It is therefore all 
but certain that lie wrote on all the Books from I to VI at 
least. The text of the eleven Books of Theon's commentary 
was published at Basel by Joachim Camerarius in 1538, but 
it is rare and, owing to the way in which it is printed, with 
insufficient punctuation marks, gaps in places, and any number 
of misprints, almost unusable ; accordingly little attention has 
so far been paid to it except as regards the first two Books, 
which were included, in a more readable form and with a Latin 
translation, by Halma in his edition of Ptolemy. 

Translations and editions. 

The ttyniaxis was translated into Arabic, first (we are told) 
by translators unnamed at the instance of Yahya b. Khalid b. 
Barmak, then by.al-Hajjaj, the translator of Euclid (about 
786-835), and again by the famous translator Lshaq b. Hunain 
(d. 910), whose translation, as improved by Thabit b. Qurra 
1 Pappus, viii, p. 1106. 13. 



PTOLEMY'S S7NTAXI8 275 

(died 901), is extant in part, as well as the version by Nasirad- 
din at-Tusi (1201-74). 

The first edition to be published was the Latin translation 
made by Gherard of Cremona from the Arabic, which was 
finished in 1175 but was not published till 1515, when it was 
brought out, without the author's name, by Peter Liechten- 
stein at Venice. A translation from the Greek had been made 
about 1160 by an unknown writer for a certain Henricus 
Aristippus, Archdeacon of Catania, who, having been sent by 
William I, King of Sicily, on a mission to the Byzantine 
Emperor Manuel I. Comnenus in 1158, brought back with 
him a Greek manuscript of the Syntaxis as a present; this 
translation, however, exists only in manuscripts in the Vatican 
and at Florence. The first Latin translation from the Greek 
to be published was that made by Georgius ' of Trebizond ' for 
Pope Nicolas V in 1451 ; this was revised and published by 
Lucas Gauricus at Venice in 1528. The editio princeps of the 
Greek text was brought out by Grynaeus at Basel in 1538. 
The next complete edition was that of Halma published 
1813-16, which is now rare. All the more welcome, there- 
fore, is the definitive Greek text of the astronomical works 
of Ptolemy edited by Heiberg (1899-1907), to which is now 
added, so far as the Syntaxis is concerned, a most valuable 
supplement in the German translation (with notes) by Manitius 
(Teubner, 1912-13). 

Summary of Contents. 

The Syntaxis is most valuable for the reason that it con- 
tains very full particulars of observations and investigations 
by Hipparchus, as well as of the earlier observations recorded 
by him, e.g. that of a lunar eclipse in 721 B.C. Ptolemy 
based himself very largely upon Hipparchus, e.g. in the 
preparation of a Table of Chords (equivalent to sines), the 
theory of eccentrics and epicycles, &c. ; and it is questionable 
whether he himself contributed anything of great value except 
a definite theory of the motion of the five planets, for which 
Hipparchus had only collected material in the shape of obser- 
vations made by his predecessors and himself. A very short 
indication of the subjects of the different Books is all that can 



276 TRIGONOMETRY 

bo given here. Book I: Indispensable preliminaries to the 
study of the Ptolemaic system, general explanations of 
the different motions of the heavenly bodies in relation to 
the earth as centre, propositions required for the preparation 
of Tables of Chords, the Table itself, some propositions in 
spherical geometry leading to trigonometrical calculations of 
the relations of arcs of the equator, ecliptic, horizon and 
meridian, a * Table of Obliquity ', for calculating declinations 
for each degree-point on the ecliptic, and finally a method of 
finding the right ascensions for arcs of the ecliptic equal to 
one-third of a sign or 10. Book II: The same subject con- 
tinued, i.e. problems on the sphere, with special reference to 
the differences between various latitudes, the length of the 
longest day at any degree of latitude, and the like. Book III : 
On the length of the year and the motion of the sun on the 
eccentric and epicycle hypotheses. Book IV : The length of the 
months and the theory of the moon. Book V : The construc- 
tion of the astrolabe, and the theory of the moon continued, 
the diameters of the sun, the moon and the earth's shadow, 
the distance of the sun and the dimensions of the sun, moon 
and earth. Book VI : Conjunctions and oppositions of sun 
and moon, solar and lunar eclipses and their periods. Books 
VII and VIII are about the fixed stars and the precession of 
the equinoxes, and Books IX-XIII are devoted to the move- 
ments of the planets. 

Trigonometry in Ptolemy. 

What interests the historian of mathematics is the trigono- 
metry in Ptolemy. It is evident that no part of the trigono- 
metry, or of the matter preliminary to it, in Ptolemy was new. 
What he did was to abstract from earlier treatises, and to 
condense into the smallest possible space, the minimum of 
propositions necessary to establish the methods and formulae 
used. Thus at the beginning of the preliminaries to the 
Table of Chords in Book I he says : 

' We will first show how we can establish a systematic and 
speedy method of obtaining the lengths of the chords based on 
the uniform use of the^ smallest possible number of proposi- 
tions, so that we may not only have the lengths of the chords 



PTOLEMY'S NYNTAXia 277 

set out correctly, but may be in possession of a ready proof of 
our method of obtaining them based on geometrical con- 
siderations/ l 

He explains that he will use the division (1) of the circle into 
360 equal parts or degrees and (2) of the diameter into 120 
equal parts, and will express fractions of these parts on the 
sexagesimal system. Then come the geometrical propositions, 
as follows. 

(a) Lemma for finding sin 18 and sin 36. 

To find the side of a pentagon and decagon inscribed in 
a circle or, in other words, the chords subtending arcs of 72 
and 36 respectively. 

Let AH be the diameter of a circle, O the centre, OC the 
radius perpendicular to A B. 

Bisect OB at JJ, join DC, and measure 
I)E along DA equal to DC. Join EC. 

Then shall OK be the side of the in- 
scribed regular decagon, and EC the side 
of the inscribed regular pentagon. 

For, since OB is bisected at />, 

)*= DE* 




Therefore BE. EO = 0(7 2 = OS 2 , 

and BE is divided in extreme und mean ratio. 

But (Eucl. XIII. 9) the sides of the regular hexagon and the 
regular decagon inscribed in a circle when placed in a straight 
line with one another form a straight line divided in extreme 
and mean ratio at the point of division. 

Therefore, BO being the side of the hexagon, EO i^the side 
of the decagon. 

Also (by Eucl. XIII. 10) 

(side of pentagon) 2 = (side of hexagon) 2 + (side of decagon) 2 



therefore EC is the side of the regular pentagon inscribed 
in the circle. 

1 Ptolemy, Syutaxi*) i. 10, pp. 31 2. 



278 TRIGONOMETRY 



The construction in fact easily leads to the results 



where a is the radius of the circle. 

Ptolemy does not however use these radicals, but calculates 
the lengths in terms of ' parts ' of the diameter thus. 

DO = 30, and DO 2 = 900 ; 00 = 60 and 00* = 3600 ; 
therefore DE* = DO 2 = 4500, and DE = 67* 4' 55" nearly ; 
therefore side of decagon or (cixl.36) = /J#-jDO = 37''4'55" 
Again 0# 2 = (37/'4' 55")* = 1375 .4' 15", and 0(7 2 =3600; 
therefore CE* = 4975 . 4' 15", and CE = 70'' 32' 3" nearly, 
i.e. side of pentagon or (crd. 72) = 70'' 32' 3". 

The method of extracting the square root is explained by 
Theon in connexion with the first of these cases, \/4500 (see 
above, vol. i, pp. 61-3). 

The chords which are the sides of other regular inscribed 
figures, the hexagon, the square and the equilateral triangle, 
are next given, namely, 

crd. 60 = 60'', 

crd. 90 = \/(2 . 60-) = >/(7200) = 84'' 51' 10", 

crd. 120 = V(3 . 60 2 ) = \/( 10800) = 103'' 55' 23". 



(/3) Equivalent of sin 2 -f cos- = 1. 
It is next observed that, if x be any arc, 

(crd. a) 2 + {crd. (180 -a;)} 2 = (diam.) 2 = 120 2 , 

a formula which is of course equivalent to sin 2 Q + cos 2 6 = 1. 

We c&i therefore, from crd. 72, derive crd. 108, from 
crd. 36, crd. 144, and so on. 

(y) 'Ptolemy's theorem', giving the equivalent of 
sin (6 (f>) = sin cos cos 6 sin 0. , 

The next step is to find a formula which will give us 
crd. (a j8) when crd. a and crd. ft are given. (This for 
instance enables us to find crd. 12 from crd. 72 and crd. 60.) 



PTOLEMY'S SYNTAX1S 



279 



The proposition giving the required formula depends upon 
a lemma, which is the famous ' Ptolemy's theorem '. 

Given a quadrilateral A BCD inscribed in a circle, the 
diagonals being AC, BD, to prove that 

. DC+ AD . EC. 



AC.BD = 

The proof is well known. Draw BE so that the angle ABE 
is equal to the angle DBC, and let BE 
meet AC in E. 

Then the triangles ABE, DBC are 
equiangular, and therefore 

AB:AE=BD:DC, I I \ X ^*> 

or AB.DC=AE.BD. (1) 

Again, to each of the equal angles 
ABE, DBC add the angle EBD ; 

then the angle ABD is equal to the angle EBC, and the 
triangles ABD, EBC are equiangular ; 




therefore BC : CE = BD : DA, 

or AD.BC=CE.BD. 

By adding (1) and (2), we obtain 

AB . DC+AD . BC = AC. BD. 



(2) 



Now let AB, AC be two arcs terminating at A, the extremity 
of the diameter A I) of a circle, and let 
AC (= a) be greater than AB ( = /3,. 
Suppose that (crd. AC) and (crd. AB) 
are given: it is required to find 
(crd. BC). 

Join BD, CD. 

Then, by the above theorem, 




Now AB, AC are given; therefore BD = crd. (180 -AB) 
and CD = crd. (180 ^4(7) are known. And AD is known. 
Hence the remaining chord BG (crd. BC) is known. 



280 TRIGONOMETRY 

The equation in fact gives the formula, 

{crd. (a- )} . (crd. 180) = (crd. a) . {crd. (180-/3)} 

-(crd. /3).{crd.(180-a)}, 

which is, of course, equivalent to 

sin (0 0) = sin 6 cos cos 6 sin 0, where a = 20, ft = 20. 
By means of this formula Ptolemy obtained 

crd. 12 = crd. (72- 60) = 12'' 32' 36". 



(5) Equivalent of siii^d = (1 cos 0). 

But, in order to get the chords of smaller angles still, we 
want a formula for finding the chord of half an arc when the 
chord of the arc is given. This is the subject of Ptolemy's 
next proposition. 

Let BG be an arc of a circle with diameter AC, and let the 
arc BC be bisected at D. Given (crd. EG), it is required to 
find (crd. DC). 

Draw DF perpendicular to AC, 
and join AB, AD, BD, DC. Measure 
AE along AC equal to A Li, and join 
DK. 

Then shall FC be equal to EF, or 
FO shall be half the difference be- 
tween AC and AB. 

For the triangles ABD, AED are 
equal in all respects, since two sides 

of the one are equal to two sides of the other and the included 
angles BAD, BAD, standing on equal arcs, are equal. 

Therefore ED = BD = DC, 

and the right-angled triangles DEF, DCF are equal in all 
respects, whence EF = FC, or CF = %(AC-AB). 

Now AC. CF = CD*, 

whence (crd. CD)* = \AG (AC-AB) 




= 4 (crd. 180).{(crd.l80 )-~(crd.l80 ~^a)j. 
This is, of course, equivalent to the formula 

= (l-cos 0). 



PTOLEMY'S tiYNTAXIS 281 

By successively applying this formula, Ptolemy obtained 
(crd. 6), (crd. 3) and finally (crd. 1|) = 12' 34' 15" and 
(crd. |) = OP 47' 8". But we want a table going by half- 
degrees, and hence two more things are necessary ; we have to 
get a value for (crd. 1) lying between (crd. l) and (crd. |), 
and we have to obtain an addition formula enabling us when 
(crd. a) is given to find {crd. (a + ^)}, and so on. 

(e) Equivalent of cos (6 + 0) = cos 6 cos sin sin <f>. 

To find the addition formula. Suppose AD is the diameter 
of a circle, and AB, BC two arcs. Given (crd. AB) and 
(crd. BC), to find (crd. AC), Draw the diameter BOE, and 
join CE, CD, DE, BD. 

Now, (crd, AB) being known, 
(crd. BD) is known, and therefore 
also (crd. DE), which is equal to 
(crd. AB)] and, (crd. BC) being 
known, (crd. CE) is known. 

And, by Ptolemy's theorem, 

BD . CE = BC . DE + BE . CD. 

The diameter BE and all the chords in this equation except 
CD being given, we can find CD or crd. (180 AC). We have 
in fact 




(crd. 180) . [crd. 

= !crd.(180-4#)].{c^ 

thus crd. (180 AC) and therefore (crd. AC) is known. 
If AB = 20, BC =20, the result is equivalent to 

cos (6 + 0) = cos cos -sin Q sin 0. 

(0 Method of interpolation based on formula 
sin a /sin ft < a//3 (where % TT > a > /9). 

Lastly we have to find (crd. 1), having given (crd. 1|) and 
(crd. |). 

Ptolemy uses an ingenious method of interpolation based on 
a proposition already assumed as known by Aristarchus. 

If AB, BC be unequal chords in a circle, BC being the 



282 



TRIGONOMETRY 




greater, then shall the ratio of CB to BA be less than the 

ratio of the arc CB to the arc BA. 

Let BD bisect the angle ABC, meeting AC in E and 

the circumference in D. The arcs 
AD, DC are then equal, and so are 
the chords AD, DC. Also CE>EA 
(since CB:BA = CE:EA). 

Draw DF perpendicular to AC; 
then AD>DE>DF, so that the 
circle with centre D and radius DE 
will meet D A in G and DF produced 
in//. 

Now FE: EA = A FED : &AED 

< (sector IIED) : (sector QED) 

< LFDE-.LEDA. 

Componeiido, FA : AE < L FDA : L ADE. 
Doubling the antecedents, we have 

CA : A E < L CD A : L A DE, 
and, separando, CE : EA < L CDE: L EDA ; 
therefore (since CB-.BA = CE-.EA) 

CB-.BA < LODE: LEVA 

< (arc CB) : (arc BA), 

i. e. (crd. CB) : (crd. BA) < (arc CB) : (arc BA ). 

[This is of course equivalent to sin a : sin /8 < a : ft, where 



It follows (1) that (crd. 1 ) : (crd. |) < 1 : | , 
and (2) that (crd. l) : (crd. 1) <!$:!. 

That is, | . (crd. |) > (crd. 1 ) > f . (crd. l). 

But (crd. |) = OP 47' 8", so that |(crd. |) = \v 2' 50" 
nearly (actually V 2' 50%"); 

and (crd. l) = V 34' 15", so that |(crd. l) = 1?' 2' 60". 

Since, then, (crd. 1) is both less and greater than a length 
which only differs inappreciably from IP 2' 50", we may say 
that (crd. 1) = 1?' 2' 50" as nearly as possible. 



PTOLEMY'S SYNTAX IS 283 

(rj) Table of Chord*. 

From this Ptolemy deduces that (crd.f ) is very nearly 
O/' 3V 25", and by the aid of the above propositions he is in 
a position to complete his Table of Chords for arcs subtending 
angles increasing from to 180 by steps of ; in other 
words, a Table of Sines for angles from J to 90 by steps 
of i. 

(0) Further use of pro r portioiial increase. 

Ptolemy carries further the principle of proportional in- 
crease as a method of finding approximately the chords of 
arcs containing an odd number of minutes between 0' and 20'. 
Opposite each chord in the Table he enters in a third column 
- 3 ^o th of the excess of that chord over the one before, i.e. the 
chord of the arc containing 30' less than the chord in question. 
For example (crd. 2) is stated in the second column of the 
Table as W 37' 4". The excess of (crd. 2) over (crd.2) in the 
Table is 0'' 31' 24": ^th of this is 0*' l' 2" 48'", which is 
therefore the amount entered in the third column opposite 
(crd. 2J). Accordingly, if we want (crd. 2 25'), we take 
(crd. 2) or 2P 5' 40" and add 25 times OP l'2"48'"; or we 
take (crd. 2) or 2/' 37' 4" and subtract 5 times 0'' l' 2" 48"'. 
Ptolemy adds that if, by using the approximation for 1 and 
^, we gradually accumulate an error, we can check the calcu- 
lation by comparing the chord with that of other related arcs, 
e.g. this double, or the supplement (the difference between the 
arc and the semicircle). 

Some particular results obtained from the Table may be 
mentioned. Since (crd. 1) = 1 P 2' 50", the whole circumference 
= 360 (1^2' 50"), nearly, and, the length of the diameter 
being 1201', the value of TT is 3 (1 + A + *Sro) = 3 + *Tr + rahy> 
which is the value used later by Ptolemy and is equivalent to 
3-14166... Again, </3 = 2 sin 60 and, 2 (crd. 120) being 
equal to 2 (103>> 55' 23"), we have ^3 = ^ (103 + ff + ?ffs) 

43 55 23 



which is correct to 6 places of decimals. Speaking generally, 



284 



TRIGONOMETRY 



the sines obtained from Ptolemy's Table are correct to 5 
places. 

(i) Plane trigonometry iu effect used. 

There are other cases in Ptolemy in which plane trigono- 
metry is in effect used, e.g. in the determination of the 
eccentricity of the sun's orbit. 1 Suppose that ACBD is 
the eccentric circle with centre 0, 
and AB, CD are chords at right 
angles through E, the centre of the 
earth. To find OE. The arc BU 
is known ( = a, say) as also the arc 
GA (=0). If BF be the chord 
parallel to CD, and CG the chord 
parallel to AB, and if JV, P be the 
middle points of the arcs BF, GC, 
Ptolemy finds (1) the arc BF 
( = a + 0-180 ), then the chord BF, 

crd. (a +0-180), then the half of it, (2) the arc GO 
= arc (a + 02/8) or arc (a 0), then the .chord 76', and 
lastly half of it. He then adds the squares on the half- 
chords, i.e. he obtains 




that is, OJP/v* = cos 2 J (a + j8) + sin 2 i (a - 0). 

He proceeds to obtain the angle OEC from its sine OR/UE, 
which he expresses as a chord of double the angle in the 
circle on OE as diameter in relation to that diameter. 

Spherical trigonometry: formulae in solution of 
spherical triangles. 

In spherical trigonometry, as already stated, Ptolemy 
obtains everything that he wants by using the one funda- 
mental proposition known as 'Meiielaus's theorem' applied 
to the sphere (Menelaus III. 1), of which he gives a proof 
following that given by Menelaus of the first case taken in 
his proposition. Where Ptolemy has occasion for other pro- 
positions of Menelaus's Sphaerica, e.g. III. 2 and 3, he does 

1 Ptolemy, Syntaxia, iii. 4, vol. i, pp. 234-7. 



PTOLEMY'S SYNTAX IS 285 

not quote those propositions, as he might have done, but proves 
them afresh by means of Menelaus's theorem. 1 The appli- 
cation of the theorem in other cases gives in effect the 
following different formulae belonging to the solution of 
a spherical triangle ABC right-angled at G', viz. 

sin a = sine sin -4, 
tan a = sin b tan A, 

cos c = cos a cos 6, 

tan b '= tan r cos A. 

One illustration of Ptolemy's procedure will be sufficient. 2 
Let 1 1 Air be the horizon, PEZH the meridian circle, EE' 
the equator, '//// the ecliptic, F an 
equinoctial point. Let EK f , %// 
cut the horizon in A, B. Let P be 
the pole, and let the great circle 
through P, B cut the equator at (!. 
Now let it be required to find the 
time which the arc FB of the ecliptic 
takes to rise; this time will be 
measured by the arc FA of the 
equator. (Ptolemy has previously found the length of the 
arcs EC, the declination, and FC. the right ascension, of B, 
I. 14, 16.) 

By Menelaus's theorem applied to the arcs AE', E'P cut by 
the arcs All' , PC which also intersect one another in 7?. 

crd.27 J 7/' __ crd. 2P# crd.26M 
crd. 2 U'K' ~ crd.TTTC' ' crd. 2 AE'* 

sinP//' sinP7i sin GA 
tnat> is. -. y Tf r ^. . ~ r7-< * "" * i T/ * 

sm/TA' am /ft' sin -4 A 




Now sinP7/' = cos//'A T ', sinP=cos C f , and si 
therefore cot H' E' = cot 6' . sin 6!A , 

in other words, in the triangle ABC right-angled at (7, 

cot A = cot a sin 6, 
or tan a = sin ft tan -4. * 

1 Syntoxis, vol. i, p. 169 and pp. 126-7 respectively. 
9 J6. f vol. i, pp. 121-2. 



286 TRIGONOMETRY 

Thus AC is found, and therefore FC-AC or FA. 

The lengths of BO, FC are found in I. 14, 16 by the same 
method, the four intersecting great circles used in the figure 
being in that case the equator EE', the ecliptic ZZ' 9 the great 
circle PBCP' through the poles, and the great circle PKLP' 
passing through the poles of both the ecliptic and the equator. 
In this case the two arcs PL, AE' are cut by the intersecting 
great circles PC, FK, and Menelaus's theorem gives (1) 

sin PL _ sin CP sinUP 
sin KL "" siiTBO ' sin FK " 

But sinPZ=l, sinKL = smBFC, sin OP =1, *inFK = l, 
and it follows that 

sin BO = sin #Psin BFC, 
corresponding to the formula for a triangle right-angled at C, 

sin a = sin c sin A. 

(2) We have 

sin PK _ sinPjS sin OP 

sin KL ~~ sin sin PA ' 

and sin PK = cos A'7, = cos JMY7, sin P# = cos BG, sin FL=l, 
so that tan jBC = sin C'Ptan BFT!, 

corresponding to the formula 

tan a = sin b tan A. 

While, therefore, Ptolemy's method implicitly gives the 
formulae for the solution of right-angled triangles above 
quoted, he does not speak of right-angled triangles at all, but 
only of arcs of intersecting great circles. The advantage 
from his point of view is that he works in sines and cosines 
only, avoiding tangents as such, and therefore he requires 
tables of only one trigonometrical ratio, namely the sine (or, 
as he has it, the chord of the double arc). 

The Ana/cmma. 

Two other works of Ptolemy should be mentioned here. 
The first is the Analemma. The object of this is to explain 
a method of representing on one plane the different points 



THE ANALEMMA OF PTOLEMY 287 

and arcs of the heavenly sphere by means of orthogonal 
projection upon three planes mutually at right angles, the 
meridian, the horizon, and the ' prime vertical '. The definite 
problem attacked is that of showing the position of the sun at 
any given time of the day, and the use of the method and 
of the instruments described in the book by Ptolemy was 
connected with the construction of sundials, as we learn from 
Vitruvius. 1 There was another dvaXr^ifjia besides that of 
Ptolemy ; the author of it was Diodorus of Alexandria, a con- 
temporary of Caesar and Cicero (' Diodorus, famed among the 
makers of gnomons, tell me the time ! ' says the Anthology 2 ), 
and Pappus wrote a commentary upon it in which, as h3 tells 
us, 3 he used the conchoid in order to trisect an angle, a problem 
evidently required in the Aiialemma in order to divide any 
arc of a circle into six equal parts (hours). The word 
ar/aXiy/z/za evidently means ' taking up ' (' Aufiiahme ') in the 
sense of ' making a graphic representation ' of something, in 
this case the representation on a plane of parts of the heavenly 
sphere. Only a few fragments remain of the Greek text of 
the A nalemma of Ptolemy; these are contained in a palimpsest 
(Ambros. Gr. L. 99 sup., now 491) attributed to the seventh 
century but probably earlier. Besides this, we have a trans- 
lation by William of Moerbeke from an Arabic version. 
This Latin translation was edited with a valuable commentary 
by the indefatigable Commandinus (Rome, 1562); but it is 
now available in William of Moerbeke's own words, Heiberg 
having edited it from Cod. Vaticanus Ottobon. lat. 1 850 of the 
thirteenth century (written in William's own hand), and in- 
cluded it with the Greek fragments (so tar as they exist) in 
parallel columns in vol. ii of Ptolemy's works (Teubner, 1907). 
The figure is referred to three fixed planes (1) the meridian, 
(2) the horizon, (3) the prime vertical; these planes are the 
planes of the three circles APZB y ACB, ZQO respectively 
shown in the diagram below. Three other great circles are 
used, one of which, the equator with pole P, is fixed; the 
other two are movable and were called by special names; 
the first is the circle represented by any position of the circle 
of the horizon as it revolves round COG' as diameter (GSM in 

1 Vitruvius, De architect, ix. 4. 2 Anil. Palat. xiv. 139. 

8 Pappus, iv, p. 246. 1. 



288 



TRIGONOMETRY 



the diagram is one position of it, coinciding with the equator), 
and it was called tKrrjpopos KVK\O$ (' the circle in six parts ') 
because the highest point of it above the horizon corresponds 
to the lapse of six hours ; the second, called the hour-circle, is 
the circle represented by any position, as BtiQA, of the circle 
of the horizon as it revolves round BA as axis. 

The problem is, as above stated, to find the position of the 
sun at a given hour of the day. In order to illustrate 
the method, it is sufficient, with A. v. Braunmuhl, 1 to take the 
simplest case where the sun is on the equator, i.e. at one of 
the equinoctial points, so that the hectemoron circle coincides 
with the equator. 

Let S be the position of the sun, lying on the equator M8C, 
P the pole, MZA the meridian, BOA the horizon, JitiQA the 
hour-circle, and let the vertical great circle ZtiV be drawn 
through S, and the vertical great circle ZQC through Z the 
zenith and C the east-point. 

We are given the arc S6 / =90 f, where / is the hofcr- 
angle, and the arc MB = 90 0, where is the elevation of 
the pole ; and we have to find the arcs >S'F (the sun's altitude). 




VC, the 'ascensional difference', SQ and QO. Ptolemy, in 
fact, practically determines the position of S in terms of 
certain spherical coordinates. 

Draw the perpendiculars, SF to the plane of the meridian, 
8H to that of the horizon, and SE to the plane of the prime 

1 Braunmuhl, Gescli. der Trigonomrtrie, i, pp. 12, 13, 



THE ANALEMMA OF PTOLEMY 



289 



vertical ; and draw FG perpendicular to BA, and ET to OZ. 
Join 7/G, and we have FG = 8U, GH = J%' = #21 

We now represent tiF in a separate figure (for clearness' 
sake, as Ptolemy uses only one figure), where B'Z'A' corre- 
sponds to BZA, P' to P and Q'M' to OM. Set off the arc 
P'S' equal to OS (= 90 -i), and draw S'F' perpendicular 
to O'M'. Then tf'J/'= 8M, and B'F'= 8F; it is as if in the 
original figure we had turned the quadrant MSC round MO 
till it coincided with the meridian circle. 

In the two figures draw IFK, 1'F'K' parallel to BA, B'A\ 
and LFG, L'F'G' parallel to OZ. O'//. 

Then (1) arc XI = areas' = arc (90 -NF), because if we 
turn the quadrant %8V about %0 till it coincides with the 




meridian, S falls on /, and V on It. It follows that the 
required arc #J r = arc /?'/' in the second figure. 

(2) To find the arc VC, set off G'X (in the second figure) 
along W equal to Fti or JW, and draw O'X through to 
meet the circle in X'. Then arc #',Y' = arc IT/; for it is as if 
we had turned the quadrant HVC about BO till it coincided 
with the meridian, when (since G'X = Ftf = (r7/) // would 
coincide with A" and K with X'. Therefore BFis also equal 
to JX'. 

(3) To find QC or ^, set off along T'F* in the second figure 
T'Y equal to F'N', and draw O'F through to F' on the circle. 

Then arc -B'F'ss arc Qf/; for it is as if we turned the prime 
vertical ZQC about ZO till it coincided with the meridian, 
when (since !TF= W= TE) E would fall on F, the radius 
OEQ on 0'FF X and Q on F'. 

(4) Lastly, arc BS = arc BL = arc B'L', because S, L are 

IS93.9 U 



290 TRIGONOMETRY 

both in the plane L8HG at right angles to the meridian; 
therefore arc 8Q = arc L'Z'. 

Hence all four arcs 8V 9 VC, QC, QS are represented in the 
auxiliary figure in one plane. 

So far the procedure amounts to a method of graphically 
constructing the arcs required as parts of an auxiliary circle 
in one plane. But Ptolemy makes it clear that practical 
calculation followed on the basis of the figure. 1 The lines 
used in the construction are SF= sin t (where the radius =1), 
FT=OFsiu(f>, FG = OF8in(9Q-<l)), and this was fully 
realized by Ptolemy. Thus he shows how to calculate the 
arc M, the zenith distance (= d, say) or its complement 8V, 
the height of the sun (~ Ji, say), in the following way. He 
says in effect: Since G is .known, and LF'Q'tf = 90 0, the 
ratios (YF' : F'T' and <YF' : Q'T' are known. 

O'F' D 

[In fact jpp = _- i - 8o0 _- a _, where D is the diameter 

of the sphere.] 

Next, since the arc MS or M'ti' is known [ = t\, and there- 
fore the arc 1*8' [= 90-*], the ratio of O'F' to /> is known 
[in fact O'F' /D- {crd. (I80-2<)}/2/>. 

It follows from these two results that 



Lastly, the arc #V( = h) being equal to /f /', the angle h is 
equal to the angle O'T'T' in the triangle I' O'T'. And in this 
triangle O'l', the radius, is known, while O'T' has been found ; 
and we have therefore 

O'T' crd.(2/t) crd. (180- 2t) crd. (180- 2 A) 

m , = p = ,y -- - * , fromabove. 

[In other words, sin h = cos t cos <f> ; or, if u = 8C = 90 /, 
sin h = sin u cos 0, the formula for finding sin h in the right- 
angled spherical triangle 8V CJ] 

For the azimuth o> (arc BV = arc B'X'), the figure gives 



_ _ fl^' _ S _ i 

tan o> - Qr (y - jfj, - Q , fj ; ff/ ^ - tan . ^ ^ , 

1 See Zeuthen in BiNiotheoa mathemaiica, i :5 , 1900, pp. 23-7. 



THE ANALEMMA OF PTOLEMY 291 

or tan VG tanSCcoaSCV in the right-angled spherical 
triangle SVG. 
Thirdly, 

ci/ ET/ ci/ rv /v tv i 

J^ ./^ \J J? A 

?> = 7TTo> TT/T/T/ = tan t . ; 

0\r 02 cos<A 

r 

that is, ^-^rrrr = ^"rrTr 7 which is Menelaus, Sphcterica, 
9 tan&lf am/Mf ^ 

III. 3, applied to the right-angled spherical triangles ZBQ, 
MBS with the angle B common. 

Zeuthen points out that later in the same treatise Ptolemy 
finds the arc 2oc described above the horizon by a star of 
given declination 8', by a procedure equivalent to the formula 

cos a = tan 5' tan 0, 

and this is the same formula which, as we have seen, 
Hipparchus must in effect have used in his Commentary on 
the Phaenomena of Eudoxus ami Aratus. 

Lastly, with regard to the calculations of the height h and 
the azimuth CD in the general case where the sun's declination 
is 8', Zeuthen has shown that they may be expressed by the 
formulae 

sin h = (cos 5' cos t sin #' tan <f>) cos 0, 

, , cos 8' sin t 
and tan co= . -~,~~ > 

' + (cos 8' cos t sin 8* tan <b) sin 

COS0 V r/ ^ 

cos 8' sin t 

Q| _ . 

sin ^ cos + cos 8* cos t sin <f> 

The statement therefore of A. v. Braunmuhl 1 that the 
Indians were the first to utilize the method of projection 
contained in the A nalemma for actual trigonometrical calcu- 
lations with the help of the Table of Chords or Sines requires 
modification in so far as the Greeks at all events showed the 
way to such use of the figure. Whether the practical applica- 
tion of the method of the Analemma for what is equivalent 
to the solution of spherical triangles goes back as far as 
Hipparchus is not certain ; but it is quite likely that it does. 

1 Braunmtthl, i, pp. 13, 14, 38-41, 



292 TRIGONOMETRY 

seeing that Diodorus wrote his Analemmtt in the next cen- 
tury. The other alternative source for Hipparchus's spherical 
trigonometry is the Menelaus-theorem applied to the sphere, 
on which alone Ptolemy, as we have seen, relies in his 
Syntaxis. In any case the Table of Chords or Sines was in 
full use in Hipparchus's works, for it is presupposed by either 
method. 

The Planisphaerium. 

With the Analemma of Ptolemy is associated another 
work of somewhat similar content, the Planisphaerium. 
This again has only survived in a Latin translation from an 
Arabic version made by one Maslama b. Ahmad al-Majiiti, of 
Cordova (born probably at Madrid, died 1007/8) ; the transla- 
tion is now found to be, not by Rudolph of Bruges, but by 
'Hermannus Secundus', whose pupil Rudolph was; it was 
first published at Basel in 1536, and again edited, with com- 
mentary, by Commandinus (Venice, 1558). It has 'been 
re-edited from the manuscripts by Heiberg in vol. ii. of his 
text of Ptolemy. The book is an explanation of the system 
of projection known as stereoyraphic, by which points on the 
heavenly sphere are represented on the plane of the equator 
by projection from one point, a pole ; Ptolemy naturally takes 
the south pole as centre of projection, as it is the northern 
hemisphere which he is concerned to represent on a plane. 
Ptolemy is aware that the projections of all circles on the 
sphere (great circles other than those through the poles 
which project into straight lines and small circles either 
parallel or not parallel to the equator) are likewise circles. 
It is curious, however, that he does not give any general 
proof of the fact, but is content to prove it of particular 
circles, such as the ecliptic, the horizon, &c. This is remark- 
able, because it is easy to show that, if a cone be described 
with the pole as vertex and passing through any circle on the 
sphere, i.e. a circular cone, in general oblique, with that circle 
as base, the section of the cone by the plane of the equator 
satisfies the criterion found for the ' subcontrary sections ' by 
Apollonius at the beginning of his Conies, and is therefore a 
circle. The fact that the method of stereographic projection is 
so easily connected with the property of subcontrary sections 



THE PLAN1SPHAERIUM OF PTOLEMY 293 

of oblique circular cones has led to the conjecture that Apollo- 
nius was the discoverer of the method. But Ptolemy makes no 
mention of Apollonius, and all that we know is that Syuesius 
of Gyrene (a pupil of Hypatia, and born about A.D. 365-370) 
attributes the discovery of the method and its application tc 
Hipparchus ; it is curious that he does not mention Ptolemy's 
treatise on the subject, but speaks of himself alone as having 
perfected the theory. While Ptolemy is fully aware that 
circles on the sphere become circles in the projection, he says 
nothing about the other characteristic of this method of pro- 
jection, namely that the angles on the sphere are represented 
by equal angles 011 the projection. 

We must content ourselves with the shortest allusion to 
other works of Ptolemy. There are, in the first place, other 
minor astronomical works as follows : 

(1) $acrei? a7rAai/a>*> aorrlpaw of which only Book II sur- 
vives, (2) 'TnoOtaeis rS>v TrAayoo/iej/eoi/ in two Books, the first 
of which is extant in Greek, the second in Arabic only, (3) the 
inscription in Canobus, (4) IIpo\*ip<QV xavovw &ara<n? Kal 
tyrjQotpopia. All these are included in Heiberg's edition, 
vol. ii. 

The Optics. 

Ptolemy wrote an Ujttic* in five Books, which was trans- 
lated from an Arabic version into Latin in the twelfth 
century by a certain Admiral Eugenius Siculus 1 ; Book I, 
however, and the end of Book V are wanting. Books I, II 
were physical, and dealt with generalities; in Book III 
Ptolemy takes up the theory of mirrors, Book IV deals with 
concave and composite mirrors, and Book V with refraction. 
The theoretical portion would suggest that the author was 
not very proficient in geometry. Many questions are solved 
incorrectly owing to the assumption of a principle which is 
clearly false, namely that * the image of a point on a mirror is 
at the point of concurrence of two lines, one of which is drawn 
from the luminous point to the centre of curvature of the 
mirror, while the other is the line from the eye to the point 

1 See G. Govi, L'ottica di Claudia Tolomeo di Euyenio Ammiraylio d* 
Sicilia, ... Torino, 1884; and particulars in G. Loria, Le sciense e#atte 
neW antica Grecia, pp. 570, 571. 



294 TRIGONOMETRY 

on the mirror where the reflection takes place ' ; Ptolemy uses 
the principle to solve various special cases of the following 
problem (depending in general on a biquadratic equation and 
now known as the problem of Alhazen), * Given a reflecting 
surface, the position of a luminous point, and the position 
of a point through which the reflected ray is required to pass, 
to find the point on the mirror where the reflection will take 
place/ Book V is the most interesting, because it seems to 
be the first attempt at a theory of refraction. It contains 
many details of experiments with different media, air, glass, 
and water, and gives tables of angles of refraction (r) corre- 
sponding to different angles of incidence (i) ; these are calcu- 
lated on the supposition that r and i are connected by an 
equation of the following form, 

r = aibi*, 

where a, 6 are constants, which is worth noting as the first 
recorded attempt to state a law of refraction. 

The discovery of Ptolemy's Optics in the Arabic at once 
made it clear that the work De S'peculis formerly attributed 
to Ptolemy is not his, and it is now practically certain that it 
is, at least in substance, by Heron. This is established partly 
by internal evidence, e.g. the style and certain expressions 
recalling others which are found in the same author's Auto- 
mata and Dloptra, and partly by a quotation by Damianus 
(0/6 hypotheses in Optics, chap. 14) of a proposition proved by 
' the mechanician Heron in his own Catoptrica ', which appears 
in the work in question, but is not found in Ptolemy's Optics, 
or in Euclid's. The proposition in question is to the effect 
that of all broken straight lines from the eye to the mirror 
and from that again to the object, that particular broken line 
is shortest in which the two parts make equal angles with the 
surface of the mirror; the inference is that, as nature does 
nothing in vain, we must assume that, in reflection from a 
mirror, the ray takes the shortest course, i.e. the angles of 
incidence and reflection are equal. Except for the notice in 
Damianus and a fragment in Olympiodorus * containing the 
proof of the proposition, nothing remains of the Greek text ; 

1 Olympiodorus on Aristotle, Meteor, iii. 2, ed. Ideler, ii, p. 96, ed. 
Stiive,pp.212. 5-213. 20. 



THE OPTICS OF FTOLEMY 295 

but the translation into Latin (now included in the Teubner 
edition of Heron, ii, 1900, pp. 316-64), which was made by 
William of Moerbeke in 1269, was evidently made from the 
Greek and not from the Arabic, as is shown by Graecisms in 
the translation. 

A mechanical work, lie pi fon&v. 

There are allusions in Simplicius 1 and elsewhere to a book 
by Ptolemy of mechanical content, nepl pojrSiv, on balancings 
or turnings of the scale, in which Ptolemy maintained as 
against Aristotle that air or water (e.g.) in their own ' place ' 
have no weight, and, when they are in their own * place ', either 
remain at rest or rotate simply, the tendency to go up or to 
fall down being due to the desire of things which are not in 
their own places to move to them. Ptolemy went so far as to 
maintain that a bottle full of air was not only not heavier 
than the same bottle empty (as Aristotle held), but actually 
lighter when inflated than when empty. The same work is 
apparently meant by the ' book on the elements ' mentioned 
by Simplicius. 2 Suidas attributes to Ptolemy three Books of 
Mechauica. 

Simplicius 3 also mentions a single book, Trepi &aoracrea>y, 
'On divieiisioii', i.e. dimensions, in which Ptolemy tried to 
show that the possible number of dimensions is limited to 
three. 

Attempt to prove the Parallel-Postulate. 

Nor should we omit to notice Ptolemy's attempt to prove 
the Parallel-Postulate. Ptolemy devoted a tract to this 
subject, and Proclus 4 has given us the essentials of the argu- 
ment used. Ptolemy gives, first, a proof of Eucl. I. 28, and 
then an attempted proof of I. 29, from which he deduces 
Postulate 5. 

1 Simplicius 011 Arist. De caelo, p. 710. 14, Heib. (Ptolemy, ed. Heib., 
vol. ii, p. 263). 

2 Ib., p. 20. 10 sq. 

3 /&., p. 9. 21 sq., .(Ptolemy, ed. Heib., vol. ii, p. 265). 

4 Proclus on Eucl. I, pp. 362. Usq., 365. 7-367. 27 (Ptolemy, ed. Heib., 
vol. ii, pp. 266-70). 



296 TRIGONOMETRY 

I. To prove I. 28, Ptolemy takes two straight lines AB, CD, 
and a transversal EFGH. We have to prove that, if the sum 





C /G D 

H 

of the angles BFG, FGD is equal to two right angles, the 
straight lines AB, CD are parallel, i.e. non-secant. 

Since AFG is the supplement of BFG, and FGC of FGD, it 
follows that the sum of the angles AFG, FGC is also equal to 
two right angles. 

Now suppose, if possible, that FB, GD, making the sum of 
the angles FG, FGD equal to two right angles, meet at K ; 
then similarly FA, GG making the sum of the angles AFG, 
FGC equal to two right angles must also meet, say at L. 

[Ptolemy would have done better to point out that not 
only are the two sums equal but the angles themselves are 
equal in pairs, i.e. AFG to FGD and FGC to BFG, and we can 
therefore take the triangle KFG and apply it to FG on the other 
side so that the sides FK, GK may lie along G(J, FA respec- 
tively, in which case GG, FA will meet at the point where 
K falls.] 

Consequently the straight lines LABK, LCDK enclose a 
space : which is impossible. 

It follows that AB, CD cannot meet in either direction ; 
they are therefore parallel. 

II. To prove I. 29, Ptolemy takes two parallel lines AB, 
CD and the transversal FG, and argues thus. It is required 
to prove that Z AFG + Z GGF = two right angles. 

For, if the sum is not equal to two right angles, it must be 
either (1) greater or (2) less. 

(1) If it is greater, the sum of the angles on the other side, 
BFG, FGD, which are the supplements of the first pair of 
angles, must be less than two right angles. 

But AF, GG are no more parallel than FB, GD, so that, if 
FG makes one pair of angles AFG, FGC togeilier greater than 



PTOLEMY ON THE PARALLEL-POSTULATE 297 

two rigid angles, it must also make the other pair BFG, FGD 
together greater than two right angles. 

But the latter pair of angles were proved less than two 
right angles : which is impossible. 

Therefore the sum of the angles AFG, FGC cannot be 
greater than two right angles. 

(2) Similarly we can show that the sum of the two angles 
AFGy FGC cannot be less than two right angles. 

Therefore L A FG + Z CGF - two right angles. 

[The fallacy here lies in the inference which I have marked 
by italics. When Ptolemy says that Ab\ CG are no more 
parallel than FB, GD, he is in effect assuming that through 
any o)ie point only one pandlel can bedrawn toagiven straight 
line, which is an equivalent for the very Postulate he is 
endeavouring to prove. The alternative Postulate is known 
as ' Playfair's axiom ', but it is of ancient origin, since it is 
distinctly enunciated in Proclus's note on Eucl. I. 31.] 

III. Post. 5 is now deduced, thus. 

Suppose that the straight lines making with a transversal 
angles the sum of which is less than two right angles do not 
meet on the side on which those angles are. 

Then, a fortiori, they will not meet on the other side on 
which are the angles the sum of whicli is greater than two 
right angles. [This is enforced by a supplementary proposi- 
tion showing that, if the lines met on that side, Eucl. I. 16 
would be contradicted.] 

Hemse the straight lines cannot meet in either direction : 
they are therefore parallel. 

But in that case the angles made with the transversal are 
eyual to two right angles : which contradicts the assumption. 

Therefore the straight lines will meet. 



XVIII 
MENSURATION: HERON OF ALEXANDRIA 

Controversies as to Heron's date. 

THE vexed question of Heron's date lias perhaps called 
forth as much discussion as any doubtful point in the history 
of mathematics. In the early stages of the controversy much 
was made of the supposed relation of Heron to Ctesibius. 
The Belopoewa of Heron has, in the best manuscript, the 
heading "JH/oau/oy Krrjo-ifiiov JBeAoTrouVca, and from this, coupled 
with an expression used by an anonymous Byzantine writer 
of the tenth century, 6 'Ao-Kprjvbs KTrjo-ifiios 6 rov X\gav8p<D$ 
"Hpou'oy KadrjyrjTrjs, 'Ctesibius of Ascra, the teacher of Heron 
of Alexandria *, it was inferred that Heron was a pupil of 
Ctesibius. The question then was, wlien did Ctesibius live ? 
Martin took him to be a certain barber of that name who 
lived in the time of Ptolemy Euergetes II, that is, Ptolemy VII, 
called Physcon (died 117 B.C.), and who is said to have made 
an improved water-organ l ; Martin therefore placed Heron at 
the beginning of the first century (say 126-60) B.C. But 
Philon of Byzantium, who repeatedly mentions Ctesibius by 
name, says that the first mechanicians (Ttyvlrai.) had the 
great advantage of being under kings who loved fame and 
supported the arts.' 2 This description applies much better 
to Ptolemy II Philadelphus (285-247) *ind Ptolemy III Euer- 
getes I (247-222). It is more probable, therefore, that Ctesibius 
was the mechanician Ctesibius who is mentioned by Athenaeus 
as having made an elegant drinking-horn in the time of 
Ptolemy Philadelphus 3 ; a pupil then of Ctesibius would 
probably belong to the end of the third and the beginning of 
the second century B.C. But in truth we cannot safely con- 
clude that Heron was an immediate pupil of Ctesibius. The 
Byzantine writer probably only inferred this from the title 

1 Athenaeus, Deipno-Soph. iv. c. 75, p. 174 b-e : cf. Vitruvius, x. 9, 13. 

2 Philon, Mechan St/nt., p. 50. 38, ed. SchOne. 

3 Athenaeus, xi. c. 97, p. 497 b-e. 



CONTROVERSIES AS TO HERON'S DATE 299 

above quoted ; the title, however, in itself need not imply 
more than that Heron's work was a new edition of a similar 
work by Ctesibius,and the Krrj(nftiov may even have been ad'ded 
by some well-read editor who knew both works and desired to 
indicate that the greater part of the contents of Heron's work 
was due to Ctesibius. One manuscript has^Hpan/os *A\*av- 
5pc<y BeAoTTou/ca, which corresponds to the titles of the other 
works of Heron and is therefore more likely to be genuine. 

The discovery of the Greek text of the Metrica by R. Schone 
in 1896 made it possible to fix with certainty an upper limit. 
In that work there are a number of allusions to Archimedes, 
three references to the x&piov aTroTopri of Apollonius, and 
two to ' the (books) about straight lines (chords) in a circle ' 
(StScLKTai 8e i/ roiy Trepi rS>v kv KVK\O> tvQti&v). Now, although 
the first beginnings of trigonometry may go back as far as 
Apollonius, we know of no work giving an actual Table of 
Chords earlier than that of Hipparchus. We get, therefore, 
at once the date 150 B.C. or thereabouts as the terminus post 
quein. A terminus aide quern is furnished by the date of the 
composition of Pappus's Collect ion] for Pappus alludes to, and 
draws upon, the works of Heron. As Pappus was writing in 
the reign of Diocletian (A.D. 284-305), it follows that Heron 
could not be much later than, say, A.D. 250. In speaking of 
the solutions by * the old geometers ' (ol naXcuol yeoyecrpa*) of 
the problem of finding the two mean proportionals, Pappus may 
seem at first sight to include Heron along with Eratosthenes, 
Nicomedes and Philon in that designation, and it has been 
argued, on this basis, that Heron lived long before Pappus. 
But a close examination of the passage 1 shows that this is 
by no means necessary. The relevant words are as follows : 

* The ancient geometers were not able to solve the problem 
of the two straight lines [the problem of finding two mean 
proportionals to them] by ordinary geometrical methods, since 
the problem is by nature '* solid " . . . but by attacking it with 
mechanical means they managed, in a wonderful way, to 
reduce the question to a practical and convenient construction, 
as may be seen in the Mesolaboii of Eratosthenes and in the 
mechanics of Philon and Heron . . . Nicomedes also solved it 
by means of the cochloid curve, with which he also trisected 
an angle.' 

1 Pappus, iii, pp. 54-6. 



300 HERON OF ALEXANDRIA 

Pappus goes on to say that he will give four solutions, one 
of which is his own ; the first, second, and third he describes 
as those of Eratosthenes, Nicomedes and Heron. But in the 
earlier sentence he mentions Philon along with Heron, and we 
know from Eutocius that Heron's solution is practically the 
same as Philon's. Hence we may conclude that by the third 
solution Pappus really meant Philon's, and that he only men- 
tioned Heron's Mechanics because it was a convenient place in 
which to find the same solution. 

Another argument has been based on the fact that the 
extracts from Heron's Mechanics given at the end of Pappus's 
Book VIII, as we have it, are introduced by the author with 
a complaint that the copies of Heron's works in which he 
found them were in many respects corrupt, having lost both 
beginning and end. 1 But the extracts appear to have been 
added, not by Pappus, but by some later writer, and the 
argument accordingly falls to the ground. 

The limits of date being then, say, 160 B.C. to A. D. 250, our 
only course is to try to define, as well as possible, the relation 
in time between Heron and the other mathematicians who 
come, roughly, within the same limits. This method has led 
one of the most recent writers on the subject (Tittel-) to 
place Heron jiot much later .than 100 B.C., while another/* 
relying almost entirely on a comparison between passages in 
Ptolemy and Heron, arrives at the very different conclusion 
that Heron was later than Ptolemy ^ncl belonged in fact to 
the second century A. D. 

In view of the difference between these results, it will be 
convenient to summarize the evidence relied on to establish 
the earlier date, and to consider how far it is or is not con- 
clusive against the later. We begin with the relation of 
Heron to Philon. Philon is supposed to come not more than 
a generation later than Ctesibius, because it would appear that 
machines for throwing projectiles constructed by Ctesibius 
and Philon respectively were both available at one time for 
inspection by experts on the subject 4 ; it is inferred that 

1 Pappus, viii, p. 1116. 4-7. 

2 Art. * Heron von Alexandreia ' in Pauly-Wissowa's JKeal-JKncyclop&die 
der class. Altertumswisnenschaft, vol. 8. 1, 1912. 

8 I. Hammer-Jensen in Hermes, vol. 48, 1913, pp. 224-35. 
4 Philon, Meek. Synt. iv, pp. 68. 1, 72. 86, 



CONTROVERSIES AS TO HERON'S DATE 3Q1 

Philon's date cannot be later than the end of the second 
century B.C. (If Ctesibius flourished before 247 B.C. the argu- 
ment would apparently suggest rather the beginning than the 
end of the second century.) Next, Heron is supposed to have 
been a younger contemporary of Philon, the grounds being 
the following. (1) Heron mentions a ' stationary-automaton ' 
representation by Philon of the Nauplius-story, 1 and this is 
identified by Tittel with a representation of the same story by 
some contemporary of Heron's (ol KaO 9 ^/xay 2 ). But a careful 
perusal of the whole passage seems to me rather to suggest 
that the latter representation was not Philon's, and that 
Philon was included by Heron among the 'ancient* auto- 
maton-makers, and not amonghis contemporaries. 3 (2) Another 
argument adduced to show that Philon was contemporary 

1 Heron, Autom., pp. 404. 11-408. 9. - /&., p. 412. 13. 

3 The relevant remarks of Heron are as follows. (1) He says that ho 
has found no arrangements of * stationary automata 1 better or more 
instructive than those described by Philon of Byzantium (p. 404. 11). 
As an instance he mentions Philon's setting of the Nauplius-story, in 
which he found everything good except two things (a) the mechanism 
for the appearance of Athene, which was too difficult ^pyadc'o-rcpor), and 
(/>) the absence of an incident promised by Philon in his description, 
namely the falling of a thunderbolt on Ajax with a sound of thunder 
accompanying it (pp. 404. 15-408. 9). This latter incident Heron could 
not find anywhere in Philon, though he had consulted a great number 
of copies of his work. He continues (p. 408. 9-13) that we are not to 
suppose that he is running down Philon or charging him with not being 
capable of carrying out what he promised. On the contrary, the omission 
was probably due to a slip of memory, for it is easy enough to make 
stage-thunder (he proceeds to show how to do it). But the rest of 
Philon's arrangements seemed to him satisfactory, and this, he says, is 
why he has not ignored Philon's work : ' for I think that my readers will 
get the most benefit if they are shown, first what has been well said by 
the ancients and then, separately from this, what the ancients overlooked 
or what in their work needed improvement ' (pp. 408. 22-410. 6). (2) The 
next chapter (pp. 410. 7-412. 2) explains generally the sort of thing the 
automaton-picture has to show, and Heron says he will give one example 
which he regards as the best. Then (3), after drawing a contrast between 
the simpler pictures made by * the ancients ', which involved three different 
movements only, and the contemporary (nl *#' fjp.as) representations of 
interesting stories by means of more numerous and varied movements 
(p. 412. 3-15), he proceeds to describe a setting of the Nauplius-story. 
This is the representation which Tittel identifies with Philon's. But it 
is to be observed that the description includes that of the episode of the 
thunderbolt striking Ajax (c.30, pp. 448. 1-452. 7) which Heron expressly 
says that Philon omitted. Further, the mechanism for the appearance 
of Athene described in c. 29 is clearly not Philon's 'more difficult' 
arrangement, but the simpler device described (pp. 404. 18-408. 5) as 
possible and preferable to Philon's (cf. Heron, vol. i, ed. Schmidt, pp. 
Ixviii-lxix). 



302 HERON OF ALEXANDRIA 

with Heron is the fact that Philon has some criticisms of 
details of construction of projectile-throwers which are found 
in Heron, whence it is inferred that Philon had Heron's work 
specifically in view. But if Heron's BeXorrouKa was based on 
the work of Ctesibius, it is equally possible that Philon may 
be referring to Ctesibius. 

A difficulty in the way of the earlier date is the relation in 
which Heron stands to Posidonius. In Heron's Mechanics, 
i. 24, there is a 'definition of ' centre of gravity ' which is 
attributed by Heron to 'Posidonius a Stoic'. But this can 
hardly be Posidonius of Apamea, Cicero's teacher, because the 
next sentence in Heron, stating a distinction drawn by Archi- 
medes in connexion with this definition, seems to imply that 
the Posidonius referred to lived before Archimedes. But the 
Definitions of Heron do contain definitions of geometrical 
notions which are put down by Proclus to Posidonius of 
Apamea or Rhodes, and, in particular, definitions of ' figure ' 
and of ' parallels'. Now Posidonius lived from 135 to 51 B.C., 
and the supporters of the earlier date for Heron can only 
suggest that either Posidonius was not the first to give these 
definitions, or alternatively, if he was, and if they were 
included in Heron's Definitions by Heron himself and not by 
some later editor, all that this obliges us to admit is that 
Heron cannot have lived before the first century B. o. 

Again, if Heron lived at the beginning of the first cen- 
tury B.C., it is remarkable that he is nowhere mentioned by 
Vitruvius. The De architecture* was apparently brought out 
in 14 B.C. and in the preface to Book VII Vitruvius gives 
a list of authorities on machinationes from whom he made 
extracts. The list contains twelve names and has every 
appearance of being scrupulously complete ; but, while it 
includes Archytas (second), Archimedes (third), Ctesibius 
(fourth), and Philon of Byzantium (sixth), it does not men- 
tion Heron. Nor is it possible to establish interdependence 
between Heron and Vitruvius ; the differences seem, on the 
whole, to be more numerous than the resemblances. A few of 
the differences may be mentioned. Vitruvius uses 3 as the 
value of TT, whereas Heron always uses the Archimedean value 
3^. Both writers make extracts from the Aristotelian 
MT]%aviKa TrpojSATj/xara, but their selections are different. The 



CONTROVERSIES AS TO HERON'S DATE 303 

machines used by the two for the same purpose frequently 
differ in details ; e. g. in Vitru vius's hodometer a pebble drops 
into a box at the end of each Roman mile, 1 while in Heron's 
the distance completed is marked by a pointer. 2 It is indeed 
pointed out that the water-organ of Heron is in many respects 
more primitive than that of Vitruvius; but, as the instru- 
ments are altogether different, this can scarcely be said to 
prove anything. 

On the other hand, there are points of contact between 
certain propositions of Heron and of the Roman agrimen- 
sores. Columella, about A.D. 62, gave certain measurements of 
plane figures which agree with the formulae used by Heron, 
notably those for the equilateral triangle, the regular hexagon 
(in this case not only the formula but the actual figures agree 
with Heron's) and {-he segment of a circle which is less than 
a semicircle, the formula in the last case being 



where s is the chord and h the height of the segment. Here 
there might seem to be dependence, one way or the other ; 
but the possibility is not excluded that the two writers may 
merely have drawn from a common source ; for Heron, in 
giving the formula for the area of the segment of a circle, 
states that it was the formula used by 'the more accurate 
investigators' (oi a/cpi/Seorepoj/ e^jrij/corer). 3 

We have, lastly, to consider the relation between Ptolemy 
and Heron. If Heron lived about 100 B.C., he was 200 years 
earlier than Ptolemy (A.D. 100178). The argument used to 
prove that Ptolemy came some time after Heron is based on 
a passage of Proclus where Ptolemy is said to have remarked 
on the untrustworthiness of the method in vogue among the 
"more ancient ' writers of measuring the apparent diameter of 
the sun by means of water-clocks. 4 Hipparchus, says Pro- 
clus, used his dioptra for the purpose, and Ptolemy followed 
him. Proclus proceeds : 

'Let us then set out here not only the observations of 
the ancients but also the construction of the dioptra of 

1 Vitruvius, x. 14. 2 Heron, Dioptra, c. 34. 

3 Heron, Metrica, i. 31, p. 74. 21. 

4 Proclus, Hypohtpo^ pp. 120. 9-15, 124, 7-26. 



304 HEKON OF ALEXANDRIA 

Hipparchus. And first we will show how we can measure an 
interval of time by means of the regular efflux of water, 
a procedure which was explained by Heron the mechanician 
in his treatise on water-clocks.' 

Theon of Alexandria has a passage to a similar effect. 1 He 
first says that the most ancient mathematicians contrived 
a vessel which would let water flow out uniformly through a 
small aperture at the bottom, and then adds at the end, almost 
in the same words as Proclus uses, that Heron showed how 
this is managed in the first book of his work on wjiter- 
clocks. Theon's account is from Pappus's Commentary on 
the Syntaods, and this is also Proclus's source, as is shown by 
the fact that Proclus gives a drawing of the water-clock 
which appears to have been lost in Theon J s transcription from 
Pappus, but which Pappus must have reproduced from the 
work of Heron. Tittel infers that Heron must have ranked 
as one of the 'more ancient' writers as compared with 
Ptolemy. But this again does not seem to be a necessary 
inference. No doubt Heron's work was a convenient place to 
refer to for a description of a water-clock, but it does not 
necessarily follow that Ptolemy was referring to Heron's 
clock rather than some earlier form of the same instrument. 

An entirely different conclusion from that of Tittel is 
reached in the article c Ptolemaios and Heron ' already alluded 
to. 2 The arguments are shortly these. (1) Ptolemy says in 
his Geography (c. 3) that his predecessors had only been able 
to measure the distance between two places (as an arc of a 
great circle on the earth's circumference) in the case where 
the two places are on the same meridian. He claims that he 
himself invented a way of doing this even in the case where 
the two places are neither on the same meridian nor on the 
same parallel circle, provided that the heights of the pole at" 
the two places respectively, and the angle between the great 
circle passing through both and the meridian circle through 
one of the places, are known. -Now Heron in his Dioptra 
deffls with the problem of measuring the distance between 
two places by means of the dioptra, and takes as an example 

1 Theon, Comm. on the Syntaxin, Basel, 1538, pp. 261 sq. (quoted in 
Proclus, Hypotyposis, ed. Manitius, pp. 309-11). 

2 Hammer-Jensen, op. dt. 



CONTROVERSIES AS TO HERON'S DATE 305 

the distance between Rome and Alexandria. 1 Unfortunately 
the text is in places corrupt and deficient, so that the method 
cannot be reconstructed in detail. But it involved the obser- 
vation of the same lunar eclipse at Rome and Alexandria 
respectively and the drawing of the aiuilemma for Rome'. 
That is to say, the mathematical method which Ptolemy- 
claims to have invented is spoken of by Heron as a thing 
generally known to experts and not more remarkable than 
other technical matters dealt with in the same book. Conse- 
quently Heron must have been later than Ptolemy. (It is 
right to add that some hold that the chapter of the Dioptra 
in question is not germane to the subject of the treatise, and 
was probably not written by Heron but interpolated by some 
later editor ; if this is so, the argument based upon it falls to 
the ground.) (2) The dioptra described in Heron's work is a 
fine and accurate instrument, very much better than anything 
Ptolemy had at his disposal. If Ptolemy had been aware of 
its existence, it is highly unlikely that ho would have taken 
the trouble to make his separate and imperfect ' parallactic ' 
instrument, since it could easily have been grafted on to 
Heron's dioptra. Not only, therefore, must Heron have been 
later than Ptolemy but, seeing that the technique of instru- 
ment-making had made such strides in the interval, he must 
have been considerably later. (3) In his work ntpl poTrtov 2 
Ptolemy, as we have seen, disputed the view of Aristotle that 
air has weight even when surrounded by air. Aristotle 
satisfied himself experimentally that a vessel full of air is 
heavier than the same vessel empty ; Ptolemy, also by ex- 
periment, convinced himself that the former is actually the 
lighter. Ptolemy then extended his argument to water, and 
held that water with water round it has no weight, and that 
the diver, however deep he dives, does not feel the weight of 
the water above him. Heron :>> asserts that water has no 
appreciable weight and has no appreciable power of com- 
pressing the ail' in a vessel inverted and forced clown into 
the water. In confirmation of this he cites the case of the 
driver, who is not prevented from breathing when far below 

1 Heron, Dioptra, c. 35 (vol. iii, pp. 302-6). 

2 Simplicius on De caelo, p. 710. 14, Heib. (Ptolemy, vol. ii, p. 263). 

3 Heron, Pneumatica, i. Pref. (vol. i, p. 22. 14 sq.). 

W3.2 X 



306 HERON OF ALEXANDRIA 

the surfo.ee. He then inquires what is the reason why the 
diver is not oppressed ^ though he has an unlimited weight of 
water on his back. He accepts, therefore, the view of Ptolemy 
as to the fact, however strange this may seem. But he is not 
satisfied with the explanation given : ' Some say ', he goes on, 
' it is because water in itself is uniformly heavy (/cro/Sapcy avrb 
KaO* ai)ro) ' this seems to be equivalent to Ptolemy's dictum 
that water in water has no weight * but they give no ex- 
planation whatever why divers . . .' He himself attempts an 
explanation based on Archimedes. It is suggested, therefore, 
that Heron's criticism is directed specifically against Ptolemy 
and no one else. (4) It is suggested that the Dionysius to whom 
Heron dedicated his Definitions is a certain Dionysius who 
was praefectus urbi at Rome in A.D. 301. The grounds are 
these (a) Heron addresses Dionysius as AIOVVCTM Aa/nrporare, 
where Aa/nr/>6raroy obviously corresponds to the Latin clarissi- 
mus, a title which in the third century and under Diocletian 
was not yet in common use. Further, 'this Dionysius was 
curatoi* aquarnm and curator operum publicorwni, so that he 
was the sort of person who would have to do with the 
engineers, architects and craftsmen for whom Heron wrote. 
Lastly, he is mentioned in an inscription commemorating an 
improvement of water supply and dedicated ' to Tiberinus, 
father of all waters, and to the ancient inventors of marvel- 
lous constructions 1 (repertoribus admirab ilium fabricawim 
precis viris), an expression which is not found in any other 
inscription, but which recalls the sort of tributq that Heron 
frequently pays to his predecessors. This identification of the 
two persons named Dionysius is an ingenious conjecture, but 
the evidence is not such as to make it anything more. 1 

The result of the whole investigation just summarized is to 
place Heron in the third century A.D., and perhaps little, if 
anything, earlier than Pappus. Heiberg accepts this conclu- 
sion, 2 which may therefore, I sup'pose, be said to hold the field 
fo* the present. 

1 Dionysius was of course a very common name. Diophantus dedicated 
his Arithmetica to a person of this name V Tifud>TaT /ioi Atoi/ucm), whom he 
praised for his ambition to learn the solutions of arithmetical problems. 
This Dionysius must have lived in the second half of the third century 
A.D., and if Heron also belonged to this time, is it not possible that. 



Heron's Dioriysius was the same person ? 
2 Heron, vol. v f p. ix. 



CONTROVERSIES AS TO HERON'S DATE 307 

Heron was known as 6 'AXtgavSpevs (e.g. by Pappus) or 
6 priyaviKos (mechanicus), to distinguish him from other 
persons of the same name ; Proclus and Damianus use the 
latter title, while Pappus also speaks of ol nepl TOV "Hpcova 



Character of works. 

Heron was an almost encyclopaedic writer on mathematical 
and physical subjects. Practical utility rather than theoreti- 
cal completeness was the object aimed at ; his environment in 
Egypt no doubt accounts largely for this. His Metrica begins 
with the old legend of the traditional origin of geometry in 
Egypt, and in the Dioptra we find one of the very problems 
which geometry was intended to solve, namely that of re- 
establishing Ixmndaries of lands when the flooding of the 
Nile had destroyed the land-marks: ' When the boundaries 
of an area have become obliterated to such an extent that 
only two or three marks remain, in addition to a plan of the 
area, to supply afresh the remaining marks/ 1 Heron makes 
little or no claim to originality ; he often quotes authorities, 
but, in accordance with Greek practice, he more frequently 
omits to do so, evidently without any idea of misleading any 
one; only when he has made what is in his opinion any 
slight improvement on the methods of his predecessors does 
he trouble to mention the fact, a habit which clearly indi- 
cates that, except in these cases, he is simply giving the best 
traditional methods in the form which seemed to him easiest 
of comprehension and application. The Melrica seems to be 
richest in definite references to the discoveries of prede- 
cessors ; the names mentioned are Archimedes, Dionysodorus, 
Eudoxus, Plato ; in the Dioptra Eratosthenes is quoted, and 
in the introduction to the (fatoptricu Plato and Aristotle are 
mentioned. 

The practical utility of Heron's manuals being so great, it 
was natural that they should have great vogue, and equally 
natural that the most popular of them at any rate should be 
re-edited, altered and added to by later writers; this was 
inevitable with books which, like the Elements of Euclid, 
were in regular use in Greek, Byzantine, Roman, and Arabian 
J Heron, Dioptra, c. 25, p. 268. 17-19. 
X 'i 



808 HERON OF ALEXANDRIA 

education for centuries. The geometrical or mensurational 
books in particular gave scope for expansion by multiplication 
of examples, so that it is difficult to disentangle the genuine 
Heron from the rest of the collections which have come down 
to us under his name. Hultsch's considered criterion is as 
follows: 'The Heron texts which have come down to our 
time are authentic in so far as they bear the author's name 
and have kept the original design and form of Heron's works, 
but are unauthentic in so far as, being constantly in use for 
practical purposes, they were repeatedly re-edited and, in the 
course of re-editing, were rewritten with a view to the 
particular needs of the time.' 

List of Treatises. 

Such of the works of Heron as have survived have reached 
us in very different ways. Those which have come down in 
the Greek are : 

I. The Metrica, first discovered in 1896 in a manuscript 
of the eleventh (or twelfth) century at Constantinople by 
R. Scheme and edited by his son, H. Schone (Heronis Opera, iii, 
Teubner, 1903). 

II. On the Dioptra, edited in an Italian version by Venturi 
in 1814 ; the Greek text was first brought out by A. J. H. 
Vincent 1 in 1858, and the critical edition of it by H. Scheme is 
included in the Teubner vol. iii just mentioned. 

III. The Pneumatica, in two Books, which appeared first in 
a Latin translation by Commandinus, published after his 
death in 1575; the Greek text was first edited by Th^venot 
in Veterum mathematicorum opera Graece et Latitte cdita 
(Paris, 1693), and is now available in Iferonis Opera, i (Teub- 
ner, 1899), by W. Schmidt. 

IV. On the art of constructing automata (irpt avro^aro" 
TTOtijrjjcijy), or The automaton-theatre, first edited in an Italian 
translation by B. Baldi in 1589 ; the Greek text was included 
in Th^venot's Vet. math., and now forms part of Heroiiis 
Opera, vol. i, by W. Schmidt. 

V. Belopoewa (on the construction of engines of war), edited 

1 Notices et extraits des manuscrits de la Bibliothtque impMale, xix, Dt. 2. 
pp. 157-337. P 



LIST OF TREATISES 309 

by B. Baldi (Augsburg, 1616), TWvenot (Vet. math.), Kochly 
and Rustow (1853) and by Wescher (Poliorc&tique dcs Grecs, 
1867, the first critical edition). 

VI. The Cheirobalistra ("Hpawo? x L P^ a ^ crT P a ^ wravKevt} 
Kal a-vfifjiTpia (?)), edited by V. Prou, Notices et extraits> xxvi. 2 
(Paris, 1877). 

VII. The geometrical works, Defmitiones, Geometria, Geo- 
daesia, Stereometrica I and 11, Meumrae, Liber Geeponicu8> 
edited by Hultsch with Variae collectioues (Hero MB Alexan- 
drini (jeometriourum et stereometricortim relicfuiae, 1864). 
This edition will now be replaced by that of Heiberg in the 
Teubner collection (vols. iv, v), which contains much addi- 
tional matter from the Constantinople manuscript referred to, 
but omits the Liber Geeponicus (except a few extracts) and the 
Geodae^ia (which contains only a few extracts from the 
Geometry of Heron). 

Only fragments survive of the Greek text of the Mechanics 
in three Books, which, however, is extant in the Arabic (now 
edited, with German translation, in lleroius Opera, vol. ii, 
by L. Nix and W. Schmidt, Teubner, 1901). 

A smaller separate mechanical treatise, the Bapov\Ko$, is 
quoted by Pappus. 1 The object of it was ' to move a given 
weight by means of a given force ', and the machine consisted 
of an arrangement of interacting toothed wheels with different 
diameters. 

At the end of the Dioptra is a description of a hodometer for 
measuring distances traversed by a wheeled vehicle, a kind of 
taxameter, likewise made of a combination of toothed wheels. 

A work on Water-clock* (nept vSpicw copocr/eoTmW) is men- 
tioned in the Pneiimatica as having contained four Books, 
and is also alluded to by Pappus. 2 Fragments are preserved 
in Proclus (Hypotyposix, chap. 4) and in Pappus's commentary 
on Book V of Ptolemy's tiyntaxis reproduced by Theon. 

Of Heron's Commentary on Euclid's Elements only very 
meagre fragments survive in Greek (Proclus), but a large 
number of extracts are fortunately preserved in the Arabic 
commentary of un-Nairizi, edited (1) in the Latin version of 
Gherard of Cremona by Curtze (Teubner, 1899), and (2) by 

1 Pappus, viii, p. 1060. 5* 3 /&., p. 1026. 1. 



310 HERON OF ALEXANDRIA 

Besthorn and Heiberg (Codex Leidensis 399. 1, five parts of 
which had appeared up to 1910). The commentary extended 
as far as Elem. VIII. 27 at least. 

The Catoptrica, as above remarked under Ptolemy, exists in 
a Latin translation from the Greek, presumed to be by William 
of Moerbeke, and is included in vol. ii of Heronis Opera, 
edited, with introduction, by W. Schmidt. 

Nothing is known of the Camarica (' on vaultings ') men- 
tioned by Eutocius (on Archimedes, Sphere and Cylinder), the 
Zygia (balancings) associated by Pappus with the Automata, 1 
or of a work on the use of the astrolabe mentioned in the 
Fihrist. 

We are in this work concerned with the treatises of mathe- 
matical content, and therefore can leave out of account such 
works as the Pneumatica, the Automata, and the Belopoeica. 
The Pneumatica and Automata have, however, an interest to 
the historian of physics in so far as they employ the force of 
compressed air, water, or steam. In the P neumatica the 
reader will find such things as siphons, ' Heron's fountain ', 
' penny -in-the-slot ' machines, a fire-engine, a water-organ, and 
many arrangements employing the force of steam. 

Geometry. 

(a) Commentary OR Euclid's Elements. 

In giving an account of the geometry and mensuration 
(or geodesy) of Heron it will be well, I think, to begin 
with what relates to the elements, and first the Commen- 
tary on Euclid's Elements, of which we possess a number 
of extracts in an-Nairizi and Proclus, enabling us to form 
a general idea of the character of the work. Speaking 
generally, Heron's comments do not appear to have contained 
much that can be called important. They may be classified 
as follows : 

(1) A few general notes, e.g. that Heron would not admit 
more than three axioms. 

(2) Distinctions of a number of particular coxes of Euclid's 
propositions according as the figure is drawn in one way 
or another. 

1 Pappus, viii, p. 1024. 28. 



GEOMETRY 311 

Of this class are the different cases of I. 35, 36, III. 7, 8 
(where the chords to be compared are drawn on different sides 
of the diameter instead of on the same side), III. 12 (which is 
not Euclid's at all but Heron's own, adding the case of 
external to that of internal contact in III. 11*, VI. 19 (where 
the triangle in which an additional line is drawn is taken to 
be the smaller of the two), VII. 19 (where the particular case 
is given of three numbers in continued proportion instead of 
four proportionals). 

(3) Alternative proofs. 

It appears to be Heron who first introduced the easy but 
uninstructive semi-algebraical method of proving the proposi- 
tions II. 2-10 which is now so popular. On this method the 
propositions are proved ' without figures ' as consequences of 
II. 1 corresponding to the algebraical formula 

a (b + c + d + . . .) = ab + oc + ad + . . . 

Heron explains that it is not possible to prove II. 1 without 
drawing a number of lines (i.e. without actually drawing the 
rcctanglcH), but that the following propositions up to II. 10 
can be proved by merely drawing one line. He distinguishes 
two varieties of the method, one by dissolutio, the other by 
composltio, by which he seems to mean splittiiig-u-p of rect- 
angles and squares and combination, of them into others. 
But in his proofs he sometimes combines the two varieties. 

Alternative proofs are given () of some propositions of 
Book III, namely III. 25 (placed after III. 30 and starting 
from the arc instead of the chord), III. 10 (proved by means 
of III. 9), III. 13 (a proof preceded by a lemma to the effect 
that a straight line cannot meet a circle in more than two 
points). 

A class of alternative proof is (6) that which is intended to 
meet a particular objection (eixrrao-is) which had been or might 
be raised to Euclid's constructions. Thus in certain cases 
Heron avoids product ny a certain straight line, where Euclid 
produces it, the object being to meet the objection of one who 
should deny our right to assume that there is any space 
available. Of this class are his proofs of I. 11, 20 and his 
note on 1. 16. Similarly in I. 48 lie supposes the right-angled 



312 HERON OF ALEXANDRIA 

triangle which is constructed to be constructed on the same 
side of the common side as the given triangle is. 

A third class (c) is that which avoids reciuct'io ad ab&ardum, 
e.g. a direct proof of I. 19 (for which he requires and gives 
a preliminary lemma) and of I. 25. 

(4) Heron supplies certain converses of Euclid's propositions 
e.g. of II. 12, 13 and VIII. 27. 

(5) A few additions to, and extensions of, Euclid's propositions 
are also found. Some are unimportant, e. g, the construction 
of isosceles and scalene triangles in a note on 1. 1 and the 
construction of two tangents in III. 17. The most important 
extension is that of III. 20 to the case where the angle at the 
circumference is greater than a right angle, which gives an 
easy way of proving the theorem of III. 22. Interesting also 
are the notes on I. 37 (on I. 24 in Proclus), where Heron 
proves that two triangles with two sides of the one equal 
to two sides of the other and with the included angles swpple- 
mentcM'y are equal in area, and compares the areas where the 
sum of the included angles (one being supposed greater than 
the other) is less or greater than two right angles, and on I. 47, 
where there is a proof (depending on preliminary lemmas) of 
the fact that, in the figure of Euclid's proposition ^see next 
page), the straight lines AL, BG, CE meet in a point. This 
last proof is worth giving. First come the lemmas. 

(1) If in a triangle ABG a straight line I)E be drawn 
parallel to the base BG cutting the sides AB, AC or those 

sides produced in D, E, and if F be the 

middle point of BC y then the straight line 

AF (produced if necessary) will also bisect 
DE. (UK is drawn through A parallel to 
DE, and HDL, KEM through J), E parallel 

to AF meeting the base in L, M respec- 

8 L F M C tively. Then the triangles ABF, AFG 
between the same parallels are equal. So are the triangles 
DBF, EFC. Therefore the differences, the triangles ADF, 
AEF t are equal and so therefore are the parallelograms HF, 
KF. Therefore LF '= FM 9 or DG = GE.) 

(2) is the converse of Eucl. 1. 43. If a parallelogram is 




GEOMETRY 



313 




cut into four others ADGE, DF, FGCB, CE, so that DF, CE 
are equal, the common vertex G will lie on the diagonal AB. 

Heron produces AG to meet OF in H, and then proves that 
AHB is a straight line. 

Since DF, CE are equal, so are 
the triangles DGF, ECG. A elding 
the triangle GCF, we have the 
triangles EOF, DCF equal, and 
DE, OF are parallel. 

But (by I. 34, 29, 26) the tri- 
angles AKE, GKJ) are congruent, 
HO that EK=KD\ and by lemma (1) it follows that CH=HF. 

Now, in the triangles F11B, CHG, two sides (BF, FH and 
GO, Gil) and the included angles are equal; therefore the 
triangles are congruent, and the angles BllF, G11C are equal. 

Add to each the angle GHF, and 
Z B1IF+ L FUG = Z CHG + Z GHF = two right angles. 

To prove his substantive proposition Heron draws AKL 
perpendicular to J3G, and joins JKO meeting AK in M. Then 
we have only to prove that liMG is a straight line. 




Complete the parallelogram FAllO* and draw the diagonals 
OA, FH meeting in F. Through M draw PQ, UR parallel 
respectively to BA, AG. 



314 HERON OF ALEXANDRIA 

Now the triangles FAH, BAG are equal in all respects ; 
therefore LEFA == LAEG 

= L CAK (since AK is at right angles to BC). 

But, the diagonals of the rectangle FH cutting one another 
in F, we have FY = YA and LHFA = ZO^l^; 

therefore LOAF LCAK, and OJ. is in a straight line 
with AKL. 

Therefore, OM being the diagonal of SQ y SA = 4Q, and, if 
we add AM to each, FM = J/fT. 

Also, since J<7 is the diagonal of FN 9 FM = J/JV. 

Therefore the parallelograms JMH, MM are equal ; and 
hence, by the preceding lemma, BMG is a straight line. Q.E.D. 

(ft) The DtfiiwtioH*. 

The elaborate collection of Definitions is dedicated to one 
Dionysius in a preface to the following effect : 

'In setting out for you a sketch, in the shortest possible 
form, of the technical terms premised in the elements of 
geometry, I shall take as my point of departure, and shall 
base my whole arrangement upon, the teaching of Euclid, the 
author of the elements of theoretical geometry ; for by this 
means I think that I shall give you a good general under- 
standing not only of Euclid's doctrine but of many other 
works in the domain of geometry. I shall begin then with 
the point. 9 

He then proceeds to the definitions of the point, the line, 
the different sorts of lines, straight, circular, ' curved ' and 
'spiral-shaped* (the Archimedean spiral and the cylindrical 
helix), Defs. 1-7 ; surfaces, plane and not plane, solid body, 
Defs. 8-11; angles and their different kinds, plane, solid, 
rectilinear and not rectilinear, right, acute and obtuse angles, 
Defs. 12-22; figure, boundaries of figure, varieties of figure, 
plane, solid, composite (of homogeneous or non-homogeneous 
parts) and incomposite, Defs. 23-6. The incomposite plane 
figure is the circle, and definitions follow of its parts, segments 
(which are composite of non-homogeneous parts), the semi- 
circle, the &tyi$ (less than a semicircle), and the segment 
greater than a semicircle, angles in segments, the sector, 



THE DEFINITIONS 315 

4 concave' and 'convex', lune, garland (these last two are 
composite of homogeneous parts) and axe (mXtKvs), bounded by 
four circular arcs, two concave and two convex, Defs. 27-38. 
Rectilineal figures follow, the various kinds of triangles and 
of quadrilaterals, the gnomon in a parallelogram, and the 
gnomon in the more general sense of the figure which added 
to a given figure makes the whole into a similar figure, 
polygons, the parts of figures (side, diagonal, height of a 
triangle), perpendicular, parallels, the three figures which will 
fill up the space round a point, Defs. 39-73. Solid figures are 
next classified according to the surfaces bounding them, and 
lines on surfaces are divided into (1) simple and circular, 
(2) mixed, like the conic and spiric curves, Defs. 74, 75. The 
sphere is then defined, with its parts, and stated to be 
the figure which, of all figures having the same surface, is the 
greatest in content, Defs. 76-82. Next the cone, its different 
species and its parts are taken up, with the distinction 
between the three conies, the section of the acute-angled cone 
(' by some also called ellipse ') and the sections of the right- 
angled and obtuse-angled cones (also called parabola and 
hyperbola), Defs. 83-94; the cylinder, a section in general, 
the spire or tore in its three varieties, open, continuous (or 
just closed) and ' crossing-itself ', which respectively have 
sections possessing special properties, ' square rings ' which 
are cut out of cylinders (i.e. presumably rings the cross-section 
of which through the centre is two squares), and various other 
figures cut out of spheres or mixed surfaces, Defs. 95-7 ; 
rectilineal solid figures, pyramids, the five regular solids, the 
semi-regular solids of Archimedes two of which (each with 
fourteen faces) were known to Plato, Defs. 98-104; prisms 
of different kinds, parallelepipeds, with the special varieties, 
the cube, the beam, SOKO? (length longer than breadth and 
depth, which may be equal), the brick, Tr\iv6i$ (length less 
than breadth and depth), the <r0?/t>(Woy or jSoo/zicr/coy with 
length, breadth and depth unequal, Defs. 105-14. 

Lastly come definitions of relations, equality of lines, sur- 
faces, and solids respectively, similarity of figures, c reciprocal 
figures', Defs. 115-18; indefinite increase in magnitude, 
parts (which must be, homogeneous with the wholes, so that 
e. g. the horn-like angle is not a part or submultiple of a right 



316 HERON OF ALEXANDRIA 

or any angle), multiples, Defs. 119-21 ; proportion in magni- 
tudes, what magnitudes can have a ratio to one another, 
magnitudes in the same ratio or magnitudes in proportion, 
definition of greater ratio, Defs. 122-5; transformation of 
ratios (coniponendo, separando, convertendo, alternando, in- 
vertendo and ex aequali), Defs. 126-7; commensurable and 
incommensurable magnitudes and straight lines, Defs. 128, 
129. There follow two tables of measures, Defs, 130-2. 

The Definitions are very valuable from the point of view of 
the historian of mathematics, for they give the different alter- 
native definitions of the fundamental conceptions; thus we 
find the Archimedean 'definition' of a straight line, other 
definitions which we know from Proclus to be due to Apol- 
lonius, others from Posidonius, and so on. No doubt the 
collection may have been recast by some editor or editors 
after Heron's time, but it seems, at least in substance, to go 
back to Heron or earlier still. So far as it contains original 
definitions of Posidonius, it cannot have been compiled earlier 
than the first century B.C.; but its content seems to belong in 
the main to the period before the Christian era. Heiberg 
adds to his edition of the Definitions extracts from Heron's 
Geometry, postulates and axioms from Euclid, extracts from 
Geminus on the classification of mathematics, the principles 
of geometry, &c., extracts from Proclus or some early collec- 
tion of scholia on Euclid, and extracts from Anatolius and 
Theon of Smyrna, which followed the actual definitions in the 
manuscripts. These various additions were apparently collected 
by some Byzantine editor, perhaps of the eleventh century. 

Mensuration. 

The Metrica, Geometrica, Stereometrua, Geodaesia, 
Mensurae. 

We now come to the mensuration of Heron. Of the 
different works under this head the Metr'uu is the most 
important from our point of view because it seems, more than 
any of the others, to have preserved its original form. It is 
also more fundamental in that it gives the theoretical basis of 
the formulae used, and is not a mere application of rules to 
particular examples. It is also more akin to theory in that it 



MENSURATION 317 

does not use concrete measures, but simple numbers or units 
which may then in particular cases be taken to be i eet, cubits, 
or any other unit of measurement. Up to 1896, when a 
manuscript of it was discovered by R. Schone at Constanti- 
nople, it was only known by an allusion to it in Eutocius 
(on Archimedes's Measurement of a Circle), who states that 
the way to obtain an approximation to the square root of 
a non-square number is shown by Heron in his Metrica, as 
well as by Pappus, Theon, and others who had commented on 
the Syntaxis of Ptolemy. 1 Tannery 2 had already in 1894 
discovered a fragment of Heron's Metrica giving the particular 
rule in a Paris manuscript of the thirteenth century contain- 
ing Prolegomena to the Syntaxis compiled presumably from 
the commentaries of Pappus and Theon. Another interesting 
difference between the Metrica and the other works is that in 
the former the Greek way of writing fractions (which is our 
method) largely preponderates, the Egyptian form (which 
expresses a fraction as the sum of diminishing submultiples) 
being used comparatively rarely, whereas the reverse is the 
case in the other works. 

In view of the greater authority of the Metrica, we shall 
take it as the basis of our account of the mensuration, while 
keeping the other works in view. It is desirable at the 
outset to compare broadly the contents of the various collec- 
tions. Book I of the Metrica contains the mensuration of 
squares, rectangles and triangles (chaps. 1-9), parallel-trapezia, 
rhombi, rhomboids and quadrilaterals with one angle right 
(10-16), regular polygons from the equilateral triangle to the 
regular dodecagon (17-25), a ring between two concentric 
circles (26), segments of circles (27-33), an ellipse (34), a para- 
bolic segment (35), the surfaces of a cylinder (36), an isosceles 
cone (37), a sphere (38) and a segment of a sphere (39). 
Book II gives the mensuration of certain solids, the solid 
content of a cone (chap. 1), a cylinder (2), rectilinear solid 
figures, a parallelepiped, a prism, a pyramid and a frustum, 
&c. (3-8), a frustum of a cone (9, 10), a sphere and a segment 
of a sphere (11, 12), a vpire or tore (13), the section of a 
cylinder measured in Archimedes's Method (14), and the solid 

1 Archimedes, vol. iii, p. 232. 1S-17. 

2 Tannery, M&noires scientijiques, ii, 1912, pp. 447-54. 



318 HERON OF ALEXANDRIA 

formed by the intersection of two cylinders with axes at right 
angles inscribed in a cube, also measured in the Method (15), 
the five regular solids (16-19). Book III deals with the divi- 
sion of figures into parts having given ratios to one another, 
first plane figures (1-19), then solids, a pyramid, a cone and a 
frustum, a sphere (20-3). 

The Geometria or Geometrumena is a collection based upon 
Heron, but not his work in its present form. The addition of 
a theorem due to Patrjcius * and a reference to him in the 
Stereometric^ (1. 22) suggest that Patricias edited both works, 
but the date of Patricius is uncertain. Tannery identifies 
him with a mathematical professor of the tenth century, 
Nicephorus Patricius ; if this is correct, he would be contem- 
porary with the Byzantine writer (erroneously called Heron) 
who is known to have edited genuine works of Heron, and 
indeed Patricius and the anonymous Byzantine might be one 
and the same person. The mensuration in the Geometry has 
reference almost entirely to the same figures as those 
measured in Book I of the Metrica, the difference being that 
in the Geometry (1) the rules are not explained but merely 
applied to examples, (2) a large number of numerical illustra- 
tions are given for each figure, (3) the Egyptian way of 
writing fractions as the sum of submultiples is followed, 
(4) lengths and areas are given in terms of particular 
measures, and the calculations are lengthened by a consider- 
able amount of conversion from one measure into another. 
The first chapters (1-4) are of the nature of a general intro- 
duction, including certain definitions and ending with a table 
of measures. Chaps. 6-99, Hultsch ( = 5-20, 14, Heib.), though 
for the most part corresponding in content to Metrica I, 
seem to have been based on a different collection, because 
chaps. 100-3 and 105 (= 21, 1-25, 22, 3-24, Heib.) are clearly 
modelled on the Metrica^ arid 101 is headed 'A definition 
(really ' measurement ') of a circle in another book of Heron '. 
Heiberg transfers to the Geometrica a considerable amount of 
the content of the so-called Liber Geeponicus, a badly ordered 
collection consisting to a large extent of extracts from the 
other works. Thus it begins with 41 definitions identical 
with the same number of the Definitiovies. Some sections 
1 Geometrica, 21 26 (vol. iv, p. 386. 23). 



MENSURATION 319 

Heiberg puts side by side with corresponding sections of the 
Geometrica in parallel columns ; others he inserts in suitable 
places ; sections 78. 79 contain two important problems in 
indeterminate analysis (= Geom. 24, 1-2, Heib.). Heiberg 
adds, from the Constantinople manuscript containing the 
Metrica, eleven more sections (chap. 24, 3-13) containing 
indeterminate problems, and other sections (chap. 24, 14-30 and 
37-51 ) giving the mensuration, mainly, of figures inscribed in or 
circumscribed to others, e.g. squares or circles in triangles, 
circles in squares, circles about triangles, and lastly of circles 
and segments of circles. 

The titereometrica I has at the beginning the title Eicra- 
ycoyat rS>v orepo//rpoi;/*i>a>i> "Hpcoj/o? but, like the Geometrica, 
seems to have been edited by Patricius. Chaps. 1-40 give the 
mensuration of the geometrical solid figures, the sphere, the 
cone, the frustum of a cone, the obelisk with circular base, 
the cylinder, the 'pillar', the cube, the crfyrjvicrKo? (also called 
owg), the fjittovpov 7rpoeavcapt0i7io/oi>, pyramids, and frusta. 
Some portions of this section of the book go back to Heron ; 
thus in the measurement of the sphere chap. 1 = Metrica 
II. 11, and both here and elsewhere the ordinary form of 
fractions appears. Chaps. 41-54 measure the contents of cer- 
tain buildings or other constructions, e.g. a theatre, an amphi- 
theatre, a swimming-bath, a well, a ship, a wine-butt, and 
the like. 

The second collection, titereometrica II, appears to be of 
Byzantine origin and contains similar matter to Stereometrica I, 
parts of which are here repeated. Chap. 31 (27, Heib.) gives 
the problem of Thales. to find the height of a pillar or a tree 
by the measurement of shadows ; the last sections measure 
various pyramids, a prism, a fi&pfcKos (little altar). 

The Geodaesia is nob an independent work, but only con- 
tains extracts from the Geometry ; thus chaps. 116 = Geom. 
5-31, Hultsch ( = 5, 2-12, 32, Heib.); chaps. 17-19 give the 
methods of finding, in any scalene triangle the sides of which 
are given, the segments of the base made by the perpendicular 
from the vertex, and of finding the area direct by the well- 
known ' formula of Heron ' ; i.e. we have here the equivalent of 
Metrica I. 5-8. 

Lastly, the /zerpqcmy, or Mensurae, was attributed to Heron 



320 HERON OF ALEXANDRIA 

in an Archimedes manuscript of tho ninth century, but can- 
not in its present form be due to Heron, although portions of 
it have points of contact with the genuine works. Sects. 2-27 
measure all sorts of objects, e.g. stones of different shapes, 
a pillar, a tower, a theatre, a ship, a vault, a hippodrome ; but 
sects. 28-35 measure geometrical figures, a circle and segments 
of a circle (cf. Metrica I), and sects. 36-48 on spheres, segments 
of spheres, pyramids, cones and frusta are closely connected 
with Stereom. I and Meti4ca II ; sects. 49-59, giving the men- 
suration of receptacles and plane figures of various shapes, 
seem to have a different origin. We can now take up the 

Contents of the Metrics. 
Book I. Measurement of Areas. 

The preface records the tradition that the first geometry 
arose out of the practical necessity of measuring and dis- 
tributing land (whence the name ' geometry '), after which 
extension to three dimensions became necessary in order to 
measure solid bodies. Heron then mentions P]udoxus and 
Archimedes as pioneers in the discovery of difficult measure- 
ments, Eudoxus having been the first to prove that a cylinder 
is three times the cone on the same base and of equal height, 
and that circles are to one another as the squares on their 
diameters, while Archimedes first proved that the surface of 
a sphere is equal to four times the area of a great circle in it, 
and the volume two-thirds of the cylinder circumscribing it. 

(a) Area of scale tie trianyle. 

After the easy cases of the rectangle, the right-angled 
triangle and the isosceles triangle, Heron gives two methods 
of finding the area of a scalene triangle (acute-angled or 
obtuse-angled) when the lengths of the three sides are given. 

The first method is based on Eucl. II. 12 and 13. If ct, b, c 
be the sides of the triangle opposite to the angles A, Jl, C 
respectively, Heron observes (chap. 4) that any angle, e.g. C, is 
acute, right or obtuse according as c 2 < = or > a 2 + // 2 , and this 
is< the criterion determining which of the two propositions is 
applicable. The method is directed to determining, first the 
segments into which any side is divided by the perpendicular 



AREA OF SCALENE TRIANGLE 



321 



from the opposite vertex, and thence the length of the per- 
pendicular itself. We have, in the cases of the triangle acute- 
angled at C and the triangle obtuse-angled at C respectively, 



or 



67) = 





whence AD* (= b* CD*) is found, so that we know the area 



In the cases given in Melrica I. 5, the sides are (14, 15, 13) 
and (11, 13, 20) respectively, and AD is found to be rational 
(=12). But of course both CD (or HD) and AD may be surds, 
in which case Heron gives approximate values. Cf. Geom. 
53, 54, Hultsch (15, 1-4, Heib.) 3 where we have a triangle 
in which a = 8, b = 4, c = 6, so that a 2 + '> 8 fl 2 = 44 and 
C7) = 44/10 = 2J. Thus 4/) 2 = 16-(2i) a = 16-7^ 
= 8J I r ^, and .47)= \/(8 f T ^) = 2 approximately, whence 
the area = 4x2|J=llj|. Heron then observes that we get 
a nearer result still if we multiply AD 2 by (|a) 2 before 
extracting the square root, for the area is then \/(16 x 8| ^) 
or \/(135), which is very nearly 11| -^ ^ T or ll^f- 

So in Metrlca I. 9, where the triangle is 10, 8, 12 (10 being 
the base), Heron finds the perpendicular to be \/63, but he 
obtains the area as \/(\AD" . #(*). or \/(1575), while observing 
that we can, of course, take the approximation to \/63, or 
7 I iV anc ' multiply it by half 10. obtaining 39| | y 1 ^ as 
the area. 



Proof of the formula A = \/{ t s(s a) (s b) (s c)}. 

The second method is that known as the * formula of 
Heron ', namely, in our notation, A = \/{s (s a) (s b) (s c) } . 
The proof of the formula is given in Metrlca I. 8 and also in 

1B23.2 Y 



322 



HERON OF ALEXANDRIA 



chap. 30 of the Dioptra ; but it is now known (from Arabian 
sources) that the proposition is due to Archimedes. 

Let the sides of the triangle ABC be given in length. 

Inscribe the circle DEF, and let be the centre. 

Join AO, KG, CO, DO, EG, FO. 

Then BG . 01) = 2 A 50(7, 

CA.GE= 2&COA, 

AB.GF=2AAOB; 




whence, by addition, 



where p is the perimeter. 
Produce CB to H, so that BU = AF. 
Then, since AE = AF, BF = BD, and CE = CD, we have 



CH.OD = A ABC. 



Therefore 



But CH.OD is the 'side' of the product C1I*.OD 2 , i.e. 
. OD*), 



so that 



PROOF OF THE ' FORMULA OF HERON' 323 

Draw OL at right angles to 00 cutting BO in K 9 and BL at 
right angles to BO meeting OL in L. Join OL. 

Then, since each of the angles COL, OBL is right, COBL is 
a quadrilateral in a circle. 

Therefore L GO B + L CLB = 2 R. 

But /.COB + AOF = 27?, because AO, BO, 00 bisect the 
angles round 0, and the angles 00 B, AOF are together equal 
to the angles AOC, BOF, while the sum of all four angles 
is equal to 4 R. 

Consequently L AOF = Z OLB. 

Therefore the right-angled triangles AOF, OLB are similar ; 
therefore BO : BL = AF: FO 

= Bll : 07), 
and, alternately, CB : Bit = BL : OD 

= BK : KD ; 
whence, componendo, OH: HB = BD : DK. 

It follows that 
Oil* : OH . HB = BD . DO : C7) . DK 

= 7*7) . 7)<7 : 07) 2 , since the angle 00 K is right. 
Therefore (A ABC}- = 6'7/ 2 . 07) 2 (from above) 

= (777 . 7/7* . BD . DC 
= 8 (s a) (s - />) (s c). 

(/J) Method of approximating to the squre root of 
a non-square number. 

It is a propos of the triangle 7, 8, 9 that Heron gives the 
important statement of his method of approximating to the 
value of a surd, which before the discovery of the passage 
of the Metrica had been a subject of unlimited conjecture 
as bearing on the question how Archimedes obtained his 
approximations to v^3. * 

In this case s = 12, 8 a = 5, 8 ?> = 4, 8 c = 3, so that 

A = \/(12 .5.4.3) = \/(720). 



324 HERON OF ALEXANDRIA 

' Since', says Heron, 1 c 720 has not its side rational, we can 
obtain its side within a very small difference as follows. Since 
the next succeeding square number is 729, which has 27 for 
its side, divide 720 by 27. This gives 26. Add 27 to this. 
making 53f , and take half of this or 26^. The side of 720 
will therefore be very nearly 26^ ^. In fact, if we multiply 
26^^ by itself , the product is 720^, so that the difference (in 
the square) is -f e . 

1 If we desire to make the difference still smaller than 3^, we 
shall take 720^ instead of 729 [or rather we should take 
26| instead of 27], and by proceeding in the same way we 
shall find that the resulting difference is much less than ^/ 

In other words, if we have a non-square number A, and 2 
is the nearest square number to it, so that A = f/ 2 + />, then we 
have, as the first approximation to \/A. 



for a second approximation we take 

= a (i+-)' < 2 > 

and so on. 2 

1 Metrica, i. 8, pp. 18. 22-20. 5. 

2 The method indicated by Heron was known to Barlaam and Nicolas 
Rhabdas in the fourteenth century. The equivalent of it was used by 
Luca Paciuolo (fifteenth-sixteenth century), and it was known to the other 
Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave 
2J, 2^ and 2f t ffa as successive approximations to -/$ He obtained 

2 (2 1 ) 2 6 

the first as 2+ 5 o the second as 2* -- ^ , and the third as 

It . Ct Z . Z., 

**~ "' The above rule Kives -H 2+ i) = 2 5. *(5+7> ! ) = 2 5 )> 



The formula of Heron was again put forward, in modern times, by 
Buzengeiger as a im i ans of accounting for the Archimedean approxima- 
tion to v3, apparently without knowing its previous history. 13ertrand 
also stated it in a treatise on arithmetic (1853). The method, too, by 
which Oppermann and Alexeieff sought to account for Archimedea's 
approximations is in reality the same. The latter method depends on 
the formula 



Alexeieff separated A into two factors o , 7' , and pointed out that if, say. 

>Z> . 



then, l(a.+ Jb)>v^> or 



APPROXIMATIONS TO SURDS 325 

Substituting in (1) the value a?b for A, we obtain 



Heron does riot seem to have used this formula with a nega- 
tive sign, unless in titereoru. I. 33 (34, Hultsch), where \/(63) 
and again, if J (</ -t & ) = j , 2 ^/Ia -f b ) = ^ , 



and so on. 

Now suppose that, ^n Heron's formulae, we put a = A" , A/ a = 
otj = .Xj, i/tfi = t r p and so on. We then have 

f A \ , , ^ ^ A A 2AVo 

- 



^ i f A \ , , 
A, = i + -J - J 



that is, X lt ^ aie, lespectively, the arithmetic and harmonic means 
between X , x X 2 , jc^ are the arithmetic and harmonic means between 
X 19 x l9 and so on, exactly as in AlexeiefTs formulae. 

Let us now try to apply the method to Archiniedes's case, \/3, and we 
shall see to what extent it serves to give what we want. Suppose 
we begin with 3 >\/3 > 1. We then have 



1), or 2 
and iiom this we derive successively 



But, if we start trom $, obtained by the formula + v , <v / (a 2 -f 
we obtain the following approximations by excess, 



The second process then gives one of Archiuiedes's results, V 
neither of the two processes gives the other, , directly. The latter 

can, however, be obtained by using the formula that, if .-<-, then 

U (J, 

a ma + nc c 
b mb -f- nd d 

For we can obtain '{'i$ from J J and ^V* thus : = , or from 

u . .... 11.97-7 1060 265 . 

Si and i thus: n 56 ^j gl2 = ^ ; and so on. Or again *&# can 

t , A - j^ IMHT- i - .i 18817497 18914 1351 
be obtained from !8Ji and $4 thus: 108644 . 56 = 1U920 = ^gg-- 



The advantage of the method is that, as compared with that of con- 
tinued fractions, it is a veiy rapid way of arriving at a close approxi- 
mation. G anther has shown that the (wM l)th approximation obtained 
by Heron's formula is the 2"'tk obtained by continued fractions. ('Die 
quadratischen Iriationahtaten der Alten und deren Entwickelungs- 
methodeii in Abhandlunyen zur Gcxch. d. Math. iv. 1882, pp. 83-6.) 



326 HERON OF ALEXANDRIA 

is given as approximately 8 T V In Metrica I, 9, as we 
have seen, \/(63) is given as 7| T ^, which was doubtless 
obtained from the formula (1) as 



The above seems to be the only classical rule which has 
been handed down for finding second and further approxi- 
mations to the value of a surd. But, although Heron thus 
shows how to obtain a second approximation, namely by 
formula (2), he does not seem to make any direct use of 
this method himself, and consequently the question how the 
approximations closer than the first which are to be found in 
his works were obtained still remains an open one. 

(y) Quadrilaterals. 

It is unnecessary to give in detail the methods of measuring 
the areas of quadrilaterals (chaps. 11-16). Heron deals with 
the following kinds, the parallel- trapezium (isosceles or non- 
isosceles), the rhombus and rhomboid, and the quadrilateral 
which has one angle right and in which the four sides have 
given lengths. Heron points out that in the rhombus or 
rhomboid, and in the general case of the quadrilateral, it is 
necessary to know a diagonal as well as the four sides. The 
mensuration in all the cases reduces to that of the rectangle 
and triangle. 

(8) The regular 'polygons with 3, 4, 5, 6', 7, #, ,9, 10 ', 11, 
or 12 sides. 

Beginning with the equilateral triangle (chap. 17), Heron 
proves that, if a be the side and p the perpendicular from 
a vertex on the opposite side, a 2 :y a = 4 : 3, whence 

a>:^ 2 a 2 = 4:3 = 16: 12, 
so that a 4 : (A ABC)* =16:3, 

and (AABCJ* = T \a 4 . In the particular case taken a = 10 
and A 2 = 1875, whence A = 43^ nearly. 

Another method is to use an approximate value for </3 in 
the formula \/3 . & 2 /4. This is what is done in the Geometrica 
14 (10, Heib.), where we are told that the area is (? 



THE REGULAR POLYGONS 327 




now J + ^o = if = i"(fS)> so ^h&i ^ e Approximation used by 
Heron for \/3 is here ff . For the side 10, the method gives 
the same result as above, for %% . 100 = 43. 

The regular pentagon is next taken (chap. 18). Heron 
premises the following lemma. 

Let ABC be a right-angled triangle, with the angle A equal 
to f A Produce AC to so that CO = AC 
If now AO is divided in extreme and 
mean ratio, AB is equal to the greater 
segment. (For produce AB to D so that 
AD = 40, and join BO, DO. Then, since 
A DO is isosceles and the angle at A=.\R, 
ZA/)0 = AOJ) = $R t and, from the 
equality of the triangles 4 #(7, OBG, 
LAOE = Z7J40 = f B. It follows that 
the triangle 4Z)0 is the isosceles triangle of Eucl. IV. 10, and 
AD is divided in extreme and mean ratio in B.) Therefore, 
says Heron, (BA + ACf = 5 AC*. [This is Eucl. XIII. 1.] 

Now, since Z BOG = f jK, if /JG f be produced to # so that 
CE = -BO, 5A 7 subtends at an angle equal to \ R, and there- 
fore BE is the side of a regular pentagon inscribed in the 
circle with as centre and OB as radius. (This circle also 
passes through 7), and BJ) is the side of a regular decagon in 
the same circle.) If now BO = AB = r, OC = p, BE = a, 
we have from above, (r-hy;) 2 = 5^, whence, since \/5 is 
approximately f , we obtain approximately r = f />, and 
a = />, so that /> = |a. Hence ?pa = %a\ and the area 
of the pentagon = fa 2 . Heron adds that, if we take a closer 
approximation to \/5 than |, we shall obtain the area still 
more exactly. In the Geometry l the formula is given as ^a 2 . 

The regular hejcayon (chap. 19) is simply 6 times the 
equilateral triangle with the same side. If A be the area 
of the equilateral triangle with side a, Heron has proved 
that A 2 = ^a 4 (Metrica I. 17), hence (hexagon) 2 = a^a 4 . If, 
e.g. a = 10, (hexagon) 2 = 67500, and (hexagon) = 259 nearly. 
In the Geometry 2 the formula is given as ^a-, while 'another 
book' is quoted as giving 6 (J + T 1 Ty )c( 2 ; it is added that the 
latter formula, obtained from the area of the triangle, (^ + Yfr)tt 2 , 
represents the more accurate procedure, and is fully set out by 

1 Geom. 102 (21, 14, Heib.). 2 Ib. 102 (21, 16, 17, Heib.). 



&28 HERON OF ALEXANDRIA 



Heron. As a matter of fact, however, 6 (^ + ^) = -^ exactly, 
and only the Metrioa gives the more accurate calculation. 

The regular heptagon. 

Heron assumes (chap. 20) that, if a be the side and r the 
radius of the circumscribing circle, a = |r, being approxi- 
mately equal to the perpendicular from the centre of the 
circle to the side of the regular hexagon inscribed in it (for - 
is the approximate value of ^ \/3). This theorem is quoted by 
Jordanus Nemorarius (d. 1237) as an 'Indian rule'; he pro- 
bably obtained it from Abu'l Wafa (940-98). The Metrica 
shows that it is of Greek origin, and, if Archimedes really 
wrote a book on the heptagon in a circle, it may be due to 
him. If then p is the perpendicular from the centre of the 
circle on the side (a) of the inscribed heptagon, r/(%a) = S/3% 
or 16/7, whence p 2 /(^a}' 2 = - 2 ^V-, and p/^i = (approxi- 
mately) HJ/7 or 43/21. Consequently the area of the 
heptagon = 7 . %pa = 7 . f|a- = f f a'-. 




The regular octagon, decagon and dodecagon. 

In these cases (chaps. 21, 23, 25) Heron finds p by drawing 
the perpendicular 00 from 0, the centre of the 
circumscribed circle, on a side AB> and then making 
the angle OA D equal to the angle AOD. 
For the octagon, 

/.ADC = %R, and p = a(l+ v/2) = a(l + |) 
A C B 

or | a . f approximately. 

For the decagon, 

LADG = f JJ, and -AD : DC =5:4 nearly (see preceding page) ; 
hence AD : AC =5:3, and p = %a (| +f ) = f r*. 

For the dodecagon, 
LADG = ^Ji, and 7; = Ja(2-h V3) = Ja(2 + J) = V~ a 

approximately. 

Accordingly A% = - 2 ^-(t 2 , ^1 10 = \ 5 -a 2 , J. 12 = -\ 5 -^ 2 ) where a is 
the side in each ca&e. 

The regular enneagon and hendecagon. 

In these cases (chaps. 22, 24) the Table of Chords (i e. 




THE REGULAR POLYGONS 329 

presumably Hipparchus's Table) is appealed to. If AB be the 
side (a) of an enneagon or hendecagon inscribed in a circle, AC 
the diameter through A 9 we are told that the Table of Chords 
gives and -fa as the respective approximate values of the 
ratio AB/AC. The angles subtended at the centre by the 
side AB are 10 and 32^ 8 T respec- 
tively, and Ptolemy's Table gives, 
as the chords subtended by angles of 
40 and 33 respectively, 41?' 2' 33" 
and 34^ 4' 55" (expressed in 120th 
parts of the diameter) ; Heron's 
figures correspond to 40^ and 33^ 
36' respectively. For the euueayoit 
AC* = SAB*, whence BC* = BAB* 
or approximately */-AB*, and 
BC = ~y a ; therefore (area of 
enueagon) = S . &ABC=- s a*. For 

the hetulecayoib AC* = $/-AB* and BC* = - 5 J/-AB~, so that 
BC = tya, and urea of hendecagon = ^- . A ABC = - 6 ^a 2 . 

An ancient formula for the ratio between the side of any 
regular polygon and the diameter of the circumscribing circle 
is preserved in (jet^m. 147 sq. (= Pscudo-Dioph. 23-41), 

namely d n = /t- n . Now the ratio iut n /d n tends to TT as the 

o 

number ( H) of sides increases, and the formula indicates a time 
when TT was generally taken its = 3. 

(6) The Circle. 

Coming to the circle (Metrica 1. 26) Heron uses Archi- 
medes's value for TT, namely - 2 T ^, making the circumference of 
a circle -\ 4 -r and the area y-J(f 2 , where / is the radius and d the 
diameter. It is here that he gives the more exact limits 
for TT which he says that Archimedes found in his work On 
Piinthides and Cylinder &> but which are not convenient for 
calculations. The limits, as we have seen, are given in the 
text as ^^^\-<rr < V^?i~ au( l w ^h Tannery's alteration to 
* < VaS?!- are c l uite satisfactory. 1 

1 See vol. i, pp. 232-3. 



330 HERON OF ALEXANDRIA 

() Segment of a circle. 

According to Heron (Metrica I. 30) the ancients measured 
the area of a segment rather inaccurately, taking the area 
to be ^ (6 + h) h, where b is the base and h the height. He 
conjectures that it arose from taking TT = 3, because, if we 
apply the formula to the semicircle, the area becomes ^ . 3 r 2 , 
where r is the radius. Those, he says (chap. 31), who have 
investigated the area more accurately have added T 1 ^(^^) i 
to the above formula, making it %(b + h)h + -r*(i^) 2 ' an ^ ^ s 
seems to correspond to the value 3^ for TT, since, when applied 
to the semicircle, the formula gives ^(3r 2 + ^r 2 ). He adds 
that this formula should only be applied to segments of 
a circle less than a semicircle, and not even to all of these, but 
only in cases where b is not greater than 3 h. Suppose e.g. 
that b = 60, h = 1 ; in that case even T 1 ^(|6) 2 = ^ . 900 = 64f , 
which is greater even than the parallelogram with 60, 1 as 
sides, which again is greater than the segment. Where there- 
fore 6 > 3 h, he adopts another procedure. 

This is exactly modelled on Archimedes's quadrature of 
a segment of a parabola. Heron proves (Metrica I. 27-29, 32) 
that, if ADB be a segment of a circle, and 1) the middle point 

of the arc, and if the arcs AD, DB be 
similarly bisected at E, F, 

A ADB < 4 ( A AED + A DFB). 

A Similarly, if the same construction be 

made for the segments AED, BFD, each 

of them is less than 4 times the sum of the two small triangles 
in the segments left over. It follows that 

(area of segmt. ADB) > A ADB { \ * i + (i) 2 + ...} 




' If therefore we measure the triangle, and add one-third of 
it, we shall obtain the area of the segment as nearly as 
possible.' That is, for segments in which b > 3ft, Heron 
takes the area to be equal to that of the parabolic segment 
with the same base and height, or f bh. 

In addition to these three formulae for 8, the area of 
a segment, there are yet others, namely 

8 = i (6 + h) h(l+ T ), Mensurae 29, 
l + , 31. 



SEGMENT OF A CIRCLE 331 

The first of these formulae is applied to a segment greater 
than a semicircle, the second to a segment less than a semi- 
circle. 

In the Metrica the area of a segment greater than a semi- 
circle is obtained by subtracting the area of the complementary 
segment from the area of the circle. 

From the Geometrica l we find that the circumference of the 
segment less than a semicircle was taken to be V(h* + 4 h?) + $h 

or alternatively V(l?+W)+ \ v/(/; 2 -f 4/< 2 )-&}f- 

(77) Ellipse, parabolic wjiueut, surface ofcyliiuter, right 
cone, sphere awl segment of sphere. 

After the area of an ellipse (Metrica I. 34) and of a parabolic 
segment (chap. 35), Heron gives the surface of a cylinder 
(chap. 36) and a right cone (chap. 37) ; in both cases he unrolls 
the surface on a plane so that the surface becomes that of a 
parallelogram in the one case and a sector of a circle in the 
other. For the surface of a sphere (chap. 38) and a segment of 
it (chap. 39) he simply uses Archimedes' s results. 

Book I ends with a hint how to measure irregular figures, 
plane or not. If the figure is plane and bounded by an 
irregular curve, neighbouring points are taken on the curve 
such that, if they are joined in order, the contour of the 
polygon so formed is not much different from the curve 
itself, and the polygon is then measured by dividing it into 
triangles. If the surface of an irregular solid figure is to be 
found, you wrap round it pieces of very thin paper or cloth, 
enough to cover it, and you then spread out the paper or 
cloth and measure that. 

Book II. Measurement of Volumes. 

The preface to Book II is interesting as showing how 
vague the traditions about Archimedes had already become. 

* After the measurement of surfaces, rectilinear or not, it is 
proper to proceed to the solid bodies, the surfaces of which we 
have already measured in the preceding book, surfaces plane 
and spherical, conical and cylindrical, and irregular surfaces 
as well. The methods of dealing with these solids are, in 
1 Cf. Geom., 94, 95 (19. 2, 4, Heib.), 97. 4 (20. 7, Heib.). 



332 HERON OF ALEXANDRIA 

view of their surprising character, referred to Archimedes by 
certain writers who give the traditional account of their 
origin. But whether they belong to Archimedes or another, 
it is necessary to give a sketch of these methods as well/ 

The Book begins with generalities about figures all the 
sections of which parallel to the base are equal to the base 
and similarly situated, while the centres of the sections are on 
a straight line through the centre of the base, which may be 
either obliquely inclined or perpendicular to the base ; whether 
the said straight line (' the axis ') is or is not. perpendicular to 
the base, the volume is equal to the product of the area of the 
base and the perpendicular height of the top of the figure 
from the base. The term ' height ' is thenceforward restricted 
to the length of the perpendicular from the top of the figure 
on the base. 

(a) Cone, cylinder, parallelepiped (prism), pyramid, and 

frustum. 

II. 1-7 deal with a cone, a cylinder, a ' parallelepiped ' (the 
base of which is not restricted to the parallelogram but is in 
the illustration given a regular hexagon, so that the figure is 
more properly a prism with polygonal bases), a triangular 
prism, a pyramid with base of any form, a frustum of a 
triangular pyramid ; the figures are in general oblique. 



(ft) Wedge-shaped solid (/Soo/LuV/coy or 

II. 8 is a case which is perhaps worth giving. It is that of 
a rectilineal solid, the base of which is a rectangle ABCD and 
has opposite to it another rectangle EFGH, the sides of which 
are respectively parallel but not necessarily proportional to 
those of ABCD. Take AK equal to EF, and BL equal to FO. 
Bisect BK, GL in V, W, and draw KRPU, VQOM parallel to 
AD, and LQRN, WOPT parallel to AB. Join FK, GR, LG, 
GU, HN. 

Then the solid is divided into (1) the parallelepiped with 
AR, EG as opposite faces, (2) the prism with KL as base and 
FG as the opposite edge, (3) the prism with NU as base and 
GH as opposite edge, and (4) the pyramid with RLCU as base 
and G as vertex. Let h be the * height ' of the figure. Now 



MEASUREMENT OF SOLIDS 



333 



the parallelepiped (1) is on A R as base and has height h ; the 
prism (2) is equal to a parallelepiped on KQ as base and with 
height h ; the prism (3) is equal to a parallelepiped with NP 
as base and height h; and finally the pyramid (4) is equal to 
a parallelepiped of height h and one-third of RC as base. 




w 



Therefore the whole solid is equal to one parallelepiped 
with height h and base equal to (AR + KQ + NP + RO + $RO) 
or AO + $RO. 

Now, if AB = a, #(7 = ft, EF = r, FG = d, 

AV = * ( 4- c), 4T 7 = J(6 + rf), Q = i(c<-c), RP = *(6-d). 
Therefore volume of solid 



The solid in (juestion is evidently the true jSoycuriror (' little 
altar'), for the formula is used to calculate the content of 
a ftwfjLio-Kos in titereom. II. 40 (68, Heib!) It is also, I think, 
the <r<l>r)vl<rKo$ (' little wedge '), a measurement of which is 
given in tftereom. 1. 26 (25, Heib.) It is true that the second 
term of the first factor -fa (a c) (& d) is there neglected, 
perhaps because in the case taken (a = 7, 6 = 6, c = 5, d = 4) 
this term (= ) is small compared with the other (= 30). A 
particular a-^rj^ia-Ko^ in which either c = a or d = 6, was 
called &vv\ the second term in the factor of the content 
vanishes in this case, and, if e.g. c = a, the content is ^(6 + d)ah. 
Another /Ja>/z/ovco9 is measured in Stereom. I. 35 (34, Heib.), 
where the solid is inaccurately called 'a pyramid oblong 
and truncated (/c^Aoupoy) or half -perfect '. 



334 HERON OF ALEXANDRIA 

The method is the same mutatis mutandis as that used in 
II. 6 for the frustum of a pyramid with any triangle for base, 
and it is applied in II. 9 to the case of a frustum of a pyramid 
with a square base, the formula for which is 



where a, a' are the sides of the larger and smaller bases 
respectively, and h the height; the expression is of course 
easily reduced to ^ h(a? + aa' + a' 2 ). 

(y) Frustum of cone, sphere, and segment thereof. 

A. frustum of a cone is next measured in two ways, (1) by 
comparison with the corresponding frustum of the circum- 
scribing pyramid with square base, (2) directly as the 
'difference between two cones (chaps. 9, 10). The volume of 
the frustum of the cone is to that of the frustum of the 
circumscribing pyramid as the area of the base of the cone to 
that of the base of the pyramid ; i.e. the volume of the frus- 
tum of the cone is TT, or ^J, times the above expression for 
the frustum of the pyramid with a 2 , a' 2 as bases, and it 
reduces to ^nh (d* + aa' + a' 2 ), where a, a' are the diameters 
of the two bases. For the sphere (chap, in Heron uses 
Archimedes's proposition that the circumscribing cylinder is 
1^ times the sphere, whence the volume of the sphere 
= f .tZ.-H^ 2 or Ir^ 3 ; f r a segment of a sphere (chap. 12) he 
likewise uses Archimedes's result (On the Sphere and Cylinder, 
II. 4). 

(8) Anchor-ring or tore. 

The anchor-ring or tore is next measured (chap. 13) by 
means of a proposition which Heron quotes from Dionyso- 
dorus, and which is to the effect that, if a be the radius of either 
circular section of the tore through the axis of revolution, and 
c the distance of its centre from that axis, 

?ra 2 : ac = (volume of tore) : ?rc 2 . 2 a 

[whence volume of tore = 2ir 2 ca 2 ]. In the particular case 
taken a = 6, c = 14, and Heron obtains, from the proportion 
113^:84 = F:7392, F= 9956f But he shows that he is 
aware that the volume is the product of the area of the 



MEASUREMENT OF SOLIDS 335 

describing circle and the length of the path of its centre* 
For, he says, since 1 4 is a radius (of the path of the centre), 
28 is its diameter and 88 its circumference. 'If then the tore 
be straightened out and made into a cylinder, it will have 88 
for its length, and the diameter of the base of the cylinder is 
12; so that the solid content of the cylinder is, as we have 
seen, 9950$' (= 88 .-^ 



(e) The tivo special solids of Archimedes 8 ' Method '. 

Chaps. 14,15 give the measurement of the two remarkable 
solids of Archimedes's Method, following Archimedes's results. 

() The Jive regular solids. 

In chaps. 16-18 Heron measures the content of the five 
regular solids after the cube. He has of course in each case 
to find the perpendicular from the centre of the circumscrib- 
ing sphere on any face. Let p be this perpendicular, a the 
edge of the solid, r the radius of the circle circumscribing any 
face. Then (1) for the tetrahedron 

a 2 = 3r 2 , p* = a 2 -a* = 2 . 

(2) In the case of the octahedron, which is the sum of two 
equal pyramids on a square base, the content is one-third 
of that base multiplied by the diagonal of the figure, 
i.e. \ .a 2 . */2a or \/2.a 3 ; in the case taken a = 7, and 
Heron takes 10 as an approximation to \/(2. 7 2 ) or \/98, the 
result being |. 10.49 or 163J. (3) In the case of the icosa- 
hedron Heron merely says that 

p : a = 93 : 127 (the real value of the ratio is $ 7 7 + 3 s/5\ ^ 
(4) In the case of the dodecahedron, Heron says that 
p:a = 9 : 8 (the true value is ^ / > and, if \/5 is 

put equal to f , Heron's ratio is readily obtained). 

Book II ends with an allusion to the method attributed to 
Archimedes for measuring the contents of irregular bodies by 
immersing them in water and measuring the amount of fluid 
displaced. 



336 HERON OF ALEXANDRIA 

Book III. Divisions of figures. 

This book has much in common with Euclid's book On <Uri- 
sions (of figures), the problem being to divide various figures, 
plane or solid, by a straight line or plane into parts having 
a given ratio. In III. 1-3 a triangle is divided into two parts 
in a given ratio by a straight line (1) passing through a vertex, 
(2) parallel to a side, (3) through any point on a side. 
III. 4 is worth description: 'Given a triangle ARC, to cut 
out of it a triangle DEF (where D, E, F are points on the 
sides respectively) given in magnitude and such that the 
triangles AEF, BFD, CED may be equal in area/ Heron 
assumes that, if 7), E, F divide the sides so that 

AF:FB=BD:DC = CE:EA, 

the latter three triangles are equal in area. 

He then has to find the value of 
each of the three ratios which will 
result in the triangle DEF having a 
given area. 

Join AD. 

Since BD:CD = CE:EA, 
BC:CD=CA:AE 9 
and &ABC : &ADC= &ADC: &ADE. 
Also &ABC: &ABD = &ADC-.AEDC. 

But (since the area of the triangle DEF is given) A EDO is 
given, as well as A ABC. Therefore AABD x A A DC is given. 
Therefore, if AH be perpendicular to BC, 

AH*. BD. DC is given; 

therefore BD . DC is given, and, since BC is given, D is given 
in position (we have to apply to BC a rectangle equal to 
BD . DC and falling short by a square). 

As an example Heron takes AB = 13, BC = 14, CA = 15, 

ADEF = 24. &ABC is then 84, and AH = 1 2. 
t 

Thus A EDO = 20, and AH 2 . BD . DC = 4 . 84 . 20 = 6720 ; 
therefore BD.DC^ 6720/144 or 46 (the text omits the ). 
Therefore, says Heron, BD = 8 approximately. For 8 we 




DIVISIONS OF FIGURES 337 

should Apparently have 8j, since DO is immediately stated to 
be 5^ (not 6). That is, in solving the equation 



= 0, 

which gives # = 7 ^(2^), Heron apparently substituted 2J or 
| for 2^, thereby obtaining 1^ as an approximation to the 
surd. 

(The lemma assumed in this proposition is easily proved. 
Let m : be the ratio AF: FB = RD : DC = CE:EA. 

Then AF mc/(m + it), FB = nc/(m + n), CE = mb/(m + 71), 

KA = nb/(m + n), &c. 
Hence 



< = --I-! 1 :.- =&BDF/AABC= ACDE/AAfid, 
(m + n) 

and the triangles AFE, BDF, CDE are equal. 

Pappus 1 has the proposition that the triangles A BC, DEF 
have the same centre of gravity.) 

Heron next shows how to divide a parallel-trapezium into 
two parts in a given ratio by a straight line (1) through the 
point of intersection of the non-parallel sides, (2) through a 
given point on one of the parallel sides, (3) parallel to the 
parallel sides, (4) through a point on one of the non-parallel 
sides (III. 5-8). III. 9 shows how to divide the area of a 
circle into parts which have a given ratio by means of an 
inner circle with the same centre. For the problems begin- 
ning with III. 10 Heron says that numerical calculation alone 
no longer suffices, but geometrical methods must be applied. 
Threw problems are reduced to. problems solved by Apollonius 
in his treatise On, mdting off an area. The first of these is 
III. 10, to cut off from the angle of a triangle a given 
proportion of the triangle by a straight line through a point 
on the opposite side produced. III. 11, 12, 13 show how 
to cut any quadrilateral into parts in a given ratio by a 
straight line through a point (1) on a side (a) dividing the 
side in the given ratio, (6) not so dividing it, (2) not on any 
side, (a) in the case where the quadrilateral is a trapezium, 
i.e. has two sides parallel, (6) in the case where it is not; the 
last case (/>) is reduced (like III. 10) to the ; cutting-off* of an 

1 Pappus, viii, pp. 1034-8. Of. pp. 430 2 post. 




358 HERON OF ALEXANDRIA 

area*. These propositions are ingenious and interesting. 

III. 11 shall be given as a specimen. 

Given any quadrilateral ABCD and a point E on the side 

AD, to draw through E a straight line EF which shall cut 

the quadrilateral into two parts in 
the ratio of AE to ED. (We omit 
the analysis.) Draw (76?. parallel 
to DA to meet AB produced in G. 

Join BE, and draw GH parallel 
to BE meeting BC in H. 

A i D Join CE, EH, EG. 

Then &GBE= AHBE and, adding &ABE to each, we have 

A AGE= (quadrilateral ABHE). 
Therefore (quadr. ABIIE) : &CED = A GAE: ACED 

= AE:ED. 

But (quadr. ABHE) and AC ED are parts of the quadri- 
lateral, and they leave over only the triangle EHC. We have 
therefore only to divide &E1W in the same ratio AK-.ED by 
the straight line EF. This is done by dividing HO at F in 
the ratio AE: ED and joining EF. 

The next proposition (III. 1 2) is easily reduced to this. 
If AE: ED is not equal to the given ratio, let F divide AD 

in the given ratio, and through F 
draw FG dividing the quadri- 
lateral in the given ratio (III. 11). 
Join EG, and draw FH parallel 
to EG. Let FH meet BC in 7/, 
and join EH. 

A FED Then is EH the required straight 

line through E dividing the quad- 
rilateral in the given ratio. 

For &FGE = AHGE. Add to each (quadr. GEDC). 
Therefore (quadr. CGFD) = (quadr. CHED). 
Therefore EH divides the quadrilateral in the given ratio, 
just as FG does. 

The case (III. 13) where E is hot on a side of the quadri- 
lateral [(2) above] takes two different forms according as the 




DIVISIONS OF FIGURES 



339 



two opposite sides which the required straight line cuts are 
(a) parallel or (6) not parallel. In the first case (a) the 
problem reduces to drawing a straight line through E inter- 
secting the parallel sides in points F, G such that BF+AG 




is equal to a given length. In the second case (b) where 
BC, AD are not parallel Heron supposes them to meet in //. 
The angle at // is then given, and the area ABH. It is then 
a question of cutting off from a triangle with vertex // a 
triangle IIF(f of given area by a straight line drawn from E, 
which is again a problem in Apollonius's Cutting-off of an 




HAG D 

area. The auxiliary problem in case (a) is easily solved in 
III. 16. Measure AH equal to the given length. Join BH 
and bisect it at M. Then EM meets BO, AD in points such 
that BF+ AG= the given length. For. by congruent triangles, 
BF = Gil. 

The same problems are solved for the case of any polygon 
in III. 14, 15. A sphere is then divided (III. 17) into segments 
such that their surfaces are in a given ratio, by means of 
Archimedes, On the Sphere and Cylinder, II. 3, just as, in 
III. 23, Prop. 4 of the same Book is used to divide a sphere 
into segments having their volumes in a given ratio. 

III. 18 is interesting because it recalls an ingenious pro- 
position in Euclid's book On Divisions. Heron's problem is 
* To divide a given circle into three equal parts by two straight 

Z 2 




340 HERON OF ALEXANDRIA 

lines ', and he observes that, ' as the problem is clearly not 
rational, we shall, for practical convenience, make the division, 

as exactly as possible, in the follow- 
ing way/ AE is the side of an 
equilateral triangle inscribed in the 
circle. Let CD be the parallel 
diameter, the centre of the circle, 
and join AO, BO, AD, DB. Then 
shall the segment ABD be very 
nearly one-third of the circle. For, 
since AB is the side of an equi- 
lateral triangle in the circle, the 
sector OAEB is one-third of the 

circle. And the triangle AOB forming part of the sector 
is equal to the triangle ADB\ therefore the segment AEB 
plus the triangle ABD is equal to one-third of the circle, 
and the segment ABD only differs from this by the small 
segment on BD as base, which may be neglected. Euclid's 
proposition is to cut off one-third (or any fraction) of a circle 
between two parallel chords (see vol. i, pp. 429-30). 

III. 19 finds a point D within any triangle ABC such that 
the triangles DBC 9 DCA, DAB are all equal ; and then Heron 
passes to the division of solid figures. 

The solid figures divided in a given ratio (besides the 
sphere) are the pyramid with base of any form (III. 20), 
the cone (III. 21) and the frustum of a cone (III. 22), the 
cutting planes being parallel to the base in each case. These 
problems involve the extraction of the cube root of a number 
which is in general not an exact cube, and the point of 
interest is Heron's method of approximating to the cube root 
in such a case. Take the case of the cone, and suppose that 
the portion to be cut off at the top is to the rest of the cone as 
m to n. We have to find the ratio in which the height or the 
edge is cut by the plane parallel to the base which cuts 
the cone in the given ratio. The volume of a cone being 
^7rc 2 A, where c is the radius of the base and h the height, 
we have to find the height of the cone the volume of which 

is - .virc 2 h, and, as the height hf is to the radius cf of 

* h 



its base as h is to c, we have simply to find // where 



DIVISIONS OF FIGURES 341 

= m/(m + 7&). Or, if we take the edges e, e' instead 
of the heights, e' 3 /e 3 = m/(m + 7i). In the case taken by 
Heron m : u = 4 : 1, and e = 5. Consequently e' 3 = . 5 3 = 100. 
Therefore, says Heron, e 4 = 4^ approximately, and in III. 20 
he shows how this is arrived at. 

Approximation to the cube root of a iwu-cube number. 

' Take the nearest cube numbers to 100 both above and 
below; these are 125 and 64. 

Then 125-100 = 25. 

and 100- 64 = 36. 

Multiply 5 into 36; this gives 180. Add 100, making 280. 
(Divide 180 by 280); this gives T \. Add this to the side of 
the smaller cube : this gives 4 X \. This is as nearly as possible 
the cube root ("cubic side") of 100 units.' 

We have to conjecture Heron's formula from this example. 
Generally, if a :i < A < (a+ l) :i , suppose that A a? = d l9 and 
(a+l)* A = t/ 2 . The best suggestion that has been made 
is Wertheim's, 1 namely that Heron's formula for the approxi- 

mate cube root was a+ .*-", The 5 multiplied 

(u + ljc^ + arfg r 

into the 36 might indeed have been the square root of 25 or 
v/6/,,, and the 100 added to the 180 in the denominator of the 
fraction might have been the original number 100 (A) and not 
4 .25 or ad^ but Wertheim's conjecture is the more satisfactory 
because it can be evolved out of quite elementary considera- 
tions. This is shown by G. Enestrom as follows. 2 Using the 
same notation, Enestrom further supposes that x is the exact 
value of J/A , and that U a)- 1 = 8 V <* + 1 x)* = 5 2 . 

Thus 
5j = # 3 3 it 2 a + 3 tea* a j , and 3ax(x a) = x 3 a :l 8 l = d l 8 l . 

Similarly from S 2 = (a+1 x)* we derive 



Therefore 

c/ 2 -5 2 _ 3(a4 

~~ 



5 a) ~" (a? a) 

_ a + 1_ a + 1 
"~a(j a) a 

1 Zeitschr. f. Math. n. Physik, xliv, 1899, hist.-litt. Abt., pp. 1-3, 

2 Bibliotheca Mathematica, viii s , 1907-8, pp. 412-13. 



342 HERON OF ALEXANDRIA 

and, solving for a; a, we obtain 



xa = 



or 



(a 



Since 8 V 8 2 are in any case the cubes of fractions, we may 
neglect them for a first approximation, and we have 



(a 



c 

/i\ 

Xl \ 




III. 22, which shows how to cut a frustum of a cone in a given 
ratio by a section parallel to the bases, shall end our account 
of the Metrica. I shall give the general formulae on the left 
and Heron's case on the right. Let ABED be the frustum, 
let the diameters of the bases be a, a', and the height h. 
Complete the cone, and let the height of ODE be x. 

Suppose that the frustum has to be cut by a plane FG in 
such a way that 

(frustum DG) : (frustum FB) = m : u. 
In the case taken by Heron 

a = 28, a' = 21, h = 12, m = 4, n = 1. 
Draw DH perpendicular to AB. 



DIVISIONS OF FIGURES 



343 



Since (DG) : (FB) = m : n, 



Now 

(DB) = Y$7rh(Q? + (ji(Jif' 
* 

and (DG) = 

Let ?/ be the height (CM) of the 
cone CFG. 

Then DH:AH = CK:KA, 
or h:%(a ') = 

whence ic is known. 



(DG):( J ra) = 4:l, 



cone CAS= (CM) + (/)). 
Now, says Heron, 



[He might have said simply 



This gives y or CM, 

whence LM is known. 

Now 



so that AD is known. 

Therefore DF 
known. 



. 

A 



s 



(DB) = 5698, 
(DG) = 4558|. 



i 14.12 
x + h= - 1 - = 48, 
d 5 

and x = 48 -12 = 36. 
(cone /).#) = 4158, 

(cone CFG) = 4158 + 4558=8716|, 
(cone CAB) = 4158 + 5698 = 9856. 



'' 



8716 



9856 + 4158 
= 87 16|.-^,Y 4 4= 97805, 
whence y = 46 approximately. 
Therefore XJ/ = y-x = 10. 

4 ^ = (3^ + 1 2- 
= 156}, 

and AD =12$. 
Therefore 



344 HERON OF ALEXANDRIA 

Quadratic eq f uuttions solved in, Heron. 

We have already met with one such equation (in Metrica 
III. 4), namely a? 2 14# + 46 = 0, the result only (x = 8|) 
being given. There are others in the Geometrica where the 
process of solution is shown. 

(1) Geometrica 24, 3 (Heib.). 'Given a square such that the 
sum of its area and perimeter is 896 feet: to separate the area 
from the perimeter ' : i.e. x 2 4- 4 a; = 896. Heron takes half of 
4 and adds its square, completing the square on the left side. 

(2) Geometrica 21, 9 and 24, 46 (Heib.) give one and the same 
equation, Geom. 24, 47 another like it. 'Given the sum of 
the diameter, perimeter and area of a circle, to find each 
of them/ 

The two equations are 



and 

Our usual method is to begin by dividing by ^ throughout, 
so as to leave d 2 as the first term. Heron's is to Multiply by 
such a number as will leave a square as the first term. In this 
case he multiplies by 154, giving H 2 d 2 + 58 . lid = 212 . 154 
or 67^,154 as the case may be. Completing the square, 
he obtains (lld-f 29) 2 = 32648 + 811 or 10395 + 841. Thus 
lld + 29 = V(33480) or ^(11236), that is, 183 or 106. 

Thus lid = 154 or 77, and d = 14 or 7, as the case may be. 



Indeterminate problems in the Geometrica. 

Some very interesting indeterminate problems are now 
included by Heiberg in the Geometrical Two of them (chap. 
24, 1-2) were included in the Geeponicus in Hultsch's edition 
(sections 78, 79;; the. rest are new, having been found in the 
Constantinople manuscript from which Schone edited the 
Metrica. As, however, these problems, to whatever period 
they belong, are more akin to algebra than to mensuration, 
they will be more properly described in a later chapter on 
Algebra. 

1 Heronin Alexandrini opera, vol. iv, p. 414. 28 sq. 



THE DIOPTRA 345 

The Dioptra (rrepl SiOTTTpas). 

This treatise begins with a careful description of the 
dioptra, an instrument which served with the ancients for 
the same purpose as a theodolite with us (chaps. 1-5). The 
problems with which the treatise goes on to deal are 
(a) problems of ' heights and distances ', (&) engineering pro- 
blems, (c) problems of mensuration, to which is added 
(chap. 34) a description of a ' hodometer', or taxameter, con- 
sisting of an arrangement of toothed wheels and endless 
screws on the same axes working on the teeth of the next 
wheels respectively. The book ends with the problem 
(chap. 37), 'With a given force to move a given weight by 
means of interacting toothed wheels', which really belongs 
to mechanics, and was apparently added, like some other 
problems (e.g. 31, 'to measure the outflow of, i.e. the volume 
of water issuing from, a spring '), in order to make the book 
more comprehensive. The essential problems dealt with are 
such as the following. To determine the difference of level 
between two given points (6), to draw a straight line connect- 
ing two points the one of which is not visible from the other 
(7), to measure the least breadth of a river (9), the distance of 
two inaccessible points (10), the height of an inaccessible point 
(12), to determine the difference between the heights of two 
inaccessible points and the position of the straight line joining 
them (13), the depth of a ditch (14) ; to bore a tunnel through 
a mountain going straight from one mouth to the other (15), to 
sink a shaft through a mountain perpendicularly to a canal 
flowing underneath (1(5) ; given a subterranean canal of any 
form, to find on the ground above a point from which a 
vertical shaft must be sunk in order to reach a given point 
on the canal (for the purpose e.g. of removing an obstruction) 
(20) ; to construct a harbour on the model of a given segment 
of a circle,^giveii the ends (17), to construct a vault so that it 
may have a spherical surface modelled on a given segment 
(18). The mensuration problems include the following: to 
measure an irregular area, which is done by inscribing a 
rectilineal figure and then drawing perpendiculars to the 
sides at intervals to meet the contour (23), or by drawing one 
straight line across the area and erecting perpendiculars from 



346 HERON OF ALEXANDRIA 

that to meet the contour on both sides (24) ; given that all 
the boundary stones of a certain area have disappeared except 
two or three, but that the plan of the* area is forthcoming, 
to determine the position of the lost boundary stones (25). 
Chaps. 26-8 remind us of the Metrica: to divide a given 
area into given parts by straight lines drawn from one point 
(26) ; to measure a given area without entering it, whether 
because it is thickly covered with trees, obstructed by houses, 
or entry is forbidden! (27) ; chaps. 28-30 = Metrica III. 7, 
III. 1, and I. 7, the last of these three propositions being the 
proof of the ' formula of Heron ' for the area of a triangle in 
terms of the sides. Chap. 35 shows how to find the distance 
between Rome and Alexandria along a great circle of the 
earth by means of the observation of the same eclipse at 
the two places, the analemma for Rome, and a concave hemi- 
sphere constructed for Alexandria to show the position of the 
sun at the time of the said eclipse. It is here mentioned that 
the estimate by Eratosthenes of the earth's circumference in 
his book On the Measurement of the Earth was the most 
accurate that had been made up to date. 1 Some hold that 
the chapter, like some others which have no particular con- 
nexion with the real subject of the Dioptra (e.g. chaps. 31, 34, 
37-8) were probably inserted by a later editor, ' in order to 
make the treatise as complete as possible V 2 

The Mechanics. 

It is evident that the Mechanics, as preserved in the Arabic, 
is far from having kept its original form, especially in 
Book I. It begins with an account of the arrangement of 
toothed wheels designed to solve the problem of moving a 
given weight by a given force ; this account is the same as 
that given at the end of the Greek text of the Diojrtra, and it 
is clearly the same description as that which Pappus 3 found in 
tho work of Heron entitled Bapov\KO$ ('weight-lifter') and 
himself reproduced with a ratio of force to weight altered 
from 5 : 1000 to 4 : 160 and with a ratio of 2 : 1 substituted for 
5 : 1 in the diameters of successive wheels. It would appear 
that the chapter from the Bapov\Ko$ was inserted in place of 

1 Heron, vol. iii, p. 302. 18-17. 2 Ib , p. 302. I). 

3 Pappus, viii, p. 1060 sq. 



THE MECHANICS 347 

the first chapter or chapters of the real Mechanics which had 
been lost. The treatise would doiibtless begin with generalities 
introductory to mechanics such as we find in the (much 
interpolated) beginning of Pappus, Book VIII. It must then 
apparently have dealt with the properties of circles, cylinders, 
and spheres with reference to their importance in mechanics ; 
for in Book II. 21 Heron says that the circle is of all figures 
the most movable and most easily moved, the same thing 
applying also to the cylinder and sphere, and he adds in 
support of this a reference to a proof ' in the preceding Book '. 
This reference may be to I. 21, but at the end of that chapter 
he says that * cylinders, even when heavy, if placed on the 
ground so that they touch it in one line only, are easily 
moved, and the same is true of spheres also, a matter which 
ive have already discussed'; the discussion may have come 
earlier in the Book, in a chapter now lost. 

The treatise, beginning with chap. 2 after the passage 
interpolated from the Bapov\Ko$, is curiously disconnected. 
Chaps. 2-7 discuss the motion of circles or wheels, equal or 
unequal, moving on different axes (e.g. interacting toothed 
wheels), or fixed on the same axis, much after the fashion of 
the Aristotelian Median teal problems. 

Aristotle's Wheel. 

In particular (chap. 7) Heron attempts to explain the puzzle 
of the ' Wheel of Aristotle ', which remained a puzzle up to quite 
modern times, and gave rise to the proverb, c rotam Aristotelis 
magis torquere, quo magis torqueretur '. l ' The question is ', says 
the Aristotelian problem 24, 6 why does the greater circle roll an 
equal distance with the lesser circle when they are placed about 
the same centre, whereas, when they roll separately, as the 
size of one is to the size of the other, so are the straight lines 
traversed by them to one another ? ' 2 Let AC, BJ) be quadrants 
of circles with centre bounded by the same radii, and draw 
tangents AE, BF at A and B. In the first case suppose the 
circle BD to roll along BF till D takes the position //; then 
the radius ODC will be at right angles to AE, and C will be 
at (?, a point such that AG is equal to BH. In the second 

1 See Van Capelle, Aristotelis quaestiones mechanicae, 1812, p. 263 sq. 

2 Avist. Mechanica, 855 a 28. 



348 



HERON OF ALEXANDRIA 



case suppose the circle AC to roll along AE till ODG takes 
the position O'FE; then D will be at F where AE = BF. 
And similarly if a whole revolution is performed and OB A is 
again perpendicular to AE. Contrary, therefore, to the prin- 
ciple that the greater circle moves quicker than the smaller on 
the same axis, it would appear that the movement of the 




smaller in this case is as quick as that of the greater, since 
BH = AG, and BF '= AE. Heron's explanation is that, e.g. 
in the case where the larger circle rolls on AE, the lesser 
circle maintains the same speed as the greater because it has 
tivo motions; for if we regard the smaller circle as merely 
fastened to the larger, and not rolling at all, its centre will 
move to 0' traversing a distance 00' equal to AE and BF\ 
-hence the greater circle will take the lesser with it over an 
equal distance, the rolling of the lesser circle having no effect 
upon this. 

The parallelogram of velocities. 

Heron next proves the parallelogram of velocities (chap. 8) ; 

he takes the case of a rectangle, but the proof is applicable 

generally. 

The way it is put is this. A 
point moves with uniform velocity 
along a straight line AB, from A 
to JB, while at the same time AB 
moves with uniform velocity always 
parallel to itself with its extremity 
A describing the straight line AG. 

Suppose that, when the point arrives at B, the straight line 




THE PARALLELOGRAM OF VELOCITIES 349 

reaches the position CD. . Let EF be any intermediate 
position of AB, and G the position at the same instant 
of the moving point on it. Then clearly AE:AC=EG:EF; 
therefore AE:EG = AC:EF= AC:CD, and it follows that 
6r lies on the diagonal AD, which is therefore the actual path 
of the moving point. 

Chaps. 9-19 contain a digression on the construction of 
plane and solid figures similar to given figures but greater or 
less in a given ratio. Heron observes that the case of plane 
figures involves the finding of a mean proportional between 
two straight lines, and 'the case of solid figures the finding of 
two mean proportionals ; in chap. 1 1 he gives his solution of 
the latter problem, which is preserved in Pappus and Eutocius 
as well, and has already been given above (vol. i, pp. 262-3). 

The end of chap. 19 contains, quite inconsequently, the con- 
struction of a toothed wheel to move on an endless screw, 
after which chap. 20 makes a fresh start with some observa- 
tions on weights in equilibrium on a horizontal plane but 
tending to fall when the plane is inclined, and on the ready 
mobility of objects of cylindrical form which touch the plane 
in one line only. 

Motion on an inclined plane. 

When a weight is hanging freely by a rope over a pulley, 
no force applied to the other end of the rope less than the 
weight itself will keep it up, but, if the weight is placed on an 
inclined plane, and both the plane and the portion of the 
weight in contact with it are smooth, the case is different. 
Suppose, e.g., that a weight in the form of a cylinder is placed 
on an inclined plane so that the line in which they touch is 
horizontal; then the force required to be applied to a rope 
parallel to the line of greatest slope in the plane in order to 
keep the weight in equilibrium is less than the weight. For 
the vertical plane passing through the line of contact between 
the cylinder and the plane divides the cylinder into two 
unequal parts, that on the downward side of the plane being 
the greater, so that the cylinder will tend to roll down ; but 
the force required to support the cylinder is the ' equivalent ', 
not of the weight of the whole cylinder, but of the difference 



350 HERON OF ALEXANDRIA 

between the two portions into which the vertical plane cuts it 
(chap. 23). 

On the centre of gravity. 

This brings Heron to the centre of gravity (chap. 24). Here 
a definition by Posidonius, a Stoic, of the c centre of gravity ' 
or ' centre of inclination ' is given, namely ' a point such that, 
if the body is hung up at it, the body is divided into two 
equal parts ' (he should obviously have said ' divided by any 
vertical plane through the point of suspension into two equal 
parts '). But, Heron says, Archimedes distinguished between 
the c centre of gravity ' and the ' point of suspension ', defining 
the latter as a point on the body such that, if the body is 
hung up at it, all the parts of the body remain in equilibrium 
and do not oscillate or incline in any direction. ' " Bodies", said 
Archimedes, " may rest (without inclining one way or another) 
with either a line, or only one point, in the body fixed "/ The 
* centre of inclination ', says Heron, c is one single p*)int in any 
particular body to which all the vertical lines through the 
points of suspension converge/ Comparing Simplicius's quo- 
tation of a definition by Archimedes in his Kei/rpojSapi/ca, to 
the effect that the centre of gravity is a certain point in the 
body such that, if the body is hung up by a string attached to 
that point, it will remain in its position without inclining in 
any direction, 1 we see that Heron directly used a certain 
treatise of Archimedes. So evidently did Pappus, who has 
a similar definition. Pappus also speaks of a body supported 
at a point by a vertical stick : if, he says, the body is in 
equilibrium, the line of the stick produced upwards must pass 
through the centre of gravity. 2 Similarly Heron says that 
the same principles apply when the body is supported as when 
it is suspended. Taking up next (chaps. 25-31) the question 
of ' supports ', he considers cases of a heavy beam or a wall 
supported on a number of pillars, equidistant or not, even 
or not even in number, and projecting or not projecting 
beyond one or both of the extreme pillars, and finds how 
much of the weight is supported on each pillar. He says 
that Archimedes laid down the principles in his ' Book on 

1 Simplicius on De caelo, p. 543. 31-4, Heib. 

2 Pappus, viii, p. 1032. 5-24. 



ON THE CENTRE OF GRAVITY 351 

Supports '. As, however, the principles are the same whether 
the body is supported or hung up, it does not follow that 
this was a different work from that known as Trepl (vy&v. 
Chaps. 32-3, which are on the principles of the lever or of 
weighing, end with an explanation amounting to the fact 
that ' greater circles overpower smaller when their movement 
is about the same centre ', a proposition which Pappus says 
that Archimedes proved in his work ncpl {vyS*?. 1 In chap. 32, 
too, Heron gives as his authority a proof given by Archimedes 
in the same work. With I. 33 may be compared II. 7, 
where Heron returns to the same subject of the greater and 
lesser circles moving about the same centre and states the 
fact that weights reciprocally proportional to their radii are 
in equilibrium when suspended from opposite ends of the 
horizontal diameters, observing that Archimedes proved the 
proposition in his work On the equalization of inclination ' 
(presumably I<roppo7riai). 

Book II. The five mechanical powers. 

Heron deals with the wheel and axle, the lever, the pulley, 
the wedge and the screw, and with combinations of these 
powers. The description of the powers comes first, chaps. 1-6, 
and then, after II. 7, the proposition above referred to, and the 
theory of the several powers based upon it (chaps. 8-20). 
Applications to specific cases follow. Thus it is shown how 
to move a weight of 1000 talents by means of a force of 
5 talents, first by the system of wheels described in the 
Bapov\Ko$, next by a system of pulleys, and thirdly by a 
combination of levers (chaps. 21-5). It is possible to combine 
the different powers (other than the wedge) to produce the 
same result (chap. 29). The wedge and screw are discussed 
with reference to their angles (chaps. 30-1). and chap. 32 refers 
to the effect of friction. 

Mechanics in daily life; queries and answers. 

After a prefatory chapter (33), a number of queries resem- 
bling the Aristotelian problems are stated and answered 
(chap. 34), e.g. 'Why do waggons with two wheels carry 
a weight more easily than those with four wheels?', 'Why 
1 Pappus, viii, p. 1068. 20-3. 



352 HERON OF ALEXANDRIA 

do great weights fall to the ground in a shorter time than 
lighter ones?', 'Why does a stick break sooner when one 
puts one's knee against i^ in the middle ? ', ' Why do people 
use pincers father than the hand to draw a tooth?', 'Why 
is it easy to move weights which are suspended?', and 
' Why is it the more difficult to move such weights the farther 
the hand is away from them, right up to the point of suspension 
or a point near it ? V Why are great ships turned by a rudder 
although it is so small ? ', * Why do arrows penetrate armour 
or metal plates but fail tp penetrate cloth spread out ? ' 

Problems on the centre of gravity, &c. 

II. 35, 36, 37 show how to find the centre of gravity of 
a triangle, a quadrilateral and a pentagon respectively. Then, 
assuming that a triangle of uniform thickness is supported by 
a prop at each angle, Heron finds what weight is supported 
by each prop, (a) when the props support the triangle only, 
(6) when they support the triangle plus a given weight placed 
at any point on it (chaps. 38, 39). Lastly, if known weights 
are put on the triangle at each angle, he finds the centre of 
gravity of the system (chap. 40) ; the problem is then extended 
to the case of any polygon (chap. 41). 

Book III deals with the practical construction of engines 
for all sorts of purposes, machines employing pulleys with 
one, two, or more supports for lifting weights, oil-presses, &c. 

The Catoptrica. 

This work need not detain us long. Several of the theoretical 
propositions which it contains are the same as propositions 
in the so-called Catoptrica of Euclid, which, as we have 
seen, was in all probability the work of Theon of Alexandria 
and therefore much later in date. In addition to theoretical 
propositions, it contains problems the purpose of which is to 
construct mirrors or combinations of mirrors of such shape 
as will reflect objects in a particular way, e.g. to make the 
right side appear as the right in the picture (instead of the 
reverse), to enable a person to see his back or to appear in 
the mirror head downwards, with face distorted, with three 
eyes or two noses, and so forth. Concave and convex 



THE CATOPTRTCA 



353 



cylindrical mirrors play a part in these arrangements. The 
whole theory of course ultimately depends on the main pro- 
positions 4 and 5 that the angles of incidence and reflection 
are equal whether the mirror is plane or circular. 

Heron's proof of equality of angles of Incidence and reflection. 

Let AB be a plane mirror, C the eye, D the object seen. 
The argument rests on the fact that nature ' does nothing in 
vain '. Thus light travels in a straight line, that is, by the 
quickest road. Therefore, even 
when the ray is a line broken 
at a point by reflection, it must 
mark the shortest broken line 
of the kind connecting the eyo 
and the object. Now, says 
Heron, I maintain that the 
shortest of the broken lines 
(broken at the mirror) which 
connect C and D is the line, as 

CAD, the parts of which make equal angles with the mirror. 
Join T) A and produce it to meet in F the perpendicular from 
C to AB. Let li be any point on the mirror other than A, 
and join FB, BD. 

Now LEAF = LBAD 




= L OAK, by hypothesis. 

Therefore the triangles AEF, AEG, having two angles equal 
and AK common, are equal in all respects. 
Therefore CA = AF, and OA + AD = DF. 
Since FE = EG, and BE is perpendicular to FC, BF = BC. 

Therefore CB + BD = FB + BD 

> FD, 
i.e. > CA+AD. 

The proposition was of course known to Archimedes. We 
gather from a scholium to the Pseudo- Euclidean Catoptrica 
that he proved it in a different way, namely by reductio ad 
absurdum, thus: Denote the angles GAE y DAB by a, /9 re- 
spectively. Then, a is > = or < ft. Suppose ot > ft. Then, 

A a 



354 HERON OF ALEXANDRIA 

reversing the ray so that the eye is at D instead of (7, and the 
object at C instead of Z), we must have /8 > a. But /8 was 
less than a, which is impossible. (Similarly it can be proved 
that a is not less than jS.) Therefore oc = /8. 

In the Pseudo-Euclidean Catopt^ca the proposition is 
practically assumed; for the third assumption or postulate 
at the beginning states in effect that, in the above figure, if A 
be the point of incidence, CE:EA = J)H:HA (where DH is 
perpendicular to AB). It follows instantaneously (Prop. 1) 
that LGAE^LDAH. 

If the mirror is the convex side of a circle, the same result 
follows a fortiori. Let CA, AD meet 
the arc at equal angles, and CB, BD at 
unequal angles. Let AE be the tan- 
gent at A, and complete the figure. 
Then, says Heron, (the angles GAG, 
BAD being by hypothesis equal), if we 
subtract the equal angles GAE, BAF 
from the equal angles OAC 9 BAD (both 
pairs of angles being ' mixed ', be it 
observed), we have LEAG = L FAD. Therefore CA+AD 
< CF+FD and a fortiori < CR + BD. 

The problems solved (though the text is so corrupt in places 
that little can be made of it) were such as the following: 
11, To construct a right-handed mirror (i.e. a mirror which 
makes the right side right and the left side left instead of 
the opposite); 12, to construct the mirror called polytheoron 
('with many images'); 16, to construct a mirror inside the 
window of a house, so that you can see in it (while inside 
the room) everything that passes in the street; 18, to arrange 
mirrors in a given place so that a person who approaches 
cannot actually see either himself or any one else but can see 
any image desired (a ' ghost-seer '). 




XIX 

PAPPUS OF ALEXANDRIA 

WE have seen that the Golden Age of Greek geometry 
ended with the time of Apollonius of Perga. But the influence 
of Euclid, Archimedes and Apollonius continued, and for some 
time there was a succession of quite competent mathematicians 
who, although not originating anything of capital importance, 
kept up the tradition. Besides those who were known for 
particular investigations, e.g. of new curves or surfaces, there 
were such men as Geminus who, it cannot be doubted, were 
thoroughly familiar with the great classics. Geminus, as we 
have seen, wrote a comprehensive work of almost encyclopaedic 
character on the classification and content of mathematics, 
including the history of the development of each subject. 
But the beginning of the Christian era sees quite a different 
state of things. Except in sphaeric and astronomy (Menelaus 
and Ptolemy), production was limited to elementary text- 
books of decidedly feeble quality. In the meantime it would 
seem that the study of higher geometry languished or was 
completely in abeyance, until Pappus arose to revive interest 
in the subject. From the way in which he thinks it necessary 
to describe the contents of the classical works belonging to 
the Treasury of Analysis, for example, one would suppose 
that by his time many of them were, if not lost, completely 
forgotten, and that the great task which he set himself was 
the re-establishment of geometry on its former high plane of 
achievement. Presumably such interest as he was able to 
arouse soon flickered out, but for us his work has an in- 
estimable value as constituting, after the works of the great 
mathematicians which have actually survived, the. most im- 
portant of all our sources. 

A a 2 



356 PAPPUS OF ALEXANDRIA 

Date of Pappus. 

Pappus lived at the end of the third century A.D. The 
authority for this date is a marginal note in a Leyden manu- 
script of chronological tables by Theon of Alexandria, where, 
opposite to the name of Diocletian, a scholium says, ' In his 
time Pappus wrote'. Diocletian reigned from 284 to 305, 
and this must therefore be the period of Pappus's literary 
activity. It is true that Suidas makes him a contemporary 
of Theon of Alexandria, adding that they both lived under 
Theodosius I (379-395). But Suidas was evidently not well 
acquainted with the works of Pappus; though he mentions 
a description of the earth by him and a commentary on four 
Books of Ptolemy's Sytitaxis, he has no word about his greatest 
work, the Synagoge. As Theon also wrote a commentary on 
Ptolemy and incorporated a great deal of the commentary of 
Pappus, it is probable that Suidas had Theon's commentary 
before him and from the association of the two names wrongly 
inferred that they were contemporaries. 

Works (commentaries) other than the Collection. 

Besides the Si/nagoge, which is the main subject of this 
chapter, Pappus wrote several commentaries, now lost except for 
fragments which have survived in Greek or Arabic. One was 
a commentary on the Elements of Euclid. This must presum- 
ably have been pretty complete, for, while Proclus (on Eucl. I) 
quotes certain things from Pappus which may be assumed to 
have come in the notes on Book I, fragments of his commen- 
tary on Book X actually survive in the Arabic (see above, 
vol. i, pp. 154-5, 209), and again Eutocius in his note on Archi- 
medes, On the Sphere and Cylinder, I. 13, says that Pappus 
explained in his commentary on the Elements how to inscribe 
in a circle a polygon similar to a polygon inscribed in another 
circle, which problem would no doubt be solved by Pappus, as 
it is by a scholiast, in a note on XII. 1. Some of the references 
by Proclus deserve passing mention. (1) Pappus said that 
the converse of Post. 4 (equality of all right angles) is not 
true, i.e. it is not true that all angles equal to a right angle are 
themselves right, since the ' angle ' between the conterminous 
arcs of two semicircles which are equal and have their 




WORKS OTHER THAN THE COLLECTION 357 

diameters at right angles and terminating at one point is 
equal to, but is not, a right angle. 1 (2) Pappus said that, 
in addition to the genuine axioms of Euclid, there were others , 
on record about unequals added to 
equals and equals added to unequals. 
Others given by Pappus are (says 
Proclus) involved by the definitions, 
e.g. that 'all parts of the plane and of 
the straight line coincide with one 
another ', that ' a point divides a line, 
a line a surface, and a surface a solid ', and that ' the infinite 
is (obtained) in magnitudes both by addition and diminution 1 . 2 
(3) Pappus gave a pretty proof of Eucl. I. 5, which modern 
editors have spoiled when introducing it into text-books. If 
AB, AC are the equal sides in an isosceles triangle, Pappus 
compares the triangles ABC and ACB (i.e. as if he were com- 
paring the triangle ABC seen from the front with the same 
triangle seen from the back), and shows that they satisfy the 
conditions of I. 4, so that they are equal in all respects, whence 
the result follows. 3 

Marinus at the end of his commentary on Euclid's Data 
refers to a commentary by Pappus on that book. 

Pappus's commentary on Ptolemy's tfyibtaxis has already 
been mentioned (p. 274) ; it seems to have extended to six 
Books, if not to the whole of Ptolemy's work. The Fihrist 
says that he also wrote a commentary on Ptolemy's Plani- 
sjtJJiaeri'U/ni, which was translated into Arabic by Thabit b. 
Qurra. Pappus himself alludes to his own commentary on 
the Anal&m/Hui of Diodorus, in the course of which he used the 
conchoid of Nicomedes for the purpose of trisecting an angle. 

We come now to Pappus's great work. 

The Synayoge or Collection, 
(a) Character of the work; luide raiuje. 

Obviously written with the object of reviving the classical 
Greek geometry, it covers practically the whole field. It is, 

1 Proclus on Eucl. I, pp. 189-90. 2 /&., pp. 197. 6-198. 15. 

3 Ib., pp. 249. 20-250. 12. 



358 PAPPUS OF ALEXANDRIA 

however, a handbook or guide to Greek geometry rather than 
an encyclopaedia ; it was intended, that is, to be read with the 
original works (where still extant) rather than to enable them 
to be dispensed with. Thus in the case of the treatises 
included in the Treasury of Analysis there is a general intro- 
duction, followed by a general account of the contents, with 
lemmas, &c., designed to facilitate the reading of the treatises 
themselves. On the other hand, where the history of a subject 
is given, e.g. that of the problem of the duplication of the 
cube or the finding of the two mean proportionals, the various 
solutions themselves are reproduced, presumably because they 
were not easily accessible, but had to bo collected from various 
sources. Even when it is some accessible classic which is 
being described, the opportunity is taken to give alternative 
methods, or to make improvements in proofs, extensions, and 
so on. Without pretending to great originality, the whole 
work shows, on the part of the author, a thorough grasp of 
all the subjects treated, independence of judgement, mastery 
of technique; the style is terse and clear; in short, Pappus 
stands out as an accomplished and versatile mathematician, 
a worthy representative of the classical Greek geometry. 

(]8) List of authors mentioned. 

The immense range of the Collection can be gathered from 
a mere enumeration of the names of the various mathematicians 
quoted or referred to in the course of it. The greatest of 
them, Euclid, Archimedes and Apollonius, are of course con- 
tinually cited, others are mentioned for some particular 
achievement, and in a few cases the mention of a name by 
Pappus is the whole of the information we possess about the 
person mentioned. In giving the list of the names occurring 
in the book, it will, I think, be convenient and may economize 
future references if I note in brackets the particular occasion 
of the reference to the writers who are mentioned for one 
achievement or as the authors of a particular book or investi- 
gation. The list in alphabetical order is : Apollonius of Perga, 
Archimedes, Aristaeus the elder (author of a treatise in five 
Books on the Elements of Conies or of ' five Books on Solid 
Loci connected with the conies'), Aristarchus of Samos (On the 



THE COLLECTION 359 

sizes and distances of the sun and moo?*,), Autolycus (0/6 the 
moving sphere), Carpus of Antioch (who is quoted as having 
said that Archimedes wrote only one mechanical book, that 
on sphere-making, since he held the mechanical appliances 
which made him famous to be nevertheless unworthy of 
written description : Carpus himself, who was known as 
mechanicus, applied geometry to other arts of this practical 
kind), Charmandrus (who added three simple arid obvious loci 
to those which formed the beginning of the Plane Loci of 
Apollonius), Conon of Samos, the friend of Archimedes (cited 
as the propounder of a theorem about the spiral in a plane 
which Archimedes proved : this would, however, seem to be 
a mistake, as Archimedes says at the beginning of his treatise 
that he sent certain theorems, without proofs, to Conon, who 
would certainly have proved them had he lived), Demetrius of 
Alexandria (mentioned as the author of a work called ' Linear 
considerations', ypa/jLfjLiKal CTrio-rao-eiy, i.e. considerations on 
curves, as to which nothing more is known), Diriostratus, 
the brother of Menaechmus (cited, with Nicornedes, as having 
used the curve of Hippias, to which they gave the name of 
quadratriM, TeTpaya>viovora, for the squaring of the circle), 
Diodorus (mentioned as the author of an A nalemma), Erato- 
sthenes (whose mean-finder, an appliance for finding two or 
any number of geometric means, is described, and who is 
further mentioned as the author of two Books ' On means ' 
and of a work entitled 'Loci witli reference to means'), 
Erycinus (from whose Pamdoxa are quoted various problems 
seeming at first sight to be inconsistent with Eucl. I. 21, it 
being shown that straight lines can be drawn from two points 
on the base of a triangle to a point within the triangle which 
are together greater than the other two sides, provided that the 
points in the base may be points other than the extremities), 
Euclid, Geminus the mathematician (from whom is cited a 
remark 011 Archimedes contained in his book ' On the classifica- 
tion of the mathematical sciences ', see above, p. 223), Heraclitus 
(from whom Pappus quotes an elegant solution of a vv<ri$ 
with reference to a square), Hermodorus (Pappus's son, to 
whom he dedicated Books VII, VIII of his Collection), Heron 
of Alexandria (whose mechanical works are extensively quoted 
from), Hierius the philosopher (a contemporary of Pappus, 



360 PAPPUS OF ALEXANDRIA 

who is mentioned as having asked Pappus's opinion on the 
attempted solution by * plane ' methods of the problem of the two 
means, which actually gives a method of approximating to 
a solution 1 ), Hipparchus (quoted as practically adopting three 
of the hypotheses of Aristarchus of Samos), Megethion (to 
whom Pappus dedicated Book V of his Collection), Menelaus 
of Alexandria (quoted as the author of Sphaerica and as having 
applied the name irapd8oo$ to a certain curve), Nicomachus 
(on three means additional to the first three), Nicomedes, 
Pandrosion (to whom Book III of the Collection is dedicated), 
Pericles (editor of Euclid's Data), Philon of Byzantium (men- 
tioned along with Heron), Philon of Tyana (mentioned as the 
discoverer of certain complicated curves derived from the inter- 
weaving of plectoid and other surfaces), Plato (with reference 
to the live regular solids), Ptolemy, Theodosius (author of the 
Sphaerica and On Days and Nights). 

(y) Translations and editions. 

4 

The first published edition of the Collection was the Latin 

translation by Commandimis (Venice 1589, but dated at the 

end 'Pisauri apud Hicronymum Concordiam 1588'; reissued 

with only the title-page changed 'Pisauri... 1602'). Up to 

1876 portions only of the Greek text had appeared, namely 

Books VII, VIII in Greek and German, by C. J. Gerharclt, 1871 

chaps. 33-105 of Book V, by Eisenmann, Paris 1824, chaps 

45-52 of Book IV in losephi Torelli Veronensis Geometrica 

1769, the remains of Book II, by John Wallis (in Open 

inathematica, III, Oxford 1699); in addition, the restorers 

of works of Euclid and Apollonius from the indication! 

furnished by Pappus give extracts from the Greek tex 

relating to the particular works, Breton le Champ on Euclid'i 

Porisms, Halley in his edition of the Conies of Apolloniu 

(1710) and in his translation from the Arabic and restoratioi 

respectively of the De sectione rationis and De sections spati 

of Apollonius (1706), Carnerer on Apolloiiius's Tactiones (1795 

Simson and Horsley in their restorations of Apollonius's Plan 

Loci and Inclinations published in the years 1749 and 177 

respectively. In the years 1876-8 appeared the only coin 

1 See vol. i, pp. 268-70. 



THE COLLECTION. BOOKS I, II, III 361 

plete Greek text, with apparatus, Latin translation, com- 
mentary, appendices and indices, by Friedrich Hultsch ; this 
great edition is one of the first monuments of the revived 
study of the history of Greek mathematics in the last half 
of the nineteenth century, and has properly formed the model 
for other definitive editions of the Greek text of the other 
classical Greek mathematicians, e.g. the editions of Euclid, 
Archimedes, Apollonius, &c., by Heiberg and others. The 
Greek index in this edition of Pappus deserves special mention 
because it largely serves as a dictionary of mathematical 
terms Uvsecl not only in Pappus but by the Greek -mathe- 
maticians generally. 

(8) tiunvinary of contents. 

At the beginning of the work, Book I and the first 13 pro- 
positions (out of 26) of Book II are missing. The first 13 
propositions of Book II evidently, like the rest of the Book, 
dealt with Apollonius's method of working with very large 
numbers expressed in successive powers of the myriad, 10000. 
This system has already been described (vol. i, pp. 40, 54-7). 
The work of Apollouius seems to have contained 26 proposi- 
tions (25 leading up to, and the 26th containing, the final 
continued multiplication). 

Book III consists of four sections. Section (1) is a sort of 
history of the problem of finding two mean proportionals, in 
continued proportion, between two given straight lines. 

It begins with some general remarks about the distinction 
between theorems and problems. Pappus observes that, 
whereas the ancients called them all alike 'by one name, some 
regarding them all as problems and others as theorems, a clear 
distinction was drawn by those who favoured more exact 
terminology. According to the latter a problem is that in 
which it is proposed to do or construct something, a theorem 
that in which, given certain hypotheses, we investigate that 
which follows from and is necessarily implied by them. 
Therefore he who propounds a theorem, no matter how he has 
become aware of the fact which is a necessary consequence of 
the premisses, must state, as the object of inquiry, the right 
result and no other. On the other hand, he who propounds 



362 PAPPUS OF ALEXANDRIA 

a problem may bid us do something which is in fact im- 
possible, and that without necessarily laying himself open 
to blame or criticism. For it is part of the solver's duty 
to determine the conditions under which the problem is 
possible or impossible, and, 'if possible, when, how, and in 
how many ways it is possible '. When, however, a man pro- 
fesses to know mathematics and yet commits some elementary 
blunder, he cannot escape censure. Pappus gives, as an 
example, the case of an unnamed person ' who was thought to 
be a great geometer' but who showed ignorance in that he 
claimed to know how to solve the problem of the two mean 
proportionals by 'plane' methods (i.e. by using the straight 
line and circle only). He then reproduces the argument of 
the anonymous person, for the purpose of showing that it 
does not solve the problem as its author claims. We have 
seen (vol. i, pp. 269-70) how the method, though not actually 
solving the problem, does furnish a series of successive approxi- 
mations to the real solution. Pappus adds a few simple 
lemmas assumed in the exposition. 

Next comes the passage 1 , already referred to, on the dis- 
tinction drawn by the ancients between (1) plane pi'oblems or 
problems which can be solved by means of the straight line 
and circle, (2) solid problems, or those which require for their 
solution one or more conic sections, (3) linear problems, or 
those which necessitate recourse to higher curves still, curves 
with a more complicated and indeed a forced or unnatural 
origin (^^La(T^vr]v) such as spirals, quadratrices, cochloids 
and cissoids, which have many surprising properties of their 
own. The problem of the two mean proportionals, being 
a solid problem, required for its solution either conies or some 
equivalent, and, as conies could not be constructed by purely 
geometrical means, various mechanical devices were invented 
such as that of Eratosthenes (the mean-finder), those described 
in the Mechanics of Philon and Heron, and that of Nicomedes 
(who used the ' cochloidal ' curve). Pappus proceeds to give the 
solutions of Eratosthenes, Nicomedes and Heron, and then adds 
a fourth which he claims as his own, but which is practically 
the same as that attributed by Eutocius to Sporus. All these 
solutions have been given above (vol. i, pp. 258-64, 266-8). 

1 Pappus, iii, p. 54. 7-22. 



THE COLLECTION. BOOK III 



363 



Section (2). The theory of means. 

Next follows a section (pp. 69-105) on the theory of the 
different kinds of means. The discussion takes its origin 
from the statement of the ' second problem ', which was that 
of 'exhibiting the three means' (i.e. the arithmetic, geometric 
and harmonic) 'in. a semicircle'. Pappus first gives a con- 
struction by which another geometer (aAXoy riy) claimed to 
have solved this problem, but he does not seem to have under- 
stood it, and returns to the same problem later (pp. 80-2). 

In the meantime he begins with the definitions of the 
three means and then shows how, given any two of three 
terms a, b, c in arithmetical, geometrical or harmonical pro- 
gression, the third can be found. The definition of the mean 
(b) of three terms a, b, c in harmonic progression being that it 
satisfies the relation a:c=a b:b c, Pappus gives alternative 
definitions for the arithmetic and geometric means in corre- 
sponding form , namely for the arithmetic mean a : a = a b : b c 
and for the geometric a : b = a b : b c. 

The construction for the harmonic mean is perhaps worth 
giving. Let AB, BG be two given straight lines. At A draw 
DAE perpendicular to AB, and make DA, AE equal. Join 
DB, BE. From G draw GF&t right 
angles to AB meeting DB in F. D 
Join EF meeting AB in C. Then 
BC is the required harmonic mean. 

For 



= EA:FG 




'C Q 



B 



= AC:CG 

= (AB-BC):(BC-BG). "" 

Similarly, by means of a like figure, we can find BG when 
AB, BO are given, and AB when 5(7, BG are given (in 
the latter case the perpendicular DE is drawn through G 
instead of A). 

Then follows a proposition that, if the three means and the 
several extremes are represented in one set of lines, there must 
be five of them at least, and, after a set of five such lines have 
been found in the smallest possible integers, Pappus passes to 



364 PAPPUS OF ALEXANDRIA 

the problem of representing the three means with the respective 
extremes by six lines drawn in a semicircle. 

Given a semicircle on the diameter AC, and B any point on 
the diameter, draw BD at right angles to A C. Let the tangent 




at D meet AC produced in G, and measure DH along the 
tangent equal to DG. Join HB meeting the radius OD in K. 
Let EF\}Q perpendicular to Of). 

Then, exactly as above, it is shown that OK is a harmonic 
mean between OF and OD. Also BD is the geometric mean 
between AB, BC, while OC (= OD) is the arithmetic mean 
between AB, BC. 

Therefore the six lines DO (= OC), OK, ^F, AB, BC, BD 
supply the three means with the respective extremes. 

But Pappus seems to have failed to observe that the ' certain 
other geometer ', who has the same figure excluding the dotted 
lines, supplied the same in Jive lines. For he said that OF 
is 'a harmonic mean'. It is in fact the harmonic mean 
between AB, BC, as is easily seen thus. 

Since ODB is a right-angled triangle, and BF perpendicular 
to OD, 

DF:BD = BD:DO, 
or DF. DO = BD* = AB . BC. 

But DO ^%(AB+BC\\ 

therefore DF . (AB + BC) = 2AB. BC. 

Therefore AB . (DF- BC) = BC.(AB- DF), 
that is, AB: BC = (AB - DF) : (DF- BC), 

and DF is the harmonic mean between AB, BC. 

Consequently the Jive lines D0( = OC), DF, AB, BC, BD 
exhibit all the three means with the extremes. 



THE COLLECTION. BOOK III 365 

Pappus does not seem to have seen this, for he observes 
that the geometer in question, though saying that DF is 
a harmonic mean, does not say how it is a harmonic mean 
or between what straight lines. 

In the next chapters (pp. 84-104) Pappus, following Nico- 
machus and others, defines seven more means, three of which 
were ancient and the last four more modern, and shows how 
we can form all ten means as linear functions of a, )8, y, where 
a, /3, y are in geometrical progression. The exposition has 
already been described (vol. i, pp. 86-9). 

Section (3). The 'Paradoxes' of Erycinus. 

The third section of Book III (pp. 104-30) contains a series 
of propositions, all of the same sort, which are curious rather 
than geometrically important. They appear to have been 
taken direct from a collection of Paradoxes by one Erycinus. 1 
The first set of these propositions (Props. 28-34) are connected 
with Eucl. I. 21, which says that, if from the extremities 
of the base of any triangle two strafght lines be drawn meeting 
at any point within the triangle, the straight lines are together 
less than the two sides of the triangle other than the base, 
but contain a greater angle. It is pointed out that, if the 
straight lines are allowed to be drawn from points in the base 
other than the extremities, their sum may be greater than the 
other two sides of the triangle. 

The first case taken is that of a right-angled triangle ABC 
right-angled at B. Draw AD to any point 7) on BC. Measure 
on it DE equal to A B, bisect A E 
in F, and join F(J. Then shall A 

DF+FC be > BA + AC. 
For EF+FC=AF+FC>AC. 

Add DE and AB respectively, 
and we have 

DF+FC> BA + AC. 

More elaborate propositions are next proved, such as the 
following. 

1. In any triangle, except an equilateral triangle or an isosceles 
1 Pappus, iii, p. 106. 5-9. 




366 PAPPUS OF ALEXANDRIA 

triangle with base less than one of the other sides, it is possible 
to construct on the base and within the triangle two straight 
lines meeting at a point, the sum of which is equal to the sum 
of the other two sides of the triangle (Props. 29, 30). 

2. In any triangle in which it is possible to construct two 
straight lines from the base to one internal point the sum 
of which is equal to the sum of the two sides of the triangle, 
it is also possible to construct two other such straight lines the 
sum of which is greater than that sum (Prop. 31). 

3. Under the same conditions, if the base is greater than either 
of the other two sides, two straight lines can be so constructed 
from the base to an internal point which are respectively 
greater than the other two sides of the triangle ; and the lines 
may be constructed so as to be respectively equal to the two 
sides, if one of those two sides is less than the other and each 
of them is less than the base (Props. 32, 33), 

4. The lines may be so constructed that their sum will bear to 
the sum of the two sides of the triangle any ratio less than 
2 : 1 (Prop. 34). 

As examples of the proofs, we will take the case of the 
scalene triangle, and prove the first and Part 1 of the third of 
the above propositions for such a triangle. 

In the triangle ABC with base BG let AB be greater 
than AC. 

Take D on BA such that BD = \ (BA + AC}. 




B H 



On DA between D and A take any point E, and draw EF 
parallel to BO. Let G be any point on EF\ draw GH parallel 
to AB and join GO. 



THE COLLECTION. BOOK III 367 

Now JSA + AC > EF+ FC 

> EG + GC and > GO, a fortiori. 

Produce GO to K so that GK = EA + AC, and with G as 
centre and GK as radius describe a circle. This circle will 
meet HC and HG, because GH = EB > BD or DA + AC and 
> GK, a fortiori. 

Then HG + GL = BE+EA+AC = BA + AC. 

To obtain two straight lines #G', <?' such that HG' + G'L 
>BA + AC, we have only to choose (?' so that HG', G'L 
enclose the straight lines IIG, GL completely. 

Next suppose that, given a triangle ABC in which BC > BA 




> AC, we are required to draw from two points on BC to 
an internal point two straight lines greater respectively than 
BA, AC. 

With B as centre and BA as radius describe the arc AEF. 
Take any point E on it, and any point 7) on BE produced 
but within the triangle. Join DC, and produce it to G so 
that DG = AC. Then with D as centre and DG as radius 
describe a circle. This will meet both BC and BD because 
BA > AC, and a fortiori DB > DG. 

Then, if L be any point on BJf, it is clear that BD, DL 
are two straight lines satisfying the conditions. 

A point ]/ on BH can be found such that /)// is equal 
to AB by marking oft* DN on DB equal to AB and drawing 
with /) as centre and DN as radius a circle meeting BH 
in //. Also, if DH be joined, 1)11 = AC. 

Propositions follow (35-9) having a similar relation to the 
Postulate in Archimedes, On the Sphere ami Cylinder, I, 
about conterminous broken lines one of which wholly encloses 



368 . PAPPUS OF ALEXANDRIA, 

the other, i.e. it is shown that broken lines, consisting of 
several straight lines, can be drawn with two points on the 
base of a triangle or parallelogram as extremities, and of 
greater total length than the remaining two sides of the 
triangle or three sides of the parallelogram. 

Props. 40-2 show that triangles or parallelograms can be 
constructed with sides respectively greater than those of a given 
triangle or parallelogram but having a less area. 

Section (4). The inscribing of tie five regular solids 
in a sphere. 

The fourth section of Book Til (pp. 132-G2) solves the 
problems of iiibcribing each of the five regular solids in a 
given sphere. After some preliminary lemmas (Props. 43-53), 
Pappus attacks the substantive problems (Props. 54-8), using 
the method of analysis followed by synthesis in the case of 
each solid. 

(a) In order to inscribe a regular pyramid or tetrahedron in 
the sphere, he finds two circular sections equal and parallel 
to one another, each of which contains^ one of two opposite 
edges as its diameter. If d be the diameter of the sphere, the 
parallel circular sections have d / as diameter, where d 2 = f cZ' 2 . 

(b) In the case of the cube Pappus again finds two parallel 
circular sections with diameter d' such that d 2 =%d' 2 ; a square 
inscribed in one of these circles is one face of the cube and 

"the square with sides parallel to those of the first square 
inscribed in the second circle is the opposite face. 

(r) In the case of the octahedron the same two parallel circular 
sections with diameter d' such that d* = f d' 2 are used ; an 
equilateral triangle inscribed in one circle is one face, and the 
opposite face is an equilateral triangle inscribed in the other 
circle but placed in exactly the opposite way. 

(d) In the case of the icosahedron Pappus finds four parallel 
circular sections each passing through three of the vertices of 
the icosahedron ; two of these are small circles circumscribing 
two opposite triangular faces respectively, and the other two 
circles are between these two circles, parallel to them, and 
equal to one another. The pairs of circles are determined in 



THE COLLECTION. BOOKS III, LV 369 

this way. If d be the diameter of the sphere, set out two 
straight lines x, y such that d, a, y are in the ratio of the sides 
of the regular pentagon,, hexagon and decagon respectively 
described in one and the same circle. The smaller pair of 
circles have r as radius where r 2 = # 2 , and the larger pair 
have r' as radius where r' 2 = %x 2 . 

(e) In the case of the dodecahedron the saw e four parallel 
circular sections are drawn as in the case of the icosahedron. 
Inscribed pentagons set the opposite way are inscribed in the 
two smaller circles ; these pentagons form opposite faces. 
Regular pentagons inscribed in the larger circles with vertices 
at the proper points (and again set the opposite way) determine 
ten more vertices of the inscribed dodecahedron. 

'The constructions are quite different from those in Euclid 
XIII. 13, 15, 14, 16, 17 respectively, where the problem is first 
to construct the particular regular solid and then to ' com- 
prehend it in a sphere ', i. e. to determine the circumscribing 
sphere in each case. I have set out Pappus's propositions in 
detail elsewhere. 1 

Book IV. 

At the beginning of Book IV the title and preface are 
missing, and the first section of the Book begins immediately 
with an enunciation. The first section (pp. 176-208) contains 
Propositions 1-12 which, with the exception of Props. 8-10, 
seem to be isolated propositions given for their own sakes and 
not connected by any general plan. 

Section (1). Extension of the theorem of Pythagoras. 

The first proposition is of great interest, being the generaliza- 
tion of Eucl. I. 47, as Pappus himself calls it, which is by this 
time pretty widely known to mathematicians. The enunciation 
is as follows. 

'If ABC be a triangle and on AB, AC any parallelograms 
whatever be described, as ABDE, ACFG, and if DE, FO 
produced meet in H and HA be joined, then the parallelo- 
grams ABDE, ACFG are together equal to the parallelogram 

1 Vide notes to Euclid's propositions in The Thirteen Books of Euclid's 
Elements, pp. 473, 480, 477, 489-91, 501-3. 

1028.2 H b 



370 



PAPPUS OF ALEXANDRIA 



contained by BC, HA in an angle which is equal to the sum of 
the angles ABC, DHA? 

Produce HA to meet BG in K, draw EL, CM parallel to KH 
meeting DE in L and FG in M, and join LNM. 

Then BLHA is a parallelogram, and HA is equal and 
parallel to EL. 




B 



Similarly HA, CM are equal and parallel ; therefore EL, CM 
are equal and parallel. 

Therefore BLMC is a parallelogram ; and its angle LBK is 
equal to the sum of the angles ABC, DHA. 

Now D ABDE = D BLHA t in the same parallels, 
= a BLNK, for the same reason. 
Similarly a ACFG = a 4(W# = D NKCM. 
Therefore, by addition, a ABDE+ a 4(7^(7 = a JBUfCL 

It has been observed (by Professor Cook Wilson 1 ) that the 
parallelograms on AB, AC need not necessarily be erected 
outwards from AB, AC. If one of them, e.g. that on AC, be 
drawn inwards, as in the second figure above, and Pappus's 
construction be made, we have a similar result with a negative 
sign, namely, 



= ABDE -O ACFG. 

Again, if both ABDE and ACFG were -drawn inwards, their 
sum would be equal to BLMC drawn outwards. Generally, if 
the areas of the parallelograms described outwards are regarded 
as of opposite sign to those of parallelograms drawn inwards, 

1 Mathematical Gazette, vii, p. 107 (May 1913). 



THE COLLECTION. BOOK IV 371 

we may say that the algebraic sum of the three parallelograms 
is equal to zero. 

Though Pappus only takes one case, as was the Greek habit, 
I see no reason to doubt that he was aware of the results 
in the other possible cases. 

Props. 2, 3 are noteworthy in that they use the method and 
phraseology of Eucl. X, proving that a certain line in one 
figure is the irrational called minor (see Eucl. X. 76), and 
a certain line in another figure is * the excess by which the 
binomial exceeds the straight line which produces with a 
rational area a medial whole ' (Eucl. X. 77). The propositions 
4-7 and 11-12 are quite interesting as geometrical exercises, 
but their bearing is not obvious : Props. 4 and 1 2 are remark- 
able in that they are cases of analysis followed by synthesis 
applied to the proof of theorems. Props. 8-10 belong to the 
subject of tangencies, being the sort of propositions that would 
come as particular cases in a book such as that of Apollonius 
On Contacts', Prop. 8 shows that, if there are two equal 
circles and a given point outside both, the diameter of the 
circle passing through the point and touching both circles 
is ' given ' ; the proof is in many places obscure and assumes 
lemmas of the same kind as those given later a propos of 
Apollonius's treatise; Prop. 10 purports to show how, given 
three unequal circles touching one another two and two, to 
find the diameter of the circle including them and touching 
all three. 



Section (2). On circles inscribed in the 
('shoemaker's knife 9 ). 

The next section (pp. 208-32), directed towards the demon- 
stration of a theorem about the relative sizes of successive 
circles inscribed in the dpftrjXos (shoemaker's knife), is ex- 
tremely interesting and clever, and I wish that I had space 
to reproduce it completely. The dpfirjXos, which we have 
already met with in Archimedes's 'Book of Lemmas', is 
formed thus. BO is the diameter of a semicircle BOO and 
BO is divided into two parts (in general unequal) at D; 
semicircles are described on BD, DO as diameters on the same 
side of BO as BOO is ; the figure included between the three 
semicircles is the &pf$ri\os. 

Bb2 



372 PAPPUS OF ALEXANDRIA 

There is, says Pappus, on record an ancient proposition to 
the follo^ng effect. Let successive circles be inscribed in the 
<fy>/3i7\oy touching the semicircles and one another as shown 
in the figure on p. 376, their centres being A, P, ... . Then, if 
Pi^ Pv Pz be the perpendiculars from the centres A, P, ... 
on BG and cZ a , c 2 , c? 3 ... the diameters of the corresponding 
circles, 

p l = d l , p 2 = 2(1 2 , ^3 = 3^3.... 

He begins by some lemmas, the course of which I shall 
reproduce as shortly as I can. 

I. If (Fig. 1) two circles with centres A, C of which the 
former is the greater touch externally at J5, and another circle 
with centre G touches the two circles at K, L respectively, 
then KL produced cuts the circle BL again in D and meets 
AC produced in a point E such that AB:BCAE:EC. 
This is easily proved, because the circular segments DL, LK 
are similar, and CD is parallel to AG. Therefore 

AB:BC = AK:CD = AE: EC. 
Also KE.EL = E&. 

For AE:EC = AB : BC = AB: CF = (AE-AB) : (EC-CF) 

= BE:EF. 




FIG 1. 

But AE : EC = KE : ED ; therefore KE : ED = BE : EF. 
Therefore KE .EL: EL. ED = BE 2 : BE . EF. 
And EL .ED = BE. EF', therefore KE . EL = EB 2 . 



THE COLLECTION. BOOK IV 



378 



II. Let (Fig. 2) BC, BD, being in one straight line, be the 
diameters of two semicircles BGC, BED, and let any circle as 
FGH touch both semicircles, A being the centre of the circle. 
Let M be the foot of the perpendicular from A on EC, r the 
radius of the circle FGH. There are two cases according 
as BD lies along BC or B lies between D and C\ i.e. in the 
first case the two semicircles are the outer and one of the inner 
semicircles of the ap/fyAoy, while in. the second case they are 
the two inner semicircles; in the latter case the circle FGH 
may either include the two semicircles or be entirely external 
to them. Now, says Pappus, it is to be proved that 



in case (1) 



BM:r = (BG+BD) : (BC-BD), 



and in case (2) BM: r = (BC-BD) : (BG+BD). 




We will confine ourselves to the first case, represented in 
the figure (Fig. 2). 

Draw through A the diameter HF parallel to BC. Then, 
since the circles BOG, HGF touch at G, and BC, HF are 
parallel diameters, GHB, GFC are both straight lines. 

Let E be the point of contact of the circles FGH and BED] 
then, similarly, BEF, HED are straight lines. 

Let HK, FL be drawn perpendicular to BC. 

By the similar triangles BGC, BKH we have 

BC:BG = BH: BK, or CB . BK = GB.BH\ 

t 

and by the similar triangles ELF, BED 

= BD:BE, or DB.BL = FB.BE. 



374 PAPPUS OF ALEXANDRIA 

But GB.BH=FB.BE\ 
therefore CB.BK=DB. BL, 

or BC:BD = BL:BK. 

Therefore (BG + BD) : (BG- BD) = (BL + BK) : (BL - BK) 

= 2BM:KL 
And KL = HF=2r; 

therefore BM:r = (BC+ BD) : (BG- BD). (a) 

It is next proved that BK .LG = AM 2 . 
For, by similar triangles BKH, FLC, 

BK : KH=FL: LG, or BK . LC = KH.FL 

= AM 2 . (6) 

Lastly, since BG : BD = BL : BK, from above, 

BC:CD = BL: KL, or BL.CD = BC.KL 

= BC.2r. (c) 

Also BD-.GD = BK:KL, or BK.GD =BD.KL 

= BD.2r. (d) 

III. We now (Fig. 3) take any two circles touching the 
semicircles BGG, BED and one another. Let their centres be 
A and P, H their point of contact, d, d' their diameters respec- 
tively. Then, if AM, PN are drawn perpendicular to BG, 
Pappus proves that 

(AM+d):d = PN:d f . 

Draw BF perpendicular to BG and therefore touching the 
semicircles BGG, BED at B. Join AP, and produce it to 
meet BF in F. 

Now, by II. (a) above, 

(BG+ BD) : (BC- BD) = BM-.AH, 
and for the same reason = BN: PH; 
it follows that AH:PH= BM : BN 

= FA:FP. 



THE COLLECTION. BOOK IV 



375 



Therefore (Lemma I), if the two circles touch the semi- 
circle BED in jR, E respectively, FRE is a straight line and 
EF. FR = FH\ 

But EF . FR = FB* ; therefore FH = FB. 

If now BH meets PN in and MA produced in 8, we have, 
by similar triangles, FH:FB = PHiPO = A& :AS, whence 



PH = PO and &4 = AH, so that 0, S are the intersections 
of PN, AM with the respective circles. 




N DM 

FIG. 3. 



BMiBN=BK:BP 



Join BP, and produce it o meet MA in AT. 
Now BM:BN=FA:FP 

= AH: PH, from above, 



And 

Therefore Kti = 
circle EUG. 

Lastly, 
that is, 
or 



, and KA = </, the diameter of the 



376 PAPPUS OF ALEXANDRIA 

IV. We now come to the substantive theorem. 
Let FGH be the circle touching all three semicircles (Fig. 4). 
We have then, as in Lemma II, 

nsi T*ir z>r* -nj 

JjLf . )/i = JjJJ JjJj, 

and for the same reason (regarding FGH as touching the 
semicircles BGC, DUG) 

BC.CL'=GD.CK. 
From the first relation we have 

BC:BD = BL:BK, 




S N K D M L C 

FIG. 4. 

whence DC:BD = KL : BK, and inversely BD : DC= BK : KL, 
while, from the second relation, EG : CD = CK : CL, 
whence BD: DC = KL : CL. 

Consequently UK : KL = KL : CL, 
or BK . LC = KL*. 

But we saw in Lemma II (6) that BK .LC = AM 2 . 

Therefore KL = AM, or p l = d r 

For the second circle Lemma III gives us 



whence, since p l d l , p z 
For the third circle 



whence p a = 3d 3 . 

And so on ad infinitum. 



THE COLLECTION. BOOK IV 377 

The same proposition holds when the successive circles, 
instead of being placed between the large and one of the small 
semicircles, come down between the two small semicircles. 

Pappus next deals with special cases (1) where the two 
smaller semicircles become straight lines perpendicular to the 
diameter of the other semicircle at its extremities, (2) where 
we replace one of the smaller semicircles by a straight line 
through D at right angles to BC, and lastly (3) where instead 
of the semicircle DUO we simply have the straight line DC 
and make the first circle touch it and the two other semi- 
circles. 

Pappus's propositions of course include as particular cases 
the partial propositions of the same kind included in the ' Book 
of Lemmas ' attributed to Archimedes (Props. 5, 6) ; cf. p. 102. 

Sections (3) and (4). Methods of squaring the circle, and of 
trisecting (or dividing in any ratio) any given angle. 

The last sections of Book IV (pp. 234-302) are mainly 
devoted to the solutions of the problems (1) of squaring or 
rectifying the circle and (2) of trisecting any given angle 
or dividing it into two parts in any ratio. To this end Pappus 
gives a short account of certain curves which were used for 
the purpose. 

(a) The Archimedean spiral. 

He begins with the spiral of Archimedes, proving some 
of the fundamental properties. His method of finding the 
area included (1) between the first turn and the initial line, 
(2) between any radius vector on the first turn and the curve, 
is worth giving because it differs from the method of Archi- 
medes. It is the area of the whole first turn which Pappus 
works out in detail. We will take the area up to the radius 
vector OJB, say. 

With centre and radius OB draw the circle A' BCD. 

Let BC be a certain fraction, say 1 / nth, of the arc BCD A', 
and CD the same fraction, OC, OD meeting the spiral in F, E 
respectively. Let A r $, 8V be the same fraction of a straight 
line KR, the side of a square KNLR. Draw ST, VW parallel 
to KN meeting the diagonal KL of the square in U, Q respec- 
tively, and draw MU> PQ parallel to KR. 



378 



PAPPUS OF ALEXANDRIA 



With as centre and OE, OF as radii dr&w arcs of circles 
meeting OF, OB in H, G respectively. 

For brevity we will now denote a cylinder in which r is the 
radius of the base and h the height by (cyl. r, h) and the cone 
with the same base and height by (cone r, h). 





N T w 



By the property of the spiral, 

OB:BG = (arc A' DOB) : (arc CB) 
= RK : KS 
= NK : KM, 
whence OB:OG = NK: NM. 

Now 
(sector OBG) : (sector OGF) = 2 : OG 2 = JVTf 2 : 

= (cyl. KN, NT) : (cyl. MN, NT). 
Similarly 
(sector 000) : (sector OEH) = (cyl. ST, TW) : (cyl. PT. TW), 

and so on. 

The sectors OBC, OCD... form the sector OA'DB, and the 
sectors OFG, OEH ... form a figure inscribed to the spiral. 
In like manner the cylinders (KN, TN), (XT, TW) ... form the 
cylinder (KN, NL), while the cylinders (MN, NT), (PT y TW) ... 
form a figure inscribed to the cone (KN> NL). 

Consequently 

(sector OA'DB) :(fig. inscr. in spiral) 

= (cyl. KN, NL) : (fig. inscr. in cone KN, NL). 



THE COLLECTION. BOOK IV 379 

We have a similar proportion connecting a figure circum- 
scribed to the spiral and a figure circumscribed to the cone. 

By increasing n the inscribed and circumscribed figures can 
be compressed together, and by the usual method of exhaustion 
we have ultimately 

(sector OA'DB) : (area of spiral) = (cyL KN, NL) : (cone KN, NL) 

= 3:1, 
or (area of spiral cut off' by OB) = (sector OA'DB). 

The ratio of the sector OA'DB to the complete circle is that 
of the angle which the radius vector describes in passing from 
the position OA to the position OB to four right angles, that 
is, by the property of the spiral, r : a, where r = OB, a = OA. 

Therefore (area of spiral cut off by OB) = r irr*. 

a 

Similarly the area of tfye spiral cut off by any other radius 

/ 

vector r' = 4 nr'*. 
a 

Therefore (as Pappus proves in his next proposition) the 
first area is to the second as r 3 to r' 3 . 

Considering the areas cut off by the radii vectores at the 
points where the revolving line has passed through angles 
of TT, TT, ITT and %TT respectively, we see that the areas are in 
the ratio of () 3 , () 3 , (|) 3 , 1 or 1, 8, 27, 64, so that the areas of 
the spiral included in the four quadrants are in the ratio 
of 1, 7, 19, 37 (Prop. 22). 

()3) The conchoid of N'icomedes. 

The conchoid of Nicomedes is next described (chaps. 26-7), 
and it is shown (chaps. 28, 29) how it can be used to find two 
geometric means between two straight lines, and consequently 
to find a cube having a given ratio to a given cube (see vol. i, 
pp. 260-2 and pp. 238-40, where I have also mentioned 
Pappus's remark that the conchoid which he describes is the 
first conchoid, while there also exist a second, a third and a 
fourth which are of use for other theorems). 

(y) The quadratrix. 

The quadratrix is taken next (chaps. 30-2), with Sporus's 
criticism questioning the construction as involving a petitio 



380 



PAPPUS OF ALEXANDRIA 



prinoipii. Its use for squaring the circle is attributed to 
Dinostratus and Nicomedes. The whole substance of this 
subsection is given above (vol. i, pp. 226-30). 



Tivo constructions for the quadratrix by means of 
1 surface-loci '. 

In the next chapters (chaps. 33, 34, Props. 28, 29) Pappus 
gives two alternative ways of producing the quadratrix ' by 
means of surface-loci', for which he claims the merit that 
they are geometrical rather than c too mechanical' as the 
traditional method (of Hippias) was. 

(1) The first method uses a cylindrical helix thus. 

Let ABC be a quadrant of a circle with centre B, and 

let BD be any radius. Suppose 
that EF, drawn from a point E 
on the radius BD perpendicular 
to BC, is (for all such radii) in 
a given ratio to the arc DC. 

' I say ', says Pappus, c that the 
locus of E is a certain curve.' 

Suppose a right cylinder 
erected from the quadrant and 
a cylindrical helix CGH drawn 
upon its surface. Let DH be 

the generator of this cylinder through J) 9 meeting the helix 
in H. Draw BL, El at right angles to the plane of the 
quadrant, and draw HIL parallel to BD. 

Now, by the property of the helix, EI( = DH) is to the 
arc CD in a given ratio. Also EF : (arc CD) = a given ratio. 
Therefore the ratio EF : El is given. .And since EF, El are 
given in position, Fl is given in position. But FI is perpen- 
dicular to BG. Therefore FI is in a plane given in position, 
and so therefore is 7. 

But I is also on a certain surface described by the line LH 9 
which moves always parallel to the plane ABC, with one 
extremity L on BL and the other extremity // on the helix. 
Therefore / lies on the intersection of this surface with the 
plane through FI. 




THE COLLECTION. BOOK IV 381 

Hence / lies on a certain curve. Therefore E, its projection 
on the plane ABC, also lies on a curve. 

In the particular case where the given ratio of EF to the 
arc CD is equal to the ratio of BA to the arc CA, the locus of 
E is a quadratrix. 

[The surface described by the straight line LH is a plectold. 
The shape of it is perhaps best realized as a continuous spiral 
staircase, i.e. a spiral staircase with infinitely small steps. 
The quadratrix is thus produced as the orthogonal projection 
of the curve in which the plectoid is intersected by a plane 
through BC inclined at a given angle to the plane ABC. It is 
not difficult to verify the result analytically.] 

(2) The second method uses a right cylinder the base <jf which 
is aji Archimedean spiral. 

Let ABC be a quadrant of a circle, as before, and EF, per- 
pendicular at F to BC, a straight 
line of such length that EF is 
to the arc DC as AB is to the 
arc ADC. 

Let a point on AB move uni- 
formly from A to B while, in the 
same time, AB itself revolves 

uniformly about B from the position BA to the position BC. 
The point thus describes the spiral A OB. If the spiral cuts 
BD in #, 

BA:BG = (arc ADC) : (arc DC), 

or BG : (arc DC) = BA : (arc ADC). 

Therefore BG = EF. 

Draw GK at right angles to the plane ABC and equal to BG. 
Then GK, and therefore K, lies on a right cylinder with the 
spiral as base. 

But BK also lies on a conical surface with vertex B such that 
its generators all make an angle of %TT with the plane ABC. 

Consequently K lies on the intersection of two surfaces, 
and therefore on a curve. 

Through K draw LKl parallel to BD y and let BL 9 El be at 
right angles to the plane ABC. 

Then LKl, moving always parallel to the plane ABC, with 
one extremity on BL and passing through K on a certain 




382 



PAPPUS OF ALEXANDRIA 



curve, describes a certain plectoid, which therefore contains the 
point J. 

Also IE = EF, IF is perpendicular to BC, and hence IF, and 
therefore /, lies on a fixed plane through BC inclined to ABO 
at an angle of ^TT. 

Therefore I, lying on the intersection of the plectoid and the 
said plane, lies on a certain curve. So therefore does the 
projection of / on ABC, i.e. the point E. 

The locus of E is clearly the quadratrix. 

[This result can also be verified analytically.] 

(S) Digression: a spiral on a sphere. 

Prop. 30 (chap. 35) is a digression on the subject of a certain 
spiral described on a sphere, suggested by the discussion of 
a spiral in a plane. 

Take a hemisphere bounded by the great circle KLM, 
with H as pole. Suppose that the quadrant of a great circle 
HNK revolves uniformly about the radius HO so that K 
describes the circle KLM and returns to its original position 
at K, and suppose that a point moves uniformly at the same 



N 




time from H to K at such speed that the point arrives at K 
at the same time that HK resumes its original position. The 
point will thus describe a spiral on the surface of the sphere 
between the points H and K as shown in the figure. 

Pappus then sets himself to prove that the portion of the 
surface of the sphere cut off towards the pole between the 
spiral and the arc HNK is to the surface of the hemisphere in 



THE COLLECTION. BOOK IV 



383 



a certain ratio hown in the second figure where ABC i$ 
a quadrant of a circle equal to a great circle in the sphere, 
namely the ratio of the segment ABC to the sector DABC. 




Draw the tangent CF to the quadrant at C. With C as 
centre and radius CA draw the circle A EF meeting CF in F. 

Then the sect&r CAF is equal to the sector ADC (since 
CA* = 2AD\ while Z ACF = Z 4(7). 

It is required, therefore, to prove that, if S be the area cut 
off by the spiral as above described, 

8 : (surface of hemisphere) = (segmt. A BC) : (sector CAF). 

Let KL be a (small) fraction, say l/?ith, of the circum- 
ference of the circle KLM, and let HPL be the quadrant of the 
great circle through H, L meeting the spiral in P. Then, by 
the property of the spiral, 

(arc HP) : (arc HL) = (arc KL) : (circumf . of KLM) 

= 1 : 71. 

Let the small circle NPQ passing through P be described 
about the pole H. 

Next let FE be the same fraction, I/nth, of the arc FA 
that KL is of the circumference of the circle KLM y and join EC 
meeting the arc ABC in B. With C as centre and CB as 
radius describe the arc BG meeting GF in G. 

Then the arc CB is the same fraction, I/ nth, of the arc 
CBA that the arc FE is of FA (for it is easily seen that 
LFGE = \LBDC, while LFCA = %CDA). Therefore, since 
(arc CBA) = (arc HPL), (arc CB) = (arc HP), and chord CB 
= chord HP. 



384 PAPPUS OF ALEXANDRIA 

Now (sector HPN on sphere) : (sector HKL on sphere) 

= (chord //P) 2 : (chord 7//,) 2 

(a consequence of Archimedes, On Sphere and Cylinder, I. 42). 
And HP 2 : HL 2 = OB 2 : GA 2 



Therefore 
(sector HPN) : (sector HKL) = (sector f77?(?) : (sector 

Similarly, if the arc LI/ be taken equal to the arc KL and 
the great circle through H, L / cuts the spiral in P', and a small 
circle described about H and through P' meets the arc HPL 
in p ; and if likewise the arc BB' is made equal to the arc Jt( \ 
and CB' is produced to meet AF in E', while again a circular 
arc with G as centre and CB' as radius meets CE in 6, 

(sector /fP'?; on sphere) : (sector HLL' on sphere) 

= (sector CB'b) : (sector CE'K). 
And so on. 

Ultimately then we shall get a figure consisting of sectors 
on the sphere circumscribed about the area H of the spiral and 
a figure consisting of sectors of circles circumscribed about the 
segment GBA ; and in like manner we shall have inscribed 
figures in each case similarly made up. 

The method of exhaustion will then give 

$: (surface of hemisphere) = (segint. ABC) : (sector CAP) 

= (segmt. ABC) : (sector DAG). 

[We may, as an illustration, give the analytical equivalent 
of this proposition. If p, o> be the spherical coordinates of P 
with reference to H as pole and the arc HNK as polar axis, 
the equation of Pappus's curve is obviously o> = 4 p. 

If now the radius of the sphere is taken as unity, we have as 
the element of area 



dA = dco(l cosp) = 4rfp(l cosp). 

pi* 
Therefore A = 4dp (1 cosp) = 27r 4. 

Jo 



THE COLLECTION. BOOK IV 385 

Therefore 

A 27r-4 -4--^ 



(surface of hemisphere) "~~ 2?r 



__ (segment ABC) ^ 
~ "(sector DABC) "J 



The second part of the last section of Book IV (chaps. 36-41, 
pp. 270-302) is mainly concerned with the problem of tri- 
secting any given angle or dividing it into parts in any given 
ratio. Pappus begins with another account of the distinction 
between plane, solid and linear problems (cf. Book III, chaps. 
20-2) according as they require for their solution (1) the' 
straight line and circle only, (2) conies or their equivalent, 
(3) higher curves still, ' which have a more complicated and 
forced (or unnatural) origin, being produced from more 
irregular surfaces and involved motions. Such are the curves 
which, are discovered in the so-called loci on surfaces, as 
well as others more complicated still and many in number 
discovered by Demetrius of Alexandria in his Linear con- 
siderations and by Philoii of Tyana by means of the inter- 
lacing of plectoids and other surfaces of all sorts, all of which 
curves -possess many remarkable properties peculiar to them. 
Some of these curves have been thought by the more recent 
writers to be worthy of considerable discussion ; one of them is 
that which also received from Menelaus the name of the 
jxirado.rical curve. Others of the Fame class are flpirals, 
cjuadratrices, cochloids and cissoids.' He adds the often-quoted 
reflection -on the error committed by geometers w.hen they 
solve a -problem by means of an 'inappropriate class* (of 
curve or its equivalent), illustrating this by the use in 
Apollonius, Book V, of a rectangular hyperbola for finding the 
feet of normals to a parabola passing through one point 
(where a circle would serve the purpose), and by the assump- 
tion by Archimedes of a tolid i/ei/(7*9 in his book On Spirals 
(see above, pp. 65-8). 

Trisection (or division in any ratio] of any angle. 

The method of trisecting any angle based on a certain i/cOcny 
is next described, with the solution of the vevvis itself by 

1MS ? C C 



386 PAPPUS OF ALEXANDRIA 

means of a hyperbola which has to be constructed from certain 
data, namely the asymptotes and a certain point through 
which the curve must pass (this easy construction is given in 
Prop. 33, chap. 41-2). Then the problem is directly solved 
(chaps. 43, 44) by means of a hyperbola in two ways prac- 
tically equivalent, the hyperbola being determined in the one 
case by the ordinary Apollonian property, but in the other by 
means of the focus-directrix property. Solutions follow of 
the problem of dividing any angle in a given ra^tio by means 
(1) of the quadratrix, (2) of the spiral of Archimedes (chaps. 
45, 46). All these solutions have been sufficiently described 
above (vol. i, pp. 235-7, 241-3, 225-7). 

Some problems follow (chaps. 47-51) depending on these 
results, namely those of constructing an isosceles triangle in 
which either of the base angles has a given ratio to the vertical 
angle (Prop. 37), inscribing in a circle a regular polygon of 
any number of sides (Prop. 38), drawing a circle the circum- 
ference of which shall be equal to a given straight line (Prop. 
39), constructing on a given straight line AB a segment of 
a circle such that the arc of the segment inay have a given 
ratio to the base (Prop. 40), and constructing an angle incom- 
mensurable with a given angle (Prop. 41). 



Section (5). Solution of the j/cCo-iy of Archimedes, c On Spirals ', 
Prop. 8, by means of conies. 

Book IV concludes with the solution of the vevtns which, 
according ,to Pappus, Archimedes unnecessarily assumed in 
On Spirals, Prop. 8. Archimedes's assumption is this. Given 
a circle, a chord (BC) in it less than the diameter, and a point 
A on the circle the perpendicular from which to BC cuts BC 
in a point D such that BD > DC and meets the circle again 
in E, it is possible to draw through A a straight line ARP 
cutting BC in R and the circle in P in such a way that RP 
shall be equal to DE (or, in the phraseology of vcvo-ei?, to 
place between the straight line BC and the circumference 
of the circle a straight line equal to DE and verging 
towards A). 

Pappus makes the problem rather more general by not 
requiring PR to be equal to DE, but making it of any given 



THE COLLECTION. BOOK IV 



387 



length (consistent with a real solution). The problem is best 
exhibited by means of analytical geometry. 

If BD = a, DC = 6, AD = c (so that DE = ab/c), we have 




to find the point R on BC such that AR produced solves the 
problem by making PR equal to k, say. 

Let DR = x. Then, since BR . RC = PR.RA,we have 



when 
(I) 



(a-x)(b + x) = A 

An obvious expedient is to put T/ for \/(' 2 
we have 

(a-x) (b + x) = ky, 

and 



if = c.' 2 + x 2 . (2) 

These equations represent a parabola and a hyperbola 
respectively, and Pappus does in fact solve the problem by 
means of the intersection of a parabola and a hyperbola ; one 
of his preliminary lemmas is, however, again a little more 
general. In the above figure y is represented by RQ. 

The first lemma of Pappus (Prop. 42, p. 298) states that, if 
from a given point A any straight line be drawn meeting 
a straight line BC given in position in R, and if RQ be drawn 
at right angles to'JSC and of length bearing a given ratio 
to AR, the locus of Q is a hyperbola. 

For draw AD perpendicular to BC and produce it to A ' 
so that 

QR:RA = A'D:DA = the given ratio. 
cc 2 



388 PAPPUS OF ALEXANDRIA 

Measure DA" along DA equal to DA'. 

Then, if QN be perpendicular to AD, 

(AR*-AD*):(QR*-A'D*) = (const.), 
that is, QN 2 : A'N .A"N = (const.), 

and the locus of Q is a hyperbola. 

The equation of the hyperbola is clearly 



where p, is a constant. In the particular case taken by 
Archimedes QR = RA 9 or p, = 1, and the hyperbola becomes 
the rectangular hyperbola (2) above. 

The second lemma (Prop. 43, p. 300) proves that, if BC is 
given in length, and Q is such a point that, when QR is drawn 
perpendicular to BC, BR . RC = k . QR, where k is a given 
length, the locus of Q is a 'parabola. 

Let be the middle point of BC, and let OK be drawn at 
right angles to BC and of length such that 



Let QN' be drawn perpendicular to OK. 
Then QN'* = OR 2 



= k . (KO - QJJ), by hypothesis, 
= k . KN'. 

Therefore the locus of Q is a parabola. 
The equation of the parabola referred to DB, DE as axes of 
x and y is obviously 



which easily reduces to 

(a x) (& + #) = ky, as above (1). 
In Archimedes's particular case k = ab/c. 

To solve the problem then we have only to draw the para- 
bola and hyperbola in question, and their intersection then 
gives Q, whence R, and therefore ARP, is determined. 



THE COLLECTION. BOOKS IV, V 389 

Book V. Preface on the Sagacity of Bees. 

It is characteristic of the great Greek mathematicians that, 
whenever they were free from the restraint of the technical 
language of mathematics, as when for instance they had occa- 
sion to write a preface, they were able to write in language of 
the highest literary quality, comparable with that of the 
philosophers, historians, and poets. We have only to recall 
the introductions to Archimedes' s treatises and the prefaces 
to the different Books of Apollonius's Conies. Heron, though 
severely practical, is no exception when he has any general 
explanation, historical or other, to give. We have now to 
note a like case in Pappus, namely the preface to Book V of 
the Collection. The editor, Hultsch, draws attention to the 
elegance and purity of the language and the careful writing ; 
the latter is illustrated by the studied avoidance of hiatus. 1 
The subject is one which a writer of taste arid imagination 
would naturally find attractive, namely the practical intelli- 
gence shown by bees in selecting the hexagonal form for the 
cells in the honeycomb. Pappus does not disappoint us ; the 
passage is as attractive as the subject, and deserves to be 
reproduced. 

4 It is of course to men that God has given the best and 
most perfect notion of wisdom in general and of mathematical 
science in particular, but a partial share in these things he 
allotted to some of the unreasoning animals as well. To men, 
as being endowed with reason, he vouchsafed that they should 
do everything in the light of reason and demonstration, but to 
the other animals, while denying them reason, he granted 
that each of them should, by virtue of a certain natural 
instinct, obtain just so much as is needful to support life. 
This instinct may be observed to exist in very many other 
species of living creatures, but most of all in bees. In the first 
place their orderliness and their submission to the queens who 
rule in their state are truly admirable, but much more admirable 
still is their emulation, the cleanliness they observe in the 
gathering of honey, and the forethought and housewifely care 
they devote to its custody. Presumably because they know 
themselves to be entrusted with the task of bringing from 
the gods to the accomplished portion of mankind a share of 

1 Pappus, vol. iii, p. 1233. 



390 PAPPUS OF ALEXANDRIA 

ambrosia in this form, they do not think it proper to pour it 
carelessly on ground or wood or any other ugly and irregular 
material ; but, first collecting the sweets of the moat beautiful 
flowers which grow on the earth, they make from them, for 
the reception of the honey, the vessels which we call honey- 
combs, (with cells) all equal, similar and contiguous to one 
another, and hexagonal in form. And that they have con- 
trived this by virtue of a certain geometrical forethought we 
may infer in this way. They would necessarily think that 
the figures must be such as to be contiguous to one another, 
that is to say, to have their sides common, in order that no 
foreign matter could enter the interstices between them and 
so defile the purity of their produce. Now only three recti- 
lineal figures would satisfy the condition, I mean regular 
figures which are equilateral and equiangular; for the 'bees 
would have none of the figures which are not uniform. . . . 
There being then three figures capable by themselves of 
exactly filling up the space about the same point, the bees by 
reason of their instinctive wisdom chose for the construction 
of the honeycomb the figure which has the most angles, 
because they conceived that it would contain more honey than 
either of the two others. 

' Bees, then, know just this fact which is of service to them- 
selves, that the hexagon is greater than the square and the 
triangle and will hold more honey for the same expenditure of 
material used in constructing the different figures. We, how- 
ever, claiming as we do a greater share in wisdom than bees, 
will investigate a problem of still wider extent, namely that, 
of all equilateral and equiangular plane figures having an 
equal perimeter, that which has the greater number of angles 
is always greater, and the greatest plane figure of all those 
which have a perimeter equal to that of the polygons is the 
circle/ 

Book V then is devoted to what we may call isoperivietry, 
including in the term not only the comparison of the areas of 
different plane figures with the same perimeter, but that of the 
contents of different solid figures with equal surfaces. 

Section (1). Isoperimetry after Zetiodorus. 

The first section of the Book relating to plane figures 
(chaps. 1-10, pp. 308-34) evidently followed very closely 
the exposition of Zenodorus Trepi /<ro/*erpo>j/ cryrffidr^if (see 
pp. 207-13, above) ; but before passing to solid figures Pappus 
inserts the proposition that of all circular segments having 




THE COLLECTION. BOOK V 391 

the sa'fne circumference the semicircle is the greatest, with some 
preliminary lemmas which deserve notice (chaps. 15, 16). 

(1) ABC is a triangle right-angled, at B. With C as centre 

and radius CA describe the arc 

AD cutting CB produced in D. 

To prove that ( R denoting a right 

angle) 

(sector CA D) : (area ABD) 

>R:/.BCA. 

Draw A F at right angles to CA meeting CD produced in F 9 
and dfaw BH perpendicular to AF. With A as centre and 
AB as radius describe the arc GBE. 

Now (area EBF) : (area EBH) > (area EBF) : (.sector ABE), 
and, component, &FBH:(EBH) > &ABF: (ABE). 

But (by an easy lemma which has just preceded) 

AFBU:(E/UI) = &ABF:(ABD), 
whence &ABF: (ABD) > &ABF: (ABE), 

and (ABE) > (ABD). 

Therefore (ABE) : (ABG) > (ABD) : (ABG) 

> (ABD) :^ ABC, a fortiori. 

Therefore L BAF: L BAC > (ABD) : &ABC, 
whence, inversely, A ABC: (ABD) > L BAC: L BAF. 
and, comiwneiuto, (sector ACD) : (A BD) > R : L BCA. 

[If a be the circular measure of LEG A, this gives (if AC=b) 

i 
aft 2 :(a& 2 ---| sin a cosa.fr 2 ) > TT :a, 

or 2a:(2a sin2a) > 7r:2a; 

that is, 0/(0-sin0) > ir/0, where < < TT.] 

(2) ABC is again a triangle right-angled at B. With C as 
centre and CA as radius draw a circle AD meeting BC pro- 
duced in D. To prove that 

(sector CAD) : (area ABD) > R : LAGD. , 



392 



PAPPUS OF ALEXANDRIA 



Draw AE at right angles to AC.' With A as centre and 
AC as radius describe the circle FOE meeting AB produced 
in F&ud AEmE. 

Then, since LACD > LVAE, (sector ACD) > (sector ACE). 
Therefore (AGO) : &ABC > (ACE) : &ABG 

> (ACE) : (ACF), a fortiori, 

> LEAC-.LGAB. 
Inversely, 

A ABC: (ACD) < LGAB^EAC, 
and, componendoy 

(ABD) : (ACD) < L EAB : L EAC. 

Inversely, (ACD) : (ABD) > Z EAC : L EAB 




We come now to the application of these lemmas to the 
proposition comparing the area of a semicircle with that of 
other segments of equal circumference (chaps. 17, 18). 

A semicircle is the greatest of all segments of circles which 
have the same circumference. 

Let ABC be a semicircle with centre 0, and DEF another 
segment of a circle such that the circumference DEF is equal 






to the circumference ABC. I say that the area of ABC is 
greater than the area of DEF. 

Let H be the centre of the circle DEF. Draw EUK, BG at 
right angles to DF, AC respectively. Join D1I, and draw 
LHM parallel to DF. 



THE COLLECTION. BOOK V 393 

Then LH:AG = (arc LE) : (arc AB) 



= (sector LHE) : (sector 
Also Lll* : AG* = (sector LHE) : (sector 

Therefore the sector LHE is to the sector AGB in the 
ratio duplicate of that which the sector LHE has to the 
sector DUE. 

Therefore 

(sector LHE) : (sector DUE) = (sector DHE) : (sector AGB). 

Now (1) in the case of the segment less than a semicircle 
and (2) in the case of the segment greater than a semicircle 

(sector EDH) : (EDK) > R : Z DHE, 

by the lemmas (1) and (2) respectively. 
That is, 

(sector EDH ) : (EDK) > Z LHE: Z DHE 

> (sector LHE) : (sector DHE) 

> (sector EDH) : (sector AGB), 

from above. 

Therefore the half segment EDK is less than the half 
semicircle A(1B, whence the semicircle ABC is greater than 
the segment DEh\ 

We have already described the content of Zenodorus's 
treatise (pp. 207-13, above) to which, so far as plane figures 
are concerned, Pappus added nothing except the above pro- 
position relating to segments of circles. 

JSection (2). Comparison of volumes of solids liaviny their 
surfaces equal. Case of the spfiere. 

The portion of Book V dealing with solid figures begins 
(p. 350. 20) with the statement that the philosophers who 
considered that the creator gave the universe the form of a 
sphere because that was the most beautiful of all shapes also 
asserted that the sphere is the greatest of all solid figures 



394 PAPPUS OF ALEXANDRIA 

which have their surfaces equal ; this, however, they had not 
proved, nor could it be proved without a long investigation. 
Pappus himself does not attempt to prove that the sphere is 
greater than all solids with the same surface, but only that 
the sphere is greater than any of the five regular solids having 
the same surface (chap. 19) and also greater than either a cone 
or a cylinder of equal surface (chap. 20). 

Section (3). Digression on the semi-regular solids 
of Archimedes. 

He begins (chap. 19) with an account of the thirteen tsenii- 
regular solids discovered by Archimedes, which are contained 
by polygons all equilateral and all equiangular but not all 
similar (see pp. 98-101, above), and he shows how to determine 
the number of solid angles and the number of edges which 
they have respectively ; he then gives them the go-by for his 
present purpose because they are not completely regular ; still 
less does he compare the sphere with any irregular solid 
having an equal surface. 

The sphere is greater than any of the regular solids which 
has its surface equal to that of the sphere. 

The proof that the sphere is greater than any of the regular 
solids with surface equal to that of the sphere is the same as 
that given by Zenodorus. Let P be any one of the regular solids, 
S the sphere with surface equal to that of P. To prove that 
S>P. Inscribe in the solid a sphere s, and suppose that r is its 
radius. Then the surface of P is greater than the surface of s, 
and accordingly, if It is the radius of /S, R > r. But the 
volume of 8 is equal to the cone with base equal to the surface 
of S y and therefore of P, and height equal to R ; and the volume 
of P is equal to the cone with base equal to the surface of P 
and height equal to r. Therefore, since R >?*, volume of tf > 
volume of P. 

Section (4). Propositions on the lilies of Archimedes, 
' On the Sphere and Cylinder '. 

For the fact that the volume of a sphere is equal to the cone 
with base equal to the surface, arid height equal to the radius, 



THE COLLECTION. BOOK V 395 

of the sphere, Pappus quotes Archimedes, On the Sphere and 
Cylinder, but thinks proper to add a series of propositions 
(chaps, 20-43, pp. 362-410) on much the same lines as those of 
Archimedes and leading to the same results as Archimedes 
obtains for the surface of a segment of a sphere and of the whole 
sphere (Prop. 28), and for the volume of a sphere (Prop. 35). 
Prop. 36 (chap. 42) shows how to divide a sphere into two 
segments such that their surfaces are in a given ratio and 
Prop. 37 (chap. "43) proves that the volume as well as the 
surface of the cylinder circumscribing a sphere is 1^ times 
that of the sphere itself. 

Among the lemmatic propositions in this section of the 
Book Props. 21, 22 may be mentioned. Prop. 21 proves that, 
if C, E be two points on the tangent at H to a semicircle such 
that CH = HE, and if CD, EF be drawn perpendicular to the 
diameter AB y then (CD + EF) CE-AB. DF\ Prop. 22 proves 
a like result where C, E are points on the semicircle, CD, EF 
are as before perpendicular to AB, and EH is the chord of 
the circle subtending the arc which with CE makes up a semi- 
circle ; in this case (CD + EF) CE = EH . DF. Both results 
are easily seen to be the equivalent of the trigonometrical 
formula 

sin (x + y) + sin (x y) = 2 sin x cos y y 
or, if certain different angles be taken as x, y, 

sin x 4- sin y , , , v 

= cot 4(8 y). 

cos // cos x 

Section (5). Of regular solufs with surfaces equal, that is 
greater which has more faces. 

Returning to the main problem of the Book, Pappus shows 
that, of the five regular solid figures assumed to have their 
surfaces equal, that is greater which has the more faces, so 
that the pyramid, the cube, the octahedron, the dodecahedron 
and the icosahedron of equal surface are, as regards solid 
content, in ascending order of magnitude (Props. 3-56). 
Pappus indicates (p. 410. 27) that 'some of the ancients' had 
worked out the proofs of these propositions by the analytical 
method; for himself, he will give a method of his own by 



396 PAPPUS OF ALEXANDRIA 

synthetical deduction, for which he claims that it is clearer 
and shorter. We have first propositions (with auxiliary 
lemmas) about the perpendiculars from the centre of the 
circumscribing sphere to a face of (a) the octahedron, (6) the 
icosahedron (Props. 39, 43), then the proposition that, if a 
dodecahedron and an icosahedron be inscribed in the same 
sphere, the same small circle in the sphere circumscribes both 
the pentagon of the dodecahedron and the triangle of the 
icosahedron (Prop. 48) ; this last is the proposition proved by 
Hypsicles in the so-called ' Book XIV of Euclid ', Prop. 2, and 
Pappus gives two methods of proof, the second of which (chap. 
56) corresponds to that of Hypsicles. Prop. 49 proves that 
twelve of the regular pentagons inscribed in a circle are together 
greater than twenty of the equilateral triangles inscribed in 
the same circle. The final propositions proving that the cube 
is greater than the pyramid with the same surface, the octa- 
hedron greater than the cube, and so on, are Props. 52-6 
(chaps. 60-4). Of Pappus's auxiliary propositions, Prop. 41 
is practically contained in Hypsicles's Prop. 1, and Prop. 44 
in Hypsicles's last lemma; but otherwise the exposition is 
different. 



Book VI. 

On the contents of Book VI we can be brief. It is mainly 
astronomical, dealing with the treatises included in the so- 
called Little Astronomy, that is, the smaller astronomical 
treatises which were studied as an introduction to the great 
tiyntaxis of Ptolemy. The preface says that many of those 
who taught the Treasury of Astronomy > through a careless 
understanding of the propositions, added some things as being 
necessary and omitted others as unnecessary. Pappus mentions 
at this point an incorrect addition to Theodosius, Sphaerica, 
III. 6, an omission from Euclid's Phaenomena, Prop. 2, an 
inaccurate representation of Theodosius, On Days and Nights, 
Prop. 4, and the omission later of certain other things as 
bein unnecessary. His object is to put these mistakes 
right. Allusions are also found in the Book to Menelaus's 
Spftaertca, e.g. the statement (p. 476. 16) that Menelaus in 
his Sphaerica called a spherical triangle TpiirXtvpov, three-side. 



THE COLLECTION. BOOKS V, VI 397 

The Sphaerica of Theodosius is dealt with at some length 
(chaps. 1-26, Props. 1-27), and so are the theorems of 
Autolycus On the moving Sphere (chaps. 27-9), Theodosius 
On Days and Nights (chaps. 30-6, Props. 29-38), Aristarchus 
0??, the sizes and distances of the Sun and Moon (chaps. 37-40, 
including a proposition, Prop. 39 with two lemmas, which is 
corrupt at the end and is not really proved), Euclid's Optics 
(chaps. 41-52, Props. 42-54), and Euclid's Phaenomena (chaps. 
53-60, Props. 55-61). 

Problem arising out of Euclid's 'Optws*. 

There is little in the Book of general mathematical interest 
except the following propositions which occur in the section on 
Euclid's Optics. 

Two propositions are fundamental in solid geometry, 
namely : 

(a) If from a point A above a plane AB be drawn perpen- 
dicular to the plane, and if from B a straight line BD be 
drawn perpendicular to any straight line EF in the plane, 
then will AD also be perpendicular to EF (Prop. 43). 

(/>) If from a point A above a plane A B be drawn to the plane 
but not at right angles to it, and AM be drawn perpendicular 
to the plane (i.e. if BM be the orthogonal projection of BA on 
the plane), the angle ABM is the least of all the angles which 
AB makes with any straight lines through B, as BP, in the 
plane ; the angle A BP increases as BP moves away from BM 
on either side* ; and, given any straight line BP making 
a certain angle with BA> only one other straight line in the 
plane will make the same angle with BA, namely a straight 
line Rl v on the other side of BM making the same angle with 
it that BP does (Prop. 44). 

These are the first of a series of lemmas leading up to the 
main problem, the investigation of the apparent form of 
a circle as seen from a point outside its plane. In Prop. 50 
(= Euclid, Optics, 34) Pappus proves the fact that all the 
diameters of the circle will appear equal if the straight line 
drawn from the point representing the eye to the centre of 
the circle is either (a) at right angles to the plane of the circle 
or (6), if not at right angles to the plane of the circle, is equal 



398 



PAPPUS OF ALEXANDRIA 



in length to the radius of the circle. In all other cases 
(Prop. 51 = Eucl. Optics, 35) the diameters will appear unequal. 
Pappus's other propositions carry farther Euclid's remark 
that the circle seen under these conditions will appear 
deformed or distorted (TrapcerTraoy-eli'os), proving (Prop. 53, 
pp. 588-92) that the apparent form will be an ellipse with its 
centre not, 'as some think', at the centre of the circle but 
at another point in it, determined in this way. Given a circle 
ABDE with centre 0, let the eye be at a point F above the 
plane of the circle such that FO is neither perpendicular 
to that plane nor equal to the radius of the circle. Draw FG 
perpendicular to the plane of the circle and let ADG be the 
diameter through G. Join AF, DF, and bisect the angle AFD 
by the straight line FC meeting AD in C. Through C draw 
BE perpendicular to AD, and let the tangents at B, E meet 
AG produced in K. Then Pappus proves that C (not 0) is the 
centre of the apparent ellipse, that AD, BE are its major and 
minor axes respectively, that the ordinates to AD are parallel 
to BE both really and apparently, and that the ordinates to 
BE will pass through K but will appear to be parallel to AD. 
Thus in the figure, G being the centre of the apparent ellipse, 




it is proved that, if LCM is any straight line through (7, LC is 
apparently equal to CM (it is practically assumed a proposi- 
tion proved later in Book VII, Prop. 156 that, if LK meet 
the circle again in P, and if PM be drawn perpendicular to 
AD to meet the circle again in M, LM passes through 0). 



THE COLLECTION. BOOKS VI, VII 399 

The test of apparent equality is of course that the two straight 
lines should subtend equal angles at F. 

The main points in the proof are these. The plane through 
OF, CK is perpendicular to the planes BFJB, PFM and LFR; 
hence OF is perpendicular to BE, QF to PM and HF to LR, 
whence BG and CE subtend equal angles at F: so do LH, HR, 
and PQ, QM. 

Since FG bisects the angle AFD, and AC:GD = AK : KD 
(by the polar property), Z.CFK is a right angle. And GF is 
the intersection of two planes at right angles, namely AFK 
and BFJS, in the former of which FK lies; therefore KF is 
perpendicular to the plane BFE, and therefore to FN. Since 
therefore (by the polar property) L N : NP = LK : KP, it 
follows that the angle LFP is bisected by FN] hence LN, NP 
are apparently equal. 

Again LC : CM = LN:NP = LF: FP = i^: m 

Therefore the angles LFC, GFM are equal, and LC, CM 
are apparently equal. 

Lastly LR:PM=LK:KP=LN:NP=LFiFP\ therefore 
the isosceles triangles FLR, FPM are equiangular; there- 
fore the angles PFM, LFR, and consequently PFQ, LFH, are 
equal. Hence LP, RM will appear to be parallel to AD. 

We have, based on this proposition, an easy method of 
solving Pappus's final problem (Prop, 54). * Given a circle 
ABDE and any point within it, to find outside the plane of 
the circle a point from which the circle will have the appear- 
ance of an ellipse with centre C! 

We have only to produce the diameter AD through C to the 
pole K of the chord BE perpendicular to AD and then, in 
the plane through A K perpendicular to the plane of the circle, 
to describe a semicircle on CK as diameter. Any point F on 
this semicircle satisfies the condition. 

Book VII. On the ' Treasury of A nalysis '. 

Book VII is of much greater importance, since it gives an 
account of the books forming what was called the Treasury of 
Analysis (avaXvontvos r6iro$) and, as regards those of the books 
which are now lost, Pappus's account, with the hints derivable 
from the large collection of lemmas supplied by him to each 



400 PAPPUS OF ALEXANDRIA 

book, practically constitutes our only source of information. 
The Book begins (p. 634) with a definition of analysis and 
synthesis which, as being the most elaborate Greek utterance 
on the subject, deserves to be quoted in full. 

c The so-called Hi/aAv6/zi/o9 is, to put it shortly, a special 
body of doctrine provided for the use of those who, after 
finishing the ordinary Elements, are desirous of acquiring the 
power of solving problems whicli may be set them involving 
(the construction of) lines, and it is useful for this alone. It is 
the work of three men, Euclid the author of the Elements, 
Apollonius of Perga and Aristaeus the elder, and proceeds by 
way of analysis and synthesis.' 

Definition of Analysis and Synthesis. 

c Analysis, then, takes that which is sought as if it were 
admitted and passes from it through its successive conse- 
quences to something which is admitted as the result of 
synthesis: for in analysis we assume that which is sought 
as if it were already done (yeyoi/os), and we inquire what it is 
from which this results, and again what is the antecedent 
cause of the latter, and so on, until by so retracing our steps 
we come upon something already known or belonging to the 
class of first principles, and such a method we call analysis 
as being solution backwards (avaira\iv Xvo-iv). 

1 But in synthesis, reversing the process, we take as already 
done that which was last arrived at in the analysis and, by 
arranging in their natural order as consequences what before 
were antecedents, and successively connecting them one with 
another, we arrive finally at the construction of what was 
sought ; and this we call synthesis. 

' Now analysis is of two kinds, the one directed to searching 
for the truth and called theoretical, the other directed to 
finding what we are told to find and called problematical. 

(1) In the theoretical kind we assume what is sought as if 
it were existent and true, after which we pass through its 
successive consequences, as if they too were true and established 
by virtue of our hypothesis, to something admitted: then 
(a), if that something admitted is true, that which is sought 
will also be true and the proof will correspond in the reverse 
order to the analysis, but (b), if we come upon something 
admittedly false, that which is sought will also be false. 

(2) In the jyroUematical kind we assume that which is pro- 
pounded as if it were known, after which we pass through its 



THE COLLECTION. BOOK VII 401 

vsuccessive consequences, taking them as true, up to something 
admitted : if then (a) what is admitted is possible and obtain- 
able, that is, what mathematicians call given, what was 
originally proposed will also be possible, and the proof will 
again correspond in the reverse order to the analysis, but if (b) 
we come upon something admittedly impossible, the problem 
will also be impossible/ 

This statement could hardly be improved upon except that 
it ought to be added that each step in the chain of inference 
in the analysis must be unconditionally convertible] that is, 
when in the analysis we say that, if A is true, B is true, 
we must be sure that each statement is a necessary conse- 
quence of the other, so that the truth of A equally follows 
from the truth of B. This, however, is almost implied by 
Pappus when he says that we inquire, not what it is (namely 
B) which follows from A, but what it is (B) from which A 
follows, and so on. 

List of works in the ' Treasury of Analysis \ 

Pappus adds a list, in order, of the books forming the 
Sy namely : 



* Euclid's Data, one Book, Apollonius's Cutting-off of a ratio, 
two Books, Cutting-off of an area, two Books, Determinate 
Section, two Books, Contacts, two Books, Euclid's Por/sms, 
three Books, Apollonius's Inclinations or Vergings (vvat$), 
two Books, the same author's Plane Loci, two Books, and 
Conies, eight Books, Aristaeus's Solid Loci, five Books, Euclid's 
Surface-Loci, two Books, Eratosthenes's On means, two Books. 
There are in all thirty-three Books, the contents of which up 
to the Conies of Apolloniua I have set out for your considera- 
tion, including not only the number of the propositions, the 
diorismi and the cases dealt with in each Book, but also the 
lemmaa which are required; indeed I have not, to the best 
of my belief, omitted any question arising in the study of the 
Books in question.' 

Description of the treatises. 

Then follows the short description of the contents of the 
various Books down to Apollonius's Conies; no account is 
given of Aristaeus'& Solid Loci, Euclid's Surface-Loci and 

1531.2 D d 



402 PAPPUS OF ALEXANDRIA 

Eratosthenes's On rrt^ans, nor are there any lemmas to these 
works except two on the Surface-Loci at the end of the Book. 
The contents of the various works, including those of the 
lost treatises so far as they can be gathered from Pappus, 
have been described in the chapters devoted to their authors, 
and need not be further referred to here, except for an 
addendum to the account of Apollonius's Conies which is 
remarkable. Pappus has been speaking of the ' locus with 
respect to three or four lines ' (which is a conic), and proceeds 
to say (p. 678. 26) that we may in like manner have loci with 
reference to five or six or even more lines ; these had not up 
to his time become generally known, though the synthesis 
of one of them, not by any means the most obvious, had been 
worked out and its utility shown. Suppose that there are 
five or six lines, and that p l , p 2 , #, , p^ , p 6 or p^ t p 29 p.^p 4t p r , , p Q 
are the lengths of straight lines drawn from a point to meet 
the five or six at given angles, then, if in the first case 
PiP^'Ps == ^PiP& a (where X is a constant ratio and a a given 
length), and in the second case Pip^p^ = Aj0 4 7> 6 ^ 6 , the locus 
of the point is in each case a certain curve given in position. 
The relation could not be expressed in the same form if 
there were more lines than six, because there are only three 
dimensions in geometry, although certain recent writers had 
allowed themselves to speak of a rectangle multiplied by 
a square or a rectangle without giving any intelligible idea of 
what they meant by such a thing (is Pappus here alluding to 
Heron's proof of the formula for* the area of a triangle in 
terms of its sides given on pp. 322-3, above ?). But the system 
of compounded, ratios enables it to be expressed for any 

number of lines thus, 2- 1 .*-? ^ for ? r ^\ = A. Pappus 

/> 2 PI a \ p n / 

proceeds in language not very clear (p. 680. 30) ; but the gist 
seems to be that the investigation of these curves had not 
attracted men of light and leading, as, for instance, the old 
geometers and the best writers. Yet there were other impor- 
tant discoveries still remaining to be made. For himself, he 
noticed that every one in his day was occupied with the elements, 
the first principles and the natural origin of the subject- 
matter of investigation ; ashamed to pursue such topics, he had 
himself proved propositions of much more importance and 



THE COLLECTION. BOOK VII 403 

utility. In justification of this statement and ' in order that 
he may not appear empty-handed when leaving the subject ', 
he wijl present his readers with the following. 

(Anticipation of Guldins Theorem.) 

The enunciations are not very clearly worded, but there 
is no doubt as to the sense. 

c Figures generated by a complete revolution of a plane figure 
about an axis are in a ratio compounded (1) of the ratio 
of the areas of the figures, and (2) of the ratio of the straight 
lines similarly drawn to (i.e. drawn to meet at the same angles) 
the axes of rotation from the respective centres of gravity. 
Figures generated by incomplete revolutions are in the ratio 
compounded (1) of the ratio of the areas of the figures and 
(2) of the ratio of the arcs described by the centres of gravity 
of the respective figure*, the latter ratio being itself compounded 

(a) of the ratio of the straight lines similarly drawn (from 
the respective centres of gravity to the axes of rotation) and 

(b) of the ratio of the angles contained (i.e. described) about 
the axes of revolution by the extremities of the said straight 
lines (i.e. the centres of (jravity). 9 

Here, obviously, we have the essence of the celebrated 
theorem commonly attributed to P. Guldin (1577-1643), 
'quantitas rotunda in viam rotationis ducta producit Pote- 
statem Rotundam uno grado altiorem Potestate sive Quantitate 
Rotata'. 1 

Pappus adds that 

' these propositions, which are practically one, include any 
number of theorems of all sorts about curves, surfaces, and 
solids, all of which are proved at once by one demonstration, 
and include propositions both old and new, and in particular 
those proved in the twelfth Book of these Elements.' 

Hultsch attributes the whole passage (pp. 680. 30-682. 20) 
to an interpolator, I do not know for what reason; but it 
seems to me that the propositions are quite beyond what 
could be expected from an interpolator, indeed I know of 
no Greek mathematician from Pappus's day onward except 
Pappus himself who was capable of discovering such a pro- 
position. 

1 Centrobaryca, Lib. ii, chap, viii, Prop. 3. Viennae 1641. 
Dd2 



404 PAPPUS OF ALEXANDRIA 

If the passage is genuine, it seems to indicate, what is not 
elsewhere confirmed, that the Collection originally contained, 
or was intended to contain, twelve Books. 

Lemmas to the different treatises. 

After the description of the treatises forming the Treasury 
of Analysis come the collections of lemmas given by Pappus 
to .assist the student of each of the books (except Euclid's 
Data) down to Apollonius's Conies, with two isolated lemmas 
to the Surface-Loci of Euclid. It is difficult to give any 
summary or any general idea of these lemmas, because they 
are very numerous, extremely various, and often quite diffi- 
cult, requiring first-rate ability and full command of all the 
resources of pure geometry. Their number is also greatly 
increased by the addition of alternative proofs, often requiring 
lemmas of their own, and by the separate formulation of 
particular cases where by the use of algebra and conventions 
with regard to sign we can make one proposition cover all the 
cases. The style is admirably terse, often so condensed as to 
make the argument difficult to follow without some little 
filling-out; the hand is that of a master throughout. The 
only misfortune is that, the books elucidated being lost (except 
the Conies and the Cutting-off of a ratio of Apollonius), it is 
difficult, often impossible, to see the connexion of the lemmas 
with one another and the problems of the book to which they 
relate. In the circumstances, all that I can hope to do is to 
indicate the types of propositions included in the lemmas and, 
by way of illustration, now and then to give a proof where it 
is sufficiently out of the common. 

(a) Pappus begins with Lemmas to the tiectio rationiv and 
Sectio spatii of Apollonius (Props. 1-21, pp. 684-704). The 
first two show how to divide a straight line in a given ratio, 
and how, given the first, second and fourth terms of a pro- 
portion between straight lines, to find the third term. The 
next section (Props. 3-12 and 16) shows how to manipulate 
relations between greater and less ratios by transforming 
them, e.g. componendo, convertendo, &c., in the same way 
as Euclid transforms equal ratios in Book V ; Prop. 1 6 proves 
that, according as a : b > or < c : d, ad > or < be. Props. 



THE COLLECTION. BOOK VII 405 

17-20 deal with three straight lines a, 6, c in geometrical 
progression, showing how to mark on a straight line containing 
a, 6, c as segments (including the whole among 'segments'), 
lengths equal to a + c + 2 v/(ac) ; the lengths are of course equal 
to a + c2b respectively. These lemmas are preliminary to 
the problem (Prop. 21), Given two straight lines AB, BC 
(C lying between A and B) y to find a point D on BA produced 
such that BD:DA = CD : (AB + %C-2 SAB7BCJ. This is, 
of course, equivalent to the quadratic equation (a + x):x 
= (a c + x) : (a + c 2 \/ac), and, after marking off A E along 
AD equal to the fourth term of this proportion, Pappus solves 
the equation in the usual way by application of areas. 

(ft) Lemmas to the 'Determinate Section 9 of Apollonius. 

The next set of Lemmas (Props. 22-64, pp. 704-70) belongs 
to the Determinate Section of Apollonius. As we have seen 
(pp. 180-1, above), this work seems to have amounted to 
a Theory of Involution. Whether the application of certain 
of Pappus's lemmas corresponded to the conjecture of Zeuthen 
or not, we have at all events in this set of lemmas some 
remarkable applications of * geometrical algebra '. They may 
be divided into groups as follows 

1. Props. 22, 25, 29 

If in the figure AD. DC = BD . DE, then 

= AB.BC:AE. EC. 



The proofs by proportions are not difficult. Prop. 29 is an 

alternative proof by means of Prop. 26 (see below). The 

algebraic equivalent may be expressed thus : if ax = 6y, then 

6 



II. Props. 30, 32, 34. 

If in the same figure AD . DE = BD . DC, then 

= AB.Bti:EC.CA. 



406 PAPPUS OF ALEXANDRIA 

Props. 32, 34 ar alternative proofs based on other lemmas 
(Props. 31, 33 respectively). The algebraic equivalent may be 

i. * j *u -e L *u b 

stated thus : if ax = by, then - = 

" y 

III. Props. 35, 36. 

If AB.BE=CB. BD, then AB : BE = DA . AC : CE . ED, 
and GB:BD = AC. CE: AD . DE, results equivalent to the 
following : if ax = by, then 

a _ (a y) (ab) , b _ (ab) (bx) 
x ~ (b^x^T-x) an y ~ (a-y)(y-x)' 

IV. Props. 23, 24, 31, 57, 58. 



If AB = CD, and E is any point in GD, 

AC.CD = AE . ED + BE . EC, 

and similar formulae hold for other positions of E. If E is 
between B and C, AG . GD = AE.ED-BE. EC; and if E 
is on AD produced, BE. EC = AE. ED + BD . DC. 

V. A small group of propositions relate to a triangle ABC 
with two straight lines AD, AE drawn from the vertex A to 
points on the base BC in accordance with one or other of the 
conditions (a) that the angles BAG, DAE are supplementary, 
(b) that the angles BAE, DAG are both right angles or, as we 




may add from Book VI, Prop. 12, (c) that the angles BAD, 

EAG are qual. The theorems are : 

In case (a) BG.CD: BE . ED = CA* : AE' 2 , 

(b) BG.CE-.BD. DE =CA Z :A D'\ 

(c) DC . CE : EB . BD = AC 3 : A&. 



THE COLLECTION. BOOK vii 407 

Two proofs are given of the first theorem. We will give the 
first (Prop. 26) because it is a case of theoretical analysis 
followed by synthesis. Describe a circle about ABD : produce 
EA, CA to meet the circle again in F, G, and join BF, FG. 

Substituting GC . CA for BC . CD and FE . EA for BE. ED, 
we have to inquire whether GC . CA : CA 2 = FE . EA : AE*, 

i.e. whether GC:CA = FE-.EA, 
i.e. whether GA : AC = FA : AE, 

i.e. whether the triangles GAP, CAE are similar or, in other 
words, whether GF is parallel to BC. 

But GF is parallel to BC, because, the angles BAC, DAE 
being supplementary, Z DAE = Z GAB = Z GFB, while at the 
same time Z DAE = suppt. <ALFAD = L FBD. 

The synthesis is obvious. 

An alternative proof (Prop. 27) dispenses with the circle, 
and only requires EKH to be drawn parallel to CM to meet 
AB, A Din II, K. 

Similarly (Prop. 28) for case (b) it is only necessary to draw 
FG through /) parallel to AC meeting BA in F and AE 
produced in G. 




Then, /.FAG, Z A DF (= Z /XA0) being both right angles, 

DG = 



Therefore C4 3 : 4D a = C^ :FD.DG = (CA : FD) . (CA : DG) 

= (BC:BD).(CE:DE) 
= BC.CE:BD.DE. 

In case (c) a circle is circumscribed to A DE cutting AB in F 
and AC in G. Then, since Z ^4 J!) = Z G4#, the arcs DF y EG 
are equal and therefore FG is parallel to DE. The proof is 
like that of case (a). 



408 PAPPUS OF, ALEXANDRIA 

VI. Props. 37, 38. 

If AB:BC = AD Z :DC*, whether AB be greater or less 
than AD, then 



[E in the 'figure is a point such that ED = 

(E) 09 
c B P (E) 



A (E) 096 



The algebraical equivalent is: If - = , ~ 2 , then ac=6 2 . 

c (o T c) 

These lemmas are subsidiary to the next (Props. 39, 40), 
being used in the first proofs of them. 

Props. 39, 40 prove the following : 

If AGDEB be a straight line, and if 

BA . AE: BD.DE = AC* : CD*, 
then AB.BD:AE.ED = BC* : CE* ; 

if, again, AC. CB : AE. EB = CD 2 : DE*, 

then EA .AC:CB.BE = AD* : DB\ 

If AB = a, BC = I), BD c, BE = d, the algebraic equiva- 
lents are the following. 

a-d)_(a-Z P 

~ 



c(c-<Z) ~ (b-cf (a-d)(c-d)~ (b-d 

.. (a-b)b (b- 
if - - T-. = 



,.. - - .. -- - 

and if - - JT-. = J - j- , then - p. - - = v - --' . 
(ad)d (c a) 2 bd c* 

i 

VIL Props. 41, 42, 43. 

If AD.DC=BD.DE, suppose that in Figures (1) and (2) 

0) 0___A _ C p E B 

(2) A EDO 

(3) A _ E B C 



k = AE+CB, and in Figure (3) k AE-BC, then 

= BA.AE, k.CD = BC.CE, k.BD = AB.BC, 

k.DE=AE.EC. 



THE COLLECTION. BOOK VII 409 

The algebraical equivalents for Figures (1) and (2) re- 
spectively may be written (if a = AD, b = DC, c = BD, 
d =- DE) : 

If ab = cd, then (a + d + c + b) a = (a + c) (a d), 
(ad + cb) b = (cb)(b + d), 
(ad + cb) c = (c + 



Figure (3) gives other varieties of sign. Troubles about 
sign can be avoided by measuring all lengths in one direction 
from an origin outside the line. Thus, if A = a, OB = 6, 
&c., the proposition may be as follows : 

If (d a) (d c) = (6 d) (e d) and /c = e a + 6 c, 
then k(d a) = (6 a)(e a), k(d c) = (6 c)(e c), 

d) = (& )(& f) and &(e d) = (e a)(0 c). 



VIII. Props. 45-56. 

More generally, if AD . DC = BD.DE and /k = AEBC, 
then, if J^ T be any point on the line, we have, according to the 
position of F in relation to A, B, (7, /), J, 

V + EF. FB = k. DF. 



Algebraically, if OA = a, OjS = 6 ... 0^ T = a;, the equivalent 
: If (d-tt)(ti-c) = (6-d)(6~d!), and & = (e- a) 



then (x a)(x 

By making .f = a, 6, c, e successively in this equation, we 
obtain the results of Props. 41-3 above. 

IX. Props. 59-64. 

In this group Props. 59, 60, 63 are lemmas required for the 
remarkable propositions (61, 62, 64) in which Pappus investi- 
gates ' singular and minimum ' values of the ratio 

AP.PJ):BP.PC, 

where (A, D), (B t C) are point-pairs on a straight line and P 
is another point on the straight line. He finds, not only when 
the ratio has the ' singular and minimum (or maximum) ' value, 



410 



PAPPUS OF ALEXANDRIA 



'but also what the value is, for three different positions of P in 
relation to the four given points. 

I will give, as an illustration, the first case, on account of its 
elegance. It depends on the following Lemma. AEB being 
a semicircle on AB as diameter, (7, D any two points on AB, 
and CE, DF being perpendicular to AB, let EF be joined and 




X C D B 

produced, and let BG be drawn perpendicular to EG. To 
prove that 

CB.BD = BG Z , (I) 

(2) 

0) 
Join GC, GD, FB, EB, AF. 

(1) Since the angles at G, D are right, F, G, B, D are concyclic. 
Similarly E, G, B, C are concyclic. 
Therefore 



= Z FAB 

= Z FEB, in the same segment of the semicircle, 

= Z GOB, in the same segment of the circle EGBC. 

And the triangles GOB, DGB also have the angle, CHG 
common ; therefore they are similar, and CJi : B(J = BG : BD, 

or CR.U1) = Mi*. 

(2) We have A B . BD = BF* ; 

therefore, by subtraction, AC. DB = BF*-BG* = FG*. 

(3) Similarly AB . BG = /JJ5? 2 ; 

therefore, by subtraction, from the same result (1), 
AD.BC= BE' 2 - BG* = W. 

Thus the lemma gives an extremely elegant construction for 
squares equal to each of the three rectangles. 



THE COLLECTION. BOOK VII 



411 



Now suppose (A, D), (B, C) to be two point-pairs on a 
straight line, and let P, another point on it, be determined by 
the relation 



then, says Pappus, the ratio AP . PD : BP . PC is singular and 
a minimum, and is equal to 

AD* : ( V'ACTBD- VAB . CD)\ 

On AD as diameter draw a circle, and draw BF, CG perpen- 
dicular to AD on opposite sides. 




Then, by hypothesis, A Ii . BD : AC . CD - BP- : CP- ; 
therefore HF* : CG* = BP- : CP*. 

or BF:CG = BP:CP, 

whence the triangles FBP, GCP arc similar and therefore 
equiangular. .o that FPG is a straight line. 

Produce G( ' to meet the circle in 11, join Fll, and draw DK 
perpendicular to Fll produced. Draw the diameter FL and 
join Lll. 

Now, by the lemma, VK* = AC . BD, and HK*= AB t . CD ; 

therefore FH = FK - UK = V(AC . BD) - S(AB . CD). 

Since, in the triangles FHL, PCG, the angles at //, C are 
right and LFLH LPGC> the triangles are similar, and 

GP :PC = FL:F1I = AD: FH 

= AD:{ V(AC . BD) - V(AB . CD)}. 
But GP:PC=FP:PB- 

therefore (U* .' PC- = FP . PG : BP . PC 

= AP.P1):BP.PC. 



412 



PAPPUS OF ALEXANDRIA 



Therefore 

AP. PD : BP.PC = AD 2 : { V(AC . BD)- V(AB . CD)}*. 

The proofs of Props. 62 and 64 are different, the former 
being long and involved. The results are : 

Prop. 62. If P is between C and J9, and 

AD.DB:AC.CB = DP 2 : PC 2 , 

then the ratio AP . PB : GP . PD is singular and a minimum 
and is equal to { J(AG . BD) + V(AD . BC) }* : DC 2 . 

Prop. 64. If P is on A D produced, and 

AB . BD : AC . CD = BP* : CP\ 

then the ratio AP . PD : BP . PC is singular and a maximum, 
and is equal to AD 2 : { V(AC . BD) + V(AB . CD) } 2 . 

(y) Lemmas on the Neva-eis of Apollonius. 

After a few easy propositions (e.g. the equivalent of the 
proposition that, if ax -f # 2 = by + y 2 , then, according as a > 
or < b, a + x > or < b + y), Pappus gives (Prop. 70) the 
lemma leading to the solution of the j/eSo-iy with regard to 
the rhombus (see pp. 190-2, above), and after that the solu- 
tion by one Heraclitus of the same problem with respect to 
a square (Props. 71, 72, pp. 780-4). The problem is, Given a 
square ABCD, to draw through B a straight line, meeting CD 
in H and AD produced in E, such that HE is ajual to a given 
length. 

The solution depends on a lemma to the effect that, if any 
straight line BHE through B meets CD in // and AD pro- 



H 




duced in E, and if EF be drawp perpendicular to BE meeting 
BC produced in F 9 then 



THE COLLECTION. BOOK VII 413 

Draw EG perpendicular to BF. 

Then the triangles BCH, EOF are similar and (since 
BO = EG) equal in all respects : therefore EF = BH. 



Now 
or BO . BF+ BF . FC = BH . BE+ BE.EH+ EF\ 

But, the angles HCF, HEF being right, H, C, F 9 E are 
concyclic, and BC.BF= BH . BE. 

Therefore, by subtraction, 



= FB.BC+EH*. 

Taking away the common part, BC . CF, we have 



Now suppose that we have to draw BHE through B in 
such a way that HE = k. Since BC, EH are both given, we 
have only to determine a length x such that x 2 = BC* + k 2 , 
produce BC to F so that OF = x, draw a semicircle on BF as 
diameter, produce AD to meet the semicircle in E, and join 
BE. BE is thus the straight line required. 

Prop. 73 (pp. 784-6) proves that, if 7) be the middle point 
of BC, the base of an isosceles triangle ABC, then BC is the 
shortest of all the straight lines through D terminated by 
the straight lines AB, AC, and the nearer to .66' is shorter than 
the more remote. 

There follows a considerable collection of lemmas mostly 
showing the equality of certain intercepts made on straight 
lines through one extremity of the diameter of one of two 
semicircles having their diameters in a straight line, either 
one including or partly including the other, or wholly ex- 
ternal to one another, on the same or opposite sides of the 
diameter. 



414 



PAPPUS OF ALEXANDRIA 



I need only draw two figures by way of illustration. 

In the first figure (Prop. 83), ABC, DEF being the semi- 
circles, BEKC is any straight line through C cutting both ; 
FG is made equal to AD ; AB is joined ; GU is drawn per- 
pendicular to BK produced. It is required to prove that 




L 
PIG. 1. 



BE = KH. (This is obvious when from L, the centre of the 
semicircle DEF, LM is drawn perpendicular to BK.) If E y K 
coincide in the point M' of the semicircle so that B'Gll' is 
a tangent, then B'M' = M'H' (Props. 83, 84). 

In the second figure (Prop. 91) D is the centre of the 
semicircle ABC and is also the extremity of the diameter 
of the semicircle DEF. If BEGF be any straight line through 




A C P 

FIG. 2. 

F cutting both semicircles, BE = EG. This is clear, since DE 
is perpendicular to BG. 

The only problem of any difficulty in this section is Prop. 
85 (p. 796). Given a semicircle ABC on the diameter AC 
and a point D on the diameter, to draw a semicircle passing 
through D and having its diameter along DC such that, if 
CEB be drawn touching it at E and meeting the semicircle 
ABO in JS, BE shall be equal to AD. 



THE COLLECTION. BOOK VII 415 

The problem is reduced to a problem contained in Apollo- 
nius's Determinate Section thus. 

Suppose the problem solved by the semicircle DEF> BE 
being equal to AD. Join E to the centre G of the semicircle 




F C 



DEF. Produce DA to 77, making HA equal to AD. Let K 
be the middle point of DC. 

Since the triangles ABC, GEC are similar, 



= AD*:EG\ by hypothesis, 

7r'-- 7W* (since DG = ( 



= //tr f : 
Therefore 



= AD*:2DC.GK. 

Take a straight line Z such that 4J9- = L.2DC: 
therefore //(; : DG = A : ^A r , 

or //tf . GA' = L . DG. 



Therefore, given the two straight lines //D, DK (or the 
three points 77, D, K on a straight line), we have to find 
a point G between D and 7C such that 



which is the second epitagma of the third Problem in the 
Determinate Section of Apollonius, and therefore may be 
taken as solved. (The problem is the equivalent of the 



416 PAPPUS OF ALEXANDRIA 

solution of a certain quadratic equation.) Pappus observes 
that the problem is always possible (requires no Siopurfios), 
and proves that it has only one solution, 

(8) Lemmas on the treatise On contacts ' by Apollonius. 

These lemmas are all pretty obvious except two, which are 
important, one belonging to Book I of the treatise, and the other 
to Book II. The two lemmas in question have already been set 
out & propos of the treatise of Apollonius (see pp. 1 82-5, above). 
As, however, there are several cases of the first (Props. 105, 
107, 108, 109), one case (Prop. 108, pp. 836-8), different frotoi 
that before given, may be put down here: Given a circle and 
two points D, E within it, to draiv straight lines through 7), E 
to a point A on the circumference in such a way that, if they 
meet the circle again in B, (7, BG shall be parallel to DE. 

We proceed by analysis. Suppose the problem solved and 
DA,EA drawn ('inflected') to A in such a way that, if AD, 

AE meet the circle again in B, C, 
BC is parallel to DE. 

Draw the tangent at B meeting 
ED P>dced in F. 

Then Z FBD = L AGB = LAED; 

therefore A, E, B, F are concyclic, 
and consequently 

FD.DE=AD.DB. 

But the rectangle AD . DB is given, since it depends only 
on the position of D in relation to the circle, and the circle 
is given. 

Therefore the rectangle FD . DE is given. 

And DE is given ; therefore FD is given, and therefore F. 

If follows that the tangent FB is given in position, and 
therefore B is given. Therefore EDA is given and conse- 
quently AE also. 

To solve the problem, therefore, we merely take F on ED 
produced such that FD . DE = the given rectangle made by 
the segments of any chord through D, draw the tangent FB, 
join BD and produce it to A, and lastly draw AE through to 
C ; BC is then parallel to DE. 




THE COLLECTION. BOOK VIT 417 

The other problem (Prop. 117, pp. 848-50) is, as we have 
seen, equivalent to the following: Given a circle arid three 
points D, E, F in a straight line external to it, to inscribe in 
the circle a triangle ABO such that its sides pass severally 
through the three points D, E, F. For the solution, see 
pp. 182-4, above. 

(e) The Lemmas to the Plane Loci of Apollonius (Props. 
119-26, pp. 852-64) are mostly propositions in geometrical 
algebra worked out by the methods of Eucl., Books II and VI. 
We may mention the following : 

Prop. 122 is the well-known proposition that, if D be the 
middle point of the side BG in a triangle ABC, 

BA* + AC 2 = 2 (AD* + DC*). 

PropiS. 123 and 124 are two cases of the same proposition, 
the enunciation being marked by an expression which is also 
found in Euclid's Data. Let AB : BC be a given ratio, and 

A P E c B 



let the rectangle CA .AD\w given; then, if BE is a mean 
proportional between DB, BC, 'the square on AE is greater 
by the rectangle CA . AD than in the ratio of AB to BO to the 
square on EC \ by which is meant that 



or 

The algebraical equivalent may be expressed thus (if A B= a, 



TC // - rr i.i (a + .r) 2 -(a b)c n 

If x = v/(a-c)6, then = 



Prop. 125 is remarkable : If C, D be two points on a straight 
line AB, 



AD* + . 7)52 = AC*+ AC . GB+ 



418 PAPPUS OF ALEXANDRIA 

This is equivalent to the general relation between four 
points on a straight line discovered by Simson and therefore 
wrongly known as Stewart's theorem : 

AD* . BC+BD* . CA + CD* . AB + BC.CA .AB = 0. 

(Simson discovered this theorem for the more general case 
where D is a point outside the line ABG.) 
An algebraical equivalent is the identity 

(d - a) 2 (6 - r) + (d - 6) 2 (r - a) + (d - <f ( - &) 



Pappus's proof of the last-mentioned lemma is perhaps 
worth giving. 



A C O a 

(7, D being two points on the straight line AB, take the 
point F on it such that 

FD:DB = AC:CH. (1) 

Then FB : BD = AB : BC, 

and (AB- FB) : (BC- BD) = AB : BC, 

or AF:CD = AB:BC, 

and therefore 

AF.CD:CD* = AB:BC. (2) 

From (1) we derive 

AC 



and from (2) 



We have 'now to prove that 

AD 2 + BD . DF = AC* + AC . CB + AF . CD, 
or AD* + BD . DF = CA . AB + AF. CD, 



THE COLLECTION. BOOK VII 419 

i.e. (if DA . AC be subtracted from each side) 

that AD . DC + FD . DB = AC. DB + AF. CD, 

i.e. (if AF . CD be subtracted from each side) 

that FD. DC+FD. DB = AC. DB, 

or FD.CB^AC.DB: 

which is true, since, by (1) above, FD:DB- AC: CB. 

() Lemmas fo the ' Porisms' of Endid. 

. The 38 Lemmas to the Por turns of Euclid form an important 
collection which, of course, has been included in one form or 
other in the ' restorations ' of the original treatise. Chasles l 
in particular gives a classification of them, and we cannot 
do better than use it in this place : ' 23 of the Lemmas relate 
to rectilineal figures, 7 refer to the harmonic ratio of four 
points, and 8 have reference to the circle. 

1 Of the 23 relating to rectilineal figures, 6 deal with the 
quadrilateral cut by a transversal; 6 with the equality of 
the anharmonic ratios of two systems of four points arising 
from the intersections of four straight lines issuing from 
one point with two other straight lines ; 4 may be regarded as 
expressing a property of the hexagon inscribed in two straight 
lines; 2 give the relation between the areas of two triangles 
which have two angles equal or supplementary ; 4 others refer 
to certain systems of straight lines; and the last is a case 
of the problem of the Cutting-off of an area. 9 

The lemmas relating to the quadrilateral and the transversal 
are 1, 2, 4, 5, 6 and 7 (Props. 127, 128, 130, 131, 132, 133). 
Prop. 130 is a general proposition about any transversal 




whatever, and is equivalent to one of the equations by which 
we express the involution of six points. If A, A' \ B, B'\ 
C, C' be the points in which the transversal meets the pairs of 

1 Chasles, Les trois ?iv>-es de Ponsmes d'Eitclide, Paris, 1860, pp. 74 sq. 



420 PAPPUS OF ALEXANDRIA 

opposite sides and the two diagonals respectively, Pappus's 
result is equivalent to 

AB^B'C _ CA 

A r B r .BC'~ C'A'' 

Props. 127, 128 are particular cases in which the transversal 
is parallel to a side; in Prop. 131 the transversal passes 
through the points of concourse of opposite sides, and the 
result is equivalent to the fact that the two diagonals divide 
into proportional parts the straight line joining the points of 
concourse of opposite sides; Prop. 132 is the particular case 
of Prop. 131 in which the line joining the points of concourse 
of opposite sides is parallel to a diagonal; in Prop. 133 the 
transversal passes through one only of the points of concourse 
of opposite sides and is parallel to a diagonal, the result being 
CA* = CB . CB'. 

Props. 129, 136, 137, 140, 142, 145 (Lemmas 3, 10, 11, 14, 16, 
19) establish the equality of the anharmonic ratios which 
four straight lines issuing from a point determine on two 
transversals ; but both transversals are supposed to be drawn 
from the same point on one of the four straight lines. Let 




AB, AC, AD be cut by transversals HBCD, HEFO. It is 
required to prove that 

HE.FG _HB.CD 
HQ.EF~*HD.BC 

Pappus gives (Prop. 129) two methods of proof which are 
practically equivalent. The following is the proof ' by com- 
pound ratios '. 

Draw HK parallel to AF meeting DA and AE produced 



THE COLLECTION. BOOK VII 421 

in K, L ; and draw LM parallel to AD meeting GH pro- 
duced in M. 

HE.FG _ HE FG_LU AF _ LH 
n HG . EF~ EF ' HG ~ AF ' //# ~~ UK ' 

In exactly the same way, if JDY/ produced meets LAI in it/' 
we prove that 

HB . CD LH 



Therefore 



UD.BC~ UK 
HE.FG UB.CD 



UG.EF~ HD.BC 

(The proposition is proved for HBCD and any other trans- 
versal not passing through // by applying our proposition 
twice, as usual.) 

Props. 136, 142 are the reciprocal; Prop. 137 is a particular 
case in which 6ne of the transversals is parallel to one of the 
straight lines, Prop. 140 a reciprocal of Prop. 137, Prop. 145 
another case of Prop. 129. 

The Lemmas 12, 13, 15, 17 (Props. 138, 139, 141, 143) are 
equivalent to the property of the hexagon inscribed in two 
straight lines, viz. that, if the vertices of a hexagon are 
situate, three and three, on two straight lines, the points of 
concourse of opposite sides are in a straight line; in Props. 
138, 141 the straight lines are parallel, in Props. 139, 143 not 
parallel. 

Lemmas 20, 21 (Props. 146, 147) prove that, when one angle 
of one triangle is equal or supplementary to one angle of 
another triangle, the areas of the triangles are in the ratios 
of the rectangles contained by the sides containing the equal 
or supplementary angles. 

The seven Lemmas 22, 23. 24, 25, 26, 27, 34 (Props. 148-53 
and 160) are propositions relating to the segments of a straight 
line on which two intermediate points arc marked. Thus : 

Props. 148, 150. 

If (7, D be two points on AB, then 
(a) if 2AB . CD = CB* t AD* = AC 2 + DB* ; 

A C D B 



(6) if 2AC.BD = CD\ AB* = A1P + U1P. 



422 PAPPUS OF ALEXANDRIA 

Props. 149, 161. 

If AB . BC = BD*, 

then (AD DC) BD = AD . DC, 

(ADDC)BO= DC 2 , 



B 



and (4 DC) BA = AD*. 

Props. 152, 153. 
If AB:BC=AD*:DC*, then A B . BC = BD*. 



4- 



B 



_ 5 _ _ 

Prop. 160. 

If AB : BC=AD : DC, then, if E be the middle point of AC, 

BE. ED = EC\ 

BD.DE = AD. DC, 



E D C 



The Lemmas about the circle include the harmonic proper- 
ties of the pole and polar, whether the pole is external to the 
circle (Prop. 154) or internal (Prop. 161). Prop. 155 is a 
problem, Given a segment of a circle on AB as base, to inflect 
straight lines AC, BC to the segment in a given ratio to one 
another. 

Prop. 156 is one which Pappus has already used earlier 
in the Collection. It proves that the straight lines drawn 
from the extremities of a chord (DE) to any point (F) of the 
circumference divide harmonically the diameter (AB) perpen- 
dicular to the chord. Or, if ED, FK be parallel chords, and 
EF, DK meet in (?, and EK> DF in //, then 



THE COLLECTION. BOOK VII 



423 



Since AB bisects ])E perpendicularly, farcAE) = (arc4D) 
and LEFA = Z AFD, or AF bisects the angle EFD. 




Since the angle AFB is right, FB bisects Z.1IFG, the supple- 
ment of Z EFD. 



Therefore (Eucl. VI. 3) GB : BH = GF: FH = 
and, alternately and inversely, AH : II B = AG : GB. 

Prop. 157 is remarkable in that (without any mention of 
a conic) it is practically identical with Apollonius's Conies 
III. 15 about the foci of a central conic. Pappus's theorem 
is as follows. Let AB be the diameter of a semicircle, and 




from A y B let two straight lines AM, BI) be drawn at right 
angles to AB. Let any straight line A)A T meet the two perpen- 
diculars in D, E and the semicircle in F. Further, let FG be 
drawn at right angles to DE, meeting AB produced in Or. 
It is to be proved that 

AG.GB = AE.BD 
Since F, D, G> B are concyclic, Z BDG = Z BFG. 



424 PAPPUS OF ALEXANDRIA 

And, since AFB, EFG are both right angles, LBFG^LAFE. 

But, since A, E, G, F are concyclic, LAFE = LAGE. 

Therefore L BDG = L AGE] 

and the right-angled triangles DBG, GAE are similar. 

Therefore AG:AE = BD: GB, 

or AG.GB = AE.DB. 



In Apollonius G and the corresponding point G' on 
produced which is obtained by drawing F'G' perpendicular to 
ED (where DE meets the circle again in F') are the foci 
of a central conic (in this case a hyperbola), and DE is any 
tangent to the conic ; the rectangle A E . BD is of course equal 
to the square on half the conjugate axis. 

(rj) The Lemmas to the Conies of Apollonius (pp. 918-1004) 
do not call for any extended notice. There arc a large number 
of propositions in geometrical algebra of the usual kind, 
relating to the segments of a straight line marked by a number 
of points on it ; propositions about lines divided into propor- 
tional segments and about similar figures; two propositions 
relating to the construction of a hyperbola (Props. 204, 205) 
and a proposition (208) proving that two hyperbolas with the 
same asymptotes do not meet one another. There are also 
two propositions (221, 222) equivalent to an obvious trigono- 

E' 
- 
^ * ~ i 

' _ ,0 



metrical formula. Let ABCD be a rectangle, and let any 
straight line through A meet DC produced in E and BC 
(produced if necessary) in F. 

Then EA . AF = ED . DC + CB . BF. 



THE COLLECTION. BOOK VII 425 

For EA* + AF* = ED* + DA 2 + A B* + BF* 

= #D 2 4- 5C 2 + <7D* + BF*. 
Also 4 2 + ^ ^ = EF* + 2EA.AF. 

Therefore 

2 EA . AF = AM 2 + AF*-EF* 

=-- A 1 /) 2 + BC* + CD* + J* - EF* 

= (ED* + CD*) + (BC* + BF*) - EF* 

= EC* + 2 ED . DC+ CF* + 2CB. BF-EF* 



i.e. *KA . AF = ED . DC+ CB . 

This is equivalent to sec 5 cosec = tan + cot 6. 

The algebraical equivalents of some of the results obtained 
by the usual geometrical algebra may be 'added. 

Props. 178, 179, 192-1. 



Prop. 195. 4cr = 2{ (a .f) 

Prop. 196. 
(a + 6 - a?) 2 + (a + b + x)* = (#- 6j- + (^ + 6) 2 + 2 (a + 26) a. 

Props. 1 C J7, 199, 198. 

If (x 4- y + a) a + ^- = 

or if (a: + -// -f- a) a + x- = (a + y)-, then x = 

or if 



Props. 200, 201. If (a . + b)z = 6 2 , then = Jf and 

1 v 7 a 6 x 



Prop. 207. If (a + 6)6 = 2 a 2 , then a = 6. 

(5) The two Lemmas to the Surface-Loci of Euclid have 
already been mentioned as significant. The first has the 
appearance of being a general enunciation, such as Pappus 




426 PAPPUS OF ALEXANDRIA 

is fond of giving, to cover a class of propositions. The 
enunciation may be translated as follows : ' If AB be a straight 
line, and CD a straight line parallel to a straight line given in 
position, and if the ratio AD . DB : D(7 2 be given, the point 
lies on a conic section. If now A B be no longer given in 
position, and the points A, B are no longer given but lie 
(respectively) on straight lines AE, EB given in position, the 
point raised above (the plane containing AE, EB) lies on 
a surface given in position. And this was proved/ Tannery 

was the first to explain this intelligibly ; 
and his interpretation only requires the 
very slight change in the text of sub- 
stituting v0tai$ for evdeTa in the phrase 
ytvrfrai 5e rrpoy 0trei tvOtia rats AJE, EB. 
It is not clear whether, when AB ceases 
to be given in position, it is still given 
in length. If it is given in length and A, B move on the lines 
AE, EB respectively, the surface which is the locus of C is 
/ complicated one such as Euclid would hardly have been 
in a position to investigate. But two possible cases are 
indicated which he may have discussed, (1) that in which AB 
moves always parallel to itself and varies in length accord- 
ingly, (2) hat in, which the two lines on which A, B move are 
parallel instead of meeting at a point. The loci in these two 
cases would of course be a cone and a cylinder respectively. 

The second Lemma is still more important, since it is the 
first statement on record of the focus-directrix property of 
the three conic sections. The proof, after Pappus, has been 
set out above (pp. 119-21). 

() An uutdlocated Lem f ma. 

Book VII ends (pp. 1016-18) with a lemma which is not 
given under any particular treatise belonging to the Treasury 
of Analysis, but is simply called 'Lemma to the 'AvaXvofiei'os'. 
If ABC be a triangle right-angled at B, and AB, BG be 
divided at F, G so that AF : FB = BG : GO = AB : BC, and 
if AEG, GEF be joined and BE joined and produced to D, 
then shall BD be perpendicular to AC. 

The text is unsatisfactory, for there is a long interpolation 
containing an attempt at a proof by reductio ad absurdum ; 



THE COLLECTION. BOOKS VII, VIII 427 

but the genuine proof is indicated, although it breaks off 
before it is quite complete. 



AF: FB = BG:GC, 



Since 



or AB:BC = FB:GC. 

But, by hypothesis, AB:BC=BG: GC ; 
therefore BF = BG. 

From this point the proof apparently proceeded by analysis. 
1 Suppose it done ' (yeyoi>eTa>), i.e. suppovse the proposition true, 
and BED perpendicular to AC. 




Then, by similarity of triangles, AD : DB = AB : BC ; 

therefore AF: FB = AD: DB, and consequently the angle 
ADB is bisected by DP. 

Similarly the angle BDC is bisected by DG. 

Therefore each of the angles BDF, BDG is half a right 
angle, and consequently the angle FDG is a right angle. 

Therefore J5, (7, D, F are concyclic ; and, since the angles 

FDB, BDG are equal, FB = BG. 

This is of course the result above proved. 

Evidently the interpolator tried to clinch the argument by 
proving that the angle BDA could not be anything but a right 
angle. 

Book VIII. 

Book VIII of the Collection is mainly on mechanics, although 
it contains, in addition, some propositions of purely geometrical 
interest. 



428 PAPPUS OF ALEXANDRIA 

Historical preface. 

It begins with an interesting preface on the claim of 
theoretical mechanics, as distinct from the merely practical 
or industrial, to be regarded as a mathematical subject. 
Archimedes, Philon, Heron of Alexandria are referred to as 
the principal exponents of the science, while Carpus of Antioch 
is also mentioned as having applied geometry to 'certain 
(practical) arts'. 

The date of Carpus is uncertain, though it is probable that 
he caine after Geminus; the most likely date seems to be the 
first or second century A.D. Simplicius gives the authority of 
lamblichus for the statement that Carpus squared the circle 
by means of a certain curve, which he simply called a curve 
generated by a double motion. 1 Proclus calls him ' Carpus the 
writer on mechanics (o fAriyaviKos) ', and quotes from a work of 
his on Astronomy some remarks about the relation between 
problems and theorems and the 'priority in order' of the 
former. 2 Proclus also mentions him as having held that an 
angle belongs to the category of quantity (TTOOW), since it 
represents a sort of ' distance ' between the two lines forming 
it, this distance being 'extended one way' (40* <EJ> &6orei>y) 
though in a different sense from that in which a line represents 
extension one way, so that Carpus's view appeared to be/ the 
greatest possible paradox ' ;5 ; Carpus seems in reality to have 
been anticipating the modern view of an angle as representing 
divergence rather than distance, and to have meant by <f>* <EI/ 
in oue sense (rotationally), as distinct from one way or in one 
dimension (linearly). 

Pappus tells us that Heron distinguished the logical, i.e. 
theoretical, part of mechanics from the practical or manual 
(yfipovpyiKov), the former being made up of geometry, arith- 
metic, astronomy and physics, the latter of work in metal, 
architecture, carpentering and painting; the man who had 
been trained from his youth up in the scie'nces aforesaid as well 
as practised in the said arts would naturally prove the best 
architect and inventor of mechanical devices, but, as it is diffi- 
cult or impossible for the same person to do both the necessary 

1 Simplicius on Ariat. Categ., p. 192, Kalbfleisch. 

a Proclus on EucL I, pp. 241-3. 3 lb., pp. 125. 25-126. 6. 



THE COLLECTION. BOOK VIII 429 

mathematics and the practical work, he who has not the former 
must perforce use the resources which practical experience in 
his particular art or craft gives him. Other varieties of 
mechanical work included by the ancients under the" general 
term mechanics were (1) the use of the mechanical powers, 
or devices for moving or lifting great weights by means of 
a small force, (2) the construction of engines of war for 
throwing projectiles a long distance, (3) the pumping of water 
from great depths, (4) the devices of 'wonder-workers' 
(OavfjLao-iovpyot), some depending on pneumatics (like Heron 
in the Pneumatwa), some using strings, &c., to produce move- 
ments like those of living things (like Heron in * Automata and 
Balancings'), some employ ing floating ladies (like Archimedes 
in ' Floating Bodies '), others using water to measure time 
(like Heron in his ' Water-clocks'), and lastly ' sphere-making ', 
or the construction of mechanical imitations of the movements 
of the heavenly bodies with the uniform circular motion of 
water as the motive power. Archimedes, says Pappus, was 
held to be the one person who had understood the cause and 
the reason of all these various devices, and had applied his 
extraordinarily versatile genius and inventiveness to all the 
purposes of daily life, and yet, although this brought him 
unexampled fame the world over, so that his name was on 
every one's lips, he disdained (according to Carpus) to write 
any mechanical work save a tract on sphere-making, but 
diligently wrote all that he could in a small compass of the 
most advanced parts of geometry and of subjects connected 
with arithmetic. Carpus himself, says Pappus, as well as 
others applied geometry to practical arts, and with reason: 
' for geometry is in no wise injured, nay it is by nature 
capable of giving substance to many arts by being associated 
with them, and, so far from being injured, it may be said, 
while itself advancing those arts, to be honoured and adorned 
by them in return.' 

The object of the Book. 

Pappus then describes the object of the Book, namely 
to set out the propositions which the ancients established by 
geometrical methods, besides certain useful theorems dis- 
covered by himself, but in a shorter and clearer form and 



430 PAPPUS OF ALEXANDRIA 

in better logical sequence than his predecessors had attained. 
The sort of questions to be dealt with are (1) a comparison 
between the force required to move a given weight along 
a horizontal plane and that required to move the same weight 
upwards on an inclined plane, (2) the finding of two mean 
proportionals between two unequal straight lines, (3) given 
a toothed wheel with a certain number of teeth, to find the 
diameter of, and to construct, another wheel with a given num- 
ber of teeth to work on the former. Each of these things, he says, 
will be clearly understood in its proper place if the principles 
on which the ' centrobaric doctrine ' is built up are first set out. 
It is not necessary, he adds, to define what is meant by < heavy ' 
and 'light* or upward and downward motion, since these 
matters are discussed by Ptolemy in his Mathematical but 
the notion of the centre of gravity is so fundamental in the 
whole theory of mechanics that it is essential in the first 
place to explain what is meant by the ' centre of gravity ' 
of any body. 

On the centre of gravity. 

Pappus then defines the centre of gravity as 'the point 
within a body which is such that, if the weight be conceived 
to be suspended from the point, it will remain at rest in any 
position in which it is put V The method of determining the 
point by means of the intersection, first of planes, and then of 
straight lines, is next explained (chaps. 1,2), and Pappus then 
proves (Prop. 2) a proposition of some difficulty, namely that, 
if D 9 E, F be points on the sides BC, CA, AB of a triangle ABC 
such that BD:DC=CE:EA = AF: FB, 



then the centre of gravity of the triangle ABC is also the 
centre of gravity of the triangle DEF. 

Let H, K be the middle points of BC, CA respectively; 
join AH, BK. Join HK meeting DE in L. 

Then AH, BK meet in 6r, the centre of gravity of the 
triangle ABC, and AG = 2 OH, BO = 2 OK, so that 



CA :AK = AB:HK = BO : OK = AO : OH. 
1 Pappus, viii, p. 1030. 11-13. 



THE COLLECTION. BOOK VIII 
Now, by hypothesis, 



431 



whence CA : AE = BC : CD, 

and, if we halve the antecedents, 

AK:AE=HC:CD; 
therefore A K : EK = HC : HD or BH: HD, 




whence, cvmpoiMtido, CE: EK = BD : DH. (1) 

But AF:FB= BD : DC = (BD : DH) . (DH : DC) 

= (CE:EK).(DH:DC). (2) 

Now, ELD being a transversal cutting the sides of the 
triangle KJIC, we have 

HL : KL = (CE:EK) . (DH : DC). (3) 

[This is ' Menelaus's theorem ' ; Pappus does not, however, 
quote it, but proves the relation ad hoc in an added lemma by 
drawing CM parallel to DE to meet HK produced in M. The 
proof is easy, for HL . LK = (HL . LM) (LM . LK) 

= (HD:DC).(CE:EK).] 
It follows from (2) and (3) that 

A r*. ETO TIT . T \7 

^LjT . r/> = fiJLj . Z/A, 

and, since AB is parallel to 7//i, and AH, BK are straight 
lines meeting in G, FGL is a straight line. 

[This is proved in another easy lemma by reductio ad 
absurdumJ] 



432 



PAPPUS OF ALEXANDRIA 



We have next to prove that EL = LD. 

Now [again by 'Menelaus's theorem', proved ad hoc by 
drawing CN parallel to HK to meet ED produced in N] 

EL : LD = (EK : KG) . (CH : HD). (4) 

But, by (1) above, CE:EK = BD : DH\ 
therefore OK : KE = BH : HD = CH : HD, 
so that (EK:KC).(CH:HD)=\, and therefore, from (4), 

EL = LD. 

It remains to prove that FG = 2 GL, which is obvious by 
parallels, since FG : GL = AG : Gil = 2 : 1. 

Two more propositions follow with reference to the centre 
of gravity. The first is, Given a rectangle with AB, BC as 
adjacent sides, to draw from G a straight line meeting the side 
opposite BC in a point D such that, if the trapezium A DOB is 
hung from the point D, it will rest with AD, BG horizontal. 




In other words, the centre of gravity must be in DL drawn 
perpendicular to BC. Pappus proves by analysis that 
CL 2 = 3BL 2 , so that the problem is reduced to that of 
dividing BC into parts BL, LC such that this relation holds. 
The latter problem is solved (Prop. 6) by taking a point, 
say X, in CB such that (JX = 3 XB, describing a semicircle on 
BO as diameter and drawing XY at right angles to BC to 
meet the semicircle in F, so that XY* = T \ B<7 2 , and then 
dividing CB at L so that 



CL:LB = CX : XY (= 



= v/3 : 1). 



The second proposition is this (Prop. 7). Given two straight 
lines AB, AC, and B a fixed point on AB, if CD be drawn 




THE COLLECTION. BOOK VIII 483 

with its extremities on AC, AB and so that AC : BD is a given 
ratio, then the centre of gravity of the triangle ADC will lie 
on a straight line. 

Take E y the middle point of AC, and F a point on DE such 
that DF = 2 FE. Also let // be a point on BA such that 
BH =2 HA. Draw FG parallel to AC. 
Then 4(? = ^/_), and AH=%AB\ 
therefore HG = $BD. ' * 

Also FG = AE = 4(7. Therefore, 
since the ratio AC:BD is given, the 
ratio 6r// : G F is given. 

And the angle FGH (A) is given ; 
therefore the triangle FGH is given in 
species, and consequently the angle GHF 
is given. And // is a given point. 

Therefore HF is a given straight line, and it contains the 
centre of gravity of the triangle ADC. 

The inclined plane. 

Prop. 8 is on the construction of a plane at a given inclina- 
tion to another plane parallel to the horizon, and with this 
Pappus loaves theory and proceeds to the practical part. 
Prop. 9 (p. 1054. 4 sq.) investigates the problem 'Given 
a weight which can be drawn along a plane parallel to the 
horizon by a given force, and a plane inclined to the horizon 
at a given angle, to find the force required to draw the weight 
upwards on the inclined plane'. This seems to be the first 
or only attempt in ancient times to investigate motion on 
an inclined plane, and as such it is curious, though of no 
value. 

Let A be the weight which can be moved by a force C along 
a horizontal plane. Conceive a sphere with weight equal to A 
placed in contact at L with the given inclined plane ; the circle 
OGL represents a section of the sphere by a vertical plane 
passing through E its centre and LK the line of greatest slope 
drawn through the point L. Draw EGH horizontal and there- 
fore parallel to MN in the plane of section, and draw LF 
perpendicular to EH. Pappus seems to regard the plane 
as rough, since he proceeds to make a system in equilibrium 




434 PAPPUS OF ALEXANDRIA 

about FL as if L were the fulcrum of a lever. Now the 
weight A acts vertically downwards along a straight line 
through E. To balance it, Pappus supposes a weight B 
attached with its centre of gravity at G. 

Then A:B=GF:EF 

= (EL-EF):EF 
[= (1 -sin 6): sin 6, 

where KMN = ff\\ 

and, since /.KMNis given, the ratio EF: EL, 
and therefore the ratio (EL - EF) : EF, is 

M N 

given ; thus B is found. 

Now, says Pappus, if D is the force which will move B 
along a horizontal plane, as C is the force which will move 
A along a horizontal plane, the sum of C and D will be the 
force ^ required to move the sphere upwards on the inclined 
plane. He takes the particular case where 6 = 60. Then 
sin0 is approximately f -J (he evidently uses -|.ff for -1^/3), 

and 4:5 = 16:104. 

Suppose, for example, that A = 200 talents; then B is 1300 
talents. Suppose further that C is 40 man-power; then, since 
D : C = B : A, D = 260 man-power ; and it will take D + C, or 
300 man-power, to move the weight up the plane! 

Prop. 10 gives, from Heron's Barulcus, the machine con- 
sisting of a pulley, interacting toothed wheels, and a spiral 
screw working on the last wheel and turned by a handle; 
Pappus merely alters the proportions of the weight to the 
force, and of the diameter of the wheels. At the end of 
the chapter (pp. 1070-2) he repeats his construction for the 
finding of two mean proportionals. 

Construction of a conic through jive paints. 

Chaps. 13-17 are more interesting, for they contain the 
solution of the problem of constructing a conic through five 
given points. The problem arises in this way. Suppose we 
are given a broken piece of the surface of a cylindrical column 
such that no portion of the circumference of either of its base 



THE COLLECTION. BOOK VIII 



435 



is left intact, and let it be required to find the diameter of 
a circular section of the cylinder. We take- any two points 
-A, B on the surface of the fragment and by means of these we 
find five points on the surface all lying in one plane section, 
in general oblique. This is done by taking five different radii 
and drawing pairs of circles with A, B as centres and with 
each of the five radii successively. These pairs of circles with 
equal radii, intersecting at points on the surface, determine 
five points on the plane bisecting AB at right angles. The five 
points are then represented on any plane by triangulation. 

Suppose the points are A, B, C 9 D, E and are such that 
no two of the lines connecting the different pairs are parallel. 




This case can be reduced to the construction of a conic through 
the five points A, B, D, E, F where EF is parallel to AB. 
This is shown in a subsequent lemma (chap. 16). 

For, if EF be drawn through E parallel to AB y and if CD 
meet AB in and EF in 0', we have, by the well-known 
proposition about intersecting chords, 

CO .OD:AO.OB = CO' . O'D : EO' . Q'F, 

whence O'F is known, and F is determined. 

We have then (Prop. 13) to construct a conic through A, B, 
D, E y F, where EF is parallel to AB. 

Bisect AB, EF at V, W\ then VW produced both ways 
is a diameter. Draw DR, the chord through 7) parallel 



436 PAPPUS OF ALEXANDRIA 

to this diameter. Then li is determined by means of the 
relation 

RG.GD:BG.GA = RH.HD:FH.HE (1) 

in this way. 

Join DB y RA, meeting EF in K, L respectively. 
Then, by similar triangles, 

RG . GD : BG . GA = (RH: 1IL) . (DH:HK) 

= RH.HD\KH.HL. 
Therefore, by (1), FH . HE = KH.HL, 

whence HL is determined, and therefore L. The intersection 
of AL, DH determines R. 

Next, in order to find the extremities P, P' of the diameter 
through V, W, we draw ED, RF meeting PP' in M, N respec- 
tively. 

Then, as before, 

FW. WE-.P'W. WP = FH. HE: RH. HD, by the ellipse, 

= FW.WE-.NW.WM, by similar triangles. 
Therefore P' W. WP = NW . WM ; 

and similarly we can find the value of P'V '. VP. 

Now, says Pappus, since l^W.WP and P'KFP are given 
areas and the points V, W are given, P, P 7 are given. His 
determination of P, P' amounts (Prop. 14 following) to an 
elimination of one of the points and the finding of the other 
by means of an equation of the second degree. 

Take two points Q, Q' on the diameter such that 

P'V.VP=WV.VQ, (a) 






Q, Q' are thus known, while P, P' remain to be found. 

By (a) P'V: VW= QV: VP, 

whence P'W:VW= PQ : PV. 

Therefore, by means of (j8), 

PQ:PV=Q'W:WP 9 



THE COLLECTION. BOOK VIII 437 

so that PQ:(JV=Q'W:PQ', 

or PQ.1>W=QV.VW. 

Thus P can be found, and similarly P'. 

The conjugate diameter is found by virtue of the relation 

(conjugate diam.)* : PP'* = p : PI*. 

where p is the latus rectum to PP f determined by the property 
of the curve V , P P>= AV*:PV.V1*. 

Problem, Given two conjugate diameters of an ellipse, 
to find the axes. 

Lastly, Pappus shows (Prop. 14, chap. 17) how, when we are 
given two conjugate diameters, we can find the axes. The 
construction is as follows. Let AB> CD be conjugate diameters 
(CD being the greater), E the centre. 

Produce EA to II so that 

KA.AH = DE*. 

Through A draw FG parallel to CD. Bisect Ell in K y and 
draw KL at right angles to EH meeting FG in L. 




With L as centre, and LE as radius, describe a circle cutting 
GFmG, F. 

Join EF, EG, and from A draw AM, AN parallel to EF, EG 
respectively. 



438 



PAPPUS OF ALEXANDRIA 



Take points P, R on EG, EF such that 

EP* = GE. EM, and ER* = FE. EN. 

Then EP is half the major axis, and ER half the minor axis. 
Pappus omits the proof. 

Problem of seven hexagons in a circle. 

Prop. 19 (chap. 23) is a curious problem. To inscribe seven 
equal regular hexagons in a circle in such a way that one 





is about the centre of the circle, while six others stand on its 
sides and have the opposite sides in each case placed as chords 
in the circle. 

Suppose GUKLNM to be the hexagon so described on HK, 
a side of the inner hexagon ; OKL will then be a straight line. 
Produce OL to meet the circle in P. 

Then OK = KL = LN. Therefore, in the triangle OLN, 
OL = 2LN, while the included angle OLN (= 120) is also 
given. Therefore the triangle is given in species; therefore 
the ratio ON : NL is given, and, since ON is given, the side NL 
of each of the hexagons is given. 

Pappus gives the auxiliary construction thus. Let AF bo 
taken equal to the radius OP. Let AC = AF, and on AC as 
base describe a segment of a circle containing an angle of 60. 
Take GE equal to AC, and draw EB to touch the circle at B. 



THE COLLECTION. BOOK VIII 439 

Then he proves that, if we join AB, AB is equal to the length 
of the side of the hexagon required. 

Produce BC to D so that BD = BA, and join DA. ABD 
is then equilateral. 

Since EB is a tangent to the segment, AE.EC = EB* or 
AE: EB = EB : EG, and the triangles EAB, EBC are similar. 

Therefore BA 2 : BC* = 4 A* : JEB* = A E : EC = 9 : 4 ; 
and BC = |4 = #A so that JW; = 2C7A 

But CF=2CA; therefore AC:CF= D(J:CB, and 
are parallel. 

Therefore BFi AD = BC:CD = 2:1, so that 



Also ^^J?(7 = ZJWXA = 60, so that LABF= 120, and 
the triangle ABF is therefore equal and similar to the required 
triangle NLO. 

Construction of toothed wheels and indented screw*. 

The rest of the Book is devoted to the construction (1) of 
toothed wheels with a given number of teeth equal to those of 
a given wheel, (2) of a cylindrical helix, the cocM-ias, indented 
so as to work 011 a toothed wheel. The text is evidently 
defective, and at the end an interpolator has inserted extracts 
about the mechanical powers from Heron's Mechanics. 



XX 

ALGEBRA: DIOPHANTUS OF ALEXANDRIA 
Beginnings learnt from Egypt. 

IN algebra, as in geometry, the Greeks learnt the beginnings 
from the Egyptians. Familiarity on the part of the Greeks 
with Egyptian methods of calculation is well attested. (1) 
These methods are found in operation in the Heronian writings 
and collections. (2) Psellus in the letter published by Tannery 
in his edition of Diophantus speaks of * the method of arith- 
metical calculations used by the Egyptians, by which problems 
in analysis are handled'; he adds details, doubtless taken 
from Anatolius, of the technical terms used for different kinds 
of numbers, including the powers of the unknown quantity. 
(3) The scholiast to Plato's Cliarmides 165 E says that 'parts 
of Xoyia-TiKrj, the science of calculation, are the so-called Greek 
and Egyptian methods in multiplications and divisions, and 
the additions and subtractions of fractions '. (4) Plato himself 
in the Laws 819 A-c says that free-born boys should, as is the 
practice in Egypt, learn, side by side with reading, simple 
mathematical calculations adapted to their age, which should 
be put into a form such as to combine amusement with 
instruction : problems about the distribution of, say, apples or 
garlands, the calculation of mixtures, and other questions 
arising in military or civil life. 

' Hau '-calculations. 

The Egyptian calculations here in point (apart from their 
method of writing and calculating in fractions, which, with 
the exception of , were always decomposed and written 
as the sum of a diminishing series of aliquot parts or sub- 
multiples) are the /tau-calculations. Hau, meaning a heap, is 
the term denoting the unknown quantity, and the calculations 



HAU '-CALCULATIONS 441 

in terms of it are equivalent to the solutions of simple equations 
with one unknown quantity. Examples from the Papyrus 
Rhind correspond to the following equations : 

= 19, 
x = 33, 
= 10. 



The Egyptians anticipated, though only in an elementary 
form, a favourite method of Diophantus, that of the 'false 
supposition' or 'regula falsi'. An arbitrary assumption is 
made as to the value of the unknown, and the true value 
is afterwards found by a comparison of the result of sub- 
stituting the wrong \&alue in the original expression with the 
actual data. Two examples may be given. The first, from 
the Papyrus Rhind, is the problem of dividing 100 loaves 
among five persons in such a way that the shares are in 
arithmetical progression, and one-seventh of the sum of the 
first three shares is equal to the sum of the other two. If 
a + 4d, a+3d, a + 2t/, a + d, a be the shares, then 



or d = 5ja. 

Alnnes says, without any explanation, 'make the difference, 
as it is, 5 1', and then, assuming a = 1, writes the series 
23, 17, 12, 6^, 1. The addition of these gives 60, and 100 is 
If times 60. Ahmes says simply * multiply If times' and, 
thus gets the correct values 38, 29, 20, 10, If. 

The second example (taken from the Berlin Papyrus 6619) 
is the solution of the equations 

x 2 + y 2 = 100, 
x:y= 1:, or y = #. 

x is first assumed to be 1, and #- + 7/ 2 is thus found to be f. 
In order to make 100, ff has to be multiplied by 64 or 8 2 . 
The true value of x is therefore 8 times 1, or 8. 

Arithmetical epigrams in the Greek Anthology. 

The simple equations solved in the Papyrus Rhind are just 
the kind of equations of which we find many examples in the 



442 ALGEBRA: DIOPHANTUS OF ALEXANDRIA 

arithmetical epigrams contained in the Greek Anthology. Most 
of these appear under the name of Metrodorus, a grammarian, 
probably of the time of the Emperors Anastasius I (A.D. 491- 
518) and Justin I (A.D. 518-27). They were obviously only 
collected by Metrodorus, from ancient as well as more recent 
sources. Many of the epigrams (46 in number) lead to simple 
equations, and several of them are problems of dividing a num- 
ber of apples or nuts among a certain number of persons, that 
is to say, the very type of problem mentioned by Plato. For 
example, a number of apples has to be determined such that, 
if four persons out of six receive one-third, one-eighth, one- 
fourth and one-fifth respectively of the whole number, while 
the fifth person receives 10 apples, there is one apple left over 
for the sixth person, i.e. 



= x. 



Just as Plato alludes to bowls (<f>id\at) of different metals, 
there are problems in which the weights of bowls have to 
be found. We are thus enabled to understand the allusions of 
Proclus and the scholiast on Charmides 165 E to fjLrjXiTai 
and ^ia\lraL dpidpoi, ' numbers of apples or of bowls'. 
It is evident from Plato's allusions that the origin of such 
simple algebraical problems dates back, at least, to the fifth 
century B.C. 

The following is a classification of the problems in the 
Anthology. (1) Twenty-three are simple equations in one 
unknown and of the type shown above; one of these is an 
epigram on the age of Diophantus and certain incidents of 
his life (xiv. 1 26). (2) Twelve are easy simultaneous equations 
with two unknowns, like Dioph. I. 6 ; they can of course be 
reduced to a simple equation with one unknown by means of 
an easy elimination. One other (xiv. 51) gives simultaneous 
equations in three unknowns 



and one (xiv. 49) gives four equations in four unknowns, 

40, # + 0=45, o; + u = 36, x + y + z + u = 60. 



With these may be compared Dioph. I. 16-21, as well as the 
general solution of any number of simultaneous linear equa- 



EPIGRAMS IN THE GREEK ANTHOLOGY 443 

tions of this type with the same number of unknown quantities 
which was given by Thymaridas, an early Pythagorean, and 
was called the ewdvOrjua, ' flower ' or ' bloom ' of Thymaridas 
(see vol. i, pp. 94-6). (3) Six more are problems of the usual 
type about the filling and emptying of vessels by pipes ; e.g. 
(xiv. 130) one pipe fills the vessel in one day, a second in two 
and a third in three ; how long will all three running together 
take to fill it? Another about brickmakers (xiv. 136) is of 
the same sort. 

Indeterminate equations of the first degree. 

The Anthology contains (4) two indeterminate equations of 
the first degree which can be solved in positive integers in an 
infinite number of ways (xiv. 48, 144) ; the first is a distribu- 
tion of apples, 3x in number, into parts satisfying the equation 
x 3y = T/, where y is not less than 2; the second leads to 
three equations connecting four unknown quantities : 



x l = 



the general solution of which is x = 4&, y = k, x l = 3k t 
y l = 2k. These very equations, which, however, are made 
determinate by assuming that x + y = x l + y l = 100, are solved 
in Dioph. I. 1 2. 

Enough has been said to show that Diophantus was not 
the inventor of Algebra. Nor was he the first to solve inde- 
terminate problems of the second degree. 

Indeterminate equations of second degree before 

Diophantus. 

Take first the problem (Dioph. II. 8) of dividing a square 
number into two squares, or of finding a right-angled triangle 
with sides in rational numbers. We have already seen that 
Pythagoras is credited with the discovery of a general formula 
for finding such triangles, namely, 



441 ALGEBRA: DIOPHANTUS OF ALEXANDRIA 

where n is any odd number, and Plato with another formula 
of the same sort, namely (2?&) 2 + (7& 2 I) 3 = (/& a +l) a . Euclid 
(Lemma following X. 28) finds the following more general 
formula 

m 2 M 2 ;p 2 g 2 = { 4 (>np 2 + mnq*) } 2 [ 4 (w ft^ 2 mitq*) } 2 . 

The Pythagoreans too, as we have seen (vol. i, pp. 91-3), 
solved another indeterminate problem, discovering, by means 
of the series of ' side- ' and ' diameter-numbers ', any number 
of successive integral solutions of the equations 

-7l a = +1. 



Diophantus does not particularly mention this equation, 
but from the Lemma to VI. 15 it is clear that he knew how 
to find any number of solutions when one is known. Thus, 
seeing that 2 a; 2 1 = y 2 is satisfied by x = 1, y = 1, he would 
put 

2(1 + a;) 2 1 = a square 



1)-, say; 
whence x = (4 + 2p)/(jr 2). 

Take the value p = 2, and we have x == 4, and .+ 1 =. 5; 
in this case 2 . 5 2 1 = 49 = 7 2 . Putting jj + 5 in place of a;, 
we can find a still higher value, and so on. 

Indeterminate equations in the Heronian collections. 

Some further Greek examples of indeterminate analysis are 
now available. They come from the Constantinople manuscript 
(probably of the twelfth century) from which Scheme edited 
the Metrica of Heron ; they have been published and translated 
by Heiberg, with comments by Zeuthen. 1 Two of the problems 
(thirteen in number) had been published in a less complete 
form in Hultsch's Heron (Geeponicus, 78, 79); the others 
are new. 

I. The first problem is to find two rectangles such that the 
perimeter of the second .is three times that of the first, and 
the area of the first is three times that of the second. The 

1 Bibliotheca mathematics, viii 8 , 1907-8, pp. 118-34. See now Geom. 
24. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 411-26. 



HERONIAN INDETERMINATE EQUATIONS 445 

number 3 is of course only an illustration, and the problem is 
equivalent to the solution of the equations 



(1) 

(2) xy = n.uv 
The solution given in the text is equivalent to 



U = >fc(4>6 3 ~ 2), V = 71 

Zeuthen suggests that the solution may have been obtained 
thus. As the problem is indeterminate, it would be natural 
to start with some hypothesis, e.g. to put v = n. It would 
follow from equation (1) that u is a multiple of n, say nz. 
We have then 



while, by (2), ,/?/ = M 3 0, 

whence xy = it 3 (^ + #) >r, 

or (x -/t 3 ) (2; /t a ) = /t 3 (u 3 1 ). 

An obvious solution is 

u 1 u :i = 7t :>> 1, y. /t 3 = >/ :i , 

which gives z = 2?i 3 1 -f 2 /t :j 1 = 4 H, 3 2, so that 
-u = nz = n(4 u 3 2). 

II. The second is a similar problem about two rectangles, 
equivalent to the solution of the equations 



(1) x+ y = u+v 

(2) a:?/ = u . w 
an<f the solution given in the text is 



x + y = u + ?' = >i 3 -!, (3) 

( ' 



In this case trial may have been made of the assumptions 



446 ALGEBRA: DIOPHANTUS OF ALEXANDRIA 

when equation (1) would give 

(n l)x = (7i 2 l)it, 
a solution of which is x = n 2 1, u = TI 1. 

III. The fifth problem is interesting in one respect. We are 
asked to find a right-angled triangle (in rational numbers) 
with area of 5 feet. We are told to multiply 5 by some 
square containing 6 as a factor, e.g. 36. This makes 180, 
and this is the area of the triangle (9, 40, 41). Dividing each 
side by 6, we have the triangle required. The author, then, 
is aware that the area of a right-angled triangle with sides in 
whole numbers is divisible by 6. If we take the Euclidean 
formula for a right-angled triangle, making the sides a . win, 
a . ^(m 2 ft 2 ), a , ^(m 2 4- ?6 2 ), where a is any number, and m, n 
are numbers which are both odd or both even, the area is 

^ mn (m n) (m + n) a 2 , 

and, as a matter of fact, the number mn(m n) (m + n) is 
divisible by 24, as was proved later (for another purpose) by 
Leonardo of Pisa. 

IV. The last four problems (10 to 13) are of great interest. 
They are different particular cases of one problem, that of 
finding a rational right-angled triangle such that the numerical 
sum of its area and its perimeter is a given number. The 
author's solution depends on the following formulae, where 
a, b are the perpendiculars, and c the hypotenuse, of a right- 
angled triangle, 8 its area, r the radius of the inscribed ciixsle, 
and s = 



S = rs = -|a&, r + 8 = a -f b, c = s r. 

(The proof of these formulae by means of the usual figure, 
namely that used by Heron to prove the formula 



is easy.) 

Solving the first two equations, in order to find a and ft, 
we have 



which formula is actually used by the author for finding a 



HERONIAN INDETERMINATE EQUATIONS 447 

and b. The method employed is to take the sum of the area 
and the perimeter +28, separated into its two obvious 
factors s(r+2), to put s(r+2) = A (the given number), and 
then to separate A into suitable factors to which s and r + 2 
may be equated. They must obviously be such that sr, the 
area, is divisible by 6. To take the first example where 
A 280 : the possible factors are 2 x 140, 4 x 70, 5 x 56, 7 x 40, 
8 x 35, 10 x 28, 14 x 20. The suitable factors in this case are 
?* + 2 = 8, s = 35, because r is then equal to 6, and rs is 
a multiple of 6. 

The author then says that 

rt = 4[G + 35-</{(6 + 35) 2 -8.6. 35]] ^^(41-U = 20, 

6 = 4(41 + 1)= 21, 

c = 35 6 = 29. 

The triangle is therefore (20, 21, 29) in this case. The 
triangles found in the other three cases, by the same method, 
are (9, 40, 41), (8, 15, 17) and (9, 12, 15). 

Unfortunately there is no guide to the date of the problems 
just given. The probability is that the original formulation 
of the most important of the problems belongs to the period 
between Euclid and Diophantus. This supposition best agrees 
with the fact that the problems include nothing taken from 
the great collection in the Arithmetica. On the other hand, 
it is strange that none of the seven problems above mentioned 
is found in Diophantus. The five relating to rational right- 
angled triangles might well have been included by him ; thus 
he finds rational right-angled triangles such that the area plus 
or minus one of the perpendiculars is a given number, but not 
the rational triangle which has a given area; and he finds 
rational right-angled triangles such that the area jdus or minus 
the sum of two sides is a given number, but not the rational 
triangle such that the sum of the area and the three sides is 
a given number. The omitted problems might, it is true, have 
come in the lost Books ; but, on the other hand, Book VI would 
have been the appropriate place for them. 

The crowning example of a difficult indeterminate problem 
propounded before Diophantus's time is the Cattle-Problem 
attributed to Archimedes, described above (pp. 97-8). 



448 ALGEBRA: DIOPHANTUS OF ALEXANDRIA 



Numerical solution of quadratic equations. 

The geometrical algebra of the Greeks has been in evidence 
all through our history from the Pythagoreans downwards, 
and no more need be said of it here except that its arithmetical 
application was no new thing in Diophantus. It is probable, 
for example, that the solution of the quadratic equation, 
discovered first by geometry, was applied for the purpose of 
finding numerical values for the unknown as early as Euclid, 
if not earlier still. In Heron the numerical solution of 
equations is well established, so that Diophantus was not the 
first to treat equations algebraically. What he did was to 
take a step forward towards an algebraic notation. 

The date of DIOPHANTUS can now be fixed with fair certainty. 
He was later than Hypsicles, from whom he quotes a definition 
of a polygonal number, and earlier than Theon of Alexandria, 
who has a quotation from Diophantus's definitions. The 
possible limits of date are therefore, say, 150 B.C. to A.D. 360. 
But the letter of Psellus already mentioned says that Anatolius 
(Bishop of Laodicea about A.D. 280) dedicated to Diophantus 
a concise treatise on the Egyptian method of reckoning; 
hence Diophantus must have been a contemporary, so that lie 
probably flourished A.D. 250 or not much later. 

An epigram in the Anthology gives some personal particulars: 
his boyhood lasted th of his life ; his beard grew after T ^th 
more ; he married after ^th more, and his son was born 5 years 
later; the son lived to half his father's ago, and the father 
died 4 years after his son. Thus, if x was his aye when 
he died, 

= X, 



which gives x = 84. 

Works of Diophantus. 
The works on which the fame of Diophantus rests are : 

(1) the Arithmetica (originally in thirteen Books), 

(2) a tract OH Polygonal Numbers. 



WOKKS 449 

Six Books only of the former and a fragment of the latter 
survive. 

Allusions in the Arithmetica imply the existence of 

(3) A collection of propositions under the title of Porisms ; 
in three propositions (3, 5, 16) of Book V, Diophantus quotes 
as known certain propositions in the Theory of Numbers, 
prefixing to the statement of them the words * We have it in 
the Porisms that . . / 

A scholium on a passage of lamblichus, where lamblichus 
cites a dictum of certain Pythagoreans about the unit being 
the dividing line (neQopiov) between number and aliquot parts, 
says ' thus Diopliantus in the Moriastica .... for he describes 
as "parts" the progression without limit in the direction of 
less than the unit '. The Moriastica may be a separate work 
by Diophantus giving rules for reckoning with fractions ; but 
I do not feel sure that the reference may not simply be to the 
definitions at the beginning of the Arithmetica. 

The Arithmetica. 

The seven lost Hooks and their place. 

None of the manuscripts which we possess contain more 
than six Books of the Arithmetica, the only variations being 
that some few divide the six Books into seven, while one or 
two give the fragment on Polygonal Numbers as VIII. The 
missing Books were evidently lost at a very early date. 
Tannery suggests that Hypatia's commentary extended only 
to the first six Books, and that she left untouched the remain- 
ing seven, which, partly as a consequence, were first forgotten 
and then lost (cf. the case of Apollonius's Conic*, where the 
only Books which have survived in Greek, I-IV, are those 
on which Eutocius commented). There is no sign that even 
the Arabians ever possessed the missing Books. The Fakhn, 
an algebraical treatise by Abu Bekr Muh. b. al-Hasan al- 
Karkhi (d. about 1029), contains a collection of problems in 
determinate and indeterminate analysis which not only show 
that their author had deeply studied Diopliantus but in many 
cases are taken direct from the Arithmetica, sometimes with 
a change in constants; in the fourth section of the work, 

1523.2 G g 



450 DIOPHANTUS OF ALEXANDRIA 

between problems corresponding to problems in Dioph. II 
and III, are 25 problems not found in Diophantus, but 
internal evidence, and especially the admission of irrational 
results (which are always avoided by Diophantus), exclude 
the hypothesis that we have here one of the lost Books. 
Nor is there any sign that more of the work than we possess 
was known to Abu'l Wafa al-Buzjam (A'.D. 940-98) who wrote 
a 'commentary on the algebra of Diophantus', as well as 
a ' Book of proofs of propositions used by Diophantus in his 
work'. These facts again point to the conclusion that the 
lost B6oks wore lost before the tenth century. 

The old view of the place originally occupied by the lost 
seven Books is that of Nesselmann, who argued it with great 
ability. 1 According to him (1) much less of Diophantus is 
wanting than would naturally be supposed on the basis of 
the numerical proportion of 7 lost to 6 extant Books, (2) the 
missing portion came, not at the end, but in the middle of 
the work, and indeed mostly between the first and second 
Books. Nesselmann's general argument is that, if we care- 
fully read the last four Books, from the third to the sixth, 
we shall find that Diophantus moves in a rigidly defined and 
limited circle of methods and artifices, and seems in fact to be 
at the end of his resources. As regards the possible contents 
of the lost portion on this hypothesis, Nesselmann can only 
point to (1) topics which we should expect to find treated, 
either because foreshadowed by the author himself or as 
necessary for the elucidation or completion of the whole 
subject, (2) the Porisms', under head (1) come, (a) deter- 
minate equations of the second degree, and (b) indeterminate 
equations of the first degree. Diophantus does indeed promise 
to show how to solve the general quadratic ax~ + bx c = so 
far as it has rational and positive solutions ; the suitable place 
for this would have been between Books I and II. But there 
is nothing whatever to show that indeterminate equations 
of the first degree formed part of the writer's plan. Hence 
Nesselmann is far from accounting for the contents of seven 
whole Books; and he is forced to the conjecture that the six 
Books may originally have been divided into even more than 
seven Books ; there is, however, no evidence to support this. 

1 Nesaelmann, ,AI(jebra der GHechen, pp. 264-73. 



RELATION OF WORKS 451 

Relation of tlte c Porisms ' to the Arithmetica. 

Did the Porisms form part of the Arithmetica in its original 
form ? The phrase in which they are alluded to, and which 
occurs three times, 'We have it in the Porisms that . . .' suggests 
that they were a distinct collection of propositions concerning 
the properties of certain numbers, their divisibility into a 
certain number of squares, and so on ; and it is possible that 
it was from tho same collection that Diophantus took the 
numerous other propositions which he assumes, explicitly or 
implicitly. If the collection was part of the Arithmetica, it 
would be t strange to quote the propositions under a separate 
title 'The Porisms' when it would have been more natural 
to refer to particular propositions of particular Books, and 
more natural still to say TOVTO yap TrpoStSeiKTai, or some such 
phrase, ' for this has been proved ', without any reference to 
the particular place where the proof occurred. The expression 
' We have it in the PorismH* (in the plural) would be still 
more inappropriate if the Porisms had been, as Tannery 
supposed, not collected together us one or more Books of the 
Arithmetica, but scattered about in the work as corollaries to 
particular propositions. Hence I agree with the view of 
Hultscli that the Porisms were not included in the Arith- 
metica at all, but formed a separate work. 

If this is right, we cannot any longer hold to the view of 
Nesselmann that the lost Books were in the middle and not at 
the end of the treatise ; indeed Tannery produces strong 
arguments in favour of the contrary view, that it is the last 
and most difficult Books which are lost. He replies first to 
the assumption that Diophantus could not have proceeded 
to problems more difficult than those of Book V. 'If the 
fifth or the sixth Book of the Arithmetica had been lost, who, 
pray, among us would have believed that such problems had 
ever been attempted by the Greeks ? It would be the greatest 
error, in any case in which a thing cannot clearly be proved 
to have been unknown to all the ancients, to maintain that 
it could not have been known to some Greek mathematician. 
If we do not know to what lengths Archimedes brought the 
theory of numbers (to say nothing of other things), let us 
admit our ignorance. But, between the famous problem of the 

o ir 



452 DIOPHANTUS OF ALEXANDRIA 

cattle and the most difficult of Diophantus's problems, is there 
not a sufficient gap to require seven Books to fill it? And, 
without attributing to the ancients what modern mathe- 
maticians have discovered, may not a number of the things 
attributed to the Indians and Arabs have been drawn from 
Greek sources? May not the same be said of a problem 
solved by Leonardo of Pisa, which is very similar to those of 
Diophantus but is not now to be found in the Arithmetica ? 
In fact, it may fairly be said that, when Chasles made his 
reasonably probable restitution of the Porisms of Euclid, he, 
notwithstanding that he had Pappus's lemmas to help him, 
undertook a more difficult task than he would have undertaken 
if he had attempted to fill up seven Diophantine Books with 
numerical problems which the Greeks may reasonably be 
supposed to have solved.' l 

It is not so easy to agree with Tannery's view of the relation 
of the treatise On Polygonal Numbers to the Arithmetica. 
According to him, just as Serenus's treatise on the sections 
of cones and cylinders was added to the mutilated Conies of 
Apollonius consisting of four Books only, in order to make up 
a convenient volume, so the tract on Polygonal Numbers was 
added to the remains of the Arithmetica, though forming no 
part of the larger work. 2 Thus Tannery would seem to deny 
the genuineness of the whole tract on Polygonal Numbers, 
though in his text he only signalizes the portion beginning 
with the enunciation of the problem ' Given a number, to find 
in how many ways it can be a polygonal number ' as ' a vain 
attempt by a commentator ' to solve this problem. Hultsch, 
on the other hand, thinks that we may conclude that Dio- 
phantus really solved the problem. The tract begins, like 
Book I of the Arithmetica) with definitions and preliminary 
propositions; then comes the difficult problem quoted, the 
discussion of which breaks off in our text after a few pages, 
and to these it would be easy to tack on a great variety of 
other problems. 

The name of Diophantus was used, as were the names of 
Euclid, Archimedes and Heron in their turn, for the pur- 
pose of palming oft* the compilations of much later authors. 

1 Diophantus, ed. Tannery, vol. ii, p. xx. 
/&., p. xviii. 



RELATION OF WORKS 453 

Tannery includes in his edition three fragments under the 
heading ' Diophantus Pseudepigraphus '. The first, which is 
not ' from the Arithmetic of Diophantus ' as its heading states, 
is worth notice as containing some particulars of one of * two 
methods of finding the square root of any square number'; 
we are told to begin by writing the number 'according to 
the arrangement of the Indian method', i.e. in the Indian 
numerical notation which reached us through the Arabs. The 
second fragment is the work edited by C. Henry in 1879 as 
Ojiusculum de multiplicatioiie et divisions sexagesimalibus 
Diophanto vel Pwppo attribuendum. The third, beginning 
with AiofydvTov C7n7r5o/*erpi/ca is a Byzantine compilation 
from later reproductions of the yeo>/zT/)orf/zJ>a and <rr/>o- 
HeTpovfAcva of Heron. Not one of the three fragments has 
anything to do with Diophantus. 

Commentators from Hypatia dowmvards. 

The first commentator on Diophantus of whom we hear 
is Hypatia, the daughter of Theon of Alexandria; she 
was murdered by Christian fanatics in A.D. 415. I have 
already mentioned the attractive hypothesis of Tannery that 
Hypatia's commentary extended only to our six Books, and 
that this accounts for their survival when the rest were lost 
It is possible that the remarks of Psellus (eleventh century) at 
the l>eginning of his letter about Diophantus, Anatolius and 
the Egyptian method of arithmetical reckoning were taken 
from Hypatia's commentary. 

Georgius Pachy meres (1240 to about 1310) wrote in Greek 
a paraphrase of at least a portion of Diophantus. Sections 
25-44 of this commentary relating to Book I, Dcf. 1 to Prop 
11, survive. Maximus Plamides (about 1260-1310) also wrote 
a systematic commentary on Books I, II. Arabian commen- 
tators were Abu'l Wafa al-Bfizjam (940-98), Qusta b. Liiqa 
al-Ba'labakki (d. about 912) and probably Ibn al-Huithaui 
(about 965-1039). 

Translations and editions. 

To Regiomontanus belongs the credit of being the first tc 
call attention to the work of Diophantus as being extant in 



454 DIOPHANTUS OF ALEXANDRIA 

Greek. In an Oratio delivered at the end of 1463 as an 
introduction to a course of lectures on astronomy which he 
gave at Padua in 1463-4 he observed: 'No one has yet 
translated from the Greek into Latin the fine thirteen Books 
of Diophantus, in which the very flower of the whole of 
arithmetic lies hid, the ars rei et cemus which to-day they 
call by the Arabic name of Algebra/ Again, in a letter dated 
February 5, 1464, to Bianchini, he writes that he has found at 
Venice 'Diofantus, a Greek arithmetician who has not yet 
been translated into Latin '. Rafael Bombelli was the first to 
find a manuscript in the Vatican and to conceive the idea of 
publishing the work; this was towards 1570, and, with 
Antonio Maria Pazzi, he translated five Books out of the 
seven into which the manuscript was divided. The translation 
was not published, but Bombelli took all the problems of the 
first four Books and some of those of the fifth and embodied 
them in his Algebra (1572), interspersing them with his own 
problems. 

The next writer on Diophantus was Wilhelm Holzmann, 
who called himself Xylander, and who with extraordinary 
industry and care produced a very meritorious Latin trans- 
lation with commentary (1575). Xylander was an enthusiast 
for Diophantus, and his preface and notes are often delightful 
reading. Unfortunately the book is now very rare. The 
standard edition of Diophantus till recent years was that of 
Bachet, who in 1621 published for the first time the Greek 
text with Latin translation and notes. A second edition 
(1670) was carelessly printed and is untrustworthy as regards 
the text ; on the other hand it contained the epoch-making 
notes of Fermat; the editor was S. Format, his son. The 
great blot on the work of Bachet is his attitude to Xylander, 
to whose translation he owed more than he was willing to 
avow. Unfortunately neither Bachet nor Xylander was able 
to use the best manuscripts ; that used by Bachet was Parisinus 
2379 (of the middle of the sixteenth century), with the help 
of a transcription of part of a Vatican MS. (Vat. gr. 304 of 
the sixteenth century), while Xylander's manuscript was the 
Wolfenbiittel MS. Guelferbytanus Gudianus 1 (fifteenth cen- 
tury). The best and most ancient manuscript is that of 
Madrid (Matritensis 48 of the thirteenth century) which was 



TRANSLATIONS AND, EDITIONS 455 

unfortunately spoiled by corrections made, especially in Books 
I, II, from some manuscript of the ' Planudean ' class ; where 
this is the case recourse must be had to Vat. gr. 191 which 
was copied from it befqre it had suffered the general alteration 
referred to : these are the first two of the manuscripts used by 
Tannery in his definitive edition of the Greek text (Teubner, 
1893, 1895). 

Other editors can only be shortly enumerated. In 1585 
Simon Stevin published a French version of the first four 
Books, based on Xylander. Albert Girard added the fifth and 
sixth Books, the complete edition appearing in 1625. German 
translations were brought out by Otto Schulz in 1822 and by 
G. Wertheim in 1890. Poselger translated the fragment on 
Polygonal Numbers in 1810. All these translations depended 
on the text of Bachet. 

A reproduction of Diophaiitus in modern notation with 
introduction and notes by the present writer (second edition 
1910) is based on the text of Tannery and may claim to be the 
most complete and up-to-date edition. 

My account of the Arithmetica of Diophantus will be most 
conveniently arranged under three main headings (1) the 
notation and definitions, (2) the principal methods employed, 
so far as they can be generally stated, (3) the nature of the 
contents, including the assumed Porisms, with indications of 
the devices by which the problems are solved. 



Notation and definitions. 

In his work Die Algebra der Griecheii Nesselmaim distin- 
guishes three stages in the evolution of algebra. (1) The 
first stage he calls 'Rhetorical Algebra' or reckoning by 
means of complete words. The characteristic of this stage 
is the absolute want of all symbols, the whole of the calcula- 
tion being carried on by means of complete words and forming 
in fact continuous prose. This first stage is represented by 
such writers as lamblichus, all Arabian and Persian algebraists, 
and the oldest Italian algebraists and their followers, including 
Regiomontanus. (2) The second stage Nesselmann calls the 
4 Syncopated Algebra', essentially like the first as regards 



456 DIOPHANTUS OF ALEXANDRIA 

v 

literary style, but marked by the use of certain abbreviational 
symbols for* constantly recurring quantities and operations. 
To this stage belong Diophantus and, after him, all the later 
Europeans until about the middle of the seventeenth century 
(with the exception of Vieta, who was the first to establish, 
under the name of Logistica speciosa, as distinct from Logist'ica 
numerosa, a regular system of reckoning with letters denoting 
magnitudes as well as numbers). (3) To the third stage 
Nesselmann gives the name of 'Symbolic Algebra', which 
uses a complete system of notation by signs having no visible 
connexion with the words or things which they represent, 
a complete language of symbols, which entirely supplants the 
' rhetorical ' system, it being possible to work out a solution 
without using a single word of ordinary language with the 
exception of a connecting word or two here and there used for 
clearness' sake. 



Sign for the unknown (= x), and its origin. 

Diophantus's system of notation then is merely abbrevia- 
tional. We will consider first the representation of the 
unknown quantity (our x). Diophantus defines the unknown 
quantity as c containing an indeterminate or undefined multi- 
tude of units ' (7rA?J0oy novdSoov dopurrov), adding that it is 
called aped/to?, i.e. number simply, and is denoted by a certain 
sign. This sign is then used all through the book. In the 
earliest (the Madrid) MS. the sign takes the form ij, in 
Marcianus 308 it appears as S. In the printed editions of 
Diophantus before Tannery's it was represented by the final 
sigma with an accent, $\ which is sufficiently like the second 
of the two forms. Where the symbol takes the place of 
inflected forms aptd/iop, dpi6p,ov t &c., the termination was put 
above and to the right of the sign like an exponent, e.g. $p for 
dpidjjLov as f* for rd*>, y o0 for dpiOpovl the symbol was, in 
addition, doubled in the plural cases, thus ss<, yy *, &c. The 
coefficient is expressed by putting the required Greek numeral 
immediately after it ; thus yy l la = 1 1 dpiOpoi, equivalent 
to II x, y'a = x, and so on. Tannery gives reasons for think- 
ing that in the archetype the case-endings did not appear, and 



NOTATION AND DEFINITIONS 457 

that the sign was not duplicated for the plural, although such 
duplication was the practice of the Byzantines. That the 
sign was merely an abbreviation for the word dpiOpos and no 
algebraical symbol is shown by the fact that it occurs in the 
manuscripts for aptd/*6? in the ordinary sense as well as for 
dpiOfjLos in the technical sense of the unknown quantity. Nor 
is it confined to Diophantus. It appears in more or less 
similar forms in the manuscripts of other Greek mathe- 
maticians, e.g. in the Bodleian MS. of Euclid (D'Orville 301) 
of the ninth century (in the forms 9 99, or* as a curved line 
similar to the abbreviation for /ecu), in the manuscripts of 
the Sand-reckoner of Archimedes (in a form approximat- 
ing to y), where again there is confusion caused by the 
similarity of the signs for dpiO/jiSs and tat, in a manuscript 
of the Geodaesia included in the Heronian collections edited 
by Hultsch (where it appears in various forms resembling 
sometimes , sometimes p, sometimes o, and once , with 
case-endings superposed) and in a manuscript of Theon of 
Smyrna. 

What is the origin of the sign? It is certainly not the 
final sigma, as is proved by several of the forms which it 
takes. I found that in the Bodleian manuscript of Diophantus 
it is written in the form ?c , larger than and quite unlike the 
final sigma. This form, combined with the fact that in one 
place Xylander's manuscript read ap for the full word, suggested 
to me that the sign might be a simple contraction of the first 
two letters of dpiOpos. This seemed to be confirmed by 
Gardthausen's mention of a contraction for ap, in the form vp 
occurring in a papyrus of A.D. 154, since the transition to the 
form found in the manuscripts of Diophantus might easily 
have been made through an intermediate form ^>. The loss of 
the downward stroke, or of the loop, would give a close 
approximation to the forms which we know. This hypothesis 
as to the origin of the sign has not, so far as I know, been 
improved upon. It has the immense advantage that it makes 
the sign for aptd/io? similar to the signs for the powers of 
the unknown, e.g. A Y for Swapis, K Y for /crfoy, and to the 

o 

sign M for the unit, the sole difference being that the two 
letters coalesce into one instead of being separate. 



458 DIOPHANTUS 'OF ALEXANDRIA 

ISignsfor the powers of the unknown and their reciprocals. 

The powers of the unknown, corresponding to our # 2 , x* . . . X Q , 
are defined and denoted as follows : 

x 2 is 8vva/ju$ and is denoted by A Y , 
# 3 Kvpos K Y , 



X 4 



AK , 
of* /o/jSo/cf/Jos' K K. 



Beyond the sixth power Diophantus does not go. It should 
be noted that, while the terms from KV&OS onwards may be 
used for the powers of any ordinary known number as well as 
for the powers of the unknown, Swa/us is restricted to the 
square of the unknown ; wherever a particular square number 
is spoken of, the term is rer/oayooi/oy dpiO/ios. The term 
SwapoSwaiJus occurs once in another author, namely in the 
Metrica of Heron, 1 where it is used for the fourth power of 
the side of a triangle. 

Diophantus has also terms and signs for the reciprocals of 
the various powers of the unknown, i.e. for I/ x, l/x*.... 
As an aliquot part was ordinarily denoted by the corresponding 
numeral sign with an accent, e.g. y'= ^, *a'= -rr, Diophantus 
has a mark appended to the symbols for x, x* ... to denote the 
reciprocals; this, which is used for aliquot parts as well, is 
printed by Tannery thus, *. With Diophantus then 

, denoted by s*, is equivalent to 1/ir, 



and so on. . ' 

The coefficient of the term in x, x l ... or I/ x, I/ x* ... is 
expressed by the ordinary numeral immediately following, 
e.g. AK Y *r = 26x\ A YX or = 250/a 2 . 

Diophantus does not need any signs for the operations of 
multiplication and division. Addition is indicated by mere 
juxtaposition ; thus K Y a A Y ty y e corresponds to X A + 13# 2 + 5x. 

1 Heron, Metrica, p. 48. 11, 19, Schone. 



NOTATION AND DEFINITIONS 459 

When there are units in addition, the units are indicated by 

o o 

the abbreviation M ; thus K Y aA Y ty$M corresponds to 



The sign (A) for minus and its meaning. 

For subtraction alone is a sign used. The full term for 
wanting is Aen/^9, as opposed to virapfcis, a fortlicominf/, 
which denotes a yxwitive term. The symbol used to indicate 
a ivanting, corresponding to our sign for minus, is A, which 
is described in the text as a ' ^ turned downwards and 
truncated ' (W cAAiTrey Araro) vtvov). The description is evidently 
interpolated, and it is now certain that the sign has nothing 
to do with \lr. Nor is it confined to Diophantus, for it appears 
in practically the same form in Heron's Metrical where in one 
place the reading of the manuscript is fjiovd8a>v oS T I'tf, 
74 T ^. In the manuscripts of Diophantus, when the sign 
is resolved by writing the full word instead of it, it is 
generally resolved into Xttyet, the dative of Aen/riy, but in 
other places the symbol is used instead of parts of the verb 
\tiirttv, namely AITTOJ/ or AeM/ray and once even \irr<ocri ; 
sometimes Aefyei in the manuscripts is followed by the 
accusative, which shows that in these cases the sign was 
Wrongly resolved. It is therefore a question whether Dio- 
phantus himself ever used the dative Xetyti for minus at all. 
The use is certainly foreign to classical Greek. Ptolemy has 
in two places Aetyai/ and Xtinovvav respectively followed, 
properly, by the accusative, and in one case he has TO airb 
FA Aet00j/ IJTTTO rov dirb rrjy ZF (where the meaning is 
TA*-). Hence Heron would probably have written a 
participle where the T occurs in the expression quoted above, 
say fjiovaStov 08 Acix/racrcoj/ TcrerapaKCu5eKaroj>. On the whole, 
therefore, it is probable that in Diophantus, and wherever else 
it occurred, A is a compendium for the root of the verb Ae/Tra*', 
in fact a A with I placed in the middle (cf. A, an abbreviation 
for rdXavTov). This is the hypothesis which I put forward 
in 1885, and it seems to be confirmed by the fresh evidence 
now available as shown above. 

1 Heron, Metrica, p. 156. 8, 10. 



460 DIOPHANTUS OF ALEXANDRIA 

Attached to the definition of minus is the statement that 
'a wanting (i.e, a minus) multiplied by a wanting makes 
a forthcoming (i.e. a plus) ; and a wanting (a minus) multi- 
plied by a, forthcoming (a plus) makes a wanting (a minus) '. 

Since Diophantus uses no .sign for plus, he has to put all 
the positive terms in an expression together and write all the 
negative terms together after the sign for minus', e.g. for 

# 3 5x 2 + 8x- 1 he necessarily writes K Y a s 17 A A Y M a. 

The Diophantine notation for fractions as well as for large 
numbers has been fully explained with many illustrations 
in Chapter II above. It is only necessary to add here that, 
when the numerator and denominator consist of composite 
expressions in terms of the unknown and its powers, he puts 
the numerator first followed by kv poptip or popiov and the 
denominator. 



Thus A Y M <f>x kv pop!? A Y A a M ^ A A Y 

= (60a> 2 + 2520)/(a 4 + 900-60a 2 ), [VI. 12] 

and A Y ie A M X? h /topfp A Y A a M A<r A A Y 4 |8 

[VI. 14]. 



For a terra in an algebraical expression, i.e. a power of x 
with a certain coefficient, and the term containing a certain 
number of units, Diophantus uses the word eTflos, 'species', 
which primarily means the particular power of the variable 
without the coefficient. At the end of the definitions he gives 
directions for simplifying equations until each side contains 
positive terms only, by the addition or subtraction of coeffi- 
cients, and by getting rid of the negative terms (which is done 
by adding the necessary quantities to both sides) ; the object, 
he says, is to reduce the equation until one term only is left 
on each side; 'but', he adds, 'I will show you later how, in 
the case also where two terms are left equal to one term, 
such a problem is solved '. We find in fact that, when he has 
to solve a quadratic equation, he endeavours by means of 
suitable assumptions to reduce it either to a simple equation 
or a 'jwre quadratic. The solution of the mixed quadratic 



NOTATION AND DEFINITIONS 461 

in three terms is clearly assumed in several places of the 
Arithmetica, but Diophantus never gives the necessary ex- 
planation of this case as promised in the preface. 

Before leaving the notation of Diophantus, we may observe 
that the form of it limits him to the use of one unknown at 
a time. The disadvantage is obvious. For example, where 
we can begin with any number of unknown quantities and 
gradually eliminate all but one, Diophantus has practically to 
perform his eliminations beforehand so as to express every 
quantity occurring in the problem in terms of only one 
unknown. When he handles problems which are by nature 
indeterminate and would lead in our notation to an inde- 
terminate equation containing two or three unknowns, he has 
to assume for one or other of these some particular number 
arbitrarily chosen, the effect being to make the problem 
determinate. However, in doing so, Diophantus is careful 
to say that we may for such and such a quantity put any 
number whatever, say such and such a number; there is 
therefore (as a rule) no real loss of generality. The particular 
devices by which he contrives to express all his unknowns 
in terms of one unknown are extraordinarily various and 
clever. He can, of course, use the same variable y in the 
safrne problem with different significations successively, as 
when it is necessary in the course of the problem to solve 
a subsidiary problem in order to enable him to make the 
coefficients of the different terms of expressions in x such 
as will answer his purpose and enable the original problem 
to be solved. There are, however, two cases, II. 28, 29, where 
for the proper working-out of the problem two unknowns are 
imperatively necessary. We should of course use x and y\ 
Diophantus calls the first y as usual; the second, for want 
of a term, he agrees to call in the first instance 'one unit 9 , 
i.e. 1. Then later, having completed the part of the solution 
necessary to find x, he substitutes its value and uses y over 
again for what he had originally called 1. That is, he has to 
put his finger on the place to which the 1 has passed, so as 
to substitute y for it. This is a tour deforce in the particular 
cases, and would be difficult or impossible in more complicated 
problems, 



462 DIOPHANTUS OF ALEXANDRIA 

The methods of Diophantus. 

It should be premised that Diophantus will have in his 
solutions no numbers whatever except 'rational' numbers; 
he admits fractional solutions as well as integral, but he 
excludes not only surds and imaginary quantities but also 
negative quantities. Of a negative quantity per se, i.e. with- 
out some greater positive quantity to subtract it from, he 
had apparently no conception. Such equations then as lead 
to imaginary or negative roots he regards as useless for his 
purpose; the solution is in these cases a&Ji/aroy, impossible. 
So we find him (V. 2) describing the equation 4 = 4*0 + 20 as 
LOTTOS, absurd, because it would give x = 4. He does, it is 
true, make occasional use of a quadratic which would give 
a root which is positive but a surd, but only for the purpose 
of obtaining limits to the root which are integers or numerical 
fractions ; he never uses or tries to express the actual root of 
such an equation. When therefore he arrives in the course 
of solution at an equation which would give an 'irrational' 
result, he retraces his steps, finds out how his equation has 
arisen, and how he may, by altering the previous work, 
substitute for it another which shall give a rational result. 
This gives rise in general to a subsidiary problem the solution 
of which ensures a rational result for the problem itself. 

It is difficult to give a complete account of Diophantus's 
methods without setting out the whole book, so great is the 
variety of devices and artifices employed in the different 
problems. There are, however, a few general methods which 
do admit of differentiation and description, and these we pro- 
ceed to set out under subjects. 

I. Diophantus's treatment of equations. 

(A) Determinate equations. 

Diophantus solved without difficulty determinate equations 
of the first and second degrees ; of a cubic we find only one 
example in the Arithmetica, and that is a very special case. 

(1) Pure determinate equations. 

Diophantus gives a general rule for this case without regard 
to degree. We have to take like from like on both sides of an 



DETERMINATE EQUATIONS 463 

equation ahd neutralize negative terms by adding to both 
sides, then take like from like again, until we have one term 
left equal to one term. After these operations have been 
performed, the equation (after dividing out, if both sides 
contain a power of x, by the lesser power) reduces to Ax m = JB, 
and is considered solved. Diophantus regards this as giving 
one root only, excluding any negative value as 'impossible'. 
No equation of the kind is admitted which does not give 
a ' rational ' value, integral or fractional. The value x = is 
ignored in the case where the degree of the equation is reduced 
by dividing out by any power of x. 

(2) Mixed quadratic equations. 

Diophantus never gives the explanation of the method of 
solution which he promises in the preface. That he had 
a definite method like that used in the Geometry of Heron 
is proved by clear verbal explanations in different propositions. 
As he requires the equation to be in the form of two positive 
terms being equal to one positive term, the possible forms for 
Diophantus are 



(a) mx 2 + 'px = </, (/>) m,/r = px + q, (c) mjp + q = px. 

It does not appear that Diophantus divided by in in order to 
make the first term a square ; rather he multiplied by m for 
this purpose. It is clear that he stilted the roots in the above 
capes in a form equivalent to 



m m 



The explanations which show this arc to be found in VI. 6, 
in IV. 39 and 31, and in V. 10 and VI. 22 respectively. For 
example in V. 10 he has the equation 17ar+ 17 < f2x, and he 
says ' Multiply half the coefficient of x into itself and we have 
1296; subtract the product of the coefficient of x 2 and the 
term in units, or 289. The remainder is 1007, the square root 
of which is not greater than 31. Add half the coefficient of x 
and the result is not greater than 67. Divide by the coefficient 
of a; 2 , and x is not greater than f .' In IV. 39 he has the 



464 DIOPHANTUS OF ALEXANDRIA 

equation 2x 2 > 6x+ 18 and says, 'To solve this, take the square 
of half the coefficient of x y i.e. 9, and the product of the unit- 
term and the coefficient of x 2 , i.e. 36. Adding, we have 45, 
the square root of which is not less than 7. Add half the 
coefficient of x [and divide by the coefficient of # 2 ] ; whence x 
is not less than 5.' In these cases it will be observed that 3 1 
and 7 are not accurate limits, but are the nearest integral 
limits which will serve his purpose. 

Diophantus always uses the positive sign with the radical, 
and there has been much discussion as to whether he knew 
that a quadratic equation has two roots. The evidence of the 
text is inconclusive because his only object, in every case, is to 
get one solution; in some cases the other root would be 
negative, and would therefore naturally be ignored as 'absurd' 
or ' impossible '. In yet other cases where the second root is 
possible it can be shown to be useless from Diophantus's point 
of view. For my part, I find it difficult or impossible to 
believe that Diophantus was unaware of the existence of two 
real roots 1131 such cases. It is so obvious from the geometrical 
form of solution based on Eucl. II. 5, 6 and that contained in 
Eucl. VI. 27-9; the construction of VI. 28, too, corresponds 
in fact to the negative sign before the radical in the case of the 
particular equation there solved, while a quite obvious and 
slight variation of the construction would give the solution 
corresponding to the positive sign. 

The following particular cases of quadratics occurring in 
the Arithmetics may be quoted, with the results stated by 
Diophantus. 

X* = 4o; 4 ; therefore x = 2. (IV. 22) 

i/; 2 = 3x+ 18; x = Jfa or aV (I v - 31 ) 

7o;= 7; x = -J. (VI. 6) 

a? a -7^= 7; sc = . (VI. 7) 

630ar-73;/; = 6; x = 3 \. (VI. 9) 

630# 2 + 73x = 6 ; # is rational. (VI. 8) 

5x < a; 2 r-60 < 8x; x not < 11 and not > 12. (V. 30) 
17a 2 +17 < 72#<19# 2 +19; x not >fandnot <f|. (V. 10) 
22x < x* + 6Q < 24#; x not < 19 but < 21. (V. 30) 



DETERMINATE EQUATIONS 465 

In the first and third of the last three cases the limits are not 
accurate, but are integral limits which are a fortiori safe. 
In the second f f should have been f f , and it would have been 
more correct to say that, if x is not greater than f % and not 
less than y|, the given conditions are a fortiori satisfied. 

For comparison with Diophantus's solutions of quadratic 
equations we may refer to a few of his solutions of 

(3) Simultaneous equations involving quadratics. 

In I. 27, 28, and 30 we have the following pairs of equations. 
(a) + 7; = 2a (ft) + , = 2</ (y) -17 = 2 a} 



I use the Greek letters for the numbers required to be found 
as distinct from the one unknown which Diopliantus uses, and 
which I shall call ,/. 

In (a), he says, lut -?; = 2.r ( > 77). 

It follows, by addition and subtraction, that = a + x, 
rj = r*,-;*;; 

therefore ?; = (a + *i) (a .*') = a 2 ./* 2 = ^, 



and & is found from the pure quadratic equation. 

In (/3) similarly he assumes g rj = 2x, and the resulting 

equation is f 2 -f t)~ = (a + ,<)- -f (a - #) 2 = 2 (a- + ur) = B. 
In (y) he puts f + ?; = 2^' and solves as in the case of (a). 



(4) C'tt&^J equation. 

Only one very particular case occurs, hi VI. 17 the problem 
leads to the equation 



Diophantus says simply ' whence x is found to bo 4 '. In fact 
the equation reduces to ^ 



x* + X = 4# 2 + 4. 

Diophantus no doubt detected, and divided out by, the common 
factor x 2 + 1, leaving x = 4. 



466 DIOPHANTUS OF ALEXANDRIA 

(B) Indeterminate equations. 

Diophantns says nothing of indeterminate equations of the 
first degree. The reason is perhaps that it is a principle with 
him to admit rational fractional as well as integral solutions, 
whereas the whole point of indeterminate equations of the 
first degree is to obtain a solution in integral numbers. 
Without this limitation (foreign to Diophantus) such equa- 
tions have no significance. 

(a) Indeterminate equations of tJie second deyree. 

The form in which these equations occur is invariably this : 
one or two (but never more) functions of x of the form 
Ax* + Bx + C or simpler forms are to be made rational square 
numbers by finding a suitable value for x. That is, we have 
to solve, in the most general case, one or two equations of the 
form Ax* + Bx + C = y 2 . 

(1) Single equation. 

The solutions take different forms according to the particular 
values of the coefficients. Special cases arise when one or 
more of them vanish or they satisfy certain conditions. 

1. When A or C or both vanish, the equation can always 
be solved rationally. 

Form Bx = y z . 
Form Bx + C = y 2 . 

Diophantus puts for y 2 any determinate square m 2 , and x is 
immediately found. 

Form Ax* + Bx y 2 . 

Diophantus puts for y any multiple of x, as x. 

2. The equation Ax 2 + C = y 2 can be rationally solved accord- 
ing to Diophantus : 

(a) when A is positive and a square, say a 2 ; 
in this case we put a 2 x 2 + C = (ax m) 2 , whence 

tf-m 2 

x = + 

- 2ma 

(m and the sign being so chosen as to give x a positive value) ; 



INDETERMINATE EQUATIONS 467 

()8) when C is positive and a square, say c 2 ; 
in this case Diophantus puts Ax 2 + c 2 = (w# + c) 2 , and obtains 



2 m c 

x= 4- 



. r/r 



(y) When one solution is known, any number of other 
solutions can be found. This is stated in the Lemma to 
VI. 15. It would be true not only of the cases + Ax 2 + C = y 2 , 
but of the general case Ax 2 + Bx + C = y\ Diophantus, how- 
ever, only states it of the case Ax* C = y 2 . 

His method of finding other (greater) values of x satisfy- 
ing the equation when one (x ) is known is as follows. If 
A x Q 2 C = q~, he substitutes in the original equation (X Q 
for x and (qkx) for y, where k is some integer. 

Then, since A (a* + x) 2 C = (7 ,)*, while 4aj 2 C = 
it follows by subtraction that 



whence 



and the new value of x is ;c + 

1C" ./l 

Form Ax 2 c 2 = ?/ 2 . 

Diophantus says (VI. 14) that a rational solution of this 
case is only possible when A is the sum of two squares. 

[In fact, if x = p/q satisfies the equation, and Ax 2 c 2 = /c 2 , 

we have Ap 

or A = 

Form Ax 2 + C = y 2 . 

Diophantus proves in the Lemma to VI. 12 that this equa- 
tion has an infinite number of solutions when A + C is a square, 
i.e. in the particular case where x = 1 is a solution. (He does 
not, however, always bear this in mind, for in III. 10 he 
regards the equation 52x 2 +l2 = y* as impossible though 
52 + 12 = 64 is a square, just as, in III. 11, 266# 2 -10 = y 2 
is regarded as impossible.) 

Suppose that A + C = q 2 ; the equation is then solved by 



468 DIOPHANTUS OF ALEXANDRIA 

substituting in the original equation 1 4- x for x and (q kx) 
for y, where k is some integer. 

3. Form Ax* + Ex +0 = y*. 
This can be reduced to the form in which the second term is 

wanting by replacing x by z - . 

Diophantus, however, treats this case separately and less 
fully. According to him, a rational solution of the equation 
Ax* + x + C=y* is only possible 

(a) when A is positive and a square, say a 2 ; 
() when G is positive and a square, say c 2 ; 
(y) when ^ J9 2 AC is positive and a square. 

In case (a) y is put equal to (ax m), and in case (ft) y is put 
equal to (mx c). 

Case (y) is not expressly enunciated, but occurs, as it 
were, accidentally (IV. 31). The equation to be solved is 
3 x + 1 8 x* = y 2 . Diophantus first assumes 3 s + 1 8 ./: 2 = 4 ,//-, 
which gives the quadratic 3&+ 18 = 5jj 2 ; but this 'is not 
rational '. Therefore the assumption of 4 x 2 for y 2 will not do, 
c aud we must find a square [to replace 4] such that 1 8 times 
(this square 4- 1) + (|) 2 may be a square'. The auxiliary 
equation is therefore 18(m' 2 + l) + f = ?/ 2 , or 7 2m 2 + 81= a 
square, and Diophantus assumes 72 ?// 2 + 8 1 = (8 m + 9) 2 , whence 
m= 18. Then, assuming 3;/;+ 18 x 1 = (!8) 2 ,/; 2 ,he obtains the 
equation 32 5 # 2 3i/: 18 = 0, whence x = 3 7 2 \, that is, 2 V 

(2) Double equation,. 

The Greek term is &7rAoar6r??y, 8i7rXfj lo-orr}? or SnrXfj fraxrtr. 
Two different functions of the unknown have to be made 
simultaneously squares. The general case is to solve in 
rational numbers the equations 

mrc 2 + a x + a = u 2 
b = 



The necessary preliminary condition is that each of the two 
expressions can be made a square. This is always possible 
when the first term (in x z ) is wanting. We take this simplest 
case first. 



INDETERMINATE EQUATIONS 469 



1. Double equation of the first degree. 
The equations arc 



a x -f a = u 2 , 



Diophantus lias one general method taking slightly different 
forms according to the nature of the coefficients. 

(a) First method of solution. 
This depends upon the identity 



1C the difference between the two expressions in x can he 
separated into two factors ^>, <?, the expressions themselves 
are equated to {i(^ + Q f )J 2 an( l { ^G> 7) }~ respectively. As 
Diophantus himself says in II. 1 1, we ' equate either the square 
of half the difference of the two factors to the lesser of the 
expressions, or the square of half the sum to the greater 5 . 

We will consider the general case and investigate to what 
particular classes of cases the method is applicable from 
Diophantus's point of view, remembering that the final quad- 
ratic in x must always reduce to a single equation. 

Subtracting, we have (a j8) # + ((* b) = M- ?/-. 
Separate (a P)x -f (a b) into the factors 



We write accordingly 

w = (*- 

P 

u + iv = p. 

m o . l(ot-P)x + (ab) )- 

Thus u- = ;> + (/ = J- j- - - - +^ji ; 

therefore ( (a - j8) . + a - ft + p* } 2 = 4 p 2 (a .c + a). 
This reduces to * 



+ (a - b)* - 2 v 2 (a + b) + r>* = 0. 



470 DIOPHANTUS OF ALEXANDRIA 

In order that this equation may reduce to a simple equation, 
either 

(1) the coefficient of x* must' vanish, or a j8 = 0, 
or (2) the absolute term must vanish, that is, 
p*-2p 2 (a + b) + (a-6) 2 = 0, 
or {p 2 -(a + ft)} 2 = 4ci&, 

so that ab must be a square number. 

As regards condition (1) we observe that it is really sufficient 
if an 2 = )8m 2 , since, if a x + a is a square, (a x + a) ?i 2 is equally 
a square, and, if ftx + b is a square, so is (/8# + 6)m 2 , and 
vice versa. 

That is, (1) we can solve any pair of equations of the form 



/v /YtTt * i' I ft if * \ 
LA i/v w |^ vt/ iv I 

a/i' 2 a; + 6= w-J 



Multiply by ri 2 , m~ respectively, and we have to solve tilt- 
equations 



a m 2 



Separate the difference, cm 2 6m 2 , into two factors^;, 5 and 
put 



therefore u /2 = 

and a m 2 7i 2 ^ + an 2 = i ( ^ -f g) 2 , 



and from either of these equations we get 

- (a^ 2 + 6m 2 ) 



,^ = 



since 2><? = ^ 2 6m 2 . 

Any factors p, q can be chosen provided that the resulting 
value of x is positive. 



INDETERMINATE EQUATIONS 471 

Ex. from Diophantus : 

65- 60:= r % 

- 32 ) 

' 



therefore 260 - 24 x = u /2 1 

G5-24#=iC' /2 j ' 

The difference = 195 = 15.13, Sciy ; 
therefore (15 1 3)* = 05 - 24 # ; that is, 24# = 64, and x = . 

Taking now the condition (2) that ab is a square, we see 
that the equations can be solved in the cases where either 
a and b are both squares, or the ratio of a to 6 is the ratio of 
a square to a square. If the equations are 

OLX+ (T = U 2 , 



and factors are taken of the difference between the expressions 
as they stand, then, since one factor p, as we saw, satisfies the 

equation [ /> 2 (<? + d 2 ) ] 2 = 4 <?<l*, 

we must have p = c d. 

Ex. from Diophantus: 

10# + 9 = u 2 | 

5x+ 4 = f iv*\ 

The difference is 5o; + 5 = 5(a;+l); the solution is given by 
(^#+3) 2 = 10# + 9, and x = 28. 

Another method is to multiply the equations by squares 
such that, when the expressions are subtracted, the absolute 
term vanishes. The case can be worked out generally, thus. 

Multiply by d 2 and c 2 respectively, and we have to solve 



Difference = (ad 2 |8c 2 )# = px.q say. 
Then x is found from the equation 



which gives p*x 2 + 2x(i)q 2ad 2 ) + ? 2 -4r 2 d 2 = 0, 



472 DIOPHANTUS OF ALEXANDRIA 

or, since pq = ad 2 )8c 2 , 

p*x*-2x(ocd* + pc*) + q*-*c*d 2 = 0. 

In order that this may reduce to a simple equation, as 
Diophantus requires, the absolute term must vanish, so that 
q = 2cd. The method therefore only gives one solution, since 
q is restricted to the value 2 cd. 

Ex. from Diophantus : 



(IV. 39) 

6a + 4=" 

Difference 2o?; q necessarily taken to be 2\/4 or 4; factors 
therefore \x, 4. Therefore 8# + 4 = (|x + 4) 2 , and a; =112. 

(j9) Second method of solution of a double equation of the 
first degree. 

There is only one case of this in Diophantus, the equations 
being of the form 

Jix + n 2 = u 1 } 



Suppose hx + ?i 2 = (y + n} 1 ; therefore Jut = y l -f 2 wy, 
and (A +/ )o; + n 2 = (y + n) 2 + t ( y * 



It only remains to make the latter expression a square, 
which is done by equating it to (py n)*. 

The case in Diophantus is the same as that last mentioned 
(IV. 39). Where I have used y, Diophantus as usual contrives 
to use his one unknown a second time. 

2. Double equations of the second degree. 
The general form is 

L +Bx +0 = 



but only three types appear in Diophantus, namely 

p 2 /#2 + (xx + a = u 2 } 

(1) - 7 K where, except in one case, a = b. 
/rar + pas-f 6 = w ) 



INDETERMINATE EQUATIONS 473 

, * 2 + a + a = t ~ 



(The case where the absolute terms are in the ratio of a square 
to a square reduces to this.) 

In all examples of these cases the usual method of solution 
applies. 



The usual method does not lierc serve, and a special artifice 
is required. 

Diophantus assumes u 2 = m 2 # 2 . 

Then x = a/(m 2 a) and, by substitution in the second 
equation, we have 

)S (---,- ) + - ., 3 which must l)e made a square, 
Vw,- a/ m j a A 

or a- (3 + Ixifyn? 01) must be a square. 

Wu have therefore to solve the equation 

(a/3 <xb) = 2T, 



whicli can or cannot be solved by Diophantus's methods 
according to the nature of the coefficients. Thus it can be 
solved if (a{i (xb)/a is a square, or if a/b is a square. 
Examples in VI. 12. 14. 

(/>) Indeterminate equations of a degree hiyher than the 

second. 
(1) Single equations. 

There are two classes, namely those in which expressions 
in x have to be made squares or cubes respectively. The 
general form is therefore 

Ajp + Bjp- 1 4- ... + Kx-L = #~ or y*. 

In Diophantus n does not exceed 6, and in the second class 
of cases, where the expression has to be made a cube, n does 
not generally exceed 3. 



474 DIOPHANTUS OF ALEXANDRIA 

The species of the first class found in the Arithmetica are 
as follows. 

1 . Equation Ax* + Bx 2 + Ox + c/ 2 = if. 
As the absolute term is a square, we can assume 



or we might assume y = m 2 x* + nx + d and determine m, n so 
that the coefficients of x, x* in the resulting equation both 
vanish. 

Diophantus has only one case, & 3 3x 2 + 3x + I = y 2 (VI. 18), 
and uses the first method. 

2. Equation Ax* + Bx 3 + Cx* + Dx + E = y\ where either A or 
E is a square. 

If A is a square (= a 2 ), we may assume y = ax* + - x + n, 

&(.(> 

determining n so that the term in x 2 in the resulting equa- 
tion may vanish. If E is a square (== e 2 ), we may assume 

y = mx 2 + - x + e, determining m so that the term in x 1 in the 
A 6 

resulting equation may vanish. We shall then, in either case, 
obtain a simple equation in x. 

3. Equation Ax* + Cx 2 + E = 7/ 2 , but in special cases only where 
all the coefficients are squares. 

4. Equation Ax* + E = 2/ 2 . 

The case occurring in Diophantus is # 4 + 97 = y* (V. 29). 
Diophantus tries one assumption, y = x 2 1 0, and finds that 
this gives a? 2 = -jfc , which leads to no rational result. He 
therefore goes back and alters his assumptions so that he 
is able to replace the refractory equation by x* + 337 = y\ 
and at the same time to find a suitable value for y, namely 
y = # 2 26, which produces a rational result, x = \ 2 -. 

5. Equation of sixth degree in the special form 



Putting y = x 3 + c, we have Ax 2 + B = 2c# 2 , and 

i> 

, which gives a rational solution if - A - is 

5 A + 2c 



INDETERMINATE EQUATIONS 475 

a square. Where this does not hold (in IV. 18) Diophantus 
harks back and replaces the equation a? 16# :) + oj + 64 = y/ 2 
by another, &- 128# 3 + x + 4096 = y\ 

Of expressions which have to be made cubes, we have the 
following cases. 



There are only two cases of this. First, in VI. 1, cr, 2 4x + 4 
has to be made a cube, being already a square. Diophantus 
naturally makes # 2 a cube. 

Secondly, a peculiar case occurs in VI. 17, where a cube has 
to be found exceeding a square by 2. Diophantus assumes 
(a? I) 3 for the cube and (x + I) 2 for the square. This gives 

a; 3 - 3 a; 2 + 3,0 - 1 = ;r 2 + 2x + 3, 

or ,*:' + ,*: = 4 2 + 4. We divide out by # 2 + 1, and x = 4. It 
seems evident that the assumptions were made with knowledge 
and intention. That is, Diophantus knew 'of the solution 27 
and 25 and deliberately led up to it. It is unlikely that he was 
aware of the fact, observed by Fermat, that 27 and 25 are the 
only integral numbers satisfying the condition. 

2. Ax* + Bx 2 + Cx + D = y/ 3 , where either A or D is a cube 
number, or both are cube numbers. Where A is a cube (a 3 ), 

we have only to assume y = cc+ ^, and where D is a cube 

3d 

(7 
(<7, 3 ), y= - x + d. Where A = a 3 and D = e2 3 , we can use 

O (v 

either assumption, or put y ax + d. Apparently Diophantus 
used the last assumption only in this case, for in IV. 27 he 
rejects as impossible the equation 8& 3 # 2 + 8cc 1 = y*, 
because the assumption y = 2#l gives a negative value 
x = T \, whereas either of the above assumptions gives 
a rational value. 

(2) Double equations. 

Here one expression has to be made a square and another 
a cube. The cases are mostly very simple, e.g. (VI. 19) 



thus y 3 = 2z\ and z = 2. 



476 DIOPHANTUS OF ALEXANDRIA 

More complicated is the case in VI. 21 : 



Diophantus assumes y = mx, whence x = 2/(m 2 2), and 
2 \ 3 _/ 2 \ a 2 



/ 2 \ 3 / 2 \ 

I __ 14-91 ___ 1 

W- 2/ ^ Vm 2 - 2/ 



m 2 - 2 "" " ' 

2m 4 

or - = 2 

(m*-2) 3 

We have only to make 2m 4 , or 2m, a cube. 

II. Method of Limits. 

As Diophantus often has to find a series of numbers in 
order of magnitude, and as he does not admit negative 
solutions, it is often necessary for him to reject a solution 
found in the usual course because it does not satisfy the 
necessary conditions; he is then obliged, in many cases, to 
find solutions lying within certain limits in place of those 
rejected. For example : 

1 . It is required to find a value of x such that some power of 
it, x n , shall lie between two given numbers, say a and b. 

Diophantus multiplies both a and b by 2 W , 3 W , and so on, 
successively, until some 7^th power is seen which lies between 
the two products. Suppose that c n lies between ap u and bp u ; 
then we can put x = c/p, for (c/p) n lies between a and b. 

Ex. To find a square between l and 2. Diophantus 
multiplies by a square 64; this gives 80 and 128, between 
which lies 100. Therefore (^) 2 or f| solves the problem 
(IV. 31 (2)). 

To find a sixth power between 8 and 16. The sixth powers 
of 1, 2, 3, 4 are 1, 64, 729, 4096. Multiply 8 and 16 by 64 
and we have 512 and 1024, between which 729 lies; - 7 ^ 9 is 
therefore a solution (VI. 21). 

2. Sometimes a value of x has to be found which will give 



METHOD OF LIMITS 477 

some function of x a value intermediate between the values 
of two other functions of x. 

Ex. 1. In IV. 25 a value of x is required such that 8/(# 2 + x) 
shall lie between x and x+ 1. 



One part of the condition gives 8 > x* + x 2 . Diophantus 
accordingly assumes 8 = (# + ) :i = X' } + x 2 + ^x + -f, which is 
> a? 3 + x 2 . Thus x + = 2 or 05 = f satisfies one part of 
the condition. Incidentally it satisfies the other, namely 
8/(x 2 + x) < x+l. This is a piece of luck, and Diophantus 
is satisfied with it, saying nothing more. 

Ex. 2. We have seen how Diophantus concludes that, if 



then x is not less than 11 and not greater than 12 (V. 30). 

The problem further requires that x 2 60 shall be a square. 
Assuming # 2 60 = (as m) 2 , we find x = (m 2 + 60)/2m. 
Since x > 11 and < 12, says Diophantus, it follows that 

24m > m 2 + 60 > 22 TO ; 

from which lie concludes that m lies between 19 and 21. 
Putting m = 20, he finds x = 11^. 

III. Method of approximation to Limits. 

Here we have a very distinctive method called by Diophantus 
TTapi<roTr]$ or Tra/Ho-orrjToy dya>yrj. The object is to solve such 
problems as that of finding two or three square numbers the 
sum of which is a given number, while each of them either 
approximates to one and the same number, or is subject to 
limits which may be the same or different. 

Two examples will best show the method. 

Ex. 1. Divide 13 into two squares each of which > 6 (V. 9). 

Take half of 13, i.e. 6^, and find what small fraction 1 /x 1 
added to it will give a square ; 

thus 6 1 H 55 or 26 + ^ > must be a square. 

x~ y 



478 DIOPHANTUS OF ALEXANDRIA 

Diophantus assumes 



whence 

T/ = 10, and 1/7/ = T fo f i.e. 



[The assumption of 5 + as the side is not haphazard : 5 is 

chosen because it is the most suitable as giving the largest 
rational value for yJ] 

We have now, says Diophantus, to divide 13 into two 
squares each of which is as nearly as possible equal to (f ) 2 . 

Now 13 =r 3 2 -f 2 2 [it is necessary that the original number 
shall be capable of being expressed as the sum of two squares] ; 

and 3 >fby^, 

while 2 < f by J. 

But if we took 3-^, 2+-| as the sides of two squares, 
their sum would be 2(f J) 2 = - 5 ,%% 2 -, which is > 13. 

Accordingly we assume 3 9#, 2 + 11 x as the sides of the 
required squares (so that x is not exactly ^ but near it). 

Thus (3-9#) 2 + (2 + llx) 2 = 13, 

and we find x = T T . 

The sides of the required squares arc f -, f ^f . 

Ex. 2. Divide 10 into three squares each of which > 3 
(V.ll). 

[The original number, here 1 0, must of course be expressible 
as the sum of three squares.] 

Take one-third of 10, i.e. 3^, and find what small fraction 
1/aj 2 added to it will make a square; i.e. we have to make 

19 1 

3 3+ - a square, i.e. 30-h 73- must be a square, or 30+ ^ 
xx y 

= a square, where 3 /x = 1 /y. 
Diophantus assumes 



the coefficient of y> i.e. 5, being so chosen as to make 1 /y as 
small as possible ; 



METHOD OF APPROXIMATION TO LIMITS 479 



therefore y = 2, and 1 /x* = ^; and 3^ + sV = -J^ 1 -, a square. 

We have now, says Diophantus, to divide 10 into three 
squares with sides as near as may be to - 1 /. 

Now 10 = 9 + 1 = 3 2 + (|-) 2 + (f) 2 . 

Bringing 3, f , f and ~y- to a common denominator, we have 

90 18 24 orul 55 

tf* "Sis ana So > 



and 3 >|gby, 

3 ^ 55 lv 7 37 
T < 3^ b y 50 > 

fs 55 Vvr r 31 
< w y 3o- 

If now we took 3 f , f + f , + f ^ as the sides of squares, 



the sum of .the squares would be 3 (V") 2 or ^V~> which is > 10. 
Accordingly we assume as the sides 3 35 x, f + 37 x, f + 3 1 x, 
where x must therefore be not exactly ^ ^ u ^ near ^. 

Solving (3-35j-) a + (| + 37ar) 2 + (| + Slaj) s = 10, 
or 10-116^-f 3555./; 2 = 10, 

we find x = ^5% 5 
thus the sides of the required squares are VrVs -VW"> VT^! 

thf* QniiaroQ fliarnaoKr^e nva 1745041 1651225 1658944 

tne squares tiiemseives aie -^^23;-, T^"S^^T" "^O^F^T"' 

Other instances of the application of the method will be 
found in V. 10, 12, 13, 14. 

Porisms and propositions in the Theory of Numbers. 

I. Three propositions are quoted as occurring in the Porisms 
(' We have it in the Porisms that ...'); and some other pro- 
positions assumed without proof may very likely have come 
from the same collection. The three propositions from the 
Porisms are to the following effect. 

1 . If a is a given number and x, y numbers such that 
x + a = m 2 , y 4- a = ?i 2 , then, if xy 4- a is also a square, m and n 
differ by unity (V. 3). 

[From the first two equations we obtain easily 

xy + a = w 2 7i 2 - a (m 2 + n 2 1) + a 2 , 

and this is obviously a square if m 2 + ?i 2 1 = 2mn, or 
m n = 1.] 



480 DIOPHANTUS OF ALEXANDRIA 

2. If m 2 , (m + I) 2 be consecutive squares and a third number 
be taken equal to 2{w 2 + (m + I) 2 } + 2, or 4(m 2 + m+ 1), the 
three numbers have the property that the product of any two 
plus either the sum of those two or the remaining number 
gives a square (V. 5). 

[In fact, if X, Y, Z denote the numbers respectively, 



XY+% = ( 

3) 2 , YZ+X = (2m 2 + 3m+2) 2 , 
, ZX + Y = (2m a + m+ I) 2 .] 

3. The difference of any two cubes is also the sum of two 
cubes, i.e. can be transformed into the sum of two cubes 
(V. 16). 

[Diophantus merely states this without proving it or show- 
ing how to make the transformation. The subject of the 
transformation of sums and differences of cubes was investi- 
gated by Vieta, Bachet and Fermat.] 

II. Of the many other propositions assumed or implied by 
Diophantus which are not referred to the Porisms we may 
distinguish two classes. 

1. The first class are of two sorts; some are more or less 
of the nature of identical formulae, e.g. the facts that the 
expressions {^(a + ft)} 2 a6 and a 2 (a-h l) 2 + a 2 -f (&+ I) 2 are 
respectively squares, that a (a 2 a) + a + (a 2 a) is always a 
cube, and that 8 times a triangular number plus 1 gives 
a square, i.e. 8.%x(x+l) + 1 = (2x+ I) 2 . Others are of the 
same kind as the first two propositions quoted from the 
Porisms, e.g. 



(1) If Z=a 2 # + 2tt, 7=(c6+l) 2 0+2(a+l) or, in other 
words, if xX+ 1 = (ax+ I) 2 and xY+ 1 = {(a+ !)#+!)*, 
then XY+ 1 is a square (IV. 20). In fact 



X7+1 = [a 

(2) If JTa = m 2 , Ya= (m+l) 2 ,and Z = 
then YZa, ZXa, XYa are all squares (V. 3, 4). 



PORISMS AND PROPOSITIONS ASSUMED 4&1 

In fact YZa = {(m + 1) (2m + l) + 2a} 2 , 
ZX+a, = {m(2m+ l) + 2a} 2 , 



(3) If 

X = m* + 2, F 
then the six expressions 



YZ-X, ZX-Y, XY-Z 



are all squares (V. 6). 
In fact 



2 , &c. 

2. The second class is much more important, consisting of 
propositions in the Theory of Numbers which we find first 
stated or assumed in the Arithmetica. It was in explana- 
tion or extension of these that Fermat's most famous notes 
were written. How far Diophantus possessed scientific proofs 
of the theorems which he assumes must remain largely a 
matter of speculation. 



(a) Theorems on the composition of numbers as the 
of two squares. 

(1) Any square number can be resolved into two squares in 
any number of ways (II. 8). 

(2) Any number which is the sum of two squares can be 
resolved into two other squares in any number of ways (II. 9). 

(It is implied throughout that the squares may be fractional 
as well as integral.) 

(3) If there are two whole numbers each of which is the 
sum of two squares, the product of the numbers can be 
resolved into the sum of two squares in two ways. 

In fact (a 2 + t 2 ) (c 2 + d 2 ) == (ac bdf + (ad T 6c) 2 . 

This proposition is used in III. 19, where the problem is 
to find four rational right-angled triangles with the same 



482 DIOPHANTUS OF ALEXANDRIA 

hypotenuse. The method is this. Form two right-angled 
triangles from (a, b) and (c, d) respectively, by which Dio- 
phantus means, form the right-angled triangles 

(a 2 + i 2 , a 2 -& 2 , 2ab) and (c 2 + cZ 2 , c a -d 2 , 2cd). 

Multiply all the sides in each triangle by the hypotenuse of 
the other; we have then two rational right-angled triangles 
with the same hypotenuse (a 2 + ft 2 ) (c 2 + cZ 2 ). 

Two others are furnished by the formula above; for we 
have only to ' form two right-angled triangles ' from (ac + Id, 
ad be) and from (ac bd, ad + bc) respectively. The method 
fails if certain relations hold between a, 6, c, d. They must 
not be such that one number of either pair vanishes, i.e. such 
that ad = be or ac = bd, or such that the numbers in either 
pair are equal to one another, for then the triangles are 
illusory. 

In the case taken by Diophantus a' 2 + 6 2 = 2 2 + 1 2 = 5, 
c 2 + d 2 = 3 2 + 2 2 = 1 3, and the four right-angled triangles are 

(65, 52, 39), (65, 60, 25), (65, 63, 16) and (65, 56, 33). 

On this proposition Format has a long and interesting note 
as to the number of ways in which a prime number of the 
form 4n+l and its powers can be (a) the hypotenuse of 
a rational right-angled triangle, (6) the sum of two squares. 
He also extends theorem (3) above : ' If a prime number which 
is the sum of two squares be multiplied by another prime 
number which is also the sum of two squares, the product 
will be the sum of two squares in two ways ; if the first prime 
be multiplied by the square of the second, the product will be 
the sum of two squares in three ways ; the product of the first 
and the cube of the second will be the sum of two squares 
in four ways, and so on ad iufinitum. 9 

Although the hypotenuses selected by Diophaiitus, 5 and 1 3, 
are prime numbers of the form 4?i+ 1, it is unlikely that he 
was aware that prime numbers of the form 4 n + 1 and 
numbers arising from the multiplication of such numbers are 
the only clesses of numbers which are always the sum of two 
squares ; this was first proved by Euler. 

(4) More remarkable is a condition of possibility of solution 
prefixed to V. 9, 'To divide 1 into two parts such that, if 



NUMBERS AS THE SUMS OF SQUARES 483 

a given number is added to either part, the result will be a 
square/ The condition is in two parts. There is no doubt as 
to the first, 'The given number must not be odd' [i.e. no 
number of the form 4n + 3 or 4^1 can be the sum of two 
squares] ; the text of the second part is corrupt, but the words 
actually found in the text make it quite likely that corrections 
made by Hankel and Tannery give the real meaning of the 
original, ' nor must the double of the given number plus 1 be 
measured by any prime number which is less by 1 than a 
multiple of 4 '. This is tolerably near the true condition 
stated by Fermat, ' The given number must not be odd, and 
the double of it increased by 1 , when divided by the greatest 
square which measures it, must not be divisible by a prime 
number of the form 4 n I/ 

(/8) On numbers which are tJie sum of three squares. 

In V. 11 the number 3ti + l has to be divisible into three 
squares. Diophantus says that a 'must not be 2 or any 
multiple of 8 increased by 2 '. That is, ' a number of the 
form 24n+7 cannot be the sum of three squares '. As a matter 
of fact, the factor 3 in the 24 is irrelevant here, and Diophantus 
might have said that a number of the form 8 n + 1 cannot be 
the sum of three squares. The latter condition is true, but 
does not include all the numbers which cannot be the sum of 
three squares. Fermat gives' the conditions to which a must l>e 
subject, proving that 3 a + 1 cannot be of the form 4" (24 k + 7) 
or 4 w (8fc+ 7), where k = or any integer. 

(y) Composition of numbers as the sum of four squares. 

There arc three problems, IV. 29, 30 and V. 14, in which it 
is required to divide a number into four squares. Diophantus 
states no necessary condition in this case, as he does when 
it is a question of dividing a number into three or luv squares. 
Now every number is eitfier a square or the sum of two, three 
or four squares (a theorem enunciated by Fermat and proved 
by Lagrange who followed up results obtained by Euler), and 
this shows that any number can be divided into four squares 
(admitting fractional as well as integral. squares), since any 
square number can be divided into two other squares, integral 



484 DIOPHANTUS OF ALEXANDRIA 

or fractional. It is possible, therefore, that Diophantus was 
empirically aware of the truth of the theorem of Fermat, but 
we cannot be sure of this. 



Conspectus of the Aritlimetica, with typical solutions. 

There seems to be no means of conveying an idea of the 
extent of the problems solved by Diophantus except by giving 
a conspectus of the whole of the six Books. Fortunately this 
can be done by the help of modern notation without occupying 
too many pages. 

It will be best to classify the propositions according to their 
character rather than to give them in Diophantus's order. It 
should be premised that x, y, z . . . indicating the first, second 
and third . . . numbers required do not mean that Diophantus 
indicates, any of them by his unknown (?) ; he gives his un- 
known in each case the signification which is most convenient, 
his object being to express all his required numbers at once in 
terms of the one unknown (where possible), thereby avoiding the 
necessity for eliminations. Where I have occasion to specify 
Diophantus's unknown, I shall as a rule call it , except when 
a problem includes a subsidiary problem and it is convenient 
to use different letters for the unknown in the original and 
subsidiary problems respectively, in order to mark clearly the 
distinction between them. When in the equations expressions 
are said to be = u 2 , */ 2 , ic; 2 , t* ... this means simply that they 
are to be made squares. Given numbers will be indicated by 
a, 6, c . . . m, -ii . . . and will take the place of the numbers used 
by Diophantus, which are always specific numbers. 

Where the solutions, or particular devices employed, arc 
specially ingenious or interesting, the methods of solution will 
be shortly indicated. The character of the book will bo best 
appreciated by means of such illustrations. 

[The problems marked with an asterisk are probably 
spurious.] 

(i) Equations of the first degree with one unknown. 
I. 7. # a = m(x b). 
I. 8. x + a = m(x + b). 



DETERMINATE EQUATIONS 485 

I. 9. a x = m (&&). 
I. 10. x + b = m(a #). 
I. 11. x + b = m(# a). 

I. 39. (a + x)b + (h + x}a = 2(a + &)#, \ 

or (a + 6) a; + (6 + #)a = 2 (a + x) 6, L (a > 6) 
or (a + 6)x+(a + a;)6 = 2(6 + #)a. ) 

Diophantus states this problem in this form, 'Given 
two numbers (a, 6), to find a third number (x) such that 
the numbers 

(a + x)b, (b + x)a, (a + b)x 

are in arithmetical progression/ 

The result is of course different according to the order 
of magnitude of the three expressions. If a>b (5 and 3 
are the numbers in Diophantus), then (a + x)b < (b + x)a\ 
there are consequently three alternatives, since (a + x)b 
must be either the least or the middle, and (b + x)a either 
the middle or the greatest of the three products. We may 
have 

(( / -f x) b < (d + b)x < (b + ,r) a, 

or (a + b) x < (a + x) b < (b + x) a, 
or (a + x) b < (b -f x)a < (a -f b)x, 
and the corresponding equations are as set out above. 



(ii) Determinate systems of equations of the first degree. 
I. 1. x + y = a, x y = b. 
I. 2. aj-f y = a, x = my, 
I. 4. ^ y = a, x = my. 
1. 3. # + 2/ = a, x = my + 6. 



I. 5. a; + y = a, # + - y = &, subject to necessary condition. 
* m n 

1 1 . 
I. 6. oj + y = a, x -- 7/=6, 



486 DIOPHANTUS OF ALEXANDRIA 

(I. 12. x l + x%i 
. 13. Xi + x^ i 



I. 15. # + a = m(y a), y + b = n(# 6). 

[Diophantus puts y = + a, where is his unknown.] 
I. 16. y + = a, s + # = 6, # + y = r. [Dioph. puts =a: + y + 3.] 
I. 17. y + z + tv = a, 2 + 10 -f # = 6, w + x + y = c, x + y + z = ri. 



1. 18. T/ + = a, 

[Dioph. puts 2 ^ = # + y 4- z. 
I. 19. 



1.20. aj + y + z = a, x + y = mz, y + z = TIOJ. 

I. 21. x = y+ 0, y = s+ ~#, 3 = a + - y (where 05 >y>0), 
??7/ *y t/ ^f) 

with necessary condition. 



II. 18*. a?-( 
\m 

/ 
[Solution wanting.] 

(iii) Determinate systems of equations reducible to the 

first degree. 
I. 26. ax = a 2 ^ fa = a. 

I. 29. x + y = a, a? 2 y 2 = 6. [Dioph. puts 2 = #-y.J 
I. 31. x = my, 
I. 32. x = my, 
I. 33. x = 
I. 34. x = 

I. 34. Cor. 1. x = my, #y = n(x + y). 
k Cor. 2. aj = my, #y = 71(03 y). 



DETERMINATE EQUATIONS 487 

(I. 35. x = my, y 2 = nx. 
I. 36. x = my, y 2 = My. 

I. 37. a; = my, y 2 = n(x + y). 
I. 38. x = my, y 2 = 7i(x y). 
I. 38. Cor. a; = my, x 2 = tiy. 

a; = my, # 2 = nx. 

.! = my, x* = n(x + y). 

a = my, x 1 = ^i(x~y). 

II. G*. -y = a, x 2 y 2 = x y + b. 
IV. 36. z = 



[Solved by means of Lemma : see under (vi) Inde- 
terminate equations of the first degree.] 

(iv) Determinate systems rfeducible to equations of 
second degree. 

I. 27. x + y = a, xy = I. 

[Dioph. states the necessary condition, namely that 
ci 2 6 must be a square, with the words <rri 8k TOVTO 
irXaorfjLaTtKov, which no doubt means 'this is of the 
nature of a formula (easily obtained)'. He puts 

I. 30. x y = a, xy = 6. 

[Necessary condition (with the same words) 46 + a 2 = 
a square, x + y is put = 2 .] 

I. 28. x + y = a, # 2 + y 2 = 6. 

[Necessary condition 2 b a 2 = a square, x y = 2 .] 

[Dioph. puts x y=2, whence #=^& + , y = ^6 . 
The numbers a, /> are so chosen that (a ^6 3 )/36 is 
a square.] 

IV. 2. a 3 - y'* = a, xy = ft. 



488 DIOPHANTUS OF ALEXANDRIA 

IV. 15. (y + z)x = a, (z + x)y = 6, (x + y)z = c. 

[Dioph. takes the third number z as his unknown ; 
thus # + 2/ = rt/2. 

Assume x = p/z, y = <//. Then 



These equations are inconsistent unless p q^a l). 
We have therefore to determine p, q by dividing c into 
two parts such that their difference = a 6 (cf. I. 1). 

A very interesting use of the c false hypothesis' 
(Diophantus first takes two arbitrary numbers for p, q 
such that p + q = c> and finds that the values 'taken have 
to be corrected). 

'on 
The final equation beiii" 1 --| +p = <t, where p, q are 

z* 

determined in the way described, z 1 = pq/(a> p) or 
pq/ (b q), and the numbers a, ft, c have to be such that 
cither of these expressions gives a square.] 



IV. 34. s 



[Dioph. states as the necessary condition for a rational 
solution that each of the three constants to which the 
three expressions are to be equal must be some square 
diminished by 1. x The true condition is seen in our 
notation by transforming the equations yz 

=y into 



DETERMINATE EQUATIONS 489 

whence 



and it is only necessary that (a + 1) (jS + 1) (y + 1) should 
be a square, not that each of the expressions a + 1, + 1, 
y -f 1 should be a square. 

Dioph. finds in a Lemma (see under (vi) below) a solu- 
tion kv dopto-TM (indeterminately) of acy + ( 
which practically means finding y in terms of #.] 

IV. 35. z- 



[The remarks on the last proposition apply mutatis 
mutandis. The lemma in this case is the indeterminate 
solution of xy (x + y) = &.] 



IV. 37. yz = a(v + y + s), cs = b(.r,+y + z), xy = c( 

[Another interesting case of c false hypothesis '. Dioph. 
first gives x + y + z an arbitrary value, then finds that 
the result is not rational, and proceeds to solve the new 
problem of finding a value of x + y + zto take the place of 
the first value. 

If w = x +y + z, we have x = civ/y> z = <iw/y, so that 

(LC> 

zx = acw^/y 1 = bw by hypothesis; therefore y 2 = -^-w. 

For a rational solution this last expression must be 

etc 
:i square. Suppose, therefore, that w = -y- 2 , and we have 

ac . t> ac . . 



Eliminating a?, jy, 0, we obtain = (bc + ca + ab)/ac, 
and 

# = (/>c + ca + ab)/a, y = (fa + ca -f ab)/b, 

z = (bc + ca + ab)/e.] 
Lemma to V. 8. yz = a 2 , c.r = /> 2 , xy = c 2 . 



490 DIOPHANTUS OF ALEXANDRIA 

(v) Systems of equations apparently indeterminate but 

really reduced, by arbitrary assumptions, to deter- 

minate equations of the first degree. 

I. 14. xy = m(x + y). [Value of y arbitrarily assumed.] 
II. 3*. xy = m(x + y), and acy = 
II. 1*. (cf. I. 31). x* + y 2 = 

II. 2*. (cf. I. 34). x 2 7/ 2 = m(x y). > [x assumed = 2 y.] 
II. 4*. (cf. I. 32). x* + y* =m(x-y). \ 
II. 5*. (cf. I. 33). aj 2 ~2/ 2 = m(# + 2/). j 
II. 7*. x 2 y 2 = m(# 2/) + ^- [Diopb. assumes # ?/ = 2.] 



T 

1. 22. # -- cc+-0 = 2/ -- 2/H -- a? = -- z+ -y. 
m p n m p 'Yi u 

[Value of y assumed.] 

T A 1 1 11 11 

1.23. x --- x 4- - w = y -- y H --- x = -- H ?/ 

r^ g 7i 7}i p n J 

= w -- w+ -0. [Value of y assumed.] 
<I p l J J 

1.24. x+ (y + z) = y+-(s + 

7/6 Iv 

[Value of y + z assumed.] 

I. 25. X+ 



= z + - (w + x + y) = w+ -(x + y+z). 
[Value of y + z + w assumed.] 



II. 17*. (cf. I. 22). x-l-i 

v ' \m 



= y- 

[Ratio of x to ?/ assumed.] 



INDETERMINATE ANALYSIS 491 

IV. 33. x+ -y m (y -y) , y + - x = n(x -x) 

Z ' \ Z * *^ \ / / 



[Dioph. assumes ^y =1.] 



(vi) Indeterminate equations of the first degree. 
Lemma to IV. 34. xy + (x + y) = a.^ [Solutions >^ , r< 

IV. 36. xy = m(x + y). J 



y practically found 
in terms of #.] 



(vii) Indeterminate analysis of the second degree. 
II. 8. ,/: 2 + 2/ 2 = a 2 . 

[2/ 2 = a 2 a; 2 must be a square = (mx a) 2 , say.] 

T T o '' i 2 *' 7 2 F 13 4- > 7 1 

II. 10. x 2 -2/ 2 = a. 

[Put x = 7/ + wi, choosing m such that m 2 < a.] 

(II. 11. <w + a = u 2 , ;/' -f- ft = v*. 

J Tl 1 9 ^ / ' !>- 

111. 13. a; a = u 2 , .r-6 = v 2 . 

[Dioph. solves 11. 11 and 13, (1) by means of the 
4 double equation ' (see p. 469 above), (2) without a double 
equation by putting x = 2 + a and equating ( 2 + a) + 6 
to ( m) 2 . In II. 12 he puts a; = a 2 .] 

II. 14 = III. 21. tf + y = a, x + z* = u 2 , y + 2 = ^' 2 . 

[Diophantus takes as the unknown, and puts 

u 2 = te + m) 2 v 2 =z (z+ ii) 2 . Therefore x = 2mz + wi 2 , 

\ / ' \ r ' 

y = 2nz + n*, and is found, by substitution in the first 

equation, to be 7 . In order that the solution 

1 2(m + n) 

may be rational, m, n must satisfy a certain condition. 
Dioph. takes them such that m 2 + n 2 < a, but it is suffi- 
cient, if m > n, that a + mn should be > 7i 2 .] 

II. 15 = III. 20. x + y = a, z*-x = u 2 , 2 -y = ^ 2 - 

[The solution is similar, and a similar remark applies 
to Diophantus's implied condition.] 



492 DIOPHANTUS OF ALEXANDRIA 

II. 16. x = mj/, a? + x = u 2 , a> 2 -}-2/ = ^ 2 . 
II. 19. # 2 7/ 2 = mfv 2 s 2 ). 

[Assume ?/ == 2mo; + m 2 , and one condition is satisfied.] 

[Assume x = + m, y = 2m + m 2 , and one condition 
i is satisfied.] 

/II. 22. x 2 + (x + y) - 

1[Put # + ?/ = 
II. 23. x* (x + y) = u 2 , 

II. 24. (# + 7/) 2 + # = u 2 , 

[Assume 02 = (m 2 1) 2 , y = (>// 2 1)| 2 , x + y = .] 

II. 25. (^ + 2/) 2 # = ^ 2 ; (^ + ?y) 2 ~~2/ == v "- 
II. 26. xy + x = tt 2 , xy + y = v*, u + v = c/. 
[Put y = ^ 2 aj-l.] 

ITT rtiw o > 
II V7 ^/"J/ Of* " <J/ ^ < ^K 1/ 3/ O/ -" fl)^ 'J/ I <j // 
J.J.. ^J I . t/X (^ tA; M/ , < ff '/ U , CC' ^ t, 7 I*. 



ill 9 Q S]j*ni& _ . /V^ /ll /y>i/i.J ^/^ /J}^ 

\-LJ-. 6i*J vU li ^^ \JU tl/ , i>lv f 7 ^~~ f / U 



II. 30. a?2/ + ( 

[Since m 2 + Tfc 2 + 2 m?i is a sc^uare, assume 

py = (m 2 -f 7i 2 ) ^ 2 and aj -f y = 2 mn ^ 2 ; 
put 05 = >, y = qg, where jpg = m 2 + ^ 2 ; then 



II. 31. o^ + ^ + Z/) = u 2 , xy-(x + y) = v 2 , 



[Suppose w 2 = 2 . 2 m . m, which is a square, and use 
formula (2m) 2 + m 2 + 2.2m.m = a square.] 



III. 32. 2 
TT 3 ^ 9/ 2 ^ 47 ^ ^ 2 // '^ >2 ^ 



INDETERMINATE ANALYSIS 493 

11.34. x* 



[Since {-|(m n) } 2 + mn is a square, take any number 
separable into two factors (m, n) in three ways. This 
gives three values, say, p, q, r for ^(m n). Put 

x = P&> y = d?> z = r > anc l * + V + 2 = w&7i 2 > therefore 
(2> + <7 + r) = mr<, 2 , and is found.] 

II. 35. x*- 



[Use the formula {4(m + 7i)j 2 m^ = a square and 
proceed similarly.] 

III. 1* (aj + 2 y + s)-^ = u 2 , ( M! + y + o)-^ = t; 2 , 

(a + y + s) c 2 = t(; 2 . 
III. 2*. 



III. 3* 
III. l*. a- 
III. 5. ^' + 2 

[Tlie first solution of this problem assumes 



whence x, y, z are found in terms of , and z + x y 
is then made a square. 

The alternative solution, however, is much more ele- 
gant, and can be generalized thus. 

We have to find x, y, z so that 

x + y + z = a square 
x y + z = a square 
x + y z = a square 
x + y + z = B> square 

Equate the first three expressions to a 2 , 6 2 , c 2 , being 
squares such that their sum is also a square = & 2 , say. 



494 DIOPHANTUS OF ALEXANDRIA 

Then, since the sum of the first three expressions is 
itself equal to x + y + z, we have 
# = (& 2 + c 2 ), y 

III. 6. x + y + z = 2 , 
III. 7. x-y = y-z, 
(III. 8. x + y + z + a = 2 , 2/ + 2 + C& = w 2 > 



III. 9. x + y + z a = 2 , y + z a - ii 2 , z + x a = v 2 , 

# -f 2/ a = w 2 . 
III. 10. 7/^ + tt = U?, zx + a = v 2 , ay-f a = ^ 2 . 

[Suppose 1/2; + a = m 2 , and let ?/ = (m 2 a)^, z = l/: 
also let z# + a = ii 2 ; therefore a; = (/t 2 a)^. 
We have therefore to make 

(m 2 a) (n 2 a) 2 + a a square. 

Diopliantus takes m 2 = 25, a = 12, n 2 =16, and 
arrives at 52^ 2 +12, which is to be made a square. 
Although 52 . l f -+ 12 is a square, and it Follows that any 
number of other solutions giving a square are possible 
by substituting 1 + 77 for in the expression, and so on, 
Diopliantus says that the equation could easily be solved 
if 52 was a square, and proceeds to solve the problem of 
finding two squares such that each increased by 12 will 
give a square, in which case their product also will be 
a square. In other words, we have to find m 2 and u 2 
such that m 2 a, n 2 a are both squares, which, as he 
says, is easy. We have to find two pairs of squares 
differing by a. If 



and 

let, then, m 2 = 
III. 11. yz a = u 2 , 3# a = v 2 , xy a = it; 2 . 

[The solution is like that of III. 1 mutatis mutwridis.] 

(III. 12. yz + x = u 2 , 0# + ?/ = ^ 2 j y + z = ^ i2 - 
(HI. 13. yzx= u 2 , zx y = v 2 , xy z = w 2 . 



INDETERMINATE ANALYSIS 495 

111. 14. yz + x* = u 2 , zx + y* = v 2 , xy + z* = w 2 . 

[Lemma. If a, a+1 be two consecutive numbers, 
a 2 (a + 1 ) 2 + a 2 + (a + 1 ) 2 is a square. Let 

2/ = w a , s = (m+1) 2 . 
Therefore (m 2 + 2 m + 2) aj -f (m + 1 ) 2 
and (m' 2 + l)# + m 2 

have to be made squares. This is solved as a double- 
equation; in Diophantus's problem m = 2. 

Second solution. Let x be the first number, m the 
second; then (m+l)# + m is a square = u 2 , say; there- 
fore x = (M 2 m)/ (771+ 1), while 2/ = w. We have then 

(m + \)z + m = a square 

. / 2 + 1\ , ?i 2 -m 

and ( )+ = a square 

V m -f 1 / m+1 j 

Diophantus has m = 3, H = 5, so that the expressions 
to be made squares are with him 

42 + 3 ) 



This is not possible because, of the corresponding coeffi- 
cients, neither pair are in the ratio of squares. In order to 
substitute, for 6, 4, coefficients which are in the ratio 
of a square to a square he then finds two numbers, say, 
^>, q to replace 5^, 3 such that 2 )( 1+P + ( 1 = a square, and 
(/>+!)/(</+ 1) = a square. He assumes and 4 + 3, 
which satisfies the second condition, and then solves for , 
which must satisfy 

4 2 + 8 + 3 = a square = (2 - 3) 2 , say, 
which gives = /Q, 4 + 3 =- 4|. 

He then solves, for 0, the third number, the double- 
equation 



= square] 



496 DIOPHANTUS OF ALEXANDRIA 

after multiplying by 25 and 100 respectively, making 
expressions 



130.T+ 30 

In the above equations we should only have to make 
>i 2 + 1 a square, and then multiply the first by 7i 2 + 1 and 
the second by (m+ I) 2 . 

Diophantus, with his notation, was hardly in a position 
to solve, as we should, by writing 



which gives x+1 =-/{(&*+ 1) (c- + l)/(a a + 1) }, &c.] 
III. 1 6. yz (y + z) = u\ zx(z + x) = v 2 , xy-(x + y) = ?'/ 2 . 

[The method is the same mutatis mutandis as the 
second of the above solutions.] 

III. 17. xy + (x + y) = u' 2 , a 
III. 18. x- 



III. 19. (^ 1 + a: a + ii; a + aj 4 ) S8 + aj 1 = 



^ fu 2 

4 )- a a = - , 

~ 



I/ 8 



f 
+ ^= j 



/2 

[Diophantus finds, in the way we have seen (p. 482), 
four different rational right-angled triangles with the 
same hypotenuse, namely (65, 52, 39), (65, 60, 25), (65, 
56, 33), (65, 63, 16), or, what is the same thing, a square 
which is divisible into two squares in four different ways ; 
this will solve the problem, since, if h, p, b be the three 
sides of a right-angled triangle, h 2 2pb are both squares. 



INDETERMINATE ANALYSIS 497 

Put therefore x l + x 2 + x% + x = 6 5 . 
and #! = 2. 39. 52 f 2 , o? 2 = 2. 25. 60 2 , # 3 = 2 . 33 . 56 2 , 

# 4 = 2.16.63 2 ; 

this gives 127G8f 2 = G5, and = 
| IV. 4. 
lIV. 5. 

IV. 13. x+ 1 = i 2 , y+ 1 = u 2 , # + ?/+ 1 = t; 2 , y-x + I = w 2 . 

[Put x = (m^ + I) 2 1 = 7?i 2 f 2 4- 2m ; the second and 
third conditions require us to find two squares with x as 
difference. The difference m' 2 2 + 2m is separate^! into 
the factors w' 2 -f 2m, ; the square of half the differ- 
ence = ((m 2 l) + mj-. Put this equal to ?/+!, so 
that y = J(m--l) a * + m(m a -l) + m a -l, and the 
first three conditions are satisfied. The fourth gives 
(??i 4 6 m~ + 1 ) | 2 + (m 3 3 m) + ?7i' 2 = a square, which 
we can equate to (n m) 2 .] 

IV. 1 4. x 2 + y* + z~ = (,(;- - y a ) + (//* - 2 ) + (x 2 - s 2 ). (^ > y > z.) 
IV. 16. 



[Put 47?i^ for y, and l>y means of the factors 2mg, 2 
we can satisfy the second condition by making x equal 
to half the difference, or m 1. The third condition 
is satisfied by subtracting (4m) 2 from some square, say 
(4m+l) 2 ; therefore s = 8in+l. By the first con- 
dition 13 mg must be a square. Let it be 169ij 2 ; the 
numbers are therefore 1 3 ?/ 2 1 , 5 2 ?/ 2 , 1 04 ij 2 + 1 , and 
the last condition gives 10816?/ 4 4- 22 1?/ 2 = a square, 
i.e. 10816?; 2 -f 221 = a square = (104ij + I) 2 , say. This 
gives the value of 17,. and solves the problem.] 

IV. 17. x + + z = J 2 , x*- = u\ tf-z = v 2 , z 2 -x = w 2 . 



IV. 19. ?/0+ 1 = u 2 

[We are asked to solve this indeterminately (i> ro> 
dopbmp). Put for y0 some square minus 1, say m 2 ^ 2 
+ 2m; one condition is now satisfied. Put z = , so 
that y = ?7i 2 ^ + 2m. 

1523.2 K k 



498 DIOPHANTUS OF ALEXANDRIA 

Similarly we satisfy the second condition by assuming 
zx = ?i 2 2 + 2 Tig ; therefore x = 7i 2 + 2n. To satisfy the 
third condition, we must have 



(m 2 n 2 2 + 2mn .m + ng + 4 WTI) + 1 a square. 

We must therefore have 4mn+ 1 a square and also 
mn (m + n) = mw \/ (4 mri + 1 ) . The first condition is 
satisfied by n = m + 1 , which incidentally satisfies the 
second condition also. We put therefore yz= (m + I) 2 1 
and 0#= { (m+ l) + 1 } 2 1, and assume that = , so that 
2/ = m 2 +2m, # = (m+ l) 2 + 2(m+l), and we have 
shown that the third condition is also satisfied. Thus we 
have a solution in terms of the undetermined unknown . 
The above is only slightly generalized from Diophantus.] 

IV. 20. xx+l = r 2 , ffO + 1 = s 2 , a+1 = f 2 , 



= u 2 , 



[This proposition depends on the last, x } , ;/' 2 , aJ 3 being 
determined as in that proposition. If x 3 corresponds to z 
in that proposition, we satisfy the condition # 3 # 4 -f 1 = w- 
by putting # 3 # 4 = |(m + 2) + I} 2 1, and so find x 4 in 
terms of , after which we have only two conditions more 
to satisfy. The condition x^+ 1 = square is auto- 
matically satisfied, since 



is a square, and it only remains to satisfy x 2 x^ + 1 = square 
That is, 



t= m 2 (m+ 2) 2 ^ 2 + 2 m(m + 2) (2m + 2) + 4m(m + 2) + 1 

has to be made a square, which is easy, since the coefficient 
of 2 is a square. 

With Diophantus m = 1, so that x l = 4 + 4, # 2 = + 2 
aj ;j = ^ ^4 = 9 ^ 6 and 9 2 + 24^+13 has to be mad< 
a square. He equates this to (3 4) 2 , giving = T ^.] 

IV. 21. xz = y 2 , x y = u 2 9 x z = v*, yz w*. (x>y>z 
|IV. 22. xyz + x = u 2 , xyz + y = v 2 , xyz + z = w 2 . 
lIV. 23. xyz x = u 2 , xyzy = v 2 , xyz z = w 2 . 



INDETERMINATE ANALYSIS 499 

IV. 29. x 2 + y 2 + z 2 + iv 2 + x + y + z + iv a. 
[Since x 2 + x + % is a square, 

(x 2 + x) + (y 2 + y) + (z 2 + z) + (w 2 + w) + 1 

is the sum of four squares, and we only have to separate 
a + 1 into four squares.] 

IV. 31. x + y = 1, (x + a) (y + b) = u 2 . 
IV. 32. x + y + z = a, 
IV. 39. x-y = m(y- 
IV. 40. x~ y 2 = m(y 

' \T 1 _ 2 i . 2 __ 2 ___ *' 

V. 2. #2 = ?/ 2 , # + a = t& 2 , y + a =. v 2 , z + d = iu~. 

V_ l> O . ,1 

V I // /Ji< /}/ I., // o- r* I // /^ 

* V . tV j^ v / , r I ^^ \M O , <V |^ Lv ~~~ \J , 

7/0 + a = u 2 , 2# 4- (6 = v 2 , #y 4- (6 = ^y 2 . 

V. / > . i> 

. A . i/y ~~~ C6 * ) '/ ~"^ ( v "" O , ^ ~~" LV "~ 6 , 

2/5 a = u 2 , zx a = v 2 , xy a = w 2 . 

[Solved by means of the 2 J orisnis that, if a be the 
given number, the numbers m 2 a, (m+1) 2 a satisfy 
the conditions of V. 3, and the numbers m 2 + , 
(m + I) 2 + the conditions of V. 4 (see p. 479 above). The 
third number is taken to be 2 (m 2 + a + (m+ l) 2 + a} 1, 
and the three numbers automatically satisfy two more 
conditions (sec p. 480 above). It only remains to make 

2 {m 2 + a + (m + I) 2 T a] 1 + a a square, 
or 4 m 2 + 4 ?*i + 3 a + 1 = a square, 

which is easily solved. 

With Diophantus + 3 takes the place of m in V. 3 
and | takes its place in V. 4, while a is 6 in V. 3 and 6 
in V. 4.] 

Vc '^ 2 i 2 2 22i2 2 ') o , *> o 

O. X/ Z "| iA? y , *j vi/ "y* Yy O , *A^ tf i* *f v , 



[Solved by means of the Porism numbered 2 on p. 480. 



500 DIOPHANTUS OF ALEXANDRIA 

V. 6. x-2 = r 2 , T/-2 = s 2 , z-2 = J 2 , 

yzyz = V?, 2# z x = ^ 2 > x y x y = w' 2 > 

yz x = u' 2 , zxy = v' 2 , xy z = it/ 2 . 

[Solved by means of the proposition numbered (3) on 
p. 481.] 

Lemma*! to V. 7. 



(u 
V. 7. 



[Solved by means of the subsidiary problem (Lemma 2) 
of finding three rational right-angled triangles with 
equal area. If m, n satisfy the condition in Lemma 1, 
i.e. if mn + m 2 -f n 2 = p*, the triangles are ' formed ' from 
the pairs of numbers (p, m), (p, ri), (p> m + 7i) respec- 
tively. Diophantus assumes this, but it is easy to prove. 
In his case m = 3, n = 5, so that p = 7. Now, in 
a right-angled triangle, (hypotenuse) 2 + four times area 
is a square. We equate, therefore, x + y + z to four 
times the common area multiplied by 2 , and the several 
numbers x> y, z to the three hypotenuses multiplied by , 
and equate the two values. In Diophantus's case the 
triangles are (40, 42, 58), (24, 70, 74) and (15, 112, 113), 
and 245 = 3360 2 .] 



V. 8. yz(x + y + z) = 






[Solved by means of the same three rational right- 
angled triangles found in the Lemma to V. 7, together 
with the Lemma that we can solve the equations 7/0= a 2 , 
zx = 6 2 , xy = c 2 .] 

9. (Of. 
11. 

[These are the problems of 



INDETERMINATE ANALYSIS 501 

described above (pp. 477-9). The problem is ' to divide 
unity into two (or three) parts such that, if one and the 
same given number be added to each part, the results are 
all squares '.] % 



(V. 10. x + y = 1, x + a = u z 
lV. 12. x+y+z= 1, x + a 

[These problems are like the preceding except that 
different given numbers are added. The second of the 
two problems is not worked out, but the first is worth 
reproducing. We must take the particular figures used 
by Diophantus, namely a = 2, b = 6. We have then to 
divide 9 into two squares such that one of them lies 
between 2 and 3. Take two squares lying between 2 
and 3, say ff f , ff. We have then to find a square 2 
lying between them ; if we can do this, we can make 
9 J 2 a square, and so solve the problem. 

Put 9- 2 = (3-m) 2 , say, so that = 6m/(m 2 + 1); 
and m has to be determined so that lies between 



17 o,,/| 19 
1 2 " 



17 6m 19 
Therefore t , < - < ,-5 

1 7)1~ T~ I 1 & 

Diophantus, as we have seen, finds afortioi4 integral 
limits for m by solving these inequalities, making m not 
greater than f and not less than f f (see pp. 463-5 above). 
He then takes m = 3 and puts 9- 2 = (3-3j) a , 
which gives = f f .] 

( V. 13. ;<j + 2/ + = c&, 2/ + = u 2 , 2 + a/ = 
V. 14. a; +gf + 2 + w = , x 1 4- y + * = s 2 , 



[The method is the same.] 
V. 21. aY 2 
V. 22. ory 2 
V. 23. x*-x*y*z 2 = u 2 , y 2 -x*y*z* = v 2 , z*-x 2 y*** = 2 - 

[Solved by means of right-angled triangles in rational 
numbers.] 



502 DIOPHANTUS OF ALEXANDRIA 

V. 24. 2/V+ 1 = u\ z*x*+l = v* } x*y*+l = w*. 
V. 25. 2/ 2 s 2 - 1 = u 2 , z*x 2 - 1 = v 2 , a?V- 1 = i(; 2 . 
V. 26. 1 -y 2 * 2 = u 2 , 1 -'0 2 x 2 = v 2 , 1 -a 2 ?/ 2 = it; 2 . 

[These reduce to the preceding set of three problems.] 

V. 27. 2/ 2 + s 2 + a = i6 2 , s 2 + a> 2 + tt = v 2 , a 2 + ?/ + a = 2 . 
V. 28. 2/ 2 + s 2 -a = u 2 , 
V. 30. mx + ny = u 2 , 



[This problem is enunciated thus. 'A rnan buys a 
certain number of measures of wine, some at 8 drachmas, 
some at 5 drachmas each. He pays for them a square 
number of drachmas ; and if 60 is added to this number, 
the result is a square, the side of which is equal to the 
whole number of measures. Find the number bought at 
each price.' 

Let = the whole number of measures ; therefore 
2 60 was the number of drachmas paid, and 2 60 
= a square, say ( m) 2 ; hence = (m 2 + 60)/2m. 

Now % of the price of the five-drachma measures + 
of that of the eight-drachma measures = ; therefore 
2 60, the total price, has to be divided into two parts 
such that -| of one -f | of the other = . 

We cannot have a real solution of this unless 



*>i-(f-60) and <i( 2 
therefore 5 < 2 ~60 < 8. 

Diophantus concludes, as we have seen (p. 464 above), 
that is not less than 11 and not greater than 12. 

Therefore, from above, since = (w 2 + 60)/2m, 
22m < m 2 + 60 < 24m; 

and Diophantus concludes that m is not less than 19 and 
not greater than 21. He therefore puts m = 20. 

Therefore = (m 2 + 60)/2m = ll, 2 = 132|, and 
2_60 = 72J. 

We have now to divide 72|; into two parts such that 
| of one part + 1 of the other = 1 \\. 



INDETERMINATE ANALYSIS 503 

Let the first part = 62; therefore J (second part) 
= 11^ 0, or second part = 92 Sz. 

Therefore 5z + 92 - 8z = 72%, and z = f 5 

therefore the number of five-drachma measures is f f and 
the number of eight-drachma measures ff .] 

Lemma 2 to VI. 1 2. ax 2 + b = u 2 (where a + b = c 2 ). \ ( see p 467 
Lemma to VI. 1 5. o* - &=u 2 (where ad 2 - 6 = c 2 ).) above.) 

f [III. 15]. xy+'x + y = u*, x+l = 

\ i 

\ v 

[III. 16]. iry-(x + y) = u*, 0-1 = 



(viii) Indeterminate analysis of the third degree. 
IV. 3. x*y = u, xy = u 3 . 

T \r n O <> O .> .> 

I IV. 7. or + 2/ 2 = u 2 , z 2 -I- j/ 2 = v\ 

|1V. 8. 
IV. 9. 



(really reducible 

!1V. 10. oj 3 + 2/ 3 = x + y. \ to the second 

IV 11 T*-,fl- v -,n ! degree>) 

A f . JL J. . ,(/ f 7 i</ ^~~ tl . I I 

^ the same problem. 
IV. 12. x :i + 2/ = 2/ 3 + ffJ I 

[We may give as examples the solutions of IV. 7,, 
IV. 8, IV. 11. 

IV. 7. Since 3 2 + 7/~ = a cube, suppose 
To make x* + y* a square, put x 3 = a 2 
which also satisfies # 3 jy 2 = 2 . We have then to make 
2 a& a square. Let a = , 6=2^; therefore a 2 -f 6 2 = 5 ^ 2 , 
2ab = 4^ 2 , y = 2, z = g, and we have only to make 
5 2 a cube. = 5, and # 3 = 125, 2/ 2 = 100, e 2 = 25. 



504 DIOPHANTUS OF ALEXANDRIA 

IV. 8. Suppose x = , y* = m 3 3 ; therefore u = (m + 1 ) 
must be the side of the cube m 3 3 + , and 



To solve this, we must have 3m 2 + 3m + 1 (the difference 
between consecutive cubes) a square. Put 

3m 2 + 3m +1 = (l-?irn) 2 , and m = (3 + 2n)/(n*-3). 

IV. 11. Assume x = (m-f 1), y = m, and we have 
to make (3m 3 +3m 2 + 1) 2 equal to 1, i.e. we have 
only to make 3m 2 + 3rn+ 1 a square.] * 

IV. 18. x* + y - u*, y 2 + x = v\ 
IV. 24. x + y = a, xy*~ u? u. 

[y = a x m , therefore ax x 2 has to be made a cube 
minus its side, say (mx I) 3 (mx 1). 

Therefore ax x 2 = m 3 z* 3 m 2 ,/j 2 -f 2 m#. 


To reduce this to a simple equation, we have only to 

put m = %aJ\ 

IV. 25. x + y + z = a, ,/:?/ = { (x-y) + (x-z) + (?y-^)} :i - 

(,/ > T/ > s) 



[The cube = B(x zf. Let cc = (m+ 1), = m, so 
that 2/ = 8 / (m 2 + m), and we have only to contrive that 
8/(m 2 + m) lies between m and m/ + 1. Dioph. takes the 
first limit 8 > m f ' J + m 2 , and puts 



8 = (m-f ) :5 or 

whence m/ = f ; therefore # = , / = !, 2 = |. Or, 
multiplying by 15, we have x = 40 , ?/ = 27 , s = 25 . 

The first equation then gives .] 

t 
(TV. 26. CCT/ + CC = u 3 , xy + y -= v*. 

UV. 27. a^oj = u 3 , xyy = v 3 . 
IV. 28. xy + (x + y) = u 3 , xy~(x + y) = v 3 . 

[05 + j/ = ^ (u 3 v 3 ), xy = ^ (u 3 + v 3 ) ; therefore 



which latter expression has to be made a square. 



INDETERMINATE ANALYSIS 505 

Diophantus assumes u = + 1, v = 1, whence 



must be a* square, or 



therefore 32 3 = 36 2 , and = f . Thus u, v are found, 
and then x, y. 

The second (alternative) solution uses the formula that 

t(?-t) + (?-& + = a cube. Put x = y = 2 - 
and one condition is satisfied. We then only have to 
make (*_) --(^-) or 3 -2g 2 a cube (less than 



IV. 38. ( 

( + 2/ + s) 5 = w 3 , [# + y + s = 
[Suppose a? + 2/ + z = 5 2 5 



^-^ , , -- ; 

therefore 4 = -J-u(i6 + 1) + v~ + w :l . 



Diopliantus puts 8 for up, but we may take any cube, as 
m 3 ; and he assumes t/ 2 = (^ 2 I)' 2 , for which we might 
substitute (' 2 w 2 )' 2 . We then have the triangular 
number %u(u+ 1) = 2?i 2 2 u* m 3 . Since 8 times a 
triangular number plus 1 gives a square, 

16/t 2 2 8?i 4 -8m 3 + l = a square = (4?? -A;)' 2 , say, 
and the problem is solved.] 
V. 15. 



[Let x + y + z = ^, u 3 = m 3 n , ^ a = n 3 ^ 3 , w 3 = p 3 ^ 3 ; 
therefore = { (m 3 - 1) + (7i 3 - 1) + (y> 3 - 1) } 3 ; 

and we have to find three cubes m <{ , 7i 3 , p* such that 
7^ 3 + 7i 3 +^ 3 3 = a square. Diophantus assumes as 
the sides of the cubes (k+l), (2 fc), 2; this gives 



506 DIOPHANTUS OF ALEXANDRIA 



9k+ 14 = a square = (3& Z) 2 , say ; and k is found. 
Retracing our steps, we find and therefore x, y, z.] 

V. 16. 



V. 17. x- 
V. 18. 



) :J -f z = 

[Put 

5 = (r 2 -l)g fl , whence 2 = (p 2 -! +r/ 2 -l +r a -l), so 
that p* 1 + q 2 1 + T* 1 must be made a fourth 
power. Diophantus assumes ^> 2 = (m 2 1 ) 2 , q 2 = (m + 1 ) 2 , 
r 2 = (7>i I) 2 , since m 4 2m 2 + m 2 + 2m + m 2 2m=m 4 .] 

V. 19. 



V. I9a. a; + 2/ + = i 2 , ^- 



V. 19. 1), c. x + y + z = a, (a + y + $f # = 
(./ + ?y + 0) 3 + y = w 2 , (,T 

V. 20. 



[IV. 8]. a;~?y = 1, a :i -?/ 3 = 
[IV. 9, 10]. B 8 + 7/( 

^2 

[IV. 11]. >_ = ^>-2/). 
[V. 15]. cc 3 + 2/ 3 + s 3 -3 =u 2 . 



[V. 1 6]. 3 - (* 3 + y 3 + z 3 ) = u 2 . 
[V. 17]. 



INDETERMINATE ANALYSIS 507 

(ix) Indeterminate analysis of the fourth degree. 
V. 29. o 



[' Why ', says Format, ' did not Diophantus seek tivo 
fourth powers such that their sum is a square. This 
problem is, in fact, impossible, as by my method I am 
able to prove with all rigour.' No doubt Diophantus 
knew this truth empirically. Let x* = 2 , 2/ 2 = p*, 
z* = q*. Therefore 4 + p* + (f = a square = ( 2 r) 2 , say ; 
therefore g 2 = (r 2 p* q*)/Zr, and we have to make 
this expression a square. 

Diophantus puts r = p 2 + 4, q* = 4, so that the expres- 
sion reduces to Sp^/(2jr + 8) or *p*/(fP + 4). To make 
tliis a square, let .//- + 4 = (jj + I) 2 , say ; therefore p = l, 
and p* = 2i, g 2 = 4, r=6; or (multiplying by 4) 
jp 2 = 9, j 2 = 16, r = 25, which solves the problem.] 



[V. 18]. 

(See above under V. 18.) 

(x) Problems of constructing right-angled triangles witli 

sides in rational numbers and satisfying various 

other conditions. 

[I shall in all cases call the hypotenuse z, and the 
other two sides a 1 , y y so that the condition a? 2 + y* = 2 
applies in all cases, in addition to the other conditions 
specified.] 

[Lemma to V. 7]. xy = ayj^ = # 2 ?/ 2 . 
VI. 1. z-x = u*> z-y = v\ 

[Form a right-angled triangle from g, m, so that 
z = 2 + w 2 , x = 2m, ?/ = 2 w 2 ; thus z y = 2m 2 , 
and, as this must be a cube, we put in = 2 ; therefore 
# = * 4 + 4 must be a cube, or 2=a cube, 
say /t :J , and = W + 2.] 

I. 2. 



508 DIOPHANTUS OF ALEXANDRIA 

VI. 3. 



[Suppose the required triangle to be hg,pg,bg; there- 
fore %pbg* 4- a = a square = 2 2 , say, and the ratio of a 
to ?i 2 %pb must be the ratio of a square to a square. 
To find n, p, b so as to satisfy this condition, form 

a right-angled triangle from m, > 



1 / 2a\- 

therefore -J6 = m 2 5 Assume n- = ( m+ - 

f ))l,~ \ 

therefore n% %vb = 4 a + ., ; and ( 4 < t + 

')U - \ 






or 4a 2 + - a o -9 has to be made a square. Put 
m 2 

4a 2 'm 2 + a(4a 2 + 1) = (2am + fc) 2 , and we have a solution. 
Diophantus has a = 5, leading to 100m 2 + 505 = a square 
= (10m + 5) 2 , say, which gives m = 2 /- and n = ^ 3 -. 
/*, y, 6 are thus determined in such a way that 
%pbg* + a = ^ 2 2 gives a rational solution.] 

VI. 4. 



VI. 5. tt-i#2/ = <M 2 . 

VI. 6. %xy + x = a. 

[Assume the triangle to be &, y>, />^, so that 



= a > an d for a rational solution of this equa- 
tion we must Rave (%p) 2 + a(%pty a square. Diophantus 
assumes p = 1, ?> = m, whence ^am + or 2am-fl 
= a square. 

But, since the triangle is rational, to w 2 + I = a square. 

That is, we have a double equation. Difference 
= m 2 2am = m(m 2a). Put 



2am + 1 = {i(m-m~2a) } 2 = a 2 , and>i = (a 2 - l)/2a. 
The sides of the auxiliary triangle are thus determined 
in such a way that the original equation in is solved 
rationally.] 

VI. 7. \m-x = a. 



INDETERMINATE ANALYSIS 509 



rVI. 8. 

I VI. 9. %xy-(x + y) -a. 

[With the same assumptions we have in these cases 



to make {%('p + fy}* + a(%pb) a square. Diophantus 
assumes as before 1 , m for the values of p, b, and obtains 
the double equation 

i (m + 1 ) 2 + Jam = square] 

r ? 

m 2 + l = square) 

m 2 + (2 a + 2) m + 1 = square) 
or f 9 

m 2 + 1 = square) 

solving in the usual way.] 



{VI. 10. 
VI. 11. 



[In these cases the auxiliary right-angled triangle has 
to be found such that 



= a square. 

Diophantus assumes it formed from 1, m + 1 ; thus 
J(fc+p) 2 = i [m a + 2m + 2 + m 2 + 2m} 2 = (m <2 + 2m + I) 2 , 
and a ( ^>&) = a (m + 1 ) (?>i 2 + 2 ??i) . 

therefore 

w 4 + (a + 4)m 3 + (See + 6)m 2 + (2a + 4)m + 1 

= a square 



and 7?i is found.] 
Lemma 1 to VI. 12. x = 
r VI. 12. %xy + x = u* 
I VI. 13. %xyx = tt 2 , %xyy = v 2 . 

[These problems and the two following are interesting, 
but their solutions run to some length; therefore only 
one case can here be given. We will take VI. 1 2 with 
its Lemma 1 . 



510 DIOPHANTUS OF ALEXANDRIA 

Lemma 1. If a rational right-angled triangle be formed 
from m, n, the perpendicular sides are 2mu, m 2 n 2 . 
We will suppose the greater of the two to be 2mn. 
The first two relations are satisfied by making m = 2 n. 
Form, therefore, a triangle from , 2. The third con- 
dition then gives 6 4 + 3 2 = a square or 6 2 + 3 = a 
square. One solution is = 1 (and there are an infinite 
number of others to be found by means of it). If = 1, 
the triangle is formed from 1,2. 

VI. 12. Suppose the triangle to be (&, b,p). Then 
(%pb)+p = & square = (A; ) 2 , say, and g=p/(l<?-%pty. 
This value must be such as to make (^ pfyg* + ^ a square 
also. By substitution of the value of we get 



so that bf)k? + ^p*l>(p-'l)) must be a square; or, if p, 
the greater perpendicular, is made a square number, 
hk* + %pl)(p b) has to be made a square. This by 
Lemma 2 (see p. 467 above) can be made a square if 
b + % pb(p-b) is a square. How to solve these problems, 
says Diophantus, is shown in the Lemmas. It is not 
clear how they were applied, but, in fact, his solution 
is such as to make p, JM &, and b + ^pb all squares, 
namely b = 3, p = 4, h = 5. 

Accordingly, putting for the original triangle 3, 4, 5, 
we have 

6 2 + 4 1 = a square j 

6 2 + 3 = a square) 

Assuming 6 2 + 4 = m 2 2 , we have = 4/(m 2 6), and 
the second condition gives 

96 12 

T r^ o ^7; H --- o ;. = a square, 
m 4 12m 2 + 36 m 2 6 1 ' 

or 1 2 m 2 + 2 4 = a square. 

This can be solved, since m = 1 satisfies it (Lemma 2). 
A solution is m 2 = 25, whence = T 4 .] 

(VI. 14. %xy-z = u 2 , 
I VI. 15. %xy + z = u 2 , 



INDETERMINATE ANALYSIS 511 

[The auxiliary right-angled triangle in this case must 
be such that 

m 2 /ijp %pb.p(h p) is a square. 

If, says Diophantus (VI. 14), we form a triangle from 
the numbers X 19 X% and suppose that p = 2 X l X 2 , and if 
we then divide out by (X l X 2 ) 2 , which is equal to h p, 
we must find a square /o 2 [= m*/(X l X 2 ) 2 ] such that 
Whp %pb . p is a square. 

The problem, says Diophantus, can be solved if X lt X 2 

m '2 

are ' similar plane numbers ' (numbers such as aft, - aft). 

This is stated without proof, but it can easily be verified 
that, if A; 2 = X l X< 2 , the expression is a square. Dioph. 
takes 4, 1 as the numbers, so that & 2 = 4. The equation 
for m becomes 

8 . 1 7m 2 4 . 1 5 . 8 . 9 = a square, 
or 136m 2 -4 320 = a square. 

The solution m 2 = 36 (derived from the fact that 

P - m 2 /(^i- A r 2 ) 2 , or 4 = m <2 /3 2 ) 
satisfies the condition that 

m*kp %pb . p(hp) is a square.] 

VI. 16. + ri = x 



(To find a rational right-angled triangle such that the 
number representing the (portion intercepted within 
the triangle of the) bisector of an acute angle is rational. 




Let the bisector be 5, the segment BD of the base 3, 
so that the perpendicular is 4. 

Let GB = 3 n. Then AC : AB = CD : DB, 



512 DIOPHANTUS OF ALEXANDRIA 

so that AC = 4 (n - ). Therefore (Eucl. I. 47) 



so that = 7 ft 2 / 32 u = ^\?i. [Dioph. has n = 1.] 
VI. 17. 



[Let be the area #2/, and let = & 2 . Since 
a'^/ = 2, suppose a? = 2, T/ = . Therefore 2 + fc 2 must 
be a cube. As we have seen (p. 475), Diophantus 
takes (m I) 3 for the cube and (m-h I) 2 for & 2 , giving 
m 8 3m 2 + 3m l=m 2 + 2m + 3, whence m=4. There- 
fore k = 5, and we assume %xy = ^, s; = 25 , with 
,r = 2, ?/ = I as before. Then we have to make 



. VI. 18. %wj + z = w\ x + y + z = v*. 
VI. 19. %xy + x = u 2 , 



[Here a right-angled triangle is formed from one odd 
number, say 2 + l, according to the Pythagorean for- 
mula m 2 + {|(7>i 2 I)} 2 = {^(wi 2 + l)} 2 , where m is an 
odd number. The sides are therefore 2+l, 2 2 +2, 

2 2 + 2 + 1 . Since the perimeter = a cube, 

%. 

l = a cube. 



Or, if we divide the sides by +1, 4 + 2 has to be 
made a cube. 



Again \xy + x = -^2 ^ s + -f = a sijuarc, 

which reduces to 2 + 1 = a square. 

But 4 + 2 is a cube. We therefore put 8 for the cube, 
and =1*.] 

VI. 20. %xy + x u s , x+y + z = v 2 . 
VI. 21. 



[Form a right-angled triangle from , 1, i.e. (2, 2 1, 
+ 1). Then 2^ 2 + 2 must be a square, and 3 + 2 



INDETERMINATE ANALYSIS 513 



a cube. Put 2 2 + 2 = m 2 2 , so that = 2/(m 2 ~2), 
and we have to make 

8 8 2 2m, 4 



Make 2m a cube = ?& 3 , so that 2m 4 = m 3 7, 3 , and 

g 

mzr^w 3 ; therefore = - -- * and must be made 

n 8 

greater than 1, in order that 2 1 may be positive. 

Therefore 8 < < 16 ; 

this is satisfied by M = 3ffi or w j = % 7 -, and m = |.] 

VI. 22. 



[(1) First seek a rational right-angled triangle such 
that its perimeter and its area are given numbers, 
say p, m. 

Let the perpendiculars be --, 2???^; therefore the hypo- 
tenuse = p - 3mf, and (Eucl. I. 47) 



or 



- - = + 4m- 



j 2 + 4 m = 4 mpg + -~ 

that is, ( /> 2 -f 4 771) ^ = 4 r>^> ^ 2 + 2^. 

(2) In order that this may have a rational solution, 

( \ (if + 4 m) ] 2 8p 2 7>i must be a square, 
i.e. 4 m 2 6 > 2 m + jp 4 = a square, 

or m 2 f p' 2 m + T ^p 4 = a square^ 

Also, by the second condition, m+p = a square) 

To solve this, we must take for p some number which 
is both a square and a cube (in order that it may be 
possible, by multiplying the second equation by some 
square, to make the constant term equal to the constant 

1583.2 L 1 



514 DIOPHANTUS OF ALEXANDRIA 

term in thq first). Diophantus takes p = 64, making 
the equations 

m 2 6 144m + 1048576 = a square] 
m + 64 = a square) 

Multiplying the second by 16384, and subtracting the two 
expressions, we have as the difference m 2 22528m. 

Diophantus observes that, if we take m, m 22528 as 
the factors, we obtain m = 7680, an impossible value for 
the area of a right-angled triangle of perimeter p = 64. 

We therefore take as factors llm, -j^m 2048, and, 
equating the square of half the difference ( = f f m + 1024) 
to 16384m + 1048576, we have ?>i = a |j| 4 . 

(3) Returning to the original problem, we have to 
substitute this value for m in 


and we obtain 



= 0, 

the solution of which is rational, namely = -fife (or 
Diophantus naturally takes the first value, though the 
second gives the same triangle.] 

VI. 23. 2 = 



VI. 24. z = u* + u, x v* v, y = iu\ 
[VI. 6, 7]. ($x 
[VI. 8, 9]. {$( 
[VI. 10, 11]. { 
[VI. 12.] y + (a-y) . %xy = u 2 , x = v*. (x > y.) 
[VI. 14, 16]. u*zx-%ay .x(z-x) = v 2 . (u 2 < or > 



^ treatise on Polygonal Numbers. 

The subject of Polygonal Numbers on which Diophantus 
also wrote is, as we have seen, an old one, going back to the 



THE TREATISE ON POLYGONAL NUMBERS 515 

Pythagoreans, while Philippus of Opus and Speusippus carried 
on the tradition. Hypsicles (about 170 B.C.) is twice men- 
tioned by Diophantus as the author of a ' definition ' of 
a polygonal number which, although it does not in terms 
mention any polygonal number beyond the pentagonal, 
amounts to saying that the nth a-gon (1 counting as the 
tirst) is 

M {2 + (n-l)(a-.2)j. 

Theon of Smyrna, Nicomachus and lamblichus all devote 
some space to polygonal numbers. Nicomachus in particular 
gives various rules for transforming triangles into squares, 
squares into pentagons, &c. 

1. If we put two consecutive triangles together, we get a square. 
In fact 



2. A pentagon is obtained from a square by adding to it 
a triangle the side of which is 1 less than that of the square ; 
similarly a hexagon from a pentagon by adding a triangle 
the side of which is 1 loss than that of the pentagon, and so on. 
In fact 



3. Nicomachus sets out the first triangles, squares, pentagons, 
hexagons and heptagons in a diagram thus : 

Triangles 13 6 10 15 21 28 36 15 55, 

Squares I 4 1) 16 25 36 19 64 81 100, 

Pentagons 1 5 12 22 35 51 70 92 117 145, 

Hexagons 1 G 15 28 45 66 91 120 153 190, 

Heptagons 1 7 18 34 55 81 112 148 189 235, 

and observes that : 

Each polygon is equal to the polygon immediately above it 
in the diagram plus the triangle with 1 less in its side, i. e. the 
triangle in the preceding column. 

Ll2 



516 DIOPHANTUS OF ALEXANDRIA 

4. The vertical columns are in arithmetical progression, the 
common difference being the triangle in the preceding column. 

Plutarch, a contemporary of Nicomachus, mentions another 
method of transforming triangles into squares. Every tri- 
angular number taken eight times and then increased by 1 
gives a square. 

In fact, 8.%n(n+l) + l = (2w+l) a . 

< 
Only a fragment of Diophantus's treatise On Polygonal 

Numbers survives. Its character is entirely different from 
that of the Arithmetica. The method of proof is strictly 
geometrical, and has the disadvantage, therefore, of being long 
and involved. He begins with some preliminary propositions 
of which two may be mentioned. Prop. 3 proves that, if a be 
the first and I the last term in an arithmetical progression 
of n terms, and if s is the sum of the terms, 2s = n(l + a). 
Prop. 4 proves that, if 1, 1+6, 1 + 26, ... 1 +(71 1)6 be an 
A. P., and s the sum of the terms, 



The main result obtained in the fragment as we have it 
is a generalization of the formula 8 . %n(n + 1) + 1 = (2 n + 1 ) 2 . 
Prop. 5 proves the fact stated in Hypsicles's definition and also 
(the generalization referred to) that 

8 P (a 2) + (a 4) 2 = a square, 

where P is any polygonal number with a angles. 

It is also proved that, if P be the nth a-gonal number 
(1 being the first), 



Diophantus deduces rules as follows. 

1. To find the number from its side. 

{2 + (2?t~l)(c6-2)} 2 
8(a-2) 

2. To find the side from the number. 



THE TREATISE ON POLYGONAL NUMBERS 517 
The last proposition, which breaks off in the middle, is : 

Given a number, to find in how many ways it can be 
polygonal. 

The proposition begins in a way which suggests that 
Diophantus first proved geometrically that, if 



-4) 2 = {2 + (2n- 1) (a-2)] 2 , 
then 2P = n{2+ ( n -l)(a-2)}. 

Wertheim (in his edition of Diophantus) has suggested a 
restoration of the complete proof of this proposition, and 
I have shown (in my edition) how the proof can be made 
shorter. Wertheim adds an investigation of the main pro- 
blem, but no doubt opinions will continue to differ as to 
whether Diophantus actually solved it. 



XXI 

COMMENTATORS AND BYZANTINES 

WE have come to the last stage of Greek mathematics ; it 
only remains to include in a last chapter references to com- 
mentators of more or less note who contributed nothing 
original but have preserved, among observations and explana- 
tions obvious or trivial from a mathematical point of view, 
valuable extracts from works which have perished, or 
historical allusions which, in the absence of original docu- 
ments, are precious in proportion to their rarity. Nor must 
it be forgotten that in several cases we probably owe to the 
commentators the fact that the masterpieces of the great 
mathematicians t have survived, wholly or partly, in the 
original Greek or at all. This may have been the case even 
with the works of Archimedes on which Eutocius wrote com- 
mentaries. It was no doubt these commentaries which 
aroused in the school of Isidorus of Miletus (the colleague 
of Anthemius as architect of Saint Sophia at Constantinople) 
a new interest in the works of Archimedes and caused them 
to be sought out in the various libraries or wherever they had 
lain hid. This revived interest apparently had the effect of 
evoking new versions of the famous works commented upon 
in a form more convenient for the student, with the Doric 
dialect of the original eliminated; this translation of the 
Doric into] the more familiar dialect was systematically 
carried out in those books only which Eutocius commented 
on, and it is these versions which alone survive. Again, 
Eutocius's commentary on Apollonius's Conies is extant for 
the first four Books, and it is probably owing to their having 
been commented on by Eutocius, as well as to their being 
more elementary than the rest, that these four Books alone 



SERENUS 519 

survive in Greek. Tannery, as we have seen, conjectured 
that, in like manner, the first six of the thirteen Books of 
Diophantus's Arithmetica survive because Hypatia wrote 
commentaries on these Books only and did not reach the 
others. 

The first writer who calls for notice in this chapter is one 
who was rather more than a commentator in so far as he 
wrote a couple of treatises to supplement the Conies of 
Apollonius, I mean SERENUS. Serenus came from Antinoeia 
or Antinoupolis, a city in Egypt founded by Hadrian (A. D. 
117-38). His date is uncertain, but he most probably be- 
longed to the fourth century A.D., and came between Pappus 
and Theon of Alexandria. He tells us himself that he wrote 
a commentary on the Conies of Apollonius. 1 This has 
perished and, apart from a certain proposition 'of Serenus 
the philosopher, from the Lemmas ' preserved in certain manu- 
scripts of Theon of Smyrna (to the effect that, if a number of 
rectilineal angles be subtended at a point on a diameter of a 
circle which is not the centre, by equal arcs of that circle, the 
angle nearer to the centre is always less than the angle more 
remote), we have only the two small treatises by him entitled 
On the Section of a Cylinder and On the Section of a Cone. 
These works came to be connected, from the seventh century 
onwards, with the Conies of Apollonius, on account of the 
affinity of the subjects, and this no doubt accounts for their 
survival. They were translated into Latin by Commandinus 
in 1566 ; the first Greek text was brought'out by Halley along 
with his Apollonius (Oxford 1710), and we now have the 
definitive text edited by Heiberg (Teubner 1896). 

(a) On the Section of a Cylinder. 

The occasion and the object of the tract On the Section of 
a Cylinder are stated in the preface. Serenus observes that 
many persons who were students of geometry were under the 
srroneous impression that the oblique section of a cylinder 
was different from the oblique section of a cone known as an 
ellipse, whereas it is of course the same curve. Hence he 
thinks it necessary to establish, by a regular geometrical 

1 Serenus, Opitscula, ed. Heiberg, p. 52. 25-6. 



520 COMMENTATORS AND BYZANTINES 

proof, that the said oblique sections cutting all the generators 
are equally ellipses whether they are sections of a cylinder or 
of a cone. He begins with ' a more general definition ' of a 
cylinder to include any oblique circular cylinder. ' If in two 
equal and parallel circles which remain fixed the diameters, 
while remaining parallel to one another throughout, are moved 
round in the planes of the circles about the centres, which 
remain fixed, and if they carry round with them the straight line 
joining their extremities on the same side until they bring it 
back again to the same place, let the surface described by the 
straight line so carried round be called a cylindrical surface! 
The cylinder is the figure contained by the parallel circles and 
the cylindrical surface intercepted by them; the parallel 
circles are the bases, the axis . is the straight line drawn 
through their centres; the generating straight line in any 
position is a side. Thirty-three propositions follow. Of these 
Prop. 6 proves the existence in an oblique cylinder of the 
parallel circular sections subcontrary to the series of which 
the bases are two, Prop. 9 that the section by any plane not 
parallel to that of the bases or of one of the subcontrary 
sections but cutting all the generators is not a circle ; the 
next propositions lead up to the main results, namely those in 
Props. 14 and 16, where the said section is proved to have the 
property of the ellipse which we write in the form 



and in Prop. 17, where the property is put in the Apollonian 
form involving the latus rectum, QV 2 = PV . VR (see figure 
on p. 137 above), which is expressed by saying that the square 
on the semi-ordinate is equal to the rectangle applied to the 
latus rectum PL, having the abscissa PFas breadth and falling 
short by a rectangle similar to the rectangle contained by the 
diameter PP' and the latus rectum PL (which is determined 
by the condition PL . PP'= DIY* and is drawn at right angles 
to PF). Prop. 18 proves the corresponding property with 
reference to the conjugate diameter DD' and the correspond- 
ing latus rectum, and Prop. 19 gives the main property in the 
form QF 2 : PF . P / F = Q'F' 2 : PF 7 . P*V. Then comes the 
proposition that ' it is possible to exhibit a cone and a cylinder 
which are alike cut in one and the same ellipse ' (Prop. 20). 



SERENUS 521 

Serenus then solves such problems as these: Given a cone 
(or cylinder) and an ellipse on it, to find the cylinder (cone) 
which is cut in the same ellipse as the cone (cylinder) 
(Props. 21, 22); given a cone (cylinder), to find a cylinder 
(cone) and to cut both by one and the same plane so that the 
sections thus made shall be similar ellipses (Props. 23, 24). 
Props. 27, 28 deal with similar elliptic sections of a scalene 
cylinder and cone ; there are two pairs of infinite sets of these 
similar to any one given section, the first pair being those 
which are parallel and subcontrary respectively to the given 
section, the other pair subcontrary to one another but not to 
either of the other sets and having the conjugate diameter 
occupying the corresponding place to the transverse in the 
other sets, and vice versa. 

In the propositions (29-33) from this point to the end of 
the book Serenus deals with what is really an optical pro- 
blem. It is introduced by a remark about a certain geometer, 
Peithon by name, who wrote a tract on the subject of 
parallels. Peithon, not being satisfied with Euclid's treat- 
ment of parallels, thought to define parallels by means of an 
illustration, observing that parallels are such lines as are 
shown on a wall or a roof by the shadow of a pillar with 
a light behind it. This definition, it appears, was generally 
ridiculed ; and Serenus seeks to rehabilitate Peithon, who 
was his friend, by showing that his statement is after all 
mathematically sound. He therefore proves, with regard to 
the cylinder, that, if any number of rays from a point outside 
the cylinder are drawn touching it on both sides, all the rays 
pass through the sides of a parallelogram (a section of the 
cylinder parallel to the axis) Prop. 29 and if they are 
produced farther to meet any other plane parallel to that 
of the parallelogram the points in which they meet the plane 
will lie on two parallel lines (Prop. 30) ; he adds that the lines 
will not seem parallel (vide Euclid's Optics, Prop. 6). The 
problem about the rays touching the surface of a cylinder 
suggests the similar one about any number of rays from an 
external point touching the surface of a cone ; these meet the 
surface in points on a triangular section of the cone (Prop. 32) 
and, if produced to meet a plane parallel to that of the 
triangle, meet that plane in points forming a similar triangle 



522 COMMENTATORS AND BYZANTINES 

(Prop. 33). Prop. 31 preceding these propositions is a par- 
ticular case of the constancy of the anharmonic ratio of a 
pencil of four rays. If two sides AB, AC of a triangle meet 
a transversal through D, an external point, in E, .Fantl another 
ray AG between AB and AC cuts DEF in a point G such 
that ED : DF = EG : GF, then any other transversal through 
1) meeting AB, AG, AC in K, L, M is also divided harmoni- 
cally, i.e. KD : DM = KL : LM. To prove the succeeding pro- 
positions, 32 and 33, Serenus uses this proposition and a 
reciprocal of it combined with the harmonic property of the 
pole and polar with reference to an ellipse. 

(/3) On the Section of a Gone. 

The treatise On the Section of a Cone is even less important, 
although Serenus claims originality for it. It deals mainly 
with the areas of triangular sections of right or scalene cones 
made by planes passing through the vertex and either through 
the axis or not through the axis, showing when the area of 
a certain triangle of a particular class is a maximum, under 
what conditions two triangles of a class may be equal in area, 
and so on, and solving in some easy cases the problem of 
finding triangular sections of given area. This sort of investi- 
gation occupies Props. 1-57 of the work, these propositions 
including various lemmas required for the proofs of the 
substantive theorems. Props. 58-69 constitute a separate 
section of the book dealing with the volumes of right cones 
in relation to their heights, their bases and the areas of the 
triangular sections through the axis. 

The essence of the first portion of the book up to Prop. 57 
is best shown by means of modern notation. We will call h 
the height of a right cone, r the radius of the base ; in the 
case of an oblique cone, let p be the perpendicular from the 
vertex to the plane of the base, d the distance of the foot of 
this perpendicular from the centre of the base, r the radius 
of the base. 

Consider first the right cone, and let 2x be the base of any 
triangular section through the vertex, while of course 2r is 
the base of the triangular section through the axis. Then, if 
A be the area of the triangular section with base 2x, 



SERENUS 523 

Observing that the sum of x* and r a x 2 + IP is constant, we 
see that A 2 , and therefore A, is a maximum when 

= r*-a; + A*, or a* = (r 8 + A 8 ) ; 

and, since a? is not greater than r, it follows that, for a real 
value of x (other than r), h is less than r, or the cone is obtuse- 
angled. When h is not less than r, the maximum triangle is 
the triangle through the axis and vice versa (Props. 5, 8) ; 
when h = r, the maximum triangle is also right-angled 
(Prop. 13). 

If the triangle with base 2 c is equal to the triangle through 
the axis, A a r* = c 2 (r a -c 8 + A a ), or (r a -c 2 ) (c 2 -/* 2 ) = 0, and, 
since c<r, A =c, so that h<r (Prop. 10). If x lies between 7* 
and c in this case, (r 2 - x 1 ) (a 8 - h*) > or * 2 (r 2 - jp + A 2 ) > A 2 r 2 , 
and the triangle with base 2x is greater than either of the 
equal triangles with bases 2r, 2c, or 2h (Prop. 11). 

In the case of the scalene cone Sereiius compares individual 
triangular sections belonging to one of three classes with other 
sections of the same class as regards their area. The classes 
are : 

(1) axial triangles, including all sections through the axis; 

(2) isosceles sections, i.e. the sections the bases of which are 
perpendicular to the projection of the axis of the cone on the 
plane of the base ; 

(3) a set of triangular sections the bases of which are (a) the 
diameter of the circular base which passes through the foot of 
the perpendicular from the vertex to the plane of the base, and 
(b) the chords of the circular base parallel to that diameter. 

After two preliminary propositions (15, 16) and some 
lemmas, Serenus compares the areas of the first class of 
triangles through the axis. If, as we said, p is the perpen- 
dicular from the vertex to the plane of the base, d the distance 
of the foot of this perpendicular from the centre of the base, 
and the angle which the base of any axial triangle with area 
A makes with the base of the axial triangle passing through 
p the perpendicular, 

A = 



This area is a minimum when 6 = 0, and increases with 



524 COMMENTATORS AND BYZANTINES 

until 6 = f TT when it is a maximum, the triangle being then 
isosceles (Prop. 24). 

In Prop. 29 Serenus takes up the third class of sections with 
bases parallel to d. If the base of such a section is 2x, 

A = aVr 2 - 





and, as in the case of the right cone, we must have for a real 
maximum value 

oj 2 = ^ (r 2 +p*), while x < r, 

so that, for a real value of x other than r, p must be less than 
r, and, if p is not less than r, the maximum triangle is that 
which is perpendicular to the base of the cone and has 2r for 
its base (Prop. 29). If p<r, the triangle in question is not 
the maximum of the set of triangles (Prop. 30). 

Coming now to the isosceles sections (2), we may suppose 
2 to be the angle subtended at the centre of the base by\the 
base of the section in the direction away from the projection 
of the vertex. Then 

A = r sin 0V {p* + (d + r cos 0) 2 } . 

If A be the area of the isosceles triangle through the axis, 
we have 



If A = <A , we must have for triangles on the side of the 
centre of the base of the cone towards the vertex of the cone 
(since cos 6 is negative for such triangles) 

2> 2 + <2 2 <r 2 sin 2 0, and a fortiori jp a + d a <r a (Prop. 35). 

If > 2 + d 2 ir 2 , A is always greater than A, so that A is the 
maximum isosceles triangle of the set (Props. 31, 32). 

If A is the area of any one of the isosceles triangles with 
bases on the side of the centre of the base of the cone away 
from the projection of the vertex, cos is positive and A^ is 
proved to be neither the minimum nor the maximum triangle 
of this set of triangles (Props. 36, 40-4). 

In Prop. 45 Serenus returns to the set of triangular sections 
through the axis, proving that the feet of the perpendiculars 
from the vertex of the cone on their , bases all lie on a circle 
the diameter of which is the straight line joining the centre of 



SERENUS 525 

the base of the cone to the projection of the vertex on its 
plane ; the areas of the axial triangles are therefore propor- 
tional to the generators of the cone with the said circle as 
base and the same vertex as the original cone. Prop. 50 is to 
the effect that, if the axis of the cone is equal to the radius of 
the base, the least axial triangle is a mean proportional 
between the greatest axial triangle and the isosceles triangular 
section perpendicular to the base ; that is, with the above nota- 
tion, if r = V(p* + tV), then r V(f + d 2 ) :rp = rp:p </(r* - d 3 ), 
which is indeed obvious. 

Prop. 57 is interesting because of the lemmas leading to it. 
It proves that the greater axial triangle in a scalene cone has 
the greater perimeter, and conversely. This is proved by 
means of the lemma (Prop. 54), applied to the variable sides 
of axial triangles, that if ti 2 + d? = 6 2 -f c 2 and a > b ^ c> rf, 
then a + d< b + c (a, d are the sides other than the base of one 
axial triangle, and 6, c those of the other axial triangle com- 
pared with it; and if ABC, ADEbe two axial triangles and 
the centre of the base, BA* + AC*= DA 2 + AE* because each 
of these sums is equal to 2 A O 2 + 2 BO 2 , Prop. 1 7). This proposi- 
tion again depends on the lemma (Props. 52, 53) that, if 
straight lines bo 'inflected' from the ends of the base of 
a segment of a circle to the curve (i. e. if we join the ends 
of the base to any point on the curve) the lino (i.e. the sum of 
the chords) is greatest when the point taken is the middle 
point of the arc, and diminishes as the point is taken farther 
and farther from that point. 

Let B be the middle point of the 
arc of the segment ABC, 7), E any 
other points on the curve towards 
(7; I say that 

BC>AD+I)C>AE+EC. 




With B as centre and BA as radius 
describe a circle, and produce AB, 
AD, AE to meet this circle in F, G, 
H. JoinFC,GC,HC. 

Since AB = EG = BF, we have AF = AB + BC. Also the 
angles BFC, BCF arc equal, and each of them is half of 
the angle ABC. 



526 COMMENTATORS AND BYZANTINES 
Again LAGG = LAFG = \LAEG = \LADG\ 

therefore the angles DOC, DOG are equal and DG = 

therefore AG = AD + DC. 

Similarly EH = KG and ^7/ = AE+ EC. 

But, by Eucl. III. 7 or 15, AF>AG>AIf, and so on ; 

therefore AB + BC > A D + DC > AE + EC, and so on. 

In the particular case where the segment ABC is a semi- 
circle AB* + BC 2 = AG* = AD* + DC*, &c., and the result of 
Prop. 57 follows. 

Props. 58-69 are propositions of this sort : In equal right 
cones the triangular sections through th$ axis are reciprocally 
proportional to their bases and conversely (Props. 58, 59) ; 
right cones of equal height have to one another the ratio 
duplicate of that of their axial triangles (Prop. 62); right 
cones which are reciprocally proportional to their bases have 
axial triangles which are to one another reciprocally in the 
triplicate ratio of their bases and conversely (Props. 66, 67); 
and so on. 

THEON OF ALEXANDRIA lived towards the end of the fourth 
century A.D. Suidas places him in the reign of Theodosius I 
(379-95); he tells us himself that he observed a solar eclipse 
at Alexandria in the year 365, and his notes on the chrono- 
logical tables of Ptolemy extend down to 372. 

Commentary on the Syntaxis. 

We have already seen him as the author of a commentary 
on Ptolemy's Syntaxis in eleven Books. This commentary is 
not calculated to give us a very high opinion of Theon's 
mathematical calibre, but it is valuable for several historical 
notices that it gives, and we are indebted to it for a useful 
account of the Greek method of operating with sexagesimal 
fractions, which is illustrated by examples of multiplication, 
division, .and the extraction of the square root of a non-square 
number by way of approximation. These illustrations of 
numerical calculation have already been given above (vol. i, 



THEONOF ALEXANDRIA 527 

pp. 58-63). Of the historical notices we may mention the 
following. (1) Theon mentions the treatise of Menelaus On 
Chords in a Circle, i.e. Menelau&'s Table of Chords, which came 
between the similar Tables of Hipparchus and Ptolemy. (2) A 
quotation from Diophantus f uriiishes incidentally a lower limit 
for the date of the Arithmetica. (3) It is in the commentary 
on Ptolemy that Thcon tells UH that the second part of Euclid 
VI. 33 relating to sectors in equal circles was inserted by him- 
self in his edition of the Elements, a notice which is of capital 
importance in that it enables the Theonine manuscripts of 
Euclid to be distinguished from the ante-Theonine, and is 
therefore the key to the question how far the genuine text 
of Euclid was altered in Theon's edition. (4) As we have 
seen (pp. 207 sq.), Theon, & propos of an allusion of Ptolemy 
to the theory of isoperimetric figures, has preserved for us 
several propositions from the treatise by Zenodorus on that 
subject. 

Theon's edition of Euclid's Elements. 

Wo are able to judge of the character of Theon's edition of 
Euclid by a comparison between the Theonine manuscripts 
and the famous Vatican MS. 190, which contains an earlier 
edition than Theon's, together with certain fragments of 
ancient papyri. It appears that, while Theon took some 
trouble to follow older manuscripts, it was not so much his 
object to get the most authoritative text as to make what he 
considered improvements of one sort or other, (l) He made 
alterations where he found, or thought he found, mistakes in 
the original; while he tried to remove some real blots, he 
altered other passages too hastily when a little more considera- 
tion would have shown that Euclid's words are right or could 
be excused, and offer no difficulty to an intelligent reader. 
(2) He made emendations intended to improve the form or 
diction of Euclid ; in general they were prompted by a desire 
to eliminate anything which was out of the common in expres- 
sion or in form, in order to reduce the language to one and the 
same standard or norm. (3) He bestowed, however, most 
attention upon additions designed to supplement or explain 
the original ; (a) he interpolated whole propositions where he 
thought them necessary or useful, e.g. the addition to VI. 33 



528 COMMENTATORS AND BYZANTINES 

already referred to, a second case to VI. 27, a porism or corollary 
to II. 4, a second porism to III. 16, the proposition VII. 22, 
a lemma after X. 12, besides alternative proofs here and there ; 
(6) he added words for the purpose of making smoother and 
clearer, or more precise, things which Euclid had expressed 
with unusual brevity, harshness, or carelessness ; (c) he sup- 
plied intermediate steps where Euclid's argument seemed too 
difficult to follow. In short, while making only inconsider- 
able additions to the content of the Elements, he endeavoured 
to remove difficulties that might be felt by learners in study- 
ing the book, as a modern editor might do in editing a classical 
text-book for use in schools ; and there is no doubt that his 
edition was approved by his pupils at Alexandria for whom it 
was written, as well as by later Greeks, who used it almost 
exclusively, with the result that the more ancient text is only 
preserved complete in one manuscript. 

Edition of the Optics of Euclid. 

In addition to the Elements, Theon edited the Optics of 
Euclid; Theon's recension as well as the genuine work is 
included by Heiberg in his edition. It is possible that the 
Catoptrica included by Heiberg in the same volume is also by 
Theon. 

4 

Next to Theon should be mentioned his daughter HYPATIA, 
who is mentioned by Theon himself as having assisted in the 
revision of the commentary on Ptolemy. This learned lady 
is said to have been mistress of the whole of pagan science, 
especially of philosophy and medicine, and by her eloquence 
and authority to have attained such influence that Christianity 
considered itself threatened, and she was put to death by 
a fanatical mob in March 415. According to Suidas she wrote 
commentaries on Diopharitus, on the Astronomical Canon (of 
Ptolemy) and on the Conies of 'Apollonius. These works 
have not survived, but it has been conjectured (by Tannery) 
that the remarks of Psellus (eleventh century) at the begin- 
ning of his letter about Diophantus, Anatolius, and the 
Egyptian method of arithmetical reckoning were taken bodily 
from some manuscript of Diophantus containing an ancient 
and systematic commentary which may very well have been 
that of Hypatia. Possibly her commentary may have extended 



HYPATIA. PORPHYRY 529 

only to the first six Books, in which case the fact that Hypatia 
wrote a commentary on them may account for the survival of 
these Books while the rest of the thirteen were first forgotten 
and .then lost. 

It will be convenient to take next the series of Neo- 
Platonist commentators. It does not appear that Ammonius 
Saccas (about A.D. 175-250), the founder of Neo-Platonism, or 
his pupil Plotinus (A.D. 204-69), who first expounded the 
doctrines in systematic form, had any special connexion with 
mathematics, but PORPHYRY (about 232-304), the disciple of 
Plotinus and the reviser and editor of his works, appears to 
have written a commentary on the Elements. This we gather 
from Proclus, who quotes from Porphyry comments on Eucl. 
I. 14 and 26 and alternative proofs of I. 18, 20. It is possible 
that Porphyry's work may have been used later by Pappus in 
writing his own commentary, and Proclus may have got his 
references from Pappus, but the form of these references sug- 
gests that he had direct access to the original commentary of 
Porphyry. 

IAMBLICHUS (died about A.D. 330) was the author of a com- 
mentary on the Introductio arithmetica of Nicomachus, and 
of other works which have already been mentioned. He was 
a pupil of Porphyry as well as of Anatolius, also a disciple of 
Porphyry. 

But the most important of the Neo-Platonists to the his- 
torian of mathematics is PROCLUS (A.D. 410-85). Proclus 
received his early training at Alexandria, where Olympio- 
dorus was his instructor in the works of Aristotle, and 
mathematics was taught him by one Heron (of course a 
different Heron from the ' mechauicus Hero' of the Metrica, 
&c.). He afterwards went to Athens, where he learnt the 
Neo-Platonic philosophy from Plutarch, the grandson of Nes- 
torius, and from his pupil Syrianus, and became one of its 
most prominent exponents. He speaks everywhere with the 
highest respect of his masters, and was in turn regarded with 
extravagant veneration by his contemporaries, as we learn 
from Marinus, his pupil and biographer. On the death of 
Syrianus he was put at the head of the Neo-Platonic school. 
He was a man of untiring industry, as is shown by the 



530 COMMENTATORS AND BYZANTINES 

number of books which he wrote, including a large number of 
commentaries, mostly on the dialogues of Plato (e.g. the 
Timaeus, the Republic, the Parme aides, the Gratylus). He 
was an acute dialectician and pre-eminent among his contem- 
poraries in the range of his learning; he was a competent 
mathematician ; he was even a poet. At the same time he 
was a believer in all sorts of myths and mysteries, and 
a devout worshipper of .divinities both Greek and Oriental. 
He was much more a philosopher than a mathematician. In 
his commentary on the Timaeus, when referring to the ques- 
tion whether the sun occupies a middle place among the 
planets, he speaks as no real mathematician could have 
spoken, rejecting the view of Hipparchus and Ptolemy because 
6 Qtovpyos (sc. the Chaldean, says Zeller) thinks otherwise, 
c whom it is not lawful to disbelieve '. Martin observes too, 
rather neatly, that ' for Proclus the Elements of Euclid had 
the good fortune not to be contradicted either by the Chaldean 
Oracles or by the speculations of Pythagoreans old and new '. 

Commentary on Euclid, Book I. 

For us the most important work of Proclus is his commen- 
tary on Euclid, Book I, because it is one of the main sources 
of our information as to the history of elementary geometry. 
Its great value arises mainly from the fact that Proclus had 
access to a number of historical and critical works which are 
now lost except for fragments preserved by Proclus and 
others. 

(a) Sources of the Commentary. 

The historical work the loss of which is most deeply to be 
deplored is the History of Geometry by Eudemus. There 
appears to be no reason to doubt that the work of Eudemus 
was accessible to Proclus at first hand. For the later writers 
Simplicius and Eutocius refer to it in terms such as leave no 
doubt that they had it before them. Simplicius, quoting 
Eudemus as the best authority on Hippocrates's quadratures 
of lunes, says he will set out what Eudemus says ' word for 
word ', adding only a little explanation in the shape of refer- 
ences to Euclid's Elements 'owing to the memorandum-like 
style of Eudemus, who sets out his explanations in the abbre- 



PROCLUS 531 

viated form usual with ancient writers. Now in the second 
book of the history of geometry he writes as follows '.* In 
like manner Eutocius speaks of the paralogisms handed down 
in connexion with the attempts of Hippocrates and Antiphon 
to square the circle, 'with which I imagine that all persons 
are accurately acquainted who have examined (7reovc/i/*j>ouy) 
the geometrical history of Eudemus and know the Geria 
Aristotelica '. 2 

The references by Proclus to Eudemus by name are not 
indeed numerous ; they are five in number ; but on the other 
hand he gives at least as many other historical data which can 
with great probability be attributed to Eudemus. 

Proclus was even more indebted to Geminus, from whom 
he borrows long extracts, often mentioning him by name 
there are some eighteen such references but often omitting 
to do so. We are able to form a tolerably certain judge- 
ment as to the origin of the latter class of passages on the 
strength of the similarity of the subjects treated and the views 
expressed to those found in the acknowledged extracts. As 
we have seen, the work of Geminus mainly cited seems to 
have borne the title The Doctrine or Theory of the Mathematics, 
which was a very comprehensive work dealing, in a portion of 
it, with the * classification of mathematics '. 

We have already discussed the question of the authorship 
of the famous historical summary given by Proclus. It is 
divided, as every one knows, into two distinct parts between 
which comes the remark, ' Those who compiled histories 
bring the development of this science up to this point. Not 
much younger than these is Euclid, who ', &c. The ultimate 
source at any rate of the early part of the summary must 
presumably have been the great work of Eudemus above 
mentioned. 

It is evident that Proclus had before him the original works 
of Plato, Aristotle, Archimedes and Plotinus, the SvfipiKTa of 
Porphyry and the works of his master Syrianus, as well as a 
group of works representing the Pythagorean tradition on its 
mystic, as distinct from its mathematical, side, from Philo- 
laus downwards, and comprising the more or less apocryphal 

1 Simplicius on Arist. Phys., p. 60. 28, Diels. 

2 Archimedes, ed. Heib., vol. iii, p. 228. 17-19. 



532 COMMENTATORS AND BYZANTINES 



Xoyoy of Pythagoras, the Oracles (\6yia) and Orphic 
verses. 

The following will be a convenient summary of the other 
works used by Proclus, and will at the same time give an 
indication of the historical value of his commentary on 
Euclid, Book I : 

Eudemus : History of Geometry. 

Geminus : The Theory of the Mathematical Sciences. 

Heron: Commentary on the Elements of Euclid. 

Porphyry: 

Pappus: 

Apollonius of Perga : A work relating to elementary 

geometry. 

Ptolemy : On the 'parallel-postulate. 
Posidonius : A book controverting Zeno of Sidon. 
Carpus: Astronomy. 
Syrianus: A discussion on the angle. 

(ft) Character of the Commentary. 

We know that in the Neo-Platonic school the pupils learnt 
mathematics ; and it is clear that Proclus taught this subject, 
and that this was the origin of his commentary. Many 
passages show him as a master speaking to scholars ; in one 
place he speaks of ' my hearers *. 1 Further, the pupils whom 
he was addressing were beginners in mathematics ; thus in one 
passage he says that he omits ' for the present ' to speak of the 
discoveries of those who employed the curves of Nicomedes 
and Hippias for trisecting an angle, and of those who used the 
Archimedean spiral for dividing an angle in a given ratio, 
because these things would be ' too difficult for beginners '. 2 
But there are signs that the cpmmentary was revised and 
re-edited for a larger public; he speaks for instance in one 
place of ' those who will corne across his work \ 3 There are 
also passages, e.g. passages about the cylindrical helix, con- 
choids and cissoids, which would not have been understood by 
the beginners to whom he lectured. 

1 Proclus on Eucl. I, p. 210. 19. 2 /&., p. 272. 12. 

3 Ib., p. 84. 9. 



PROCLUS 533 

The commentary opens with two Prologues. The first is 
on mathematics in general and its relation to, and use in, 
philosophy, from which Proclus passes to the classification of 
mathematics. Prologue II deals with geometry generally and 
its subject-matter according to Plato, Aristotle and others. 
After this section comes the famous summary (pp. 64-8) 
ending with a eulogium of Euclid, witli particular reference 
to the admirable discretion shown in the selection of the pro- 
positions which should constitute the Elements of geometry, 
the ordering of the whole subject-matter, the exactness and 
the collusiveness of the demonstrations, and the power with 
which every question is handled. Generalities follow, such as 
the discussion of the nature of elements, the distinction between 
theorems and problems according to different authorities, and 
finally a division of Book I into three main sections, (1) the 
construction and properties of triangles and their parts and 
the comparison between triangles in respect of their angles 
and sides, (2) the properties of parallels and parallelograms 
and their construction from certain data, and (3) the bringing 
of triangles and parallelograms into relation as regards area. 

Coming to the Book itself, Proclus deals historically and 
critically with all the definitions, postulates and axioms in 
order. The notes on the postulates and axioms are preceded 
by a general discussion of the- principles of geometry, hypo- 
theses, postulates and axioms, and their relation to one 
another ; here as usual Proclus quotes the opinions of all the 
important authorities. Again, when he comes to Prop. 1, he 
discusses once more the difference between theorems and 
problems, then sets out and explains the formal divisions of 
a proposition, the enunciation (Trporatny), the settiny-out 
(cAcflecny), the definition or specification (Siopurpos), the con- 
sti*uction (KaraaKtvri), the proof (a7ro#e*ty), the conclusion 
(crvfjLTTtpao-jjLa), and finally a number of other technical terms, 
e.g. things said to be given, in the various senses of this term, 
the lemma, the case, the porism in its two senses, the objection 
(ci/<rra<ns), the reduction of a problem, reductio ad absurdum, 
analy&is and synthesis. 

In his comments on the separate propositions Proclus 
generally proceeds in this way : first he gives explanations 
regarding Euclid's proofs, secondly he gives a few different 



534 COMMENTATORS AND BYZANTINES 

cases, mainly for the sake of practice, and thirdly he addresses 
himself to refuting objections which cavillers had taken or 
might take to particular propositions or arguments. He does 
not seem to have had any notion of correcting or improving 
Euclid; only in one place does he propose anything of his 
own to get over a difficulty which he finds in Euclid ; this is 
where he tries to prove the parallel-postulate, after giving 
Ptolemy's attempt to prove it and pointing out objections to 
Ptolemy's proof. 

The book is evidently almost entirely a compilation, though 
a compilation ' in the better sense of the term '. The onus 
probandi is on any one who shall assert that anything in it is 
Proclus's own ; very few things can with certainty be said to 
be so. Instances are (1) remarks on certain things which he 
quotes from Pappus, since Pappus was the last of the com- 
mentators whose works he seems to have used, (2) a defence 
of Geminus against Carpus, who criticized Geminus's view of 
the difference between theorems and problems, and perhaps 
(3) criticisms of certain attempts by Apollonius to improve on 
Euclid's proofs and constructions ; but the only substantial 
example is (4) the attempted proof of the parallel-postulate, 
based on an * axiom ' to the effect that, * if from one point two 
straight lines forming an angle be produced ad injinitum, the 
distance between them when so produced ad infiuitum exceeds 
any finite magnitude (i. e. length) ', an assumption which 
purports to be the equivalent of a statement in Aristotle. 1 
Philoponus says that Proclus as well as Ptolemy wrote a whole 
book on the parallel-postulate. 2 

It is still not quite certain whether Proclus continued his 
commentaries beyond Book I. He certainly intended to do so, 
for, speaking of the trisection of an angle by means of certain 
curves, he says, c we may perhaps more appropriately examine 
these things on the third Book, where the writer of the 
Elements bisects a given circumference', and again, after 
saying that of all parallelograms which have the same peri- 
meter the square is the greatest ' and the rhomboid least of 
all ', he adds, ' But this we will prove in another place, for it 
is more appropriate to the discussion of the hypotheses of the 

1 De caelo, i. 5, 271 b 28-30. 

2 Philoponus on Anal. Post. i. 10, p. 214 a 9-12, Brandis. 



PttOCLUS 535 

second Book 3 . But at the time when the commentary on 
Book I was written he was evidently uncertain whether he 
would be able to continue it, for at the end he says, ' For my 
part, if I should be able to discuss the other Books in the 
'same way, I should give thanks to the gods; but, if other 
cares should draw me away, 1 beg those who are attracted by 
this subject to complete the exposition of the other Books as 
well, following the same method and addressing themselves 
throughout to the deeper and more sharply defined questions 
involved '. l Wachsmuth, finding a Vatican manuscript contain- 
ing a collection of scholia on Books I, II, V, VI, X, headed Eh ra 
TOi\Ta Trpo\afjif$av6p.va K T&V HpoK\ov criTopdSvjv 
Kar* tTriTOfjLrjis, and seeing that the scholia on Book I were 
extracts from the extant commentary of Proclus, concluded 
that those on the other Books were also from Proclus; but 
the TT/DO- in 7r/>oXa/i/Jai>6/zj>a rather suggests that only the 
scholia to Book I are from Proclus. Heiberg found and 
published in 1903 a scholium to X. 9, in which Proclus is 
expressly quoted as the authority, but he does not regard 
this circumstance as conclusive. On the other hand, Heiberg 
has noted two facts which go against the view that Proclus 
wrote on the later Books: (1) the scholiast's copy of 
Proclus was not much better than our manuscripts ; in 
particular, it had the same lacunae in the notes to I. 36, 
37, and I. 41-3; this makes it improbable that the scholiast 
had further commentaries of Proclus which have vanished 
for us ; (2) there is no trace in the scholia of the notes 
which Proclus promised in the passages already referred to. 
All, therefore, that we can say is that, while the Wachsmuth 
scholia may be extracts from Proclus, it is on the whole 
improbable. 

Hypotyposis of Astronomical Hypotheses 

Another extant work of Proclus which should be referred 
to is his Hypotyposis of Astronomical Hypotheses, a sort of 
readable and easy introduction to the astronomical system 
of Hipparchus and Ptolemy. It has been well edited by 
Munitius (Teubner, 1909). Three things may be noted as 

1 Proclus on Eucl. I, p. 432. 9-15. 



536 COMMENTATORS AND BYZANTINES 

regards this work. It contains 1 a description of the method 
of measuring the sun's apparent diameter by means of 
Heron's water-clock, which, by comparison with the corre- 
sponding description in Theon's commentary to the Syntaxis 
of Ptolemy, is seen to have a common source with it. That 
source is Pappus, and, inasmuch as Proclus has a figure (repro- 
duced by Manitius in his text from one set of manuscripts) 
corresponding to the description, while the text of Theon has 
no figure, it is clear that Proclus drew directly on Pappus, 
who doubtless gave, in his account of the procedure, a figure 
taken from Heron's own work on water-clocks. A simple 
proof of the equivalence of the epicycle and eccentric hypo- 
theses is quoted by Proclus from one Hilarius of Antioch. 2 
An interesting passage is that in chap. 4 (p. 130, 18) where 
Sosigenes the Peripatetic is said to have recorded in his work 
'on reacting spheres' that an annular eclipse of the sun is 
sometimes observed at times of perigee; this is, so far as 
I know, the only allusion in ancient times to annular eclipses, 
and Proclus himself questions the correctness of Sosigenes's 
statement. 

Commentary on the Republic. 

The commentary of Proclus on the Republic contains some 
passages of great interest to the historian of mathematics. 
The most important is that 3 in which Proclus indicates that 
Props. 9, 10 of Euclid, Book II, are Pythagorean proposi- 
tions invented for the purpose of proving geometrically the 
fundamental property of the series of side-' and ' diameter-' 
numbers, giving successive approximations to the value of 
\/2 (see vol. i, p. 93). The explanation 4 of the passage in 
Plato about the Geometrical Number is defective and dis- 
appointing, but it contains an interesting reference to one 
Paterius, of date presumably intermediate betwefen Nestorius 
and Proclus. Paterius is said to have made a calculation, in 
units and submultiples, of the lengths of different segments of 

1 Proclus, Hypotyposis, c. 4, pp. 120-22. 
8 Ib., c. 3, pp. 76, 17 sq. 

8 Prodi Diadochi in Platonis EempuUicam Commentarii, ed. Kroll, 
vol. ii, p. 27. 
Ib., vol. ii, pp. 36-42. 



PKOCLUS. MARINUS 537 

straight lines in a figure formed by taking a triangle with 

sides 3, 4, 6 as ABC, then drawing 

BD from the right angle B perpen- 

dicular to AC, and lastly drawing 

perpendiculars DE, DF to AB, BC. 

A diagram in the text with the 

lengths of the segments shown along- 

side them in the usual numerical 

notation shows that Paterius obtained from the data AB = 3, 

BC = 4, CA = 5 the following : 




= -ye'= 3> 



This is an example of the Egyptian method of stating frac- 
tions preceding by some three or four centuries the exposition 
of the same method in the papyrus of Akhmlm. 

MARINUS of Neapolis, the pupil and biographer of Proclus, 
wrote a commentary or rather introduction to the Data of 
Euclid. 1 It is mainly taken up with a discussion of the 
question ri TO 5c5o/ici/o^, what is meant by given 1 There 
were apparently many different definitions of the term given 
by earlier and later authorities. Of those who tried to define 
it in the simplest way by means of a single differentia, three 
are mentioned by name. Apollonius in his work on i/eu<reiy 
and his 'general treatise' (presumably that on elementary 
geometry) described the given as assigned or fixed (reray- 
/xo/oj/), Diodorus called it known (yvtopipov)] others regarded 
it as rational (faroy) and Ptolemy is classed with these, rather 
oddly, because ' he called those things given the measure of 
which is given either exactly or approximately*. Others 

1 See Heiberg and Menge's Euclid, vol. vi, pp. 234-56. 



538 COMMENTATORS AND BYZANTINES 

combined two of these ideas and called it assigned OY fixed 
and procurable or capable of being found (rr6pip.oi>)\ others 
c fixed and known ', and a third class * known and procurable ' 
These various views are then discussed at length. 

DOMNINUS of Larissa, a pupil of Syrianus at the same time 
as. Proclus, wrote a Manual of Introductory Arithmetic eyxei- 
pi'Stov dpiQftrjTiKfjs /o-aya>y^y, which was edited by Boissonade 1 
and is the subject of two articles by Tannery, 2 who also left 
a translation of it, with prolegomena, which has since been 
published. 3 It is a sketch of the elements of the theory of 
numbers, very concise and well arranged, and -is interesting 
because it indicates a serious attempt at a reaction against the 
Introductio arithmetica of Nicomachus and $ return to the 
doctrine of Euclid. Besides Euclid, Nicomachus and Theon 
of Smyrna, Domninus seems to have used another sourco, 
now lost, which was also drawn upon by lamblichus. At the 
end of this work Domninus foreshadows a more complete 
treatise on the theory of numbers under the. title Elements of 
Arithmetic (apiQ^riK^ oTcux^oxny), but whether this was 
ever written or not we do not know. Another tract 
attributed to Domninus 7ro>9 GCTTI \6yov K \6yov a0eAea> 
(how a ratio can be taken out of a ratio) has been published 
with a translation by Ruelle 4 ; if it is not by Domninus, it 
probably belongs to the same period. 

A most honourable place in our history must be reserved 
for SIMPLICIUS, who has been rightly called 'the excellent 
Siinplicius, the Aristotle-commentator, to whom the world can 
never be grateful enough for the preservation of the frag- 
ments of Parmenides, Empedocles, Anaxagoras, Melissus, 
Theophrastus and others' (v. Wilamowitz-Mollendortt). He 
lived in the first half of the sixth century and was a pupil, 
first of Ammonius of Alexandria, and then of Damascius, 
the last head of the Platonic school at Athens. When in the 
year 529 the Emperor Justinian, in his zeal to eradicate 
paganism, issued an edict forbidding the teaching of philo- 

1 Anecdota Graeca, vol. iv, pp. 413-29. 

2 Mtmoires scientifiques, vol. ii, nos. 35, 40. 

8 Revue des Mudes yrecques, 1906, pp. 359- 82 ; Mtmoires 
vol. iii, pp. 256-81. 
4 Revue de Philoloyie, 1883, p. 83 sq. 



DOMNINUS. SIMPLICIUS 539 

sophy at Athens, the last members of the school, including 
Damascius and Simplicius, migrated to Persia, but returned 
about 533 to Athens, where Simplicius continued to teach for 
some time though the school remained closed. 

Extracts from Eudemus. 

To Simplicius we owe two long extracts of capital impor- 
tance for the history of mathematics and astronomy. The 
first is his account, based upon and to a large extent quoted 
textually from Eudemus's History of Geometry, of the attempt 
by Antiphon to square. the circle and of the quadratures of 
lunes by Hippocrates of Chios. It is contained in Simplicity's 
commentary on Aristotle's Physics, 1 and has been the subject 
of a considerable literature extending from 1870, the date 
when Bretsclmeider first called attention to it, to the latest 
critical edition with translation and notes by Rudio (Teubner, 
1907). It has already been discussed (vol. i, pp. 183-99). 

The second, and not less important, of the two passages is 
that containing the elaborate and detailed account of the 
system of concentric spheres, as first invented by Eudoxus for 
explaining the apparent motion of the sun, moon, and planets, 
and of the modifications made by Callippus and Aristotle. It 
is contained in the commentary on Aristotle's De caelo*\ 
Simplicius quotes largely from Sosigenes the Peripatetic 
(second century A.D.), observing that he in his turn drew 
from Eudemus, who dealt with the subject in the second 
book of his History of Astronomy. It is this passage of 
Simplicius which, along with a passage in Aristotle's Meta- 
physics,'* enabled Schiaparelli to reconstruct Eudoxus's system 
(bee vol. i, pp. 329-34). Nor must it be forgotten that it is in 
Simplicius's commentary on the Physics* that the extract 
from Geminus's summary of the Meteorologica of Posidonius 
occurs which was used by Schiaparelli to support his view 
that it was Heraclides of Pontus, not Aristarchus of Samos, 
who first propounded the heliocentric hypothesis. 

Simplicius also wrote a commentary on Euclid's Elements, 
Book I, from which an-Nairizi, the Arabian commentator, 

1 Simpl. in Phys., pp. 54-69, ed. Diels. 

2 Simpl. on Arist. De ceio, p. 488. 18-24 and pp. 493-506, ed. Heiberg. 

3 JUetaph. A. 8, 1073 b 17-1074 a 14. 

4 Simpl. in Phys., pp. 291-2, ed. Diels. 



540 COMMENTATOKS AND BYZANTINES 

made valuable extracts, including the account of the attempt of 
' Aganis' to prove the parallel -postulate (see pp. 228- 30 above). 

Contemporary with Simplicius, or somewhat earlier, was 
EUTOCIUS, the commentator on Archimedes and Apollonius, 
As he dedicated the commentary on Book I On the Sphere 
and Cylinder to Ammonius (a pupil of Proclus and teacher 
of Simplicius), who can hardly have been alive after A.D. 510, 
Eutocius was probably born about A.D. 480. His date used 
to be put some fifty years later because, at the end of the com- 
mentaries on Book II On the Sphere and Cylinder and on 
the Measurement of a Circle, there is a note to the eft'ect that 
* the edition was revised by Isidorus of Miletus, the mechanical 
engineer, our teacher 9 . But, in view of the relation to Ammo- 
nius, it is impossible that Eutocius can have been a pupil of 
Isidorus, who was younger than Anthemius of Tralles, the 
architect of Saint Sophia at Constantinople in 532, whose 
work was continued by Isidorus after Anthemius's death 
about A.D. 534. Moreover, it was to Anthemius that Eutocius 
dedicated, separately, the commentaries on the first four 
Books of Apollonius's Conies, addressing Anthemius as ' my 
dear friend '. Hence we conclude that Eutocius was an elder 
contemporary of Anthemius, and that the reference to Isidoras 
is by an editor of Eutocius's commentaries who was a pupil of 
Isidorus. For a like reason, the reference in the commentary 
on Book II OH the Sphere and Cylinder 1 to a SiaprjTrjs 
invented by Isidorus ' our teacher ' for drawing a parabola 
must be considered to be an interpolation by the same editor. 

Eutocius's commentaries on Archimedes apparently ex- 
tended only to the three works, On the Spttere and Cylinder, 
Measurement of a Circle and Plane Equilibriums, and those 
on the Conies of Apollonius to the first four Books only. 
We are indebted to these commentaries for many valuable 
historical notes. Those deserving special mention here are 
(1) the account of the solutions of the problem of the duplica- 
tion of the cube, or the finding of two mean proportionals, 
by 'Plato', Heron, Philon, Apollonius, Diocles, Pappus, 
Sporus, Menaechmus, Archytas, Eratosthenes, Nicomedes, (2) 
the fragment discovered by Eutocius himself containing the 

1 Archimedes, ed. Heiberg, vol. iii, p. 84. 8-11. 



EUTOCIUS. ANTHEMIUS 541 

missing solution, promised by Archimedes in On the Sphere 
and Cylinder, II. 4, of the auxiliary problem amounting 
to the solution by means of conies of the cubic equation 
(ax)x 2 = fcc 2 , (3) the solutions (a) by Diocles of the original 
problem of II. 4 without bringing in the cubic, (b) by Diony- 
sodorus of the auxiliary cubic equation. 

ANTHEMIUS of Tralles, the architect, mentioned above, was 
himself an able mathematician, as is seen from a fragment of 
a work of his, On Burning-mirrors. This is a document of 
considerable importance for the history of conic sections. 
Originally edited by L. Dupuy in 1777, it was reprinted in 
Westermann's IlapaSogoypdQoi (Scriptures rerum mirabilium 
Graeci), 1839, pp. 149-58. The first and third portions of 
the fragment are those which interest us. 1 The first gives 
a solution of the problem, To contrive that a ray of the sun 
(admitted through a small hole or window) shall fall in a 
given spot, without moving away at any hour and season. 
This is contrived by constructing an elliptical mirror one focus 
of which is at the point where the ray of the sun is admitted 
while the other is at the point to which the ray is required 
to be reflected at all times. Let B be the hole, A the point 
to which reflection must always take place, BA being in the 
meridian and parallel to the horizon. Let BG be at right 
angles to BA, so that OB is an equinoctial ray ; and let BD be 
the ray at the summer solstice, BE a winter ray. 

Take F at a convenient distance on BE and measure FQ 
equal to FA. Draw HFG through F bisecting the angle 
A FQ, and let BG be the straight line bisecting the angle EBC 
between the winter and the equinoctial rays. Then clearly, 
since FO bisects the angle QFA, if we have a plane mirror in 
the position HFG, the ray BFE entering at B will be reflected 
to A. 

To get the equinoctial ray similarly reflected to A, join GA t 
and with G as centre and GA as radius draw a circle meeting 
BO in K. Bisect the angle KGA by the straight line GLM 
meeting BK in L and terminated at M, a point on the bisector 
of the angle OBJ}. Then LM bisects the angle K LA also, and 
KL = LA, and KM = MA. If then GLM is a plane mirror, 
the ray BL will be reflected to A. 

1 See Bibliotheca matliematica, vii 3 , 1907, pp. 225-33. 



542 COMMENTATORS AND BYZANTINES 

By taking the point N on BD such that M N = MA, and 
bisecting the angle NMA by the straight line MOP meeting 
BD in 0, we find that, if MOP is a plane mirror, the ray BO 
is reflected to A. 

Similarly, by continually bisecting angles and making more 
mirrors, we can get any number of other points of impact. Mak- 
ing the mirrors so short as to form a continuous curve, we get 
the curve containing all points such that the sum of the distances 
of each of them from A and B is constant and equal to BQ, BK, 
or BN. ' If then ', says Anthemius, c we stretch a string passed 




H 



round the points A, B, and through the first point taken on the 
rays which are to be reflected, the said curve will be described, 
which is part of the so-called " ellipse ", with reference to 
which (i.e. by the revolution of which round BA) the surface 
of impact of the said mirror has to be constructed/ 

We have here apparently the first mention of the construc- 
tion of an ellipse by means of a string stretched tight round 
the foci. Anthemius's construction depends upon two pro- 
positions proved by Apollonius (1) that the sum of the focal 
distances of any point on the ellipse is constant, (2) that the 
focal distances of any point make equal angles with the 
tangent at that point, and also (3) upon a proposition not 
found in Apollonius, namely that the straight line joining 



ANTHEMIUS 543 

the focus to the intersection of two tangents bisects the angle 
between the straight lines joining the focus to the two points 
of contact respectively. 

In the third portion of the fragment Anthemius proves that 
parallel rays can be reflected to one single point from a para- 
bolic mirror of which the point is the focus. The directrix is 
used in the construction, which follows, mutatis mutandis, the 
same course as the above construction in the case of the ellipse." 

As to the supposition of Heiberg that Anthemius may also 
be the author of the Fraymentum mathematicum Bobiense, see 
above (p. 203). 

The Papyrus of Akhmim. 

Next in chronological order must apparently be placed 
the Papyrus of Akhmim, a manual of calculation written 
in Greek, which was found in the metropolis of Akhmim, 
the ancient Panopolis, and is now in the Mus^e du 
Gizeh. It was edited by J. Baillet 1 in 1892. Accord- 
ing to the editor, it was written between the sixth and 
ninth centuries by a Christian. It is interesting because 
it preserves the Egyptian method of reckoning, with proper 
fractions written as the sum of primary fractions or sub- 
multiples, a method which survived alongside the Greek and 
was employed, and even exclusively taught, in the East. The 
advantage of this papyrus, as compared with Ahmes's, is that 
we can gather the formulae used for the decomposition of 
ordinary proper fractions into sums of submultiples. The 
formulae for decomposing a proper fraction into the sum of 
two submultiples may be shown thus : 

,-. u 1 1 



i 1 ; 


be b -f c 7 b + c 
c. b . 








a a 








2 11 


3 1 1 


18 1 1 




Examples - - * 

11 O DO 


llO "~ 70 77 3 


323 3438 




a 1 1 






/0\ 


n 






\") 


/>< 6 + me 7 b + mc 
C/-) 


1 






c/ . 

a a 


m 





1 Memoires publics par les membres de la Mission archto1ogiquefran$aise 
au Caire, vol. ix, part 1, pp. 1-89. 



544 COMMENTATORS AND BYZANTINES 

Fx -L - 1 l l , 3 

176 , /16 + 3.1K+ /16 + 3.1K 1 77^112 



-3. 11\ 1 

"7 /3 
and again -- = + T R -.y"j-\ = 

\ L & /lo -r 7\ /lOTfl.f\l UoU 

7 ( S ) 16 (" 3"~)2 

\ o / V u / & 

(3) cdf = cd + df + . cd + df 

C 9 ~' T 

a J a 
Example. 

O Q O O 

<6O <SO 



1320 10.12.11 120 + 132 120+ 132 9099 

0. -_ . 

The object is, of course, to choose the factors of the denomi- 
nator, and the multiplier w in (2), in such a way as to make 
the two denominators on the right-hand side integral. 

When the fraction has to be decomposed into a sum of three 
or more submultiples, we take out an obvious submultiple 
first, then if necessary a second, until one of the formulae 
will separate what remains into two submultiples. Or we 
take out a part which is not a submultiple but which can be 
divided into two submultiples by one of the formulae. 

For example, to decompose ^Vfc The factors of 6 1 6 are 8.77 
or 7 . 88. Take out ^, and -/fa = -fa & = ^ -^ = -fa ^ ^ ; 
and -^ = -fa fo by formula ( 1), so that ^ F = ^ T V sV sV 

Take /$. The factors of 6460 are 85.76 or 95.68. Take 
out e 1 ^, and $r/v = ^ ^VVo Again take out ^, and we have 
F^sV^tftf or sVsVsV The actual problem here is to find 
3^-jrd of iH-sxV^ffj which latter expression reduces to 
A 239. 

The sort of problems solved in the book are (1) the division 
of a number into parts in the proportion of certain given 
numbers, (2) the solution of simple equations such as this: 
From a certain treasure we take away ^th, then from the 
remainder ^ T th of that remainder, and we find 150 units left; 



what was the treasure? \<x # rr(x x\ ...| =jR. 



THE PAPYRUS OF AKHMIM. PSELLUS 545 

(3) subtractions such as : From subtract $ rr yfr tt & ^s 
& n TG Tis- KG e* TS rr TO ?<s 5? rita TTS Answer, ^ ^ . 

The book ends with long tables of results obtained (1) by 
multiplying successive numbers, tens, hundreds and thousands 
up to 10,000 by |, , |, , $, &c., up to T V, (2) by multiplying 

all the successive numbers 1, 2, 3.. .n by -> where w is succes- 

u n 

sively 11, 12, ... and 20; the results are all arranged as the 
sums of integers and submultiples. 

The Geodaesia of a Byzantine author formerly called, with- 
out any authority, ' Heron the Younger ' was translated into 
Latin by Barocius in 1572, and the Greek text was published 
with a French translation by Vincent. 1 The place of the 
author's observations was the hippodrome at Constantinople, 
and the date apparently about 938. The treatise was modelled 
on Heron of Alexandria, especially the Dioptra, while some 
measurements of areas and volumes are taken from the 
Metrica. 

MICHAEL PSELLUS lived in the latter part of the eleventh 
century, since his latest work bears the date 1092. Though 
he was called ' first of philosophers ', it cannot be said that 
what survives of his mathematics suits this title. Xy lander 
edited in 1556 the Greek text, with a Latin translation, of 
a book purporting to be by Psellus on the four mathematical 
sciences, arithmetic, music, geometry and astronomy, but it is 
evident that it cannot be entirely Psellus's own work, since 
the astronomical portion is dated 1008. The arithmetic con- 
tains no more than the names and classification of numbers 
and ratios. The geometry has the extraordinary remark that, 
while opinions differed as to how to find the area of a circle, 
the method which found most favour was to take the area as 
the geometric mean between the inscribed and circumscribed 
squares; this gives TT = VS = 2-8284271 ! The only thing of 
Psellus which has any value for us is the letter published by 
Tannery in his edition of Diophantus. 2 In this letter Psellus 
says that both Diophantus and Anatolius (Bishop of Laodicea 
about A. p. 280) wrote on the Egyptian method of reckoning, 

1 Notices et extmits, xix, pt. 2, Paris, 1858. 
a Diophantus, vol. ii, pp. 37-42. 
1528.2 N n 



546 COMMENTATORS AND BYZANTINES 

and that Anatolius's account, which was different and more 
succinct, was dedicated to Diophantus (this enables us to 
determine Diophantus's date approximately). He also notes 
the difference between the Diophantine and Egyptian names 
for the successive powers of dpidftSs: the next power after 
the fourth (SwapoSfyapif = # 4 ), i.e. x 5 , the Egyptians called 
' the first undescribed ' ((JfAoyoy 7rp<Sroy) or the ' fifth number ' ; 
the sixth, # 6 , they apparently (like Diophantus) called the 
cube-cube ; but with them the seventh, a.' 7 , was the * second 
undescribed ' or the c seventh number ', the eighth (,r s ) was the 
'quadruple square' (reTpairXij Svvafus), the ninth (x*) the 
' extended cube ' (KV{$O$ e^eAjtfroy). Tannery conjectures that 
all these remarks were taken direct from an old commentary 
on Diophantus now lost, probably Hypatia's. 

GEORGIUS PACJIYMKRKS (1242-1310) was the author of a 
work on the Quadrivium (Hvvrayna rS>v Ttvvdpw fiadr)fidr(i)v 
or TtTpdf$ip\ov). The arithmetical portion contains, besides 
excerpts from Nicomachus and Euclid, a paraphrase of Dio- 
phantus, Book I, which Tannery published in his edition of 
Diophantus ] ; the musical section with part of the preface was 
published by Vincent, 2 and some fragments from Book IV by 
Martin in his edition of the Astronomy of Theon of Smyrna. 

MAXIMUS PLANUDES, a monk from Nicomedia, was the 
envoy of the Emperor Andronicus II at Venice in the year 
1297, and lived probably from about 1260 to 1310. He 
wrote scholia on the first two Books of Diophantus, which 
are extant and are included in Tannery's edition of Dio- 
phantus. 8 They contain nothing of particular interest except 
a number of conspectuses of the working-out of problems of 
Diophantus written in Diophantus's own notation but with 
steps in separate lines, and with abbreviations on the left of 
words indicating the operations (e.g. ?K0. = e/c0<ri9, rerp. = 
TGTpaywvKriJLos, orvvO. = crtfpfccri?, &c.); the result is to make 
the work almost as easy to follow as it is in our notation. 

Another work of Planudes is called Wrj^o^opia /car' 9 Iv8ov$, 
or Arithmetic after the Indian method, and was edited as Das 

1 Diophantus, vol. ii, pp. 78-122. 

2 Notices et extraits, xvii, 1858, pp. 362-533. 
8 Diophantus, vol. ii, pp. 125-255. 



PSELLUS. PACHYMERES. PLANUDES 547 

Rechenbuch des Maximus Planudes in Greek by Gerhardt 
(Halle, 18(55) and in a German translation by H. Waeschke 
(Halle, 1878). There was, however, an earlier book under the 
similar title 'Apxy rtjy /zeyaAijr KOL 'IvSiKrjs ^(/u^opmy (sic), 
written in 1252, which is extant in the Paris MS. Suppl. Gr. 
387 ; and Planudes seems to have raided this work. He 
begins with an account of the symbols which, he says, were 

' invented by certain distinguished astronomers for the most 
convenient and accurate expression of numbers. There are 
nine of these symbols (our 1, 2, 3, 4, 5,6, 7, 8, 9), to which is 
added another called Tzifra, (cypher), written and denoting 
aero. The nine signs as well as this one are Indian/ 

But this is, of course, not the first occurrence of the Indian 
numerals; they were known, except the zero, to Gerbert 
(Pope Sylvester II) in the tenth century, and were used by 
Leonardo of Pisa in his Liber abaci (written in 1202 and 
rewritten in 1228). Planudes used the Persian form of the 
numerals, differing in this from the writer of the treatise of 
1252 referred to, who used the form then current in Italy. 
It scarcely belongs to Greek mathematics to give an account 
of Planudes's methods of subtraction, multiplication, &c. 

Extraction of tlie square root. 

As regards the extraction of the square root, he claims to 
have invented a method different from the Indian method 
and from that of Theon. It docs not appear, however, that 
there was anything new about it. Let us try to see in what 
the supposed new method consisted. 

Planudes describes fully the method of extracting the 
square root of a number with several digits, a method which 
is essentially the same as ours. This appears to be what he 
refers to later on as ' the Indian method '. Then he tells us 
how to find a first approximation to the root when the number 
is not a complete square. 

'Take the square root of the next lower actual square 
number, and double it : then, from the number the square root 
of whicli is required, subtract the next lower square number 
so found, and to the remainder (as numerator) give as de- 
nominator the double of the square root already found.' 

N n 2 



548 , COMMENTATORS AND BYZANTINES 

* 

The example given is </(18). Since 4 2 =16 is the next 

2 
2~4 



2 
lower square, the approximate square root is 4 + or 



The formula used is, therefore, \/(a a + 6) = a + ^ approxi- 
mately. (An example in larger numbers is 

^(1690196789) = 41112 + gfff * approximately.) 

Planudes multiplies 4 by itself and obtains 18 T ^, which 
shows that the value 4 J is not accurate. He adds that he will 
explain later a method which is more exact and nearer the 
truth, a method 'jvhich I claim as a discovery made by rne 
with the help of God '. Then, coming to the method which he 
claims to have discovered, Planudes applies it to A/6, The 
object is to develop this in units and sexagesimal fractions. 
Planudes begins by multiplying the 6 by 3600, making 21600 
second-sixtieths, and finds the square root of 21600 to lie 
between 146 and 147. Writing the 146' as 2 26', he proceeds 
to find the rest of the approximate square root (2 26' 58" 9"') 
by the same procedure as that used by Theon in extracting 
the square root of 4500 and 2 28' respectively. The differ- 
ence is that in neither of the latter cases does Theon multiply 
by 3600 so as to reduce the units to second-sixtieths, but he 
begins by taking the approximate square root of 2, viz. 1, just 
as he does that of 4500 (viz. 67). It is, then, the multiplication 
by 3600, or the reduction to second-sixtieths to start with, that 
constitutes the difference from Theon's method, and this must 
therefore be what Planudes takes credit for as a new dis- 
covery. In such a case as V(2 28') or -v/3, Theon's method 
has the inconvenience that the number of minutes in the 
second term (34' in the one case and 43' in the other) cannot 
be found without some trouble, a difficulty which is avoided 
by Planudes's expedient. Therefore the method of Planudes 
had its advantage in such a case. But the discovery was not 
new. For it will be remembered that Ptolemy (and doubtless 
Hipparchus before him) expressed the chord in a circle sub- 
tending an angle of 120 at the centre (in terms of 120th parts 
of the diameter) as 103 P 55' 23", which indicates that the first 
step in calculating A/ 3 was to multiply it by 3600, making 
10800, the nearest square below which is 103 2 (= 10609). In 



PLANUDES. MOSCHOPOULOS 549 

the scholia to Eucl., Book X, the same method is applied. 
Examples have been given above (vol. i, p. 63). The supposed 
new method was therefore not only already known to the 
scholiast, but goes back, in all probability, to Hipparchus. 

Two problems. 

Two problems given at the end of the Manual of Planudes 
are worth mention. The first is stated thus : ' A certain man 
finding himself at the point of death had his desk or safe 
brought to him and divided his money among his sons with 
the following words, " I wish to divide my money equally 
between my sons : the first shall have one piece and ^th of the 
rest, the second 2 and th of the remainder, the third 3 and 
?th of the remainder/' At this point the father died without 
getting to the end either of his money or the enumeration of 
his sons. I wish to know how many sons he had and how 
much money.' The solution is given as (n I) 2 for the number 
of coins to be divided and (n1) for the number of his sons; 
or rather this is how it might be stated, for Planudes takes 
n = 7 arbitrarily. Comparing the shares of the first two we 
must clearly have 

l + -(a-l) = 2 + i{a-(l + + 2)}, 
n ' n l N n ' 

which gives x = (n 1 ) 2 ; therefore each of (/i 1) sons received 

(71-1). 

The other problem is one which we have already met with, 
that of finding two rectangles of equal perimeter such that 
the area of one of them is a given multiple of the area of 
the other. If n is the given multiple, the rectangles are 
(>* 2 1, n s n, 2 ) andl('ftl, 7& 3 n) respectively. Planudes 
states the solution correctly, but how he obtained it is not clear. 

We find also in the Manual of Planudes the ' proof by nine ' 
(i.e. by casting out nines), with a statement that it was dis- 
covered by the Indians and transmitted to us through the 
Arabs. 

MANUEL MOSCHOPOULOS, a pupil and friend of Maximus 
Planudes, lived apparently under the Emperor Andronicus II 
(1282-1328) and perhaps under his predecessor Michael VIII 
(1261-82) also. A man of wide learning, he wrote (at the 



550 COMMENTATORS AND BYZANTINES 

instance of Nicolas Rhabdas, presently to be mentioned) a 
treatise on magic squares ; he showed, that is, how the num- 
bers 1, 2, 3 ... 7i 2 could be placed in the n 2 compartments of 
a square, divided like a chess-board into n 2 small squares, in such 
a way that the sum of the numbers in each horizontal and 
each vertical row of compartments, as well as in the rows 
forming the diagonals, is always the same, namely %n (n 2 + 1). 
Moschopoulos gives rules of procedure for the cases in which 
11 = 2 m + 1 and n = 4 m respectively, and these only, in the 
treatise as we have it ; he promises to give the case where 
n = 4ra+2 also, but does not seem to have done so, as the 
two manuscripts used by Tannery have after the first two cases 
the words reAoy rov avTov. The treatise was translated by 
De la Hire, 1 edited by S. Giinther, 2 and finally edited in an 
improved text with translation by Tannery. 3 

The work of Moschopoulos was dedicated to Nicolas Arta- 
vasdus, called RHABDAS, a person of some importance in the 
history of Greek arithmetic. He edited, with some additions 
of his own, the Manual of Planudes; this edition exists in 
the Paris MS. 2428. But he is also the author of two letters 
which have been edited by Tannery in the Greek text with 
French translation. 4 The date of Rhabdas is roughly fixed 
by means of a calculation of the date of Easter ' in the current 
year ' contained in one of the letters, which shows that its 
date was 1341. It is remarkable that each of the two letters 
has a preface which (except for the words T^V 8j\a><nv T&V ci/ 
Tofr dpiOfiois faTrifjLdTow and the name or titlfr of the person 
to whom it is addressed) copies word for word the first thir- 
teen lines of the preface to Diophantus's Arithmetica, a piece 
of plagiarism which, if it does not say much for the literary 
resource of Rhabdas, may indicate that he had studied Dio- 
phantus. The first of the two letters has the heading A con- 
cise and most clear exposition of the science of calculation 
written at Byzantium of Constantine, by Nicolas Artavasdus 

1 Mem. de VAcad. Eoyale des Sciences, 1705. 

2 Vermischte Untersuchungen zur Gesch. d. Math., Leipzig, 1876. 

8 'Le traite de Manuel Moschopoulos sur les carrSs magiques' in 
Annuaire de V Association pour V encouragement des ttudes grecques, xx, 
1886, pp. 88-118. * 

4 'Notices sur les deux lettres arithnietiques de Nicolas Rhabdas' in 
Notices et extmits des tnanuscHts de la Bibliotheque Rationale, xxxii, pt. 1. 
1886, pp. 121-252. 



MOSCHOPOULOS. RHABDAS 551 

of Smyrna, arithmetician and geometer, TOV 'PafiSa, at the 
instance of the most revered Master of Requests, Georgius 
Chatzyces, and most easy for those who desire to study it/ 
A long passage, called K<f>pa<ri$ rov SaKrvXiKov ptrpov, deals 
with a method of finger-notation, in which the fingers of each 
hand held in different positions are made to represent num- 
bers. 1 The fingers of the left hand serve to represent all the 
units and tens, those of the right all the hundreds and 
thousands up to 9000 ; * for numbers above these it is neces- 
sary to use writing, the hands not sufficing to represent such 
numbers.' The numbers begin with the little fingers of each 
hand; if we call the thumb and the fingers after it the 1st, 
2nd, 3rd, 4th, and 5th fingers in the German style, the succes- 
sive signs may be thus described, premising that, where fingers 
are not either bent or ' half -closed ' (KXivopcvoi) or c cl9sed ' 
i) 9 they are supposed to be held out straight 



(a) On the left hand : 

for 1, half-close the 5th finger only; 
2, 4th and 5th fingers only ; 

3, 3rd, 4th and 5th fingers only ; 

4, 3rd and 4tli fingers only ; 

5, 3rd finger only ; 

6, 4th 

7, close the 5th finger only; 
8, 4th and 5th fingers only ; 
9, 3rd, 4th and 5th fingers only. 

(6) The same operations on the right hand give the thou- 
sands, from 1000 to 9000. 

(c) On the left hand : 

for 10, apply the tip of the forefinger to the first joint of 
the thumb so that the resulting figure resembles <r ; 




, 

as showing that he too wan acquainted with the system (The Miscellaneous 
Works of the Venerable Bede, ed. J. A. Giles, vol. vi, 1843, pp. 141-3). 



552 COMMENTATORS AND BYZANTINES 

for 20, stretch out the forefinger straight and vertical, 
keep fingers 3, 4, 5 together but separate from it 
and inclined slightly to the palm ; in this position 
touch the forefinger with the thumb ; 

30, join the tips of the forefinger and thumb ; 

40, place the thumb on the knuckle of the forefinger 
behind, making a figure like the letter f ; 

50, make a like figure with the thumb on the knuckle 
of the forefinger inside ; 

60, place the thumb inside the forefinger as for 50 and 
bring the forefinger down over the thumb, touch- 
ing the ball of it ; 

70, rest the forefinger round the tip of the thumb, 
making a curve like a spiral ; 

80, fingers 3, 4, 5 being held together and inclined 
at an angle to the pdflm, put the thumb across the 
palm to touch the third phalanx of the middle 
linger (3) and in this position bend the forefinger 
above the first joint of the thumb ; 

90, close the forefinger only as completely as possible. 

(d) The same operations on the right hand give the hun- 
dreds, from 100 to 900. 

The first letter also contains tables for addition and sub- 
traction and for multiplication and division ; as these are said 
to be the 'invention of Palamedes', we must suppose that 
such tables were in use from a remote antiquity. Lastly, the 
first letter contains a statement which, though applied to 
particular numbers, expresses a theorem to the effect that 

(a + 10^ + ... + 10 r 'XJ (&o+ 10^ + ... + 10 w ft n ) 

is not > 10+"+-, 

where , Oj ... 6 , ^ ... are any numbers from to 9. 

In the second letter of Rhabdas we find simple algebraical 
problems of the same sort as those of the Anthologia Graeca 
and the Papyrus of Akhmim. Thus there are five problems 
leading to equations of the type 

x x 

+ - + ... =a. 

f m n 



RHABDAS 553 

Rhabdas solves the equation + - = a, practically as we 

^ m n r * 

should, by multiplying up to get rid of fractions, whence he 
obtains x = mna /(m + n). Again he solves the simultaneous 
equations x + y = a, ma? = ny ; also the pair of equations 

, y x 

x + 2- = y+ -- = a. 

m y n 

Of course, m, n, a... have particular numerical values in 
all cases. 

Rhabdas' s Rule for approximating to the square root of 

a non-square number. 

We find in Rhabdas the equivalent of the Heronian formula 
for the approximation to the square root of a non-square 
number A = a* + 6, namely 

b 

a = a + ; 
2a 

he further observes that, if a be an approximation by excess, 
then o^ = A/OL is an approximation by defect, and ^(a + cXj) 
is an approximation nearer than either. This last form is of 

course exactly Heron's formula a = } (a + \ The formula 

was also known to Barlaam (presently to be mentioned), who 
also indicates that the procedure can be continued indefinitely. 
It should here be added that there is interesting evidence 
of the Greek methods of approximating to square roots in two 
documents published by Heiberg in 1899. 1 The first of 
these documents (from a manuscript of the fifteenth century 
at Vienna) gives the approximate square root of certain non- 
square numbers from 2 to 147 in integers and proper fractions. 
The numerals are the Greek alphabetic numerals, but they are 
given place-value like our numerals : thus OLTJ = 18, #= 147, 

~ = 9 and so on: is indicated by 4 or, sometimes, by . 
PTJ 28 

All these square roots, such as \/(21) = 4, \/(35) = 5^J, 
'/(1 12) = 10||, and so on, can be obtained (either exactly or, 
in a few cases, by neglecting or adding a small fraction in the 

1 ( Byzantinische Analekten' in Abh. zurGesch. d. Math. ix. Heft, 1899, 
pp. 163 sqq. 



554 COMMENTATORS AND BYZANTINES 

numerator of the fractional part of the root) in one or other 
of the following ways : 

(1) by taking the nearest square to the given number A, 
say a 2 , and using the Heronian formulae 



(2) by using one or other of the following approximations, 
where 

a 2 < A < (tt+1) 2 , and A = a* + b = 



namely, b b 7 

a+ _, f b , 
2 a 2a+ -- 

2 a 



or a combination of two of these with 

(JL C 

(3) the formula that, if y < -, > then 

o ct 

a ma + nc c 
T) mb 4- nd d 

It is clear that it is impossible to deny to the Greeks the 
knowledge of these simple formulae. 

Three more names and we have done. 

IOANNES PKDIASIMUS, also called Galenus, was Keeper of the 
Seal to the Patriarch of Constantinople in the reign of 
Andronicus III (1328-41). Besides literary works of his, 
some notes on difficult points in arithmetic and a treatise on 
the duplication of the cube by him are said to exist in manu- 
scripts. His Geometry, which was edited by Friedlein in 1866, 
follows very closely the mensuration of Heron. 

BAKLAAM, a monk of Calabria, was abbot at Constantinople 
and later Bishop of Geraci in the neighbourhood of Naples; 
he died in 1348. He wrote, in Greek, arithmetical demon- 
strations of propositions in Euclid, Book II, 1 and a Logistic in 
six Books, a laborious manual of calculation in whole numbers, 

1 Edited with Latin translation- by Dasypodius in 1564, and included 
in Heiberg and Menge's Euclid, vol. v, ad fin. 



PEDIASIMUS. BARLAAM. ARGYRUS 555 

ordinary fractions and sexagesimal fractions (printed at 
Strassburg in 1592 and at Paris in 1600). Barlaam, as we 
have seen, knew the Heronian formulae for finding successive 
approximations to square roots, and was aware that they could 
be indefinitely continued. 

ISAAC ARGYRUS, a monk, who lived before 1368, was one of 
a number of Byzantine translators of Persian astronomical 
works. In mathematics he wrote a Geodaesia and scholia to 
the first six Books of Euclid's Elements. The former is con- 
tained in the Paris MS. 2428 and is called 'a method of 
geodesy or the measurement of surfaces, exact and shortened ' ; 
the introductory letter addressed to one Colybos is followed 
by a compilation of extracts from the Geometrica and Stereo- 
metrica of Heron. He is apparently the author of some 
further additions to Rhabdas's revision of the Manual of 
Planudes contained in the same manuscript. A short tract 
of his ' On the discovery of the square i*oots of non-rational 
square numbers J is mentioned as contained in two other manu- 
scripts at Venice and Rome respectively (Codd. Marcianus Gr. 
333 and Vaticanus Gr. 1058), where it is followed by a table 
of the square roots of all numbers from 1 to 102 in sexa- 
gesimal fractions (e.^. ^2 = 1 24' 51" 48'", ^3=1 43' r^O'"). 1 



1 Heiherg, ' Byxantinische Analekten ', in Abh. zur Cesch. d. Math, ix, 
pp. 169-70. 



APPENDIX 

On Archimedes's proof of the subtangent-property of 

a spiral. 

THE section of the treatise On Spirals from Prop. 3 to 
Prop. 20 is an elaborate series of propositions leading up 
to the proof of the fundamental property of the subtangent 
corresponding to the tangent at any point on any turn of the 
spiral. Libri, doubtless with this series of propositions in 
mind, remarks (Histoire des sciences math&matiques en Italie, 
i, p. 31) that'Apr&s vingt si&cles de travaux et de d^cou- 
vertes, les intelligences les plus puissantes viennent encore 
(Schouer contre la sy&th&se difficile du Traite des Spirales 
d'Arehimfcde/ There is no foundation for this statement, 
which seems to be a too hasty generalization from a dictum, 
apparently of Fontenelle, in the Histoire de VAcadtmie des 
Sciences pour Vann&e 1704 (p. 42 of the edition of 1722), 
who says of the proofs of Archimedes in the work On 
Spirals: 'Elles sont si longues, et si difficiles a embrasser, 
que, comme on l'a pft voir dans la Preface de 1' Analyse des 
Infiniment petits, M. Bouillaud a avou qu'il ne les avoit 
jamais bien entendues, et que Vi&te les a injustement soup- 
^onn^es de paralogisme, parce qu'il n'avoit pA non plus 
parvenir & les bien entendre. Mais Routes les preuves qu'on 
peut donner de leur difficult^ et de leur obscurit^* tournent 
a la gloire d'Archimfede; car quelle vigueur d'esprit, quelle 
quantit^ de vftes diff<rentes, quelle opini&tretd de travail n'a- 
t-il pas fallu pour lier et pour disposer un raisonnement que 
quelques-uns de nos plus grands g^om^tres ne peuvent suivre, 
tout li^ et tout disposd qu'il est ? ' 

P. Tannery has observed 1 that, as a matter of fact, no 
mathematicians of real authority who have applied or ex- 
tended Archimedes's methods (such men as Huygens, Pascal, 
Roberval and Fermat, who alone could have expressed an. 
opinion worth having), have ever complained of the 

1 Bulletin des sciences mathtmatiques, 1895, Part i, pp. 265-71. 



APPENDIX 557 

' obscurity ' of Archimedes ; while, as regards Vieta, he has 
shown that the statement quoted is based on an entire mis- 
apprehension, and that, so far from suspecting a fallacy in 
Archimedes's proofs, Vieta made a special study of the treatise 
On Spirals and had the greatest admiration for that work. 

But, as in many cases in Greek geometry where the analy- 
sis is omitted or even (as Wallis was tempted to suppose) of 
set purpose hidden, the reading of the completed synthetical 
proof leaves a certain impression of mystery; for there is 
nothing in it to show why Archimedes should have taken 
precisely this line of argument, or how he evolved it. It is 
a fact that, as Pappus said, the subtangent-property can be 
established by purely 'plane 1 methods, without recourse to 
a 'solid* vtvorts (whether actually solved or merely assumed 
capable of being solved). If, then, Archimedes chose the more 
difficult method which we actually find him employing, it is 
scarcely possible to assign any reason except his definite 
predilection for the form of proof by reductio ad absurdum 
based ultimately on his famous ' Lemma ' or Axiom. 

It seerns worth while to re-examine the whole question of 
the discovery and proof of the property, and to see how 
Archimedes's argument compares with an easier f plane ' proof 
suggested by the figures of some of the very propositions 
proved by Archimedes in the treatise. 

In the first place, we may be sure that the property was 
not discovered by the steps leading to the proof avS it stands. 
I cannot but think that Archimedes divined the result by an 
argument corresponding to our use of the differential calculus 
for determining tangents. He must have considered the 
instantaneous direction of the motion of the point P describ- 
ing the spiral, using for this purpose the parallelogram of 
velocities. The motion of P is compounded of two motions, 
one along OP and the other at right angles to it. Comparing 
the distances traversed in an instant of time in the two direc- 
tions, we see that, corresponding to a small increase in the 
radius vector r, wo have a small distance traversed perpen- 
dicularly to it, a tiny arc of a circle of radius r subtended by 
the angle representing the simultaneous small increase of the 
angle 6 (AOP). Now r has a constant ratio to 6 which we call 
a (when 6 is the circular measure of the angle 6). Consequently 



558 



APPENDIX 



the small increases of r and 6 are in that same ratio. There- 
fore what we call the tangent of the angle OPT is r/a, 
i.e. OT/r = r/a; and OT = r 2 /a, or rd, that is, the arc of a 
circle of radius r subtended by the angle 0. 

To i>rove this result Archimedes would doubtless begin by 
an analysis of the following sort. Having drawn OT perpen- 
dicular to OP and of length equal to the arc ASP, he had to 
prove that the straight lino joining P to T is the tangent 
at P. He would evidently take the line of trying to show 
that, if any radius vector to the spiral is drawn, as OQ', on 
either side of OP, Q' is always on the side of TP towards 0, 
or, if OQ' meets TP in F, OQ' is always less than OF. Suppose 




that in the above figure OR' is any radius vector between OP 
and OS on the ' backward ' side of OP, and that OR meets the 
circle with radius OP in R, the tangent to it at P in (r, the 
spiral in JB', and TP in F'. We have to prove that R, R' lio 
on opposite sides of F', i.e. that RR' > RF' ; and again, sup- 
posing that any radius vector OQ' on the ' forward ' side of 
OP meets the circle with radius OP in Q, the spiral in Q' and 
TP produced in F, we have to prove that QQ' < QF. 
Archimedes then had to prove that 

(1) F'M : RO < RR': RO, and 

(2) FQ:QO>Qff:QQ. 
Now (1) is equivalent to 

F'R : RO < Yarc RP\ : Tare A8P\. since RO = PO. 



APPENDIX 559 

But (arc A HP) = OT, by hypothesis ; 

therefore it was necessary to prove, alternando, that 
(3) F'R : (arc RP) < RO : OT, or PO : OT, 

i.e. < 'PM: MO, where OM is perpendicular to SP. 

Similarly, in order to satisfy (2), it was necessary to 
prove that 

(4) /<V : (arc PQ) > PM: MO. 

Now, as a matter of fact, (3) is a fortiori satisfied if 
F'R : (chord RP) < PM:MO ; 

but in the case of (4) we cannot substitute the chord PQ for 
the arc PQ, and we have to substitute PG', where G' is the 

point in which the tangent at P to 
the circle meets OQ produced ; for 
of course PG' > (arc PQ), so that (4) 
is a fortiori satisfied if 




It is remarkable that Archimedes 
uses for his proof of the'two cases Prop. 
FIG. 1. 8 and Prop. 7 respectively, and makes 

no use of Props 6 and 9, whereas 

the above argument points precisely to the use of the figures 
of the two latter propositions only. 

For in the figure of Prop. 6 (Fig. 1), if OFP is any radius 
cutting AB in F, and if PB produced cuts OT, the parallel to 
AB through 0, in H, it is obvious, by parallels, that 

PF: (chord PB) = OP : Pll. 

Also PH becomes greater the farther P moves from B 
towards A, so that the ratio PF :P.B diminishes continually, 
while it is always less than OB : BT (where BT is the tangent 
at B and meets Oil in T), i.e. always less than BM: MO. 

Hence the relation (3) is always satisfied for any point R' of 
the spiral on the ' backward ' side of P. 

But (3) is equivalent to (1), from which it follows that F'R 
is always less than RR', so that R' always lies on the side 
of TP towards 0. 



560 APPENDIX 

Next, for the point Q' on the ' forward * side of the spiral 
from P, suppose that in the figure of Prop. 9 or Prop. 7 (Fig. 2) 
any radius OP of the circle meets AB produced in F, and 




PIG. 2. 



the tangent at B in G ; and draw BPH, BGT meeting OT, the 
parallel through to AB, in H, T. 

Then PF:BG>FG: BG, since PF > FG, 

> OG : GT, by parallels, 

> OB: BT, a fortiori, 
>BM:MO\ 

and obviously, as P moves away from B towards OT, i.e. as G 
moves away from B along BT, the ratio OG:GT increases 
continually, while, as shown, PF:BG is always > BM:MO, 
and, a fortiori, 

PJP:(arcPJ5) >BM:MO. 

That is, (4) is always satisfied for any point Q' of the spiral 
' forward ' of P, so that (2) is also satisfied, and QQ' is always 
less than QF. 

It will be observed that no i/eflcm, and nothing beyond 
' plane ' methods, is required in the above proof, and Pappus's 
criticism of Archimedes's proof is therefore justified. 

Let us now consider for a moment what Archimedes actually 
does. In Prop. 8, which he uses to prove our proposition in 
the 'backward' case (-R', JR, F'), he shows that, if PO:OV 
is any ratio whatever less