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AKCHIMEDES.

JLontom: C. J. CLAY AND SONS,

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,

AVE MARIA LANE.

263, ARGYLE STREET.

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flefo gorfc: THE MACMILLAN COMPANY.

PREFACE.

S book is intended to form a companion volume to my
edition of the treatise of Apollonius on Conic Sections
lately published. If it was worth while to attempt to make the
work of "the great geometer" accessible to the mathematician
of to-day who might not be able, in consequence of its length
and of its form, either to read it in the original Greek or in a
Latin translation, or, having read it, to master it and grasp the
whole scheme of the treatise, I feel that I owe even less of an
apology for offering to the public a reproduction, on the same
lines, of the extant works of perhaps the greatest mathematical
genius that the world has ever seen.

Michel Chasles has drawn an instructive distinction between
the predominant features of the geometry of Archimedes and
of the geometry which we find so highly developed in Apollo-
nius. Their works may be regarded, says Chasles, as the origin
and basis of two great inquiries which seem to share between
them the domain of geometry. Apollonius is concerned with
the Geometry of Forms and Situations, while in Archimedes
we find the Geometry of Measurements dealing with the quad-
rature of curvilinear plane figures and with the quadrature
and cubature of curved surfaces, investigations which "gave
birth to the calculus of the infinite conceived and brought
to perfection successively by Kepler, Cavalieri, Fermat, Leibniz,
and Newton." But whether Archimedes is viewed as the
man who, with the limited means at his disposal, nevertheless
succeeded in performing what are really integrations for the
purpose of finding the area of a parabolic segment and a

VI PREFACE.

spiral, the surface and volume of a sphere and a segnrent
of a sphere, and the volume of any segments of the solids
of revolution of the second degree, whether he is seen finding
the centre of gravity of a parabolic segment, calculating
arithmetical approximations to the value of TT, inventing a
system for expressing in words any number up to that which
we should write down with 1 followed by 80,000 billion
ciphers, or inventing the whole science of hydrostatics and at
the same time carrying it 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, the
intelligent reader cannot fail to be struck by the remarkable
range of subjects and the mastery of treatment. And if these
are such as to create genuine enthusiasm in the student of
Archimedes, the style and method are no less irresistibly
attractive. One feature which will probably most impress the
mathematician accustomed to the rapidity and directness secured
by the generality of modern methods is the deliberation with
which Archimedes approaches the solution of any one of his
main problems. Yet this very characteristic, with its incidental
effects, is calculated to excite the more admiration because the
method suggests the tactics of some great strategist who
foresees everything, eliminates everything not immediately
conducive to the execution of his plan, masters every position
in its order, and then suddenly (when the very elaboration of
the scheme has almost obscured, in the mind of the spectator,
its ultimate object) strikes the final blow. Thus we read in
Archimedes proposition after proposition the bearing of which is
not immediately obvious but which we find infallibly used later
on ; and we are led on by such easy stages that the difficulty of
the original problem, as presented at the outset, is scarcely
appreciated. As Plutarch says, "it is not possible to find in
geometry more difficult and troublesome questions, or more
simple and lucid explanations." But it is decidedly a rhetorical
exaggeration when Plutarch goes on to say that we are deceived

PREFACE. Vll

b}|the easiness of the successive steps into the belief that anyone
could have discovered them for himself. On the contrary, the
studiejd simplicity and the perfect finish of the treatises involve
at the same time an element of mystery. Though each step
depends upon the preceding ones, we are left in the dark as to
how they were suggested to Archimedes. There is, in feet,*
much truth in a remark of Wallis to the effect that he seems
" as it were of set purpose to have covered up the traces of his
investigation as if he had grudged posterity the secret of his
method of inquiry while he wished to extort from them assent
to his results." Wallis adds with equal reason that not only
Archimedes but nearly all the ancients so hid away from
posterity their method of Analysis (though it is certain that
they had one) that more modern mathematicians found it easier
to invent a new Analysis than to seek out the old. This is no
doubt the reason why Archimedes and other Greek geometers
have received so little attention during the present century and
why Archimedes is for the most part only vaguely remembered
as the inventor of a screw, while even mathematicians scarcely
know him except as the discoverer of the principle in hydro-
statics which bears his name. It is only of recent years that
we have had a satisfactory edition of the Greek text, that of
Heiborg brought out in 1880-1, and I know of no complete
translation since the German one of Nizze, published in 1824,
which is now out of print and so rare that I had some difficulty
in procuring a copy.

The plan of this work is then the same as that which I
followed in editing the Conies of Apollonius. In this case,
however, there has been less need as well as less opportunity for
compression, and it has been possible to retain the numbering
of the propositions and to enunciate them in a manner more
nearly approaching the original without thereby making the
enunciations obscure. Moreover, the subject matter is not so
complicated as to necessitate absolute uniformity in the notation
used (which is the only means whereby Apollonius can be made

viii PREFACE.

even tolerably readable), though I have tried to secure as mu/sh
uniformity as was fairly possible. My main object has been to
present a perfectly faithful reproduction of the treatises as they
have come down to us, neither adding anything nor leaving out
anything essential or important. The notes are for the most
f part intended to throw light on particular points in the text or
to supply proofs of propositions assumed by Archimedes as
known; sometimes I have thought it right to insert within
square brackets after certain propositions, and in the same type,
notes designed to bring out the exact significance of those
propositions, in cases where to place such notes in the Intro-
duction or at the bottom of the page might lead to their being
overlooked.

Much of the Introduction is, as will be seen, historical ; the
rest is devoted partly to giving a more general view of certain
methods employed by Archimedes and of their mathematical
significance than would be possible in notes to separate propo-
sitions, and partly to the discussion of certain questions arising
out of the subject matter upon which we have no positive
historical data to guide us. In these latter cases, where it is
necessary to put forward hypotheses for the purpose of explaining
obscure points, I have been careful to call attention to their
speculative character, though I have given the historical evidence
where such can be quoted in support of a particular hypothesis,
my object being to place side by side the authentic information
which we possess and the inferences which have been or may
be drawn from it, in order that the reader may be in a position
to judge for himself how far he can accept the latter as probable.
Perhaps I may be thought to owe an apology for the length of
one chapter on the so-called vevcreis, or inclinationes, which goes
somewhat beyond what is necessary for the elucidation of
Archimedes; but the subject is interesting, and I thought it
well to make my account of it as complete as possible in
order to round off, as it were, my studies in Apollonius and
Archimedes.

PREFACE. IX

jl have had one disappointment in preparing this book for
the press. I was particularly anxious to place on or opposite
the title-page a portrait of Archimedes, and I was encouraged
in this idea by the fact that the title-page of Torelli's edition
bears a representation in medallion form on which are endorsed
the words Archimedis effigies marmorea in veteri anaglypho*
Romae asservato. Caution was however suggested when I
found two more portraits wholly unlike this but still claiming to
represent Archimedes, one of them appearing at the beginning
of Peyrard's French translation of 1807, and the other in
Gronovius' Thesaurus Graecarum Antiquitatum ; and I thought
it well to inquire further into the matter. I am now informed
by Dr A. S. Murray of the British Museum that there does
not appear to be any authority for any one of the three, and
that writers on iconography apparently do not recognise an
Archimedes among existing portraits. I was, therefore, re-
luctantly obliged to give up my idea.

The proof sheets have, as on the former occasion, been read
over by my brother, Dr R. S. Heath, Principal of Mason College,
Birmingham ; and I desire to take this opportunity of thanking
him for undertaking what might well have seemed, to any one
less genuinely interested in Greek geometry, a thankless task.

T. L. HEATH.

March, 1897.

LIST OF THE PRINCIPAL WORKS CONSULTED.

JOSEPH TORELLI, Archimedis yuae supersunt omnia cum Eutocii A$ca-
lonitae commentariis. (Oxford, 1792.)

ERNST NIZZE, Archimedes van Syrakus vorhandene Werke aus dem
griechischen iibersetzt und mil erldutemden und kriti&chen Anmerk-
ungen begleitet. (Stralsund, 1824.)

J. L. HEIBERG, Archimedis opera omnia cum commenlariia Eutocii.
(Leipzig, 1880-1.)

J. L. HEIBERG, Quaestiones Archimedean. (Copenhagen, 1879.)

F. HULTSCH, Article Archimedes in Pauly-Wissowa's Real-Encycloptidie der

classischen Altertumswmeiwhaften. (Edition of 1895, n. 1, pp.

507-539.)

C. A. BRETSCHNEIDKR, Die Geometric uiid die Geometer vor Euklide*.

(Leipzig, 1870.)
M. CANTOR, Vorlesungen fiber Ges^hichte der Mathematik, Band I, zweite

Auflage. (Leipzig, 1894.)
G. FRIEDLEIN, Procti Diadochi in primum Euclidis elcmentorum libmm

commentarii. (Leipzig, 1873.)

JAMES (row, A short history of Greek Mathematics. (Cambridge, 1884.)
SIEGMUND OUNTHER, Abriis der Getchichte der Mathematik und der

Naturwissenschaften ini Altertum in I wan von Mailer's Handbuch der

klassischen Altertumswi&senschaft, \. 1.
HERMANN HANKEL, Zur Geschichte der Mathematik in Alterthum und

Mittelalter. (Leipzig, 1874.)

J. L. HEIBERG, Litterarge&chichtlichc Studien iiber Euklid. (Leipzig,
1882.)

J. L. HEIBKRG, Euclidis elemerta. (Leipzig, 1883-8.)
( F. HULTSCH, Article Arithmetica in Pauly-Wissowa^s Real- Encyclopedic,
II. 1, pp. 1066-1116.

Xll LIST OF PRINCIPAL WORKS CONSULTED.

F. HULTSCH, fferonis Alexandrini geometricorum et stereometricorum
reliquiae. (Berlin, 1864.) (

F. HULTSCH, Pappi Alexandrini collectionis quae supersunt. (Berlin,
1876-8.)

QINO LORI A, II periodo aureo della geometria greca. (Modena, 1895.)

MAXIMILIEN MARIE, Histoire des sciences mathe'matiques et physiques,

<- Tome I. (Paris, 1883.)

J. H. T. MULLER, Beitriige zur Terminologie der griechischen Mathematiker.
(Leipzig, 1860.)

Q. H. F. NESSELMANN, Die Algebra der Griechen. (Berlin, 1842.)

F. SUSEMIHL, Geschichte der griechischen Litteratur in der Alejcandrinerzeit,
Band I. (Leipzig, 1891.)

P. TANNERY, La Geome'trie grecque, Premi6re partio, Histoire ge'ne'rale de la
Geometric tltmentaire. (Paris, 1887.)

H. G. ZEUTHEN, Die Lehre von den Kegelschnitten im Altertum. (Copen-
hagen, 1886.)

H. G. ZEUTHEN, Geschichte der Mathematik im Altertum und Mittelalter.
(Copenhagen, 1896.)

CONTENTS.

INTRODUCTION.

PAGE

CHAPTER I. ARCHIMEDES xv

CHAPTER II. MANUSCRIPTS AND PRINCIPAL EDITIONS ORDER

OF COMPOSITION DIALECT LOST WORKS . Xxiii

CHAPTER III. RELATION OF ARCHIMEDES TO HIS PREDECESSORS xxxix

1. Use of traditional geometrical methods . . xl
2. Earlier discoveries affecting quadrature and

cubature xlvii

3. Conic Sections lii

4. Surfaces of the second degree .... liv
5. Two mean proportionals in continued propor-
tion Ixvii

CHAPTER IV. ARITHMETIC IN ARCHIMEDES .... Ixviii

1. Greek numeral system Ixix

2. Addition and subtraction Ixxi

3. Multiplication Ixxii

4. Division ........ Ixxiii

5. Extraction of the square root .... Ixxiv

6. Early investigations of surds or incoiumeiisii-

rables Ixxvii

7. Archimedes' approximations to v /3 . . . Ixxx
8. Archimedes' approximations to the square roots

of large numbers which are not complete

squares Ixxxiv

Note on alternative hypotheses with regard to

the approximations to ^/3 . . . . xc

xiv CONTENTS.

PAGE

CHAPTER V^ UN THE PROBLEMS KNOWN AS NEY2EI2 . . ^

1. Nevcrctff referred to by Archimedes ... c
2. Mechanical constructions : the conchoid of Nico-

medes cv

3. Pappus' solution of the vcva-ts referred to in

Props. 8, 9 On Spirals cvii

4. The problem of the two mean proportionals . ex

5. The trisection of an angle .... cxi

6. On certain plane vcvo-ets cxiii

CHAPTER VI. CUBIC EQUATIONS cxxiii

CHAPTER VII. ANTICIPATIONS BY ARCHIMEDES OF THE INTE-
GRAL CALCULUS cxlii

CHAPTER VIII. THE TERMINOLOGY OF ARCHIMEDES ... civ

THE WORKS OF ARCHIMEDES.

ON THE SPHERE AND CYLINDER, BOOK 1 1

BOOK II. ... r>6

MEASUREMENT OF A CIRCLE 91

ON CONOIDS AND SPHEROIDS 99

ON SPIRALS 151

ON THE EQUILIBRIUM OF PLANES, BOOK I. ... 189

BOOK II. ... 203

THE SAND-RECKONER 221

QUADRATURE OF THE PARABOLA 233

ON FLOATING BODIES, BOOK 1 253

BOOK II 263

BOOK OF LEMMAS . 301

THE CATTLE-PROBLEM 319

INTRODUCTION.

CHAPTER I.

ARCHIMEDES.

A LIFE of Archimedes was written by one Heracleides*, but
this biography has not survived, and such particulars as are known
have to be collected from many various sources f. According to
TzetzesJ he died at the age of 75, and, as he perished in the sack
of Syracuse (B.C. 212), it follows that he was probably born about
287 B.C. He was the son of Pheidias the astronomer, and was
on intimate terms with, if not related to. king Hieron and his

* Eutocius mentions this work in his commentary on Archimedes' Measure-
ment of the circle, ws QrjGiv 'HpaK\eiSijs tv r$ 'ApxiM^Sou* pUp. He alludes to it
again in his commentary on Apollonius' Conies (ed. Heiberg, Vol. n. p. 168),
where, however, the name is wrorn^v given as 'Hpd/tXetos. This Heracleides is
perhaps the same as the Heraciei^s mentioned by Archimedes himself in the
preface to his book On Spiral*.

t An exhaustive collection of the materials is given in Heiberg's Quaestiones
Archimedcac (1879). The preface to Torelli's edition also gives the main points,
and the same work (pp. 363 370) quotes at length most of the original
references to the mechanical inventions of Archimedes. Further, the article
Archimedes (by Hultsch) in Pauly-Wissowa's Real-JKncyclopfitlie der cfassischen
Altertunuwi**en*chaftcH gives an entirely admirable summary of all the available
information. See also Susemihl's Geschichte der gricchitchen Litteratur in der
Alexandrinerzcit, i. pp. 723 733.

t Tzetzes, Chiliad., n. 35, 105.

Pheidian is mentioned in the Sand-reckoner of Archimedes, rwv Trpartpuv
dffrpoXbywv Ev$6$ov ..<l>ct5la 8t TOV d/uou -rrarpos (the last words being the correction
of Blass for rov ' AKOVTTO.TPOS, the reading of the text). Of. Schol. Clark, in
(^regor. Nazianz. Or. 34, p. 355 a Morel. 4>et5ias rb /ueV 7^0? yv Zi'
6

XVI INTRODUCTION.

son Gelon. It appears from a passage of Diodorus* that he spent
a considerable time at Alexandria, where it may be inferred that
he studied with the successors of Euclid. It may have been at
Alexandria that he made the acquaintance of Conon oi; Samoa
(for whom he had the highest regard both as a mathematician
and as a personal friend) and of Eratosthenes. To the former
he was in the habit of communicating his discoveries before their
publication, and it is to the latter that the famous Cattle-problem
purports to have been sent. Another friend, to whom he dedicated
several of his works, was Dositheus of Pelusium, a pupil of Conon,
presumably at Alexandria though at a date subsequent to Archi-
medes' sojourn there.

After his return to Syracuse he lived a life entirely devoted
to mathematical research. Incidentally he made himself famous
by a variety of ingenious mechanical inventions. These things
were however merely the " diversions of geometry at play t," and
he attached no importance to them. In the words of Plutarch, " he
possessed so high a spirit, so profound a soul, and such treasures
of scientific knowledge that, though these inventions had obtained
for him the renown of more than human sagacity, he yet would
not 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 in whose beauty and
subtlety there is no admixture of the common needs of life J." In
fact he wrote only one such mechanical book, On Sphere-making ,
to which allusion will be made later.

Some of his mechanical inventions were used with great effect
against the Romans during the siege of Syracuse. Thus he contrived

* Diodorus v. 37, 3, oOy [roi)y KOxXtas] 'ApxtM^rjs 6 SupaKO(rtos eflpei', ore
irap{pa\fv eft Atyvirrov.

t Plutarch, Marcellw, 14.

ibid. 17.

Pappus vin. p. 1026 (ed. Hultsch). KdpTros 3 TTOI; <j>-r\<n.v 6 'Avnoxcbs
' A.pXi/j.jjdrf rbv "Zvpaufxriov tv povov fiifiXiov ffvyrerax^o-i n-rixavtKbv rb /rarA rr)v
<70aipo7roita>, T&V dt &\\wv ovtev jfcuaictvai ffwrdt-at. KO.LTOL vapa rots TroXXois tvl
fjcqxaviKy doa.a0els Kal fjLcya.\o<f>v/is m yevbfjievos 6 0av/ia0T6s Iwlpof, wore
Tapd raffcy dvOp&irois UTrep/SaXX^rws u/xi/oiJ/xevos, rwv TC irpoijyovfjitvwv
Kal dpi 0/u 777-1*77$ lx^ vlav #cuplas ret ppaxvTara. doKOvvra clvai ffirovdatus ffvvtypa<f>cv
6s (paiverat rds eipyptvas twi(TT7)(j.as oOrws dya-rrtfaas cos
a i/rats

ARCHIMEDES. XVJi

catapults so ingeniously constructed as to be equally serviceable
at ^ong or short ranges, machines for discharging showers of
missiles through holes made in the walls, and others consisting
of long^ moveable poles projecting beyond the walls which either
dropped heavy weights upon the enemy's ships, or grappled the
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*. Marcellus
is said to have derided his own engineers and artificers with the
words, " Shall we not make an end of fighting against this geo-
metrical Briareus who, sitting at ease by the sea, plays pitch and
toss with our ships to our confusion, and by the multitude of
missiles that he hurls at us outdoes the hundred-handed giants of
mythology?t"; but the exhortation had no effect, the Romans being
in such abject terror that " if they did but see a piece of rope
or wood projecting above the wall, they would cry l there it is
again,' declaring that Archimedes was setting some engine in motion
against them, and would turn their backs and run away, insomuch
that Marcellus desisted from all conflicts and assaults, putting all
his hope in a long siege J."

If we are rightly informed, Archimedes died, as he had lived,
absorbed in mathematical contemplation. The accounts of the
exact circumstances of his death differ in some details. Thus
Livy says simply that, amid the scenes of confusion that followed
the capture of Syracuse, he was found intent on some figures which
he had drawn in the dust, arid was killed by a soldier who did
not know who he was. Plutarch gives more than one version in
the following passage. " Marcellus was most of all afflicted at
the death of Archimedes ; for, as fate would have it, he was intent
on working out some problem with a diagram and, having fixed
his mind and his eyes alike on his investigation, he never noticed
the incursion of the Romans nor the capture of the city. And
when a soldier came up to him suddenly and bade him follow to

* Polybius, Hi*t. vin. 78; Livy xxiv. 34; Plutarch, Marcellus, 1517.

t Plutarch, Marcellus, 17.

ibid.

Livy xxv. 31. Cum multa irae, multa auaritiae foeda exempla ederentur,
Archimedem memoriae proditum est in tanto tumultu, quantum pauor captae
urbia in discursu diripientium militum ciere poterat, intentura formis, quas in
puluere descripaerat, ab ignaro milite quis easet interfectum ; aegre id Marcellum
tulisse sepulturaeque curam habitam, et propinquis etiam iuquisitis honori
praesidioque nomen ao memoriain eius fuisse.

H. A. b

XV111 INTRODUCTION.

Marcellus, he refused to do so until he had worked out his problem
to a demonstration ; whereat the soldier was so enraged that he-
drew his sword and slew him. Others say that the Roman ran
up to him with a drawn sword offering to kill him ; and, when
Archimedes saw him, he begged him earnestly to wait a short time
in order that he might not leave his problem incomplete and
unsolved, but the other took no notice and killed him. Again
there is a third account to the effect that, as he was carrying to
Marcellus some of his mathematical instruments, sundials, spheres,
and angles adjusted to the apparent size of the sun to the sight, some
soldiers met him and, being under the impression that lie carried
gold in the vessel, slew him*." The most picturesque version of the
story is perhaps that which represents him as saying to a Roman
soldier who came too close, " Stand away, fellow, from my diagram,"
whereat the man was so enraged that he killed him t- The addition
made to this story by Zonaras, representing him as saying Trapu
K<f>a\dv KOL fjirj Trapd ypafjifjidv, while it no doubt recalls the second
version given by Plutarch, is perhaps the most far-fetched of the
touches put to the picture by later hands.

Archimedes is said to have requested his friends and relatives
to place upon his tomb a representation of a cylinder circumscribing
a sphere within it, together with an inscription giving the ratio
which the cylinder bears to the sphere J ; from which we may
infer that he himself regarded the discovery of this ratio [fM the
Sphere and Cylinder, I. 33, 34] as his greatest achievement. Cicero,
when quaestor in Sicily, found the tomb in a neglected state and
restored it.

Beyond the above particulars of the life of Archimedes, we
have nothing left except a number of stories, which, though perhaps
not literally accurate, yet help us to a conception of the personality
of the most original mathematician of antiquity which we would
not willingly have altered. Thus, in illustration of his entire
preoccupation by his abstract studies, we are told that he would
forget all about his food and such necessities of life, and would
be drawing geometrical figures in the ashes of the fire, or, when

* Plutarch, Marcellus, 19.

t Tzetzes, Chil. n. 35, 135 ; Zonaras ix. 5.

J Plutarch, Marcellus, 17 ad fin.

Cicero, Tutc. v. 64 sq.

ARCHIMEDES. XIX

anointing himself, in the oil on his body*. Of the same kind is
the well-known story that, when he discovered in a bath the
solution of the question referred to him by Hieron as to whether
a certajn crown supposed to have been made of gold did not in
reality contain a certain proportion of silver, he ran naked through
the street to his home shouting vprjKa, cvp^xat.

According to Pappus J it was in connexion with his discovery
of the solution of the problem To move a yiven weight by a given
force that Archimedes uttered the famous saying, " Give me a
place to stand on, and I can move the earth (So's /xot TTOV <TTW KOI
Kiva> rrjv yyv)" Plutarch represents him as declaring to Hieron
that any given weight could be moved by a given force, and
boasting, in reliance 011 the cogency of his demonstration, that, if
he were given another earth, he would cross over to it and move
this one. "And when Hieron was struck with amazement and asked
him to reduce the problem to practice 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 and many men ; and loading
her with many passengers and a full freight, sitting himself the
while far off, with no great endeavour but only holding the end
of a compound pulley (TroAiWaoros) quietly in his hand and pulling
at it, he drew the ship along smoothly and safely as if she were
moving through the seaSf." According to Proclus the ship was one
which Hieron had had made to send to king Ptolemy, and, when all
the Syracusans with their combined strength were unable to launch
it, Archimedes contrived a mechanical device which enabled Hieron
to move it by himself, insomuch that the latter declared that
"from that day forth Archimedes was to be believed in every-
thing that he might say \." While however it is thus established
that Archimedes invented some nicchunical contrivance for moving
n large ship and thus gave a practical illustration of his thesis,
it is not certain whether the machine used was simply a compound

* Plutarch, Marcellut, 17.

t Vitruuus, Atchitect. ix. 3. For an explanation of the manner in which
Archimedes probably solved this problem, see the note following On floating
bodies, i. 7 (p. 259 sq.).

* Pappus VIH. p. lOtiO.

Plutarch, Marcellu*, 14.

i! Proclus, Comm. nn End. i., p. (53 (ed. Friedleiu).

XX INTRODUCTION.

pulley (TToXvo-Traoros) as stated by Plutarch; for Athenaeus*, in
describing the same incident, says that a helix was used. This
term must be supposed to refer to a machine similar to the KOX\{CL$
described by Pappus, in which a cog-wheel with obliqug teeth
moves on a cylindrical helix turned by a handle f. Pappus, how-
ever, describes it in connexion with the /3apov\.Kos of Heron, and,
while he distinctly refers to Heron as his authority, he gives no
hint that Archimedes invented either the /3apov\Kos or the par-
ticular Ko^Xtas ; on the other hand, the TroXvo-Traoros is mentioned
by Galen J, and the TptWaoros (triple pulley) by Oribasius, as one
of the inventions of Archimedes, the TptWaoros being so called
either from its having three wheels (Vitruvius) or three ropes
(Oribasius). Nevertheless, it may well be that though the ship
could easily be kept in motion, when once started, by the rpL-
o-Tracrros or TroXvo-Tracrros, Archimedes was obliged to use an appliance
similar to the /co^Xtas to give the first impulse.

The name of yet another instrument appears in connexion with
the phrase about moving the earth. Tzetzes' version is, " Give
me a place to stand on (tra /3o>), and I will move the whole earth
with a xapioTiwv || " ; but, as in another passaged he uses the word
TpicnraoTos, it may be assumed that the two words represented one
and the same thing**.

It will be convenient to mention in this place the other
mechanical inventions of Archimedes. The best known is the

* Athenaeus v. 207 a-b, KaraaKeudcras yap e'XiKa r6 TIJ\IKOVTOV <rjcd0os efs rijv
6d\a<T<rav KCLTTiyayc' irpurros 6' 'ApxiMfys evpc Tyv 777$ ?Xi*oy KaraffKfv^v. To the
same effect is the statement of Eustathius ad 11. in. p. 114 (ed. Stallb.) Xl^crcu
3 Xt Kal rt jj.rjxaJ'r)* eldos, 6 Trpwroj cvp&v 6 'Apx^dr/y fvSoKl/J.rjff^ 4>a<rt t di avrov.

t Pappus viii. pp. 1066, 1108 sq.

t Galen, fit Hippocr. De artic., iv. 47 ( = xvm. p. 747, ed. Kuhn).

Oribasius, Coll. med., XLIX. 22 (iv. p. 407, cd. Bussemaker), 'AreXX^ous j
'Apxwydovs TpiffiraurTov, described in the same passage as having been invented
vpfa rdt TUV wXoiwv KaOo\Kdt.

|| Tzetzes, Chil. 11. 130.

*| Ibid., in. 61, 6 777^ avaairJav /x>7X a ^9 T V TpiffTraffry fioCw' oira /3u> Kal (

** Heiberg compares Simplicius, Comm. in Aristot. Phys. (ed. Diels, p. 1110,
1. 2), raurrj 8t TT) ava\aylq. rov KLVOVVTO* Kal rov KIVOV/JL^OV Kal rov
r6 ffTaOtuffTiKbv Spyavov rbv KCL\OV^VOV \apiffr iwv a crvffTrjcrat 6 'A
lU\pi iravrbi rrjt dvaXoyiai irpoxwpofarit tKbuiraatv {MIVQ rb ira PW Kal KWW rav
yav.

ARCHIMEDES. XXI

wa^er-screw * (also called Ko^Xtas) which was apparently invented
by him in Egypt, for the purpose of irrigating fields. It was
also used for pumping water out of mines or from the hold of
ships.

Another invention was that of a sphere constructed so as to
imitate the motions of the sun, the moon, and the five planets
in the heavens. Cicero actually saw this contrivance and gives a
description of itf, stating 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. Hultsch conjectures that it was moved by water J.
We know, as above stated, from Pappus that Archimedes wrote
a book on the construction of such a sphere (irepl <r<cupo7rouas),
and Pappus speaks in one place of "those who understand the
making of spheres and produce a model of the heavens by means
of the regular circular motion of water." In any case it is certain
that Archimedes was much occupied with astronomy. Livy calls
him "imicus spectator caeli siderurnque." 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 (OVK tLirtXirifa) that both I and Archimedes have erred to
the extent of a quarter of a day both in the observation and in the
deduction therefrom." It appears therefore that Archimedes had
considered the question of the length of the year, as Ammianus
also states !. Macrobius says that he discovered the distances of
the planets *[. Archimedes himself describes in the Sand-reckoner
the apparatus by which he measured the apparent diameter of the
sun, or the angle subtended by it at the eye.

The story that he set the Roman ships on lire by an arrange-
ment of burning-glasses or concave mirrors is not found in any

* Diodoms i. 34, v. 37; Vitruvius x. 16 (11) ; Philo HI. p. 330 (ed. Pfeiffer);
Strabo xvn. p. 807 ; Athenaeus v. 208 f.

f Cicero, DC rep., i. 21-2*2; Tune., i. 63; DC nat. dcor., n. 88. Cf. Ovid,
Fasti, vi. 277; Lactantius, Instit., n. o, 18; Martianus Capella, n. 212, vi.
683 sq. ; Claudian, Epigr. 18 ; Scxtua Empiricus, p. 416 (ed. Bekker).

J Xeitichnftf. Math. u. Phymk (hist, hit Abth.) t xxn. (1877), 106 sq.

Ptolemy, <riWais, i. p. 153.

|| Ammiauns Marcell., xxvi. i. 8.

[ Macrobius, in Sown. ci/>., n. 3.

INTRODUCTION.

authority earlier than Lucian*; and the so-called loculus Artfii-
medins, which was a sort of puzzle made of 14 pieces of ivory of
different shapes cut out of a square, cannot be supposed to be his
invention, the explanation of the name being perhaps that* it was
only a method of expressing that the puzzle was cleverly made,
in the same way as the 7rpo/3\rjfjLa 'A/^t/x^Sttov came to be simply
a proverbial expression for something very difficult f.

* The same story is told of Proclus in Zoimras xiv. 3. For the other
references on the subject see Heiberg's Quaestione* Archimedean pp. 39-41.
t Cf. also Tzetzes, Chil. xii. 270, T

CHAPTER II.

MANUSCRIPTS AND PRINCIPAL EDITIONS ORDER OF
COMPOSITION DIALECT LOST WORKS.

THE sources of the text and versions are very fully described
by Heiberg in the Prolegomena to Vol. in. of his edition of Archi-
medes, where the editor supplements and to some extent amends
what he had previously written on the same subject in his dis-
>ertation entitled Quaestiones Archimedeae (1879). It will there-
fore suffice here to state briefly the main points of the discussion.

The MSS. of the best class all had a common origin in a MS.
which, so far as is known, is 110 longer extant. It is described
in one of the copies made from it (to be mentioned later and dating
from some time between A.D. 1499 and 1531) as 'most ancient*
(rraAatoTaroi;), and all the evidence goes to show that it was written
as early as the 9th or 10th century. At one time it was in the
possession of George Valla, who taught at Venice between the
years 14^0 and 1499; and many important inferences with regard
to its readings can be drawn from some translations of parts of
Archimedes and Eutocius made by Valla himself and published
in his book entitled de expetendis et fuyiendis rebus (Venice, 1501).
It appears to have been carefully copied from an original belonging
to some one well versed in mathematics, and it contained figures
drawn for the most part with great care and accuracy, but there
was considerable confusion between the letters in the figures and
those in the text. This MS., after the death of Valla in 1499,
became the property of Albertus Pius Carpensis (Alberto Pio,
prince of Carpi). Part of his library passed through various hands
and ultimately reached the Vatican ; but the fate of the Valla
MS. appears to have been different, for we hear of its being in
the possession of Cardinal Hodolphus Pius (Rodolfo Pio), a nephew
of Albertus, in 1544, after which it seems to have disappeared.

XXIV INTRODUCTION.

The three most important MSS. extant are:

F (= Codex Florentinus bibliothecae Laurentianae Mediceae
plutei xxvin. 4to.).

B (= Codex Parisinus 2360, olim Mediceus).

C (= Codex Parisinus 2361, Fonteblandensis).

Of these it is certain that B was copied from the Valla MS.
This is proved by a note on the copy itself, which states that the
archetype formerly belonged to George Valla and afterwards to
Albertus Pius. From this it may also be inferred that B was
written before the death of Albertus in 1531 ; for, if at the date
of B the Valla MS. had passed to Rodolphus Pius, the name of
the latter would presumably have been mentioned. The note re-
ferred to also gives a list of peculiar abbreviations used in the
archetype, which list is of importance for the purpose of com-
parison with F and other MSS.

From a note on C it appears that that MS. was written by
one Christophorus Auverus at Rome in 1544, at the expense of
Georgius Armagniacus (Georges d'Armagnac), Bishop of Rodez,
then on a mission from King Francis I. to Pope Paul III. Further,
a certain Guilelmus Philander, in a letter to Francis I. published
in an edition of Vitruvius (1552), mentions that he was allowed,
by the kindness of Cardinal Rodolphus Pius, acting at the instance
of Georgius Armagniacus, to see and make extracts from a volume
of Archimedes which was destined to adorn the library founded
by Francis at Fontainebleau. He adds that the volume had been
the property of George Valla. We can therefore hardly doubt
that C was the copy which Georgius Armagniacus had made in
order to present it to the library at Fontainebleau.

Now F, B and C all contain the same works of Archimedes
and Eutocius, and in the same order, viz. (1) two Books de sphaera
et cylindro, (2) de dimensione circuli^ (3) de conoidibus^ (4) fk
lineis spiralibus, (5) de planis aeque, ponderantibna, (6) arenarius,
(7) quadratures parabolae, and the commentaries of Eutocius on
(1) (2) and (5). At the end of the quadralura parabolne both
F and B give the following lines :

cvruxofys Ac'ov ycw/xerpa

iroAAous cts AvKajSavras tots irokv ^tArarc /movcrats.
F and C also contain mensurae from Heron and two fragments
ircpi ora^/Luov and vipl /ticrpcoi', the order being the same in both

MANUSCRIPTS. XXV

and the contents only differing in the one respect that the last
fragment Trcpt /xcVpwi/ is slightly longer in F than in C.

A short preface to C states that the first page of the archetype
was so rubbed and worn with age that not even the name of
Archimedes could be read upon it, while there was no copy at
Rome by means of which the defect could be made good, and
further that the last page of Heron's de mensuris was similarly
obliterated. Now in F the first page was apparently left blank
at first and afterwards written in by a different hand with many
gaps, while in B there are similar deficiencies and a note attached
by the copyist is to the effect that the first page of the archetype
was indistinct. In another place (p. 4 of Vol. in., ed. Heiberg)
all three MSS. have the same lacuna, and the scribe of B notes
that one whole page or even two are missing.

Now C could not have been copied from F because the last
page of the fragment Trcpi ptrpw is perfectly distinct in F; and,
on the other hand, the archetype of F must have been illegible
at the end because there is no word rcAo? at the end of F, nor any
other of the signs by which copyists usually marked the completion
of their ta>sk. Again, Valla's translations show that his MS. had
certain readings corresponding to correct readings in B and C
instead of incorrect readings given by F. Hence F cannot have
been Valla's MS. itself.

The positive evidence about F is as follows. Valla's trans-
lations, with the exception of the few readings just referred to,
agree completely with the text of F. From a letter written at
Venice in 1491 by Angelus Politianus (Angelo Poliziano) to Lau-
rentius Mediceus (Lorenzo de' Medici), it appears that the former
had found a MS. at Venice containing works by Archimedes and
Heron and proposed to have it copied. As G. Valla then lived
at Venice, the MS. can hardly have been any other but his, and
no doubt F was actually copied from it in 1491 or soon after.
Confirmatory evidence for this origin of F is found in the fact
that the form of most of the letters in it is older than the 15th
century, and the abbreviations etc., while they all savour of an
ancient archetype, agree marvellously with the description which
the note to B above referred to gives of the abbreviations used
in Valla's MS. Further, it is remarkable that the corrupt passage
corresponding to the illegible first page of the archetype just takes
up one page of F, no more and no less.

XXVI INTRODUCTION.

The natural inference from all the evidence is that F, B
C all had their origin in the Valla MS. ; and of the three F is
the most trustworthy. For (1) the extreme care with which the
copyist of F kept to the original is illustrated by a number of
mistakes in it which correspond to Valla's readings but are cor-
rected in B and C, and (2) there is no doubt that the writer of
B was somewhat of an expert and made many alterations on his
own authority, not always with success.

Passing to other MSS., we know that Pope Nicholas V. had
a MS. of Archimedes which he caused to be translated into Latin.
The translation was made by Jacobus Cremonensis (Jacopo Cas-
siani*), and one copy of this was written out by Joannes Regio-
montanus (Johann Miiller of Konigsberg, near Hassfurt, in Fran-
coma), about 1461, who not only noted in the margin a number
of corrections of the Latin but added also in many places Greek
readings from another MS. This copy by Regiomontanus is pre-
served at Xurnberg and was the source of the Latin translation
given in the editio princrps of Thomas Gechautf' Venatorius (Basel,
1544); it is called X b by Heiberg. (Another copy of the same
translation is alluded to by Regiomontanus, and this is doubtless
the Latin MS. 327 of 15th c. still extant at Venice.) From the
fact that the translation of Jacobus Cremonensis has the same
lacuna as that in F, B and C above referred to (Vol. in., ed.
Heiberg, p. 4), it seems clear that the translator had before him
either the Valla MS. itself or (more likely) a copy of it, though
the order of the books in the translation differs in one respect
from that in our MSS., viz. that the armarius comes after instead
of before the quadrat am paraf*oJae.

It is probable that the Greek MS. used by Regiomontanus was V
(= Codex Venetus Marcianus cccv. of the 15th c.), which is still extant
and contains the same books of Archimedes and Eutocius with the
same fragment of Heron as F has, and in the same order. If the
above conclusion that F dates from 1491 or thereabouts is correct,
then, as V belonged to Cardinal Bessarione, who died in 1472, it
cannot have been copied from F, and the simplest way of accounting
for its similarity to F is to suppose that it too was derived from
Valla's MS.

* Tirabochi, Storia delta Letterntura Italiana, Vol. vi. Pt. 1 (p. 858 of the
edition of 1807). Cantor (Vorlcmtngen ill. (tench, d. Math., u. p. 102) ivcB the
full name and title as Jacopo da S. Cassiano Cremonese canonico regolarc.

MANUSCRIPTS. XXV11

Jlegiomontanus mentions, in a note inserted later than the
rest and in different ink, two other Greek MSS., one of which he
calls "exemplar vetus apud magistrum Paulum." Probably the
monk Paulus (Albertini) of Venice is here meant, whose date was
1430 to 1475; and it is possible that the "exemplar vetus" is
the MS. of Valla.

The two other inferior MSS., viz. A (= Codex Parisinus 2359,
olim Mediceus) and D (= Cod. Parisinus 2362, Fonteblandensis),
owe their origin to V.

It is next necessary to consider the probabilities as to the MSS.
used by Nicolas Tartaglia for his Latin translation of certain of
the works of Archimedes. The portion of this translation published
at Venice in 1543 contained the books de centris yravium vel de
fiequerepentibus /-//, tetrayonismus [paraMae], diniemio circuli
and de insidfiutibus aquae /; the rest, consisting of Book II de
iasidentibus aquae, was published with Book I of the same treatise,
after Tartaglia's death in 15 .57, by Troianus Curtius (Venice, 1565).
Now the last-named treatise is not extant in any Greek MS. and,
as Tartaglia adds it, without any hint of a separate origin, to the
rest of the books which he says he took from a mutilated and
almost illegible Greek MS., it might easily be inferred that the
Greek MS. contained that treatise also. But it is established, by
a letter written by Tartaglia himself eight years later (1551) that
he then had no Greek text of the Books <l? insidentibus aquae, and
it would be strange if it had disappeared in so short a time without
leaving any trace. Further, Commandinus in the preface to his
edition of the same treatise (Bologna, 1565) shows that he had
never hoard of a Greek text of it. Hence it is most natural to
suppose that it reached Tartaglia from some other source and in the
Latin translation only*.

The fact that Tartaglia speaks of the old MS. which he used
as "fracti et qui vix legi poterant libri," at practically the same
time as the writer of the preface to C was giving a similar de-
scription of Valla's MS., makes it probable that the two were

* The Greek fragment of Book I., irtpl rv i>5an (^KTra^vuv ij **/* TUW
, edited by A. Mai from two Vatican MSS. (CfeiMifi duct. i. p. 426-30 ;
Vol. ii. of Heiberg's edition, pp. 35(5-8), seems to be of doubtful authenticity.
Except for the first proposition, it contains enunciations only and no proofs.
Heiberg is inclined to think that it represents an attempt at retranslation into
Greek made by some mediaeval scholar, and he compares the similar attempt
made by liivanlt.

XXV111 INTRODUCTION.

identical ; and this probability is confirmed by a considerable agree-
ment between the mistakes in Tartaglia and in Valla's versions.

But in the case of the quadratures parabolae and the dimensio
circuli Tartaglia adopted bodily, without alluding in any* way to
the source of it, another Latin translation published by Lucas
Gauricus " Tuphanensis ex regno Neapolitano " (Luca Gaurico of
Gifuni) in 1503, and he copied it so faithfully as to reproduce most
obvious errors and perverse punctuation, only filling up a few
gaps and changing some figures and letters. This translation by
Gauricus is seen, by means of a comparison with Valla's readings
and with the translation of Jacobus Cremonensis, to have been
made from the same MS. as the latter, viz. that of Pope Nicolas V.

Even where Tartaglia used the Valla MS. he does not seem
to have taken very great pains to decipher it when it was
not easily legible it may be that he was unused to deciphering
MSS. and in such cases he did not hesitate to draw from other
sources. In one place (de planor. equiUb. n. 9) he actually
gives as the Archimedean proof a paraphrase of Eutocius some-
what retouched and abridged, and in many other instances he
has inserted corrections and interpolations from another Greek
MS. which he once names. This MS. appears to have been a copy
made from F, with interpolations due to some one not unskilled
in the subject-matter; and this interpolated copy of F was ap-
parently also the source of the Nurnberg MS. now to be mentioned.

N a (= Codex Norimbergensis) was written in the 16th century
and brought from Rome to Nurnberg by Wilibaki Pirckheymer.
It contains the same works of Archimedes and Eutocius, and in
the same order, as F, but was evidently not copied from F direct,
while, on the other hand, it agrees so closely with Tartaglia's
version as to suggest a common origin. N a was used by Vena-
torius in preparing the fditio princeps, and Venatorius corrected
many mistakes in it with his own hand by notes in the margin
or on slips attached thereto ; he also made many alterations in
the body of it, erasing the original, and sometimes wrote on it
directions to the printer, so that it was probably actually used
to print from. The character of the MS. shows it to belong to
the same class as the others ; it agrees with them in the more
important errors and in having a similar lacuna at the beginning.
Some mistakes common to it and F alone show that its source was
F, though at second hand, as above indicated.

EDITIONS AND TRANSLATIONS.

ft remains to enumerate the principal editions of the Greek
text and the published Latin versions which are based, wholly or
partially, upon direct collation of the MSS. These are as follows,
in addition to Gaurico's and Tartaglia's translations.

1. The editio princeps published at Basel in 1544 by Thomas
Gechauff Venatorius under the title Archimedis opera quae quidem
exstant omnia nunc primum graece et latine in lucent edita. Adiecta
t/uoque aunt Eutocii Ascalonitae commentaria item graece et latine
nunquam antea eoccusa. The Greek text and the Latin version in
this edition were taken from different sources, that of the Greek
text being N a , while the translation was Joannes Regiomontanus'
revised copy (N b ) of the Latin version made by Jacobus Cremo-
nensis from the MS. of Pope Nicolas V. The revision by
Regiomontanus was effected by the aid of (1) another copy of
the same translation still extant, (2) other Greek MSS., one of
which was probably V, while another may have been Valla's MS.
itself.

2. A translation by F. Commaiidinus (containing the following
works, circuli dimensio, de lineis spiralibus, quadratura parabolae,
de conoidibus et sphaeroidibus, de aretiae nuniero) appeared at
Venice in 1558 under the title Archimedis opera nonnulla in
latinum conversa et commentariis illustrata. For this translation
several MSS. were used, among which was V, but none preferable
to those which we now possess.

3. D. Ri vault's edition, Archimedis opera quae exstant graece
et latine novis demonstr. et comment, illustr. (Paris, 1615), gives
only the propositions in Greek, while the proofs are in Latin and
somewhat retouched. Rivault followed the Basel editio princeps
with the assistance of B.

4. Torelli's edition (Oxford, 1792) entitled 'Ap^^Sou? ra <ro>-
6/xcva /ida ru)v EUTOKI'OU 'AovcaA.a>i/trov vTro/xny/iaTwv, Archimedis
quae supersunt omnia cum Eutocii Ascalonitae commentariis ex
recensione J. Torelli Veronensis cum nova versione latina. Acced-
unt lectiones variantes ex codd. Mediceo et Parisiensibiis. Torelli
followed the Basel editio princeps in the main, but also collated
V. The l)ook was brought out after Torelli's death by Abram
Robertson, who added the collation of five more MSS., F, A, B, C, D,
with the Basel edition. The collation however was not well done,
and the edition was not properly corrected when in the press.

XXX INTRODUCTION.

5. Last of all comes the definitive edition of Heiberg (Ajffti-
rnedis opera omnia curti commentariis Eutocii. E coclice Florentine
recensuit) Latine uertit notisque illustrauit J. L. Heiberg. Leipzig,
18801).

The relation of all the MSS. and the above editions and trans-
lations is well shown by Heiberg in the following scheme (with
the omission, however, of his own edition) :

Codex Uallae saec. ix x

Cod. N
c.

Gauricu

icolai V F
1453 c. 1491

Tartalea
a. 1543 sae

C. XV C.

EdT
a.

i:>oo

Hiualti
1615

a. ir> 14

Cod. Tartaleae n
N a saec. xvi
Ed. Basil. 1544

A, 1)
saec. xvi

Torellius 1

Commanding

s Cremonensis

c. 14()0

Cod. Uenet. 327
saec. xv

N b , c. 1401

The remaining editions which give portions of Archimedes in
Greek, and the rest of the translations of the complete works or
parts of them which appeared before Heiberg's edition, were not
based upon any fresh collation of the original sources, though some
excellent corrections of the text were made by some of the editors,
notably Wallis and Nizze. The following books may be mentioned.

Joh. Chr. Sturm, Des unvergleichlichen Archimedis Kunstbucher,
vbersetzt und erlautert (Nurnberg, 1G70). This translation em-
braced all the works extant in Greek and followed three years
after the same author's separate translation of the Sand-reckoner.
It appears from Sturm's preface that he principally used the edition
of Rivault.

Is. Barrow, Opera Archimedis^ Apoflonii Pergaei conicorum libri,
Theodosii sphaerica methodo novo illustrata et demonstrates (London,
1675).

Wallis/ Archimedis arenarius et dimensio circuli, Eutocii in hanc
commentarii cum versions et notis (Oxford, 1678), also given
in Wallis' Opera, Vol. in. pp. 509546.

Karl Friedr. Hauber, Archimeds zwei Bucher iiber Kugel und
Cylinder. Ebendeaselben Kreismeasung. Uebersetzt mit Anmerkungen
u. s. w. begleitet (Tubingen, 1798).

TRANSLATIONS ORDER OF WORKS. XXXI

F. Peyrard, CEuvres d'Archimdde, traduites litteralement, avec
un commentaire, suivies d'un memoir e du traductcur, ttur un nouveau
miroir ardent, et tffun autre inemoire de M. DelamLre, sur Varith-
vnetique des Green. (Second edition, Paris, 1808.)

Ernst Nizze, Archimedes von Syrakus vorhandene Werke, aus de/n
Griechischen ubersetzt und mit erlduternden und kritischen Aniner-
kunyen begleitet (Stralsund, 1824).

The MSS. give the several treatises in the following order.

1. TTcpi ox/xupa? KOI KvXivrjpov a /?', two Books On the Sphere
and Cylinder.

2. KVK\OV fjLTprj<ris*, Measurement of a Circle.

3. Trcpi, KojvoetoVojv Kcu ox^utpociScW, On Conoids and Spheroids.

4. jrepi eA.iKwi', Ou Spirals.

5. cTTiTTeSwi/ taoppoTTtajv a /?'f, two Books On, t/te Equilibrium
of Planes.

G. i/ra/A/xiVtys, The Sand-reckoner.

7. TCTpaytDvio-fjLos TrapaftoXrjs (a name substituted later for that
given to the treatise by Archimedes himself, which must
undoubtedly have been TTpayum<7/oio9 rrjs TOV op&oyaiviov
/cwyou To/xiysJ:), Quadrature of the Parabola.

To these should be added

8. TTcpi 6xovfjLvti>v, the Greek title of the treatise On floatiny
bodies, only preserved in a Latin translation.

* Pappus alludes (i. p. 312, ed. Hultsch) to the KVK\OV utrpijffis in the words
4v T<JJ irfpi 7775 roO KVK\OV 7re/)t0ep6/aj.

t Archimedes himself twice alludes to properties proved in Book i. as
demonstrated $v rots /zTjxcwiAcoij (Quadrature of the Parabola, Props. 6, 10).
Pappus (vin. p. 1084) quotes rd 'ApxtM^ous ircpi i<roppomwv. The beginning of
Book i. is also cited by Proclus in his Commentary on End. i., p. 181, where the
reading should be TOV a uroppoTncJv, and not ruv dvuroppoiriwv (Hultsch).

^ The name 4 parabola' was first applied to the curve by Apollouius. Archi-
medes always used the old term ' section of a right-angled cone.' Of. Eutociub
(Heiberg, vol. in., p. 342) 5^<5et*rcu tv r< rrepi TTJS TOV dpQoywiov KWVOV TO/A^S.

This title corresponds to the references to the book in Strabo i. p. 54
IP rots Trcpt roJf bxov^vwv) and Pappus YIII. p. 1024 (u>s ' Xpxwtf'n*
. The fragment edited by Mai has a longer title, rc/ri TUV 05an
v 17 Trcpi TU>V 6xoi>/n^u)v, where the first part corresponds to Tartaglia's
version, de insidentibus aquae, and to that of Commandiuus, de Us quae vehun-
tur i/i aqua. But Archimedes intentionally used the more general word vyp6t>
(fluid) instead of vdwp ; and hence the shorter title ircpi dxoi^wv, de Us quae
in humido vehun tur (Torelli and Heiberg), seems the better.

xxxii INTRODUCTION.

The books were not, however, written in the above order; and
Archimedes himself, partly through his prefatory letters and partly
by the use in later works of properties proved in earlier treatises,
gives indications sufficient to enable the chronological equence
to be stated approximately as follows :

1. On the equilibrium of planes, I.

2. Quadrature of the Parabola.

3. On the equilibrium of planes, II.

4. On the Sphere and Cylinder, I, II.

5. On Spirals.

6. On Conoids and Spheroids.

7. On floating bodies, I, II.

8. Measurement of a circle.

9. The Sand-reckoner.

It should however be observed that, with regard to (7), no
more is certain than that it was written after (G), and with regard
to (8) no more than that it was later than (4) and before (9).

In addition to the above we have a collection of Lemmas (Liber
Assumptorum) which has reached us through the Arabic. The
collection was first edited by S. Foster, Miscellanea (London, 1659),
and next by Borelli in a book published at Florence, 1661, in
which the title is given as Liber assumptorum Archimedis interprets
Thebit ben Kora et exponente doctore Almochtasso AbiUiasan. The
Lemmas cannot, however, have been written by Archimedes in
their present form, because his name is quoted in them more than
once. The probability is that they were propositions collected by
some Greek writer* of a later date for the purpose of elucidating
some ancient work, though it is quite likely that some of the
propositions were of Archimedean origin, e.g. those concerning
the geometrical figures called respectively dp/fyXost (literally

* It would seem tbat the compiler of the Liber Assumptorum must have
drawn, to a considerable extent, from the same sources as Pappus. The
number of propositions appearing substantially in the same form in both
collections is, I think, even greater than has yet been noticed. Tannery (La
Geomttrie grecque, p. 162) mentions, as instances, Lemmas 1, 4, 5, 6 ; but it
will be seen from the notes in this work that there are several other coin-
cidences.

t Pappus gives (p. 208) what he calls an 'ancient proposition' (dpxala
vpbraffis) about the same figure, which he describes as \uplov, 6 3i? *a\oiW
&ppTj\ot>. Cf. the note to Prop. 6 (p. 308). The meaning of the word is gathered

WORKS ASCRIBED TO ARCHIMEDES. XXX111

'shoemaker's knife') and a-dXwov (probably a 'salt-cellar'*), and
Prop. 8 which bears on the problem of trisecting an angle.

from the^Soholia to Nioander, Theriaca, 423: Ap^Xoi \4yovrai rb. KVK\orepr)
<rt5i}pta, oft ol <rKvror6/uot r^/xi/oucrt /cai tfou<rt rd tepfMra. Cf. Hesychius,
dvdpfttjXa, T&. fj.rj <!ecr/^fa fl^tara* Ap/S^Xoi yip TO. 07uMa.

* The best authorities appear to hold that in any case the name o-lXciw was
not applied to the figure in question by Archimedes himself but by some later
writer. Subject to this remark, I believe crdXivov to be simply a Graecised
form of the Latin word salinum. We know that a salt-cellar was an essential
part of the domestic apparatus in Italy from the early days of the Roman
Republic. "All who were raised above poverty had one of silver which
descended from father to son (Hor., Carm. n. 16, 13, Liv. xxvi. 36), and
was accompanied by a silver patella which was used together with the salt-
cellar in the domestic sacrifices (Pers. HI. 24, 25). These two articles of
silver were alone compatible with the simplicity of Roman manners in the
early times of the Republic (Plin., //. A', xxxm. 153, Val. Max. iv. 4, 3).
...In shape the salinum was probably in most cases a round shallow bowl"
[Diet, of Greek and Roman Antiquities, article salinum]. Further we have
in the early chapters of Mommsen's History of Rome abundant evidence
of similar transferences of Latin words to the Sicilian dialect of Greek. Thus
(Book i., ch. xiii.) it is shown that, in consequence of Latino-Sicilian com-
merce, certain words denoting measures of weight, libra, triens, quadrans,
sextans, uncia, found their way into the common speech of Sicily in the third
century of the city under the forms Xfrpa, rptas, Tcrpas, eas, ovyicia. Similarly
Latin law-terms (ch. xi.) were transferred ; thus mutuum (a form of loan)
became /AO?TOI>, career (a prison) Kapnapov. Lastly, the Latiii word for lard,
arvina, became in Sicilian Greek dp/3foi?, and pafum (a dish) Tra.Ta.vrj. The last
word is as close a parallel for the supposed transfer of salinum as could be
wished. Moreover the explanation of rdXurw as salinum has two obvious
advantages in that (1) it does not require any alteration in the word, and

(2) the resemblance of the lower curve to an ordinary type of salt-cellar is
evident. I should add, as confirmation of my hypothesis, that Dr A. S. Murray,
of the British Museum, expresses the opinion that we cannot be far wrong in
accepting as a sal mum one of the small silver bowls in the Roman ministerium

H. A. C

XXXIV INTRODUCTION.

Archimedes is further credited with the authorship of f the
Cattle-problem enunciated in the epigram edited by Leasing in
1773. According to the heading prefixed to the epigram it was
communicated by Archimedes to the mathematicians at Alexandria
in a letter to Eratosthenes*. There is also in the Scholia to Plato's
Charmides 165 E a reference to the problem "called by Archimedes
the Cattle-problem" (TO K\7j6w VTT 'A/o^tfwySovs /JoeiKoV Tr/oo^Ary^ta).
The question whether Archimedes really propounded the problem,
or whether his name was only prefixed to it in order to mark the
extraordinary difficulty of it, has been much debated. A complete
account of the arguments for and against is given in an article
by Krumbiegel in the Zeitschrift fur Mathematik und Physik
(Hist. Hit. Abtheilung) xxv. (1880), p. 121 sq., to which Amthor
added (ibid. p. 153 sq.) a discussion of the problem itself. The
general result of Krumbiegel's investigation is to show (1) that

at the Museum which was found at Chaourse (Aisne) in France and is of a
section sufficiently like the curve in the Salinon.

The other explanations of <rd\ivov which have been suggested are as follows.

(1) Cantor connects it with 0-dXos, " das Schwanken des liohen Meeres,"
and would presumably translate it as wave-line. But the resemblance is
not altogether satisfactory, and the termination -LVOV would need explanation.

(2) Heiberg says the word is " sine dubio ab Arabibus deprauatum," and
suggests that it should be fft\ipov, parsley ("ex similitudine frondis apii").
But, whatever may be thought of the resemblance, the theory that the word is
corrupted is certainly not supported by the analogy of &ppr)\os which is correctly
reproduced by the Arabs, as we know from the passage of Pappus referred to in
the last note.

(3) Dr Gow suggests that <rd\ivov may be a ' sieve,' comparing <rd\a. But
this guess is not supported by any evidence.

* The heading is, Hp6(3\r)iJ.a. ftirep 'A/>x i M^&7* ev irLypdfjLiJ.aat.v evp&v rots tv
' > A\c}-avdpiq. irepi TCLVTCL irpaynarevoiuLtvoii ftrfiv dirforciXcv tv rrj irp&s 'EpaTOffBtvyi/
rbv Kvpyvaiov tm<FTo\fj. Heiberg translates this as "the problem which
Archimedes discovered and sent in an epigram... in a letter to Eratosthenes."
He admits however that the order of words is against this, as is also the use of
the plural iiri.ypdnna.aw. It is clear that to take the two expressions iv
^iriypdfjiiJia<TLv and Iv ciricrTo\fj as both following dTrlareiXep is very awkward. In
fact there seems to be no alternative but to translate, as Krumbiegel does, in
accordance with the order of the words, "a problem which Archimedes found
among (some) epigrams and sent... in his letter to Eratosthenes " ; and this sense
is certainly unsatisfactory. Hultsch remarks that, though the mistake -rrpay-
fiarovfjLfpoa for irpay/jLaTvo/j.^voLS and the composition of the heading as a whole
betray the hand of a writer who lived some centuries after Archimedes, yet he
must have had an earlier source of information, because he could hardly have
invented the story of the letter to Eratosthenes.

WORKS ASCRIBED TO ARCHIMEDES. XXXV

the ipigram can hardly have been written by Archimedes in its
present form, but (2) that it is possible, nay probable, that the
problem was in substance originated by Archimedes. Hultsch* has
an ingenious suggestion as to the occasion of it. It is known that
Apollonius in his WKVTOKIOV had calculated a closer approximation to
the value of IT than that of Archimedes, and he must therefore have
worked out more difficult multiplications than those contained in
the Measurement of a circle. Also the other work of Apollonius
on the multiplication of large numbers, which is partly preserved
in Pappus, was inspired by the Hand-reckoner of Archimedes ; and,
though we need not exactly regard the treatise of Apollonius as
polemical, yet it did in fact constitute a criticism of the earlier
book. Accordingly, that Archimedes should then reply with a
problem which involved such a manipulation of immense numbers
as would be difficult even for Apollonius is not altogether outside
the bounds of possibility. And there is an unmistakable vein of
satire in the opening words of the epigram " Compute the number
of the oxen of the Sun, giving thy mind thereto, if thou hast a
share of wisdom," in the transition from the first part to the
second where it is said that ability to solve the first part would
entitle one to be regarded as " not unknowing nor unskilled in
numbers, but still not yet to be numbered among the wise," and
again in the last lines. Hultsch concludes that in any case the
problem is not much later than the time of Archimedes and dates
from the beginning of the 2nd century B.C. at the latest.

Of the extant books it is certain that in the 6th century A.D.
only three were generally known, viz. On the Sphere and Cylinder,
the Measurement of a circle, and On the equilibrium of planes. Thus
Eutocius of Ascalon who wrote commentaries on these works only
knew the Quadrature of the Parabola by name and had never seen
it nor the book On Spirals. Where passages might have been
elucidated by references to the former book, Eutocius gives ex-
planations derived from Apollonius and other sources, and he
speaks vaguely of the discovery of a straight line equal to the
circumference of a given circle "by means of certain spirals,"
whereas, if he had known the treatise On Spirals, he would have
quoted Prop. 18. There is reason to suppose that only the three
treatises on which Eutocius commented were contained in the

* Pauly-Wissowa's Real-Encyclopiidie, n. 1, pp. 534, 5.

XXXVI INTRODUCTION.

ordinary editions of the time such as that of Isidorus of Mijetus,
the teacher of Eutocius, to which the latter several times alludes.

In these circumstances the wonder is that so many more books
have survived to the present day. As it is, they havelost to a
considerable extent their original form. Archimedes wrote in the
Doric dialect*, but in the best known books (On the Sphere and
Cylinder and the Measurement of a circle) practically all traces
of that dialect have disappeared, while a partial loss of Doric forms
has taken place in other books, of which however the Sand-
reckoner has suffered least. Moreover in all the books, except the
Sand-reckoner, alterations and additions were first of all made by
an interpolator who was acquainted with the Doric dialect, and
then, at a date subsequent to that of Eutocius, the book On the
Sphere and Cylinder and the Measurement of a circle were completely
recast.

Of the lost works of Archimedes the following can be identified.

1. Investigations relating to polyhedra are referred to by
Pappus who, after alluding (v. p. 352) to the five regular polyhedra,
gives a description of thirteen others discovered by Archimedes
which are semi-regular, being contained by polygons equilateral
and equiangular but not similar.

2. A book of arithmetical content, entitled a'px<" Principles
and dedicated to Zeuxippus. We learn from Archimedes himself
that the book dealt with the naming of numbers (KaTov6p.ais TO>I/

and expounded a system of expressing numbers higher

* Thus Eutocius in his commentary on Prop. 4 of Book n. On the Sphere
and Cylinder speaks of the fragment, which he found in an old book and which
appeared to him to be the missing supplement to the proposition referred to,
as *' preserving in part Archimedes' favourite Doric dialect" (ev nfyci 81 ryv
fai <J>l\i)v AwpfSa y\u>ff(rav airtffwfrv). From the use of the expression tv
Heiberg concludes that the Doric forms had by the time of Eutocius
begun to disappear in the books which have come down to us no less than in
the fragment referred to.

t Observing that in all the references to this work in the Sand-reckoner
Archimedes speaks of the naming of numbers or of numbers which are named or have
their names (&pi0/j.ol KaTwonafffdvoi, rd, 6v6fJ.ara ^OPTCS, r ^- v Ko-rovofia^iav ^OPTCS),
Hultsch (Pauly-Wissowa's Real- Encyclopa die, n. 1, p. 511) speaks of KCLTOVO-
fj.a.^3 TWV dpiOpuv as the name of the work ; and he explains the words nvas TW
iv dpx a " <dpi0fMov> T&V KarovofJLai-lav txbvruv as meaning "some of the
numbers mentioned at the beginning which have a special name," where "at
the beginning " refers to the passage in which Archimedes first mentions r&v

LOST WORKS. XXXV11

thai* those which could be expressed in the ordinary Greek no-
tation. This system embraced all numbers up to the enormous
figure which we should now represent by a 1 followed by 80,000
billion ciphers; and, in setting out the same system in the Sand-
reckoner, Archimedes explains that he does so for the benefit of
those who had not had the opportunity of seeing the earlier work
addressed to Zeuxippus.

3. 7T/ol vyoiv, On balances or levers, in which Pappus says (vm.
p. 1068) that Archimedes proved that " greater circles overpower
(KaraKpaTovo-C) lesser circles when they revolve about the same
centre." It was doubtless in this book that Archimedes proved
the theorem assumed by him in the Quadrature of the Parabola,
Prop. 6, viz. 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.

4. KVT/oo/?aptKa, On centres of gravity. This work is mentioned
by Simplicius on Aristot. de caelo n. (Scholia in Arist. 508 a 30).
Archimedes may be referring to it when he says (On the equilibrium
of planes i. 4) that it has before been proved that the centre of
gravity of two bodies taken together lies on the line joining the
centres of gravity of the separate bodies. In the treatise On

floating bodies Archimedes assumes that the centre of gravity of a
segment of a paraboloid of revolution is on the axis of the segment
at a distance from the vertex equal to rds of its length. This
may perhaps have been proved in the Kcvrpo/fo/Hxa, if it was
not made the subject of a separate work.

Doubtless both the TTC/OI vy<3i/ and the KcvrpoftapiKoi preceded
the extant treatise On the equilibrium of planes.

5. KaTOTrrpiKOL, an optical work, from which Theon (on Ptolemy,
Synt. I. p. 29, ed. Halma) quotes a remark about refraction.
Of. Olympiodorus in Aristot. Meteor., n. p. 94, ed. Ideler.

v ev row irorl

But h dpx a ?s seems a less natural expression for " at the beginning"
than tv apxti or KCLT dpxfa would have been. Moreover, there being no
participial expression except Karovo^iav W" to be taken with tv dpxcus in
this sense, the meaning would be unsatisfactory ; for the numbers are not
named at the beginning, but only referred to, and therefore some word like
tlpvintvwv should have been used. For these reasons I think that Heiberg,
Cantor and Susemihl are right in taking dpxai to be the name of the treatise.

XXXV111 INTRODUCTION.

6. TTpl ox^aipoTToua?, On sphere-making, a mechanical worfe on
the construction of a sphere representing the motions of the
heavenly bodies as already mentioned (p. xxi).

7. <o8iov, a Method, noticed by Suidas, who says that Theo-
dosius wrote a commentary on it, but gives no further information
about it.

8. According to Hipparchus Archimedes must have written
on the Calendar or the length of the year (cf. p, xxi).

Some Arabian writers attribute to Archimedes works (1) On
a heptagon in a circle, (2) On circles touching one another, (3) On
parallel lines, (4) On triangles, (5) On the properties of right-
angled triangles, (6) a book of Data ; but there is no confirmatory
evidence of his having written such works. A book translated
into Latin from the Arabic by Gongava (Lou vain, 1548) and en-
titled antiqui scriptoris cle speculo coinburente concavitatis parabolae
cannot be the work of Archimedes, since it quotes Apollonius.

CHAPTER III.

THE RELATION OF ARCHIMEDES TO HIS PREDECESSORS.

AN extraordinarily large proportion of the subject matter of
the writings of Archimedes represents entirely new discoveries of
his own. Though his range of subjects was almost encyclopaedic,
embracing geometry (plane and solid), arithmetic, mechanics, hydro-
statics and astronomy, he was no compiler, no writer of text-
books ; and in this respect he differs even from his great successor
Apollonius, whose work, like that of Euclid before him, largely
consisted of systematising and generalising the methods used, and
the results obtained, in the isolated efforts of earlier geometers.
There is in Archimedes no mere working-up of existing materials ;
his objective is always some new thing, some definite addition to
the sum of knowledge, and his complete originality cannot fail
to strike any one who reads his works intelligently, without any
corroborative evidence such as is found in the introductory letters
prefixed to most of them. These introductions, however, are emi-
nently characteristic of the man and of his work ; their directness
and simplicity, the complete absence of egoism and of any effort
to magnify his own achievements by comparison with those of
others or by emphasising their failures where he himself succeeded :
all these things intensify the same impression. Thus his manner
is to state simply what particular discoveries made by his pre-
decessors had suggested to him the possibility of extending them
in new directions ; e.g. he says that, in connexion with the efforts
of earlier geometers to square the circle and other figures, it
occurred to him that no one had endeavoured to square a parabola,
and he accordingly attempted the problem and finally solved it.
In like manner, he speaks, in the preface of his treatise On the

xl INTRODUCTION.

Sphere and Cylinder, of his discoveries with reference to tfiose
solids as supplementing the theorems about the pyramid, the cone
and the cylinder proved by Eudoxus. He does not hesitate to
say that certain problems baffled him for a long time, and that
the solution of some took him many years to effect; and in one
place (in the preface to the book On Spirals) he positively insists,
for the sake of pointing a moral, on specifying two propositions
which he had enunciated and which proved on further investigation
to be wrong. The same preface contains a generous eulogy of
Conon, declaring that, but for his untimely death, Conon would
have solved certain problems before him and would have enriched
geometry by many other discoveries in the meantime.

In some of his subjects Archimedes had no fore-runners, e.g.
in hydrostatics, where he invented the whole science, and (so
far as mathematical demonstration was concerned) in his me-
chanical investigations. In these cases therefore he had, in laying
the foundations of the subject, to adopt a form more closely re-
sembling that of an elementary textbook, but in the later parts
he at once applied himself to specialised investigations.

Thus the historian of mathematics, in dealing with Archimedes'
obligations to his predecessors, has a comparatively easy task before
him. But it is necessary, first, to give some description of the use
which Archimedes made of the general methods which had found
acceptance with the earlier geometers, and, secondly, to refer to
some particular results which he mentions as having been previously
discovered and as lying at the root of his own investigations, or
which he tacitly assumes as known.

1. Use of traditional geometrical methods.

In my edition of the Conies of Apollonius*, I endeavoured,
following the lead given in Zeuthen's work, Die Lehre von den
Kegelschnitten im Altertum, to give some account of what has been
fitly called the geometrical algebra which played such an important
part in the works of the Greek geometers. The two main methods
included under the term were (1) the use of the theory of pro-
portions, and (2) the method of application of areas, and it was
shown that, while both methods are fully expounded in the Elements
of Euclid, the second was much the older of the two, being
attributed by the pupils of Eudemus (quoted by Proclus) to the

* Apollonius of Perga, pp. ci sqq.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xli

Pythagoreans. It was pointed out that the application of areas,
as set forth in the second Book of Euclid and extended in the
sixth, was made by Apollonius the means of expressing what he
takes as* the fundamental properties of the conic sections, namely
the properties which we express by the Cartesian equations

referred to any diameter and the tangent at its extremity as axes ;
and the latter equation was compared with the results obtained in the
27th, 28th and 29th Props, of Euclid's Book vi, which are equivalent
to the solution, by geometrical means, of the quadratic equations

ax + - x s - D.

It was also shown that Archimedes does not, as a rule, connect his
description of the central conies with the method of application of
areas, as Apollonius does, but that Archimedes generally expresses
the fundamental property in the form of a proportion

y* y'*

JJ _ is _

X . X 1 X . #/ '

and, in the case of the ellipse,

x . x l a

where x, x l are the abscissae measured from the ends of the diameter
of reference.

It results from this that the application of areas is of much less
frequent occurrence in Archimedes than in Apollonius. It is
however used by the former in all but the most general form. The
simplest form of "applying a rectangle" to a given straight line
which shall be equal to a given area occurs e.g. in the proposition On
the equilibrium of Planes n. 1 ; and the same mode of expression
is used (as in Apollonius) for the property y 9 = px in the parabola,
px being described in Archimedes' phrase as the rectangle " applied
to" (7ra/>a7ri7rTov 7rapct) a line equal to p and "having at its width"
(irAaro* X ov) the abscissa (x). Then in Props. 2, 25, 26, 29 of the
book On Conoids and Spheroids we have the complete expression
which is the equivalent of solving the equation

ax + x a = 6 2 ,
" let a rectangle be applied (to a certain straight line) exceeding by

xlii INTRODUCTION.

a square figure (7rapa7r7rTKT xwpiov V7Tp/3d\.Xov ciet
and equal to (a certain rectangle)." Thus a rectangle of this sort
has to be made (in Prop. 25) equal to what we have above called
x . x l in the case of the hyperbola, which is the same thing as
x (a + x) or ax + x 8 , where a is the length of the transverse axis.
But, curiously enough, we do not find in Archimedes the application
of a rectangle "falling short by a square figure," which we should
obtain in the case of the ellipse if we substituted x (a x) for x . x .
In the case of the ellipse the area x . x l is represented (On Conoids
and Spheroids, Prop. 29) as a gnomon which is the difference
between the rectangle h . /^ (where h, 7^ are the abscissae of the
ordinate bounding a segment of an ellipse) and a rectangle applied
to /^ - h and exceeding by a square figure whose side is h - x ; and
the rectangle h. /^ is simply constructed from the sides A, h 1 . Thus
Archimedes avoids* the application of a rec tangle falling short by a
square, using for x . x 1 the rather complicated form

h . h, - {(h, - h) (h - x) + (h - x)*\.

It is easy to see that this last expression is equal to x . x ly for it
reduces to

h.h l -{h 1 (h-x)-x(h-x)\

= x (A! -f h) - X s ,

= ax- x*, since AJ + h = a,

It will readily be understood that the transformation of rectangles
and squares in accordance with the methods of Euclid, Book n, is
just as important to Archimedes as to other geometers, and there is
no need to enlarge on that form of geometrical algebra.

The theory of proportions, as expounded in the fifth and sixth
Books of Euclid, including the transformation of ratios (denoted by
the terms componendo, divide udo, etc.) and the composition or
multiplication of ratios, made it possible for the ancient geometers
to deal with magnitudes in general and to work out relations
between them with an effectiveness not much inferior to that of
modern algebra. Tims the addition and subtraction of ratios could
be effected by procedure equivalent to what we should in algebra

* The object of Archimedes was no doubt to make the Lemma in Prop. 2
(dealing with the summation of a series of terms of the form a.rx + (rx)' 2 , where r
successively takes the values 1, 2, 3, ...) serve for the hyperboloid of revolution
and the spheroid as well.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xliii

calPbringing to a common denominator. Next, the composition or
multiplication of ratios could be indefinitely extended, and hence
the algebraical operations of multiplication and division found easy
and convenient expression in the geometrical algebra. As a par-
ticular case, suppose that there is a series of magnitudes in continued
proportion (i.e. in geometrical progression) as a 09 a lt a 2 , ... a n , so that

_ l _ _ a n-l
1~" a 2 a n

We have then, by multiplication,

=( ) , or
a

i */a n
= * / .

a V a

It is easy to understand how powerful such a method as that of
proportions would become in the hands of an Archimedes, and a few
instances are here appended in order to illustrate the mastery with
which he uses it.

1. A good example of a reduction in the order of a ratio after
the manner just shown is furnished by On the equilibrium of Planes
II. 10. Here Archimedes has a ratio which we will call a 3 /6 3 , where
a?/b 2 = c/d', and he reduces the ratio between cubes to a ratio
between straight lines by taking two lines x, y such that

c _ x - ^
x~d~y'

(c\ 2 c a 2
- ) = =
x) d b 2

a _ c

6 = 5 ;

. . a 3 /c\ 3 c ,T d c

and hence IT = ( - ) =--,.-=-.

6 3 \a5/ x d y y

2. In the last example we have an instance of the use of
auxiliary fixed lines for the purpose of simplifying ratios and
thereby, as it were, economising power in order to grapple the more
successfully with a complicated problem. With the aid of such
auxiliary lines or (what is the same thing) auxiliary fixed points in
a figure, combined with the use of proportions, Archimedes is able to
effect some remarkable eliminations.

Thus in the proposition On the Sphere and Cylinder n. 4 he obtains
three relations connecting three as ) r et undetermined points, and

Xliv INTRODUCTION.

proceeds at once to eliminate two of the points, so that the problem
is then reduced to finding the remaining point by means of one
equation. Expressed in an algebraical form, the three original
relations amount to the three equations *

3a x _ y

2a-x~ x
a + x z

x ~ 2a-x

and the result, after the elimination of y and , is stated by
Archimedes in a form equivalent to

m + n a + x 4cr

n ' a ~ (2a-xy

Again the proposition On the equilibrium of Planes n. 9 proves
by the same method of proportions that, if a, 6, c, d, x, y, are straight
lines satisfying the conditions

and

d _ X I

a~-~cl~ f(a- C y
2a + 46 -i- 6c + 3d _ ?/
5 4- 106 4- lOc + 5d ~ a - c '

then x + y \a.

The proposition is merely brought in as a subsidiary lemma to the
proposition following, and is not of any intrinsic importance ; but a
glance at the proof (which again introduces an auxiliary line) will
show that it is a really extraordinary instance of the manipulation
of proportions.

3. Yet another instcince is worth giving here. It amounts to
the proof that, if

, 2a-x . . 72

then - .y 2 (a x) + -- . y 2 (a -f x) = 4a6 .

a + x * ^ ' a-x u ^ '

A, A' are the points of contact of two parallel tangent planes to a
spheroid ; the plane of the paper is the plane through A A' and the

RELATION OF ARCHIMEDES TO HIS PREDECESSORS.

xlv

of the spheroid, and PP' is the intersection of this plane with
another plane at right angles to it (and therefore parallel to the
tangent planes), which latter plane divides the spheroid into two
segments whose axes are AN, A'N. Another plane is drawn through

the centre and parallel to the tangent plane, cutting the spheroid
into two halves. Lastly cones are drawn whose bases are the
sections of the spheroid by the parallel planes as shown in the
figure.

Archimedes' proposition takes the following form [On Conoids
and Spheroids, Props. 31, 32].

APP' being the smaller segment of the two whose common base
is the section through PP', and x, y being the coordinates of P,
he has proved in preceding propositions that

(volume of) segment APP' _ 2a + x
(volume of) cone A PP 1 ~ a + x ^ a ''

. half spheroid A BB'
and - /m

and he seeks to prove that

segment A'PP' _2a-x
cone A'PP' ~ a x '
The method is as follows.

cone ABB' a b~ a i
cone

We have

If we suppose

' ~~ a - x ' y- ~~ oT- x ' o^o- 2 '
z a

a a-x

(y).

the ratio of the cones becomes - . - - , .

a 2 x 2

xlvi INTRODUCTION.

Next, by hypothesis (a),

cone APP f __ a + x
segmt. APP' = 2a + x '
Therefore, ex aeqitali,

cone ABB' za

segmt. APP' (a - x) (2a + x) '
It follows from (/?) that

spheroid _ 4=za

~~ =

segmt. A'PP' _ 4za (a x) (2a + a;)
segmtTTP?' = "(a - xffi

Now we have to obtain the ratio of the segment A'PP' to the cone
A'PP', and the comparison between the segment APP' and the cone
A'PP' is made by combining two ratios ex aequali. Thus

segmt. APP' _ 2a + x
' ~

, cone APP' a x

and t, rt ni~ ~ -

cone A Pr a + x

Thus combining the last three proportions, ex aequali, we have

segmt. A'PP' _ z(2a -x) + (2a + x)(z-a- a;)
~ ~~~ ~

a +

_ z (2a - x) + (2a + x)(z-a- x)
z (a - x) + (2a -f- x) x '

since a a = 2 (a re), by (y).

[The object of the transformation of the numerator and denominator
of the last fraction, by which z('2a x) and z (a x) are made the

first terms, is now obvious, because - is the fraction which

a - x

Archimedes wishes to arrive at, and, in order to prove that the
required ratio is equal to this, it is only necessary to show that

2a - x _ z - (a x) ,
a x x '-'

RELATION OP ARCHIMEDES TO HIS PREDECESSORS. xlvii

j. 2a-x
Now = 1 +

a x a x

a + 2
a

z (a-x) . .
= (dividendo),

. . segmt. A'PP 1 2a-x

so that -2 -- y>-i>r = -----

cone A PP' ax

4. One use by Euclid of the method of proportions deserves
mention because Archimedes does not use it in similar circumstances.
Archimedes (Quadrature of the Parabola, Prop. 23) sums a particular
geometric series

in a manner somewhat similar to that of our text-books, whereas
Euclid (ix. 35) sums any geometric series of any number of terms by
means of proportions thus.

Suppose ,, 2 , ..., a n+ i to be (n+I) terms of a geometric
series in which a n+l is the greatest term. Then

ft/t+l _ _ a n _ ft n-i _ _ 2
/, _i a n-2 '" a i

Therefore ^-" rt ' 1 = " ^ = . . . = ^l^ 1 .

a n a ;l _! aj

Adding all the antecedents and all the consequents, we have

-f- . . . + a n aj
which gives the sum of n terms of the series.

2. Earlier discoveries affecting quadrature and cuba-
ture.

Archimedes quotes the theorem that circles are to one another as
the squares on their diameters as having being proved by earlier
geometers, and he also says that it was proved by means of a certain
lemma which he states as follows: "Of unequal lines, unequal
surfaces, or unequal solids, the greater exceeds the less by such a
magnitude as is capable, if added [continual lyj to itself, of exceeding

xlviii INTRODUCTION.

any given magnitude of those which are comparable with one another
(TO>V irpbs a\\rjXa Acyo/Ae'vwi/)." We know that Hippocrates of Chios
proved the theorem that circles are to one another as the squares on
their diameters, but no clear conclusion can be established. as to the
method which he used. On the other hand, Eudoxus (who is
mentioned in the preface to The Sphere and Cylinder as having
proved two theorems in solid geometry to be mentioned presently)
is generally credited with the invention of the met /tod of exhaustion
by which Euclid proves the proposition in question in xn. 2. The
lemma stated by Archimedes to have been used in the original proof
is not however found in that form in Euclid and is not used in the
proof of xn. 2, where the lemma used is that proved by him in
X. 1, viz. that "Given two unequal magnitudes, if from the greater
[a part] be subtracted greater than the half, if from the remainder
[a part] greater than the half be subtracted, and so on continually,
there will be left some magnitude which will be less than the lesser
given magnitude." This last lemma is frequently assumed by
Archimedes, and the application of it to equilateral polygons in-
scribed in a circle or sector in the manner of xn. 2 is referred to as
having been handed down in the Elements*, by which it is clear
that only Euclid's Elements can be meant The apparent difficulty
caused by the mention of two lemmas in connexion with the theorem
in question can, however, I think, be explained by reference to
the proof of x. 1 in Euclid. He there takes the lesser magnitude
and says that it is possible, by multiplying it, to make it some time
exceed the greater, and this statement he clearly bases on the 4th
definition of Book v. to the effect that " magnitudes are said to bear
a ratio to one another, which can, if multiplied, exceed one another."
Since then the smaller magnitude in x. 1 may be regarded as the
difference between some two unequal magnitudes, it is clear that the
lemma first quoted by Archimedes is in substance used to prove the
lemma in x. 1 which appears to play so much larger a part in the in-
vestigations in quadrature and cubature which have come down to us.
The two theorems which Archimedes attributes to Eudoxus
by namet are

(1) that any pyramid is one third part of the jrrism which has
the same base as the pyramid and equal height, and

* On the Sphere and Cylinder, i. 6.
t ibid. Preface.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. xlix

() that any cone is one third part of the cylinder which has
the same base as the cone and equal Jieight.

The other theorems in solid geometry which Archimedes quotes
as having been proved by earlier geometers are * :

(3) Cones of equal height are in the ratio of their bases, and
conversely.

(4) If a cylinder be divided by a plane parallel to the base,
cylinder is to cylinder as axis to axis.

(5) Cones which have the same bases as cylinders and equal
height with them are to one anotfwr as the cylinders.

(6) The bases of equal cones are reciprocally proportional to
their heights, and conversely.

(7) Cones the diameters of whose bases have the same ratio as
their axes are in the triplicate ratio of the diameters of their bases.

In the preface to the Quadrature of the Parabola he says
that earlier geometers had also proved that

(8) Spheres have to one another the triplicate ratio of their
diameters ; and he adds that this proposition and the first of those
which he attributes to Eudoxus, numbered (1) above, were proved
by means of the same lemma, viz. that the difference between
any two unequal magnitudes can be so multiplied as to exceed
any given magnitude, while (if the text of Heiberg is right) the
second of the propositions of Eudoxus, numbered (2), was proved
by means of "a lemma similar to that aforesaid." As a matter
of fact, all the propositions (1) to (8) are given in Euclid's twelfth
Book, except (5), which, however, is an easy deduction from (2) ;
and (1), (2), (3), and (7) all depend upon the same lemma [x. 1]
as that used in Eucl. xn. 2.

The proofs of the above seven propositions, excluding (5), as
given by Euclid are too long to quote here, but the following sketch
will show the line taken in the proofs and the order of the propo-
sitions. Suppose ABCD to be a pyramid with a triangular base,
and suppose it to be cut by two planes, one bisecting AB, AC,
AD in t\ (w y E respectively, and the other bisecting EC, BD> BA
in 77, K, F respectively. These planes are then each parallel to
one face, and they cut off two pyramids each similar to the original

* Lemmas placed between Props. 16 and 17 of Book i. On the Sphere and
Cylinder.

H. A. d

1 INTRODUCTION.

pyramid and equal to one another, while the remainder oJJo the
pyramid is proved to form two equal prisms which, taken together,

are greater than one half of the original pyramid [xn. 3]. It is
next proved [xn. 4] that, if there are two pyramids with triangular
bases and equal height, and if they are each divided in the
manner shown into two equal pyramids each similar to the whole
and two prisms, the sum of the prisms in one pyramid is to the
sum of the prisms in the other in the ratio of the bases of the
whole pyramids respectively. Thus, if we divide in the same
manner the two pyramids which remain in each, then all
the pyramids which remain, and so on continually, it follows
on the one hand, by x. 1, that we shall ultimately have
pyramids remaining which are together less than any assigned
solid, while on the other hand the sums of all the prisms
resulting from the successive subdivisions are in the ratio of
the bases of the original pyramids. Accordingly Euclid is able
to use the regular method of exhaustion exemplified in xn. 2,
and to establish the proposition [xn. 5] that pyramids with the
same height and with triangular bases are to one another as their
bases. The proposition is then extended [xn. 6] to pyramids with the
same height and with polygonal bases. Next [xn. 7] a prism with
a triangular base is divided into three pyramids which are shown
to be equal by means of xn. 5 ; and it follows, as a corollary, that
any pyramid is one third part of the prism which has the same
base and equal height. Again, two similar and similarly situated
pyramids are taken and the solid parallelepipeds are completed,
which are then seen to be six times as large as the pyramids
respectively; and, since (by XL 33) similar parallelepipeds are in
the triplicate ratio of corresponding sides, it follows that the same

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. li

is ttue of the pyramids [xn. 8]. A corollary gives the obvious
extension to the case of similar pyramids with polygonal bases.
The proposition [xn. 9] that, in equal pyramids with triangular
bases, the bases are reciprocally proportional to the heights is
proved by the same method of completing the parallelepipeds and
using xi. 34 ; and similarly for the converse. It is next proved
[xn. 10] that, if in the circle which is the base of a cylinder a
square be described, and then polygons be successively described
by bisecting the arcs remaining in each case, and so doubling the
number of sides, and if prisms of the same height as the cylinder
be erected on the square and the polygons as bases respectively,
the prism with the square base will be greater than half the
cylinder, the next prism will add to it more than half of the
remainder, and so on. And each prism is triple of the pyramid with
the same base and altitude. Thus the same method of exhaustion
as that in xn. 2 proves that any cone is one third part of the
cylinder with the same base and equal height. Exactly the same
method is used to prove [xu. 11] that cones and cylinders which
have the same height are to one another as their bases, and
[xn. 12] that similar cones and cylinders are to one another in
the triplicate ratio of the diameters of their bases (the latter
proposition depending of course on the similar proposition xn. 8
for pyramids). The next three propositions are proved without
fresh recourse to x. 1. Thus the criterion of equimultiples laid
down in Def. 5 of Book v. is used to prove [xn. 13] that, if a
cylinder be cut by a plane parallel to its bases, the resulting
cylinders are to one another as their axes. It is an easy deduction
[xn. 14] that cones and cylinders which have equal bases are
proportional to their heights, and [xn. 15] that in equal cones
and cylinders the bases are reciprocally proportional to the heights,
and, conversely, that cones or cylinders having this property are
equal. Lastly, to prove that spheres are to one another in the
triplicate ratio of their diameters [xn. 18], a new procedure is
adopted, involving two preliminary propositions. In the first of
these [xn. 16] it is proved, by an application of the usual lemma
x. 1, that, if two concentric circles are given (however nearty
equal), an equilateral polygon can be inscribed in the outer circle
whose sides do not touch the inner; the second proposition [xn. 17]
uses the result of the first to prove that, given two concentric
spheres, it is possible to inscribe a certain polyhedron in the outer

Hi INTRODUCTION.

so that it does not anywhere touch the inner, and a corollary Adds
the proof that, if a similar polyhedron be inscribed in a second
sphere, the volumes of the polyhedra are to one another in the
triplicate ratio of the diameters of the respective spheres. This
last property is then applied [xn. 18] to prove that spheres are
in the triplicate ratio of their diameters.

3. Conic Sections.

In my edition of the Conies of Apollonius there is a complete
account of all the propositions in conies which are used by Archi-
medes, classified under three headings, (1) those propositions
which he expressly attributes to earlier writers, (2) those which
are assumed without any such reference, (3) those which appear to
represent new developments of the theory of conies due to Archi-
medes himself. As all these properties will appear in this
volume in their proper places, it will suffice here to state only
such propositions as come under the first heading and a few under
the second which may safely be supposed to have been previously
known.

Archimedes says that the following propositions "are proved
in the elements of conies," i.e. in the earlier treatises of Euclid
and Aristaeus.

1. In the parabola

(a) if PV be the diameter of a segment and QVq the
chord parallel to the tangent at P 9 then QV= Vq\

(b) if the tangent at Q meet VP produced in T, then

PV=PT-,

PV-.PV'^QV* : Q'V*.

2. If straight lines drawn from the same point touch any
conic section whatever, and if two chords parallel to the respective
tangents intersect one another, then the rectangles under the
segments of the chords are to one another as the squares on the
parallel tangents respectively.

3. The following proposition is quoted as proved " in the conica."
If in a parabola p a be the parameter of the principal ordinates,

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Hii

QQ '*any chord not perpendicular to the axis which is bisected in V
by the diameter PV, p the parameter of the ordinates to PV, and
if QD be drawn perpendicular to FV, then

QV*:QD*=p: Pa .
[On Conoids and Spheroids, Prop. 3, which see.]

The properties of a parabola, PN* = p a .AN, and Q7*=p.PV,
were already well known before the time of Archimedes. In fact
the former property was used by Menaechmus, the discoverer of
conic sections, in his duplication of the cube.

It may be taken as certain that the following properties of the
ellipse and hyperbola were proved in the Conies of Euclid.

1. For the ellipse

PN* : AN. A 'JV= P'N'* : AN' . A 'N' = CB* : CA *
and QV* : PV . P'V= Q'V* : PV . P'V = CD 9 : CP*.

(Either proposition could in fact be derived from the proposition
about the rectangles under the segments of intersecting chords
above referred to.)

2. For the hyperbola

PN* : AN. A'N=P'N'* : AN'.A'N'

and QV*iPV.P'V=Q'V'*i PV'.P'V,

though in this case the absence of the conception of the double
hyperbola as one curve (first found in Apollonius) prevented Euclid,
and Archimedes also, from equating the respective ratios to those
of the squares on the parallel semidiameters.

3. In a hyperbola, if P be any point on the curve and PK,
PL be each drawn parallel to one asymptote and meeting the

other,

PK. PL--- (const.)

This property, in the particular case of the rectangular hyperbola,
was known to Menaechmus.

It is probable also that the property of the subnormal of the
parabola (NG~^p a ) was known to Archimedes' predecessors. It
is tacitly assumed, On floating bodies, II. 4, etc.

From the assumption that, in the hyperbola, AT < AN (where
N is the foot of the ordinate from P, and T the point in which the

liv INTRODUCTION.

tangent at P meets the transverse axis) we may perhaps fiifer
that the harmonic property

TP : TP' = PV:P'V,
or at least the particular case of it,
TA : TA'

was known before Archimedes' time.

Lastly, with reference to the genesis of conic sections from
cones and cylinders, Euclid had already stated in his Phaenomena
that, " if a cone or cylinder be cut by a plane not parallel to the
base, the resulting section is a section of an acute-angled cone
[an ellipse] which is similar to a flupcoV Though it is not probable
that Euclid had in mind any other than a right cone, the statement
should be compared with On Conoids and fijdieroidtt, Props. 7, 8, 9.

4. Surfaces of the second degree.

Prop. 1 1 of the treatise On Conoids and Spheroids states without
proof the nature of certain plane sections of the conicoids of revo-
lution. Besides the obvious facts (1) that sections perpendicular
to the axis of revolution are circles, and (2) that sections through
the axis are the same as the generating conic, Archimedes asserts
the following.

1. In a paraboloid of revolution any plane section parallel to
the axis is a parabola equal to the generating parabola.

2. In a hyperboloid of revolution any plane section parallel
to the axis is a hyperbola similar to the generating hyperbola.

3. In a hyperboloid of revolution a plane section through the
vertex of the enveloping cone is a hyperbola which is not similar
to the generating hyperbola

4. In any spheroid a plane section parallel to the axis is an
ellipse similar to the generating ellipse.

Archimedes adds that " the proofs of all these propositions
are manifest (^avcpot)." The proofs may in fact be supplied as
follows.

1. Section of a paraboloid of revolution by a plane parallel
to the axis.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS.

Suppose that the plane of the paper represents the plane section
through the axis AN which intersects the given plane section at right
angles, and let A'O be the line of intersection.
Let POP' be any double ordinate to AN in the
section through the axis, meeting A'O and AN
at right angles in 0, N respectively. Draw A'M
perpendicular to AN.

Suppose a perpendicular drawn from to
A '0 in the plane of the given section parallel to
the axis, and let y be the length intercepted by
the surface on this perpendicular.

Then, since the extremity of y is on the
circular section whose diameter is PP\

if-=PO.OP f .

If A '0 x, and if p is the principal parameter of the generating
parabola, we have then

= px,
so that the section is a parabola equal to the generating parabola.

2. Section of a hyperboloid of revolution by a plam parallel to
the axis.

Take, as before, the plane section through the axis which intersects

the given plane section at right angles in A'O. Let the hyperbola

Ivi INTRODUCTION.

PAP' in the plane of the paper represent the plane section through
the axis, and let G be the centre (or the vertex of the enveloping
cone). Draw CC' perpendicular to CA, and produce OA' to meet it
in C". Let the rest of the construction be as before.
Suppose that

CA=a, C'A'-^a', C'O -,r,

and let y have the same meaning as before.

Then y" - PO . OP' = PN' 2 - A'M*.

And, by the property of the original hyperbola,

PiV 2 : CJP-ai 2 = ,l'J/ a : CM/ 2 -CM 2 (which is constant).

Thus A' J/ 2 : CM* - CA* = PN* : CiV 2 - CA*

whence it appears that the section is a hyperbola similar to the
original one.

3. Section of a hyperboloid of revolution by a plane passim/
through the centre (or the vertex of tlie, enveloping cone).

I think there can be no doubt that Archimedes would have proved
his proposition about this section by means of the same general
property of conies which he uses to prove Props. 3 and 12-14 of
the same treatise, and which he enunciates at the beginning of
Prop. 3 as a known theorem proved in the "elements of conies," viz.
that the rectangles under the segments of intersecting chords are as
the squares of the parallel tangents.

Let the plane of the paper represent the plane section through
the axis which intersects the given plane passing through the
centre at right angles. Let CA'O be the line of intersection, C
being the centre, and A' being the point where CA'O meets the
surface. Suppose CAMN to be the axis of the hyperboloid, and
POp, P'O'p' two double ordinates to it in the plane section through
the axis, meeting CA'O in 0, 0' respectively; similarly let A'M be
the ordinate from A'. Draw the tangents at A and A' to the
section through the axis meeting in 1\ and let QOq, Q'O'q be the
two double ordinates in the same section which are parallel to the
tangent at A' and pass through 0, O' respectively.

Suppose, as before, that y, y' are the lengths cut off by the

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Ivii

surface from the perpendiculars at and 0' to OC in the plane of
the given section through CA'O, and that

, CO = x, CO' = x', OA = a, CA' = '.

Then, by the property of the intersecting chords, we have, since

CO = 7 ,

PO . Op : Q0*=- TA* : TA' 2

= P'0' . O'p' : Q'0'\

Also y* = PO . Op, y'^P'O 1 . O'p',

and, by the property of the hyperbola,

QO- :x'-a'* = Q'0'* :x"-' 2 .
It follows, ejL- (trquali, that

y 2 : x*-a' 2 = y'* :x'--a' 3 .................. (a),

and therefore that the section is a hyperbola.

To prove that this hyperbola is not similar to the generating
hyperbola, we draw CC' perpendicular to CJ, and C'A' parallel to
CA meeting CC' in C' and Pp in U.

If then the hyperbola (a) is similar to the original hyperbola, it
must by the last proposition be similar to the hyperbolic section
made by the plane through C'A'l" at right angles to the plane of
the paper.

Now COi-CA'^U'-C'A'

and

PO.Op<PU.Up.

Iviii INTRODUCTION.

Therefore PO . Op : CO 3 - CA'* < PU . Up : C ' U* - C 'A'\
and it follows that the hyperbolas are not similar*.

4. Section of a spheroid by a plane parallel to the axis.

That this is an ellipse similar to the generating ellipse can of
course be proved in exactly the same way as theorem (2) above
for the hyperboloid.

* I think Archimedes is more likely to have used this proof than one on the
lines suggested by Zeuthen (p. 421). The latter uses the equation of the
hyperbola simply and proceeds thus. If y have the same meaning as above,
and if the coordinates of P referred to CA, CG' as axes be z, ar, while those of O
referred to the same axes are z, x', we have, for the point P,

.c 2 =/c(* 2 -a 2 ),
where K is constant.

Also, since the angle A'CA is given, x' = az t where a is constant.

Thus ?/ 2 =;r 2 - x" 2 = (K - a 2 ) z* - KO?.

CO
Now z is proportional to CO, being in fact equal to /^ a> and the equation

becomes

which is clearly a hyperbola, since a'-</c.

Now, though the Greeks could have worked out the proof in a geometrical
form equivalent to the above, I think that it is alien from the manner in which
Archimedes regarded the equations to central conies. These he always expressed
in the form of a proportion

r &~ n

= ... in the case of the ellipse ,
L a J

and never in the form of an equation between areas like that used by
Apollonius, viz.

Moreover the occurrence of the two different constants and the necessity
of expressing them geometrically as ratios between areas and lines respectively
would have made the proof very long and complicated ; and, as a matter of fact,
Archimedes never does express the ratio y~l(x* - a 2 ) in the case of the hyperbola
in the form of a ratio between constant areas like b 2 /a 2 . Lastly, when the
equation of the given section through CA'O was found in 'the form (1), assuming
that the Greeks had actually found the geometrical equivalent, it would still
have been held necessary, I think, to verify that

before it was finally pronounced that the hyperbola represented by the equation
and the section made by the plane were one and the same thing.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. lix

We are now in a position to consider the meaning of Archimedes'
remark that " the proofs of all these properties are manifest." In
the first place, it is not likely that "manifest" means "known" as
having been proved by earlier geometers ; for Archimedes' habit is
to be precise in stating the fact whenever he uses important
propositions due to his immediate predecessors, as witness his
references to Eudoxus, to the Elements [of Euclid], and to the
"elements of conies." When we consider the remark with reference
to the cases of the sections parallel to the axes of the surfaces
respectively, a natural interpretation of it is to suppose that
Archimedes meant simply that the theorems are such as can easily
be deduced from the fundamental properties of the three conies now
expressed by their equations, coupled with the consideration that
the sections by planes perpendicular to the axes are circles. But I
think that this particular explanation of the " manifest " character
of the proofs is not so applicable to the third of the theorems
stating that any plane section of a hyperboloid of revolution
through the vertex of the enveloping cone but not through the axis
is a hyperbola. This fact is indeed no more "manifest" in the
ordinary sense of the term than is the like theorem about the
spheroid, viz. that any section through the centre but not through
the axis is an ellipse. But this latter theorem is not given along
with the other in Prop. 1 1 as being " manifest " ; the proof of it is
included in the more general proposition (14) that any section of a
spheroid not perpendicular to the axis is an ellipse, and that parallel
sections are similar. Nor, seeing that the propositions are essen-
tially similar in character, can I think it possible that Archimedes
wished it to be understood, as Zeuthen suggests, that the proposition
about the hyperboloid alone, and not the other, should be proved
directly by means of the geometrical equivalent of the Cartesian
equation of the conic, and not by means of the property of the
rectangles under the segments of intersecting chords, used earlier
[Prop. 3] witli reference to the parabola and later for the case of
the spheroid and the elliptic sections of the conoids and spheroids
generally. This is the more unlikely, I think, because the proof
by means of the equation of the conic alone would present much
more difficulty to the Greek, and therefore could hardly be called
" manifest."

It seems necessary therefore to seek for another explanation,
and I think it is the following. The theorems, numbered 1, 2, and

x INTRODUCTION.

4 above, about sections of conoids and spheroids parallel to the" axis
are used afterwards in Props. 1517 relating to tangent planes;
whereas the theorem (3) about the section of the hyperboloid by a
plane through the centre but not through the axis is nol used in
connexion with tangent planes, but only for formally proving that a
straight line drawn from any point on a hyperboloid parallel to any
transverse diameter of the hyperboloid falls, on the convex side of
the surface, without it, and on the concave side within it. Hence
it does not seem so probable that the four theorems were collected
in Prop. 1 1 on account of the use made of them later, as that they
were inserted in the particular place with special reference to the
three propositions (1214) immediately following and treating of the
elliptic sections of the three surfaces. The main object of the whole
treatise was the determination of the volumes of segments of the
three solids cut off by planes, and hence it was first necessary to
determine all the sections which were ellipses or circles and therefore
could form the bases of the segments. Thus in Props. 12-14
Archimedes addresses himself to finding the elliptic sections, but,
before he does this, he gives the theorems grouped in Prop. 11 by
way of clearing the ground, so as to enable the propositions about
elliptic sections to be enunciated with the utmost precision. Prop.
11 contains, in fact, explanations directed to defining the scope of
the three following propositions rather than theorems definitely
enunciated for their own sake ; Archimedes thinks it necessary to
explain, before passing to elliptic sections, that sections perpen-
dicular to the axis of each surface are not ellipses but circles, and
that some sections of each of the two conoids are neither ellipses nor
circles, but parabolas and hyperbolas respectively. It is as if he had
said, " My object being to find the volumes of segments of the three
solids cut off by circular or elliptic sections, I proceed to consider
the various elliptic sections ; but I should first explain that sections
at right angles to the axis are not ellipses but circles, while sections
of the conoids by planes drawn in a certain manner are neither
ellipses nor circles, but parabolas and hyperbolas respectively. With
these last sections I am not concerned in the next propositions, and
I need not therefore cumber my book with the proofs ; but, as some
of them can be easily supplied by the help of the ordinary properties
of conies, and others by means of the methods illustrated in the
propositions now about to be given, I leave them as an exercise for
the reader/' This will, I think, completely explain the assumption

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Ixi

of all the theorems except that concerning the sections of a spheroid
parallel to the axis ; and I think this is mentioned along with the
others for symmetry, and because it can be proved in the same way
as the corresponding one for the hyperboloid, whereas, if mention of
it had been postponed till Prop. 14 about the elliptic sections of a
spheroid generally, it would still require a proposition for itself, since
the axes of the sections dealt with in Prop. 14 make an angle with
the axis of the spheroid and are not parallel to it.

At the same time the fact that Archimedes omits the proofs of
the theorems about sections of conoids and spheroids parallel to the
axis as "manifest" is in itself sufficient to raise the presumption
that contemporary geometers were familiar with the idea of three
dimensions and knew how to apply it in practice. This is no matter
for surprise, seeing that we find Archytas, in his solution of the
problem of the two mean proportionals, using the intersection of a
certain cone with a curve of double curvature traced on a right
circular cylinder*. But, when we look for other instances of early
investigations in geometry of three dimensions, we find practically
nothing except a few vague indications as to the contents of a lost
treatise of Euclid's consisting of two Books entitled Surface-loci
(TOTTOL Trpos 7ri<avia)t. This treatise is mentioned by Pappus
among other works by Aristaeus, Euclid and Apollonius grouped
as forming the so-called roVos avaXvo/xci'osJ. As the other works in
the list which were on plane subjects dealt only with straight lines,
circles find conic sections, it is a priori likely that the surface-loci of

* Cf. Eutocius on Archimedes (Vol. in. pp. 98102), or Apollonius of Perga,
pp. xxii. xxiii.

t By this term we conclude that the Greeks meant "loci which are surfaces "
as distinct from loci which are lines. Cf. Proclus' definition of a locus as
"a position of a line or a surface involving one and the same property"
(ypajJLfj.^ TJ tiTHpwelas 6t<ris iroiovffa v Kal raMv <rifywrrw/*a), p. 394. Pappus
(pp. 6602) gives, quoting from the Plane Loci of Apollonius, a classification of
loci according to their order in relation to that of which they are the loci. Thus,
he says, loci are (1) tyeKTiKol, i.e. fixed, e.g. in this sense the locus of a point is
a point, of a line a line, and so on; (2) $ieo5i/coi or moving along, a line being in
this sense the locus of a point, a surface of a line, and a solid of a surface ;
(3) dvacrrpocpLKoL, turning backwards, i.e., presumably, moving backwards and
forwards, a surface being in this sense the locus of a point, and a solid of a line.
Thus a surface-locus might apparently be either the locus of a point or the
locus of a line moving in space.

J Pappus, pp. 634, 636.

Ixii INTRODUCTION.

Euclid included at least such loci as were cones, cylinders and
spheres. Beyond this, all is conjecture based upon two lemmas
given by Pappus in connexion with the treatise.

First lemma to the Surface-loci of Euclid*.

The text of this lemma and the attached figure are not satisfac-
tory as they stand, but they have been explained by Tannery in a
way which requires a change in the figure, but only the very slightest
alteration in the text, as followsf.

"If AB be a straight line and CD be parallel to a straight line
given in position, and if the ratio AD . DB : DC 2 be [given], the
point C lies on a conic section.
If now AB be no longer given in
position and A, B be no longer
given but lie on straight lines
AE, EB given in position J, the
point C raised above [the plane
containing AE, EB] is on a
surface given in position. And
this was proved."

According to this interpretation, it is asserted that, if A B moves
with one extremity on each of the lines AE, EB which are fixed,
while DC is in a fixed direction and AD . DB : DC 2 is constant,
then G lies on a certain surface. So far as the first sentence is
concerned, AB remains of constant length, but it is not made
precisely clear whether, when AB is no longer given in position, its
length may also varyg. If however AB remains of constant length
for all positions which it assumes, the surface which is the locus of
C would be a complicated one which we cannot suppose that Euclid
could have profitably investigated. It may, therefore, be that
Pappus purposely left the enunciation somewhat vague in order to
make it appear to cover several surface-loci which, though belonging
to the same type, were separately discussed by Euclid as involving

* Pappus, p. 1004.

f Bulletin den sciences math., 2 S6rie, vi. 149.

The words of the Greek text are y^ijTat te irpbs 0^<ret ci>0ta rats AE, EB,
and the above translation only requires e60efat* instead of eMeta. The figure in
the text is so drawn that AI)B, AEB are represented as two parallel lines, and
CD is represented as perpendicular to ADB and meeting AEB in E.

The words are simply "if AB be deprived of its position ((n-epi/flj} TTJS
06rews) and the points A, B be deprived of their [character of] being given"
(ffTcpr)0rj TOV do0vTOS efrou).

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Ixili

in each case somewhat different sets of conditions limiting the
generality of the theorem.

It is at least open to conjecture, as Zeuthen has pointed out*,
that two ases of the type were considered by Euclid, namely, (1)
that in which A B remains of constant length while the two fixed
straight lines on which -4, B respectively move are parallel instead
of meeting in a point, and (2) that in which the two fixed straight
lines meet in a point while AB moves always parallel to itself
and varies in length accordingly.

(1) In the first case, where the length of AB is constant and
the two fixed lines parallel, we should have a surface described by a
conic moving bodily f. This surface would be a cylindrical surface,
though it would only have been called a " cylinder " by the ancients
in the case where the moving conic was an ellipse, since the essence
of a " cylinder " was that it could be bounded between two parallel
circular sections. If then the moving conic was an ellipse, it would
not be difficult to find the circular sections of the cylinder; this
could be done by first taking a section at right angles to the axis,
after which it could be proved, after the manner of Archimedes,
On Conoids and Splwroids, Prop. 9, first that the section is an ellipse
or a circle, and then, in the former case, that a section made by
a plane drawn at a certain inclination to the ellipse and passing
through, or parallel to, the major axis is a circle. There was
nothing to prevent Euclid from investigating the surface similarly
generated by a moving hyperbola or parabola ; but there would
be no circular sections, and hence the surfaces might perhaps not
have been considered as of very great importance.

(2) In the second case, where AE, BE meet at a point and
AB moves always parallel to itself, the surface generated is of
course a cone. Some particular cases of this sort may easily have
been discussed by Euclid, but he could hardly have dealt with the
general case, where DC has any direction whatever, up to the
point of showing that the surface was really a cone in the sense
in which the Greeks understood the term, or (in other words)
of finding the circular sections. To do this it would have been
necessary to determine the principal planes, or to solve the dis-

* Zeuthen, Die Lehre von den Kegelschnitten, pp. 425 sqq.
f This would give a surface generated by a moving line, Steo5tK&s
as Pappus has it.

Ixiv -INTRODUCTION.

criminating cubic, which we cannot suppose Euclid to have done.
Moreover, if Euclid had found the circular sections in the most
general case, Archimedes would simply have referred to the fact
instead of setting himself to do the same thing in the particular
case where the plane of symmetry is given. These remarks apply
to the case where the conic which is the locus of C is an ellipse ;
there is still less ground for supposing that Euclid could have
proved the existence of circular sections where the conic was a
hyperbola, for there is no evidence that Euclid even knew that
hyperbolas and parabolas could be obtained by cutting an oblique
circular cone.

Second lemma to the Surface-loci.

In this Pappus states, and gives a complete proof of the propo-
sition, that the locus of a point whose distance from a given point
is in a given ratio to its distance from a fixed line is a conic
section, which is an ellipse, a parabola, or a hyperbola according
as the given ratio is less than, equal to, or greater than unity*.
Two conjectures are possible as to the application of this theorem
by Euclid in the treatise referred to.

(1) Consider a plane and a straight line meeting it at any angle.
Imagine any plane drawn at right angles to the straight line and
meeting the first plane in another straight line which we will call
JT. If then the given straight line meets the plane at right angles
to it in the point S, a conic can be described in that plane with
S for focus and X for directrix ; and, as the perpendicular on X
from any point on the conic is in a constant ratio to the per-
pendicular from the same point on the original plane, all points
on the conic have the property that their distances from S are in
a given ratio to their distances from the given plane respectively.
Similarly, by taking planes cutting the given straight line at right
angles in any number of other points besides S, we see that the locus
of a point whose distance from a given straight line is in a given
ratio to its distance from a given plane is a cone whose vertex is
the point in which the given line meets the given plane, while the
plane of symmetry passes through the given line and is at right
angles to the given plane. If the given ratio was such that the
guiding conic was an ellipse, the circular sections of the surface

* See Pappus, pp. 1006 1014, and Hultsch's Appendix, pp. 12701273 ; or
cf. Apolloniw of Perga, pp. xxxvi. xxxviii.

RELATION OF ARCHIMEDES TO HIS PREDECESSORS. Ixv

could, in that case at least, be found by the same method as
that used by Archimedes (On Conoids and Spfaroids, Prop. 8) in
the rather more general case where the perpendicular from the
vertex of* the cone on the plane of the given elliptic section does
not necessarily pass through the focus.

(2) Another natural conjecture would be to suppose that, by
means of the proposition given by Pappus, Euclid found the locus
of a point whose distance from a given point is in a given ratio
to its distance from a fixed plane. This would have given surfaces
identical with the conoids and spheroids discussed by Archimedes
excluding the spheroid generated by the revolution of an ellipse
about the minor axis. We are thus brought to the same point as
Chasles who conjectured that the Surface-loci of Euclid dealt with
surfaces of revolution of the second degree and sections of the
same*. Recent writers have generally regarded this theory as
improbable. Thus Heiberg says that the conoids and spheroids
were without any doubt discovered by Archimedes himself ; other-
wise he would not have held it necessary to give exact definitions
of them in his introductory letter to Dositheus ; hence they could
not have been the subject of Euclid's treatise f. I confess I think
that the argument of Heiberg, so far from being conclusive against
the probability of Chasles' conjecture, is not of any great weight.
To suppose that Euclid found, by means of the theorem enunciated
and proved by Pappus, the locus of a point whose distance from
a given point is in a given ratio to its distance from a fixed plane
does not oblige us to assume either that he gave a name to the
loci or that he investigated them further than to show that sections
through the perpendicular from the given point on the given plane
were conies, while sections at right angles to the same perpendicular
were circles ; and of course these facts would readily suggest them-
selves. Seeing however that the object of Archimedes was to
find the volumes of segments of each surface, it is not surprising
that he should have preferred to give a definition of them which
would indicate their form more directly than a description of them
as loci would have done ; and we have a parallel case in the dis-
tinction drawn between conies as such and conies regarded as loci,
which is illustrated by the different titles of Euclid's Conies and
the Solid Loci of Aristaeus, and also by the fact that Apollonius,
* Apergu hiatorique, pp. 273, 4.

Ixvi INTRODUCTION.

though he speaks in his preface of some of the theorems in his
Conies as useful for the synthesis of 'solid loci' and goes on to
mention the. ' locus with respect to three or four lines,' yet enun-
ciates no proposition stating that the locus of such and sudh a point
is a conic. There was a further special reason for defining the
conoids and spheroids as surfaces described by the revolution of
a conic about its axis, namely that this definition enabled Archi-
medes to include the spheroid which he calls 'flat' (cTrurAaTv
<r</atpotSs), i.e. the spheroid described by the revolution of an
ellipse about its minor axis, which is not one of the loci which
the hypothesis assumes Euclid to have discovered. Archimedes'
new definition had the incidental effect of making the nature of
the sections through and perpendicular to the axis of revolution
even more obvious than it would be from Euclid's supposed way
of treating the surfaces; and this would account for Archimedes'
omission to state that the two classes of sections had been known
before, for there would have been no point in attributing to Euclid
the proof of propositions which, with the new definition of the
surfaces, became self-evident. The further definitions given by
Archimedes may be explained on the same principle. Thus the
axis, as defined by him, has special reference to his definition of
the surfaces, since it means the axis of revolution, whereas the
axis of a conic is for Archimedes a diameter. The enveloping cone
of the hyperboloid, which is generated by the revolution of the
asymptotes about the axis, and the centre regarded as the point
of intersection of the asymptotes were useful to Archimedes' dis-
cussion of the surfaces, but need not have been brought into
Euclid's description of the surfaces as loci. Similarly with the
axis and vertex of a segment of each surface. And, generally, it
seems to me that all the definitions given by Archimedes can be
explained in like manner without prejudice to the supposed dis-
covery of three of the surfaces by Euclid.

I think, then, that we may still regard it as possible that
Euclid's Surface-loci was concerned, not only with cones, cylinders
and (probably) spheres, but also (to a limited extent) with three
other surfaces of revolution of the second degree, viz. the paraboloid,
the hyperboloid and the prolate spheroid. Unfortunately however
we are confined to the statement of possibilities ; and certainty
can hardly be attained unless as the result of the discovery of
fresh documents,

RELATION OF ARCHIMEDES TO HIS PREDECESSORS, kvii

5. Two mean proportionals in continued proportion.

Archimedes assumes the construction of two mean proportionals
in two propositions (On the Sphere and Cylinder n. 1, 5). Perhaps
he was content to use the constructions given by Archytas,
Menaechmus*, and Eudoxus. It is worth noting, however, that
Archimedes does not introduce the two geometric means where
they are merely convenient but not necessary ; thus, when (On the

Sphere and Cylinder i. 34) he has to substitute for a ratio (- j ,

where /3 :> y, a ratio between lines, and it is sufficient for his

(R^ i
- ] but

may be less, he takes two arithmetic means between /3, y, as 8, ,
and then assumes! as a known result that

* The constructions of Archytas and Menaechmus are given by Eutocius
[Archimedes, Vol. m. pp. 92102] ; or see Apollonius of Perga, pp. xix xxiii.

t The proposition is proved by Eutocius; see the note to On the Sphere
and Cylinder i. 34 (p. 42).

CHAPTER IV.

ARITHMETIC IN ARCHIMEDES.

Two of the treatises, the Measurement of a circle and the
Sand-reckoner, are mostly arithmetical in content. Of the Sand-
reckoner nothing need be said here, because the system for expressing
numbers of any magnitude which it unfolds and applies cannot be
better described than in the book itself ; in the Measurement of a
circle, however, which involves a great deal of manipulation of
numbers of considerable size though expressible by means of the
ordinary Greek notation for numerals, Archimedes merely gives the
results of the various arithmetical operations, multiplication, extrac-
tion of the square root, etc., without setting out any of the operations
themselves. Various interesting questions are accordingly involved,
and, for the convenience of the reader, I shall first give a short
account of the Greek system of numerals and of the methods by
which other Greek mathematicians usually performed the various
operations included under the general term AoytoriKif (the art of
calculating), in order to lead up to an explanation (1) of the way in
which Archimedes worked out approximations to the square roots of
large numbers, (2) of his method of arriving at the two approximate
values of V3 which he simply sets down without any hint as to how
they were obtained*.

* In writing this chapter I have been under particular obligations to Hultsch's
articles Arithmetica and Archimedes in Pauly-Wissowa's Real- Encyclopa die, n.
1, as well as to the same scholar's articles (1) Die Niiherungswerthe irrationaler
Quadratwurzeln lei Archimedes in the Nachrichten von dcr kgl. Gesellschaft der
Wissenschaften zu Gottingen (1893), pp. 367 sqq., and (2) Zur Kreismessung des
Archimedes in the Zeitschrift fur Math. u. Physik (Hist. litt. Abtheilung) xxxix.
(1S94), pp. 121 sqq. and 161 sqq. I have also made use, in the earlier part
of the chapter, of Nesselmann's work Die Algebra der Griechen and the histories
of Cantor and Gow.

ARITHMETIC IN ARCHIMEDES. Ixix

1. Greek numeral system.

It is well known that the Greeks expressed all numbers from 1
to 999 by means of the letters of the alphabet reinforced by the
addition of three other signs, according to the following scheme, in
which however the accent on each letter might be replaced by a
short horizontal stroke above it, as a.

a ' P'> y' S', ' *"', ', V> & are !> 2 > 3, 4, 5, 6, 7, 8, 9 respectively.

*', K', X', /, v', ', o', *', q' 10, 20, 30, 90

p', or', r', ', t', x ', i/,', a/, V,, 100, 200, 300, 900

Intermediate numbers were expressed by simple juxtaposition
(representing in this case addition), the largest number being placed
on the left, the next largest following it, and so on in order. Thus
the number 153 would be expressed by pvy' or pvy. There was no
sign for zero, and therefore 780 was I/^TT', and 306 rr' simply.

Thousands (xtXiaSes) were taken as units of a higher order, and
1,000, 2,000, ... up to 9,000 (spoken of as x^'Atoi, SurxtXun, K.T.X.) were
represented by the same letters as the first nine natural numbers
but witli a small dash in front and below the line ; thus e.g. & was
4,000, and, on the same principle of juxtaposition as before, 1,823 was
expressed by awy or ,ao>/cy, 1,007 by x a', and so on.

Above 9,999 came a myriad (/^vptas), and 10,000 and higher
numbers were expressed by using the ordinary numerals with the
substantive /AupSes taken as a new denomination (though the words
jjivpioLj Sur/LLvptoe, rpiufjivpioi, K.T.X. are also found, following the
analogy of xiAioi, 8io-\tXiot and so on). Various abbreviations were
used for the word /uvpia?, the most common being M or Mv ; and,
where this was used, the number of myriads, or the multiple of
10,000, was generally written over the abbreviation, though some-

times before it and even after it. Thus 349,450 was MQvv*.

Fractions (AeTrra) were written in a variety of ways. The most
usual was to express the denominator by the ordinary numeral with
two accents affixed. When the numerator was unity, and it was
therefore simply a question of a symbol for a single word such as

* Diophantus denoted myriads followed by thousands by the ordinary signs
for numbers of units, only separating them by a dot from the thousands. Thus
for 3,069,000 he writes rv.,0, and Vy . t a\l/ar for 331,776. Sometimes myriads
were represented by the ordinary letters with two dots above, as /> = 100 myriads
(1,000,000), and myriads of myriads with two pairs of dots, as i 1 for 10 myriad-
myriads (1,000,000,000).

Ixx INTRODUCTION.

rpiVov, J, there was no need to express the numerator, and the
symbol was y"; similarly r" = , i r = jj, and so on. When the
numerator was not unity and a certain number of fourths, fifths,
etc., had to be expressed, the ordinary numeral was used for the
numerator; thus ff La" = T \, i' oa" = iy. In Heron's Geometry the
denominator was written twice in the latter class of fractions ; thus
(Svo 7T/x7rra) was /3VV, |^ (\cirra TpiaKOtrroTpiTa Ky' or tlKocrLrpia.
TpicLKooroTpiTa) was Ky' Ay" Ay". The sign for J, ^f/juo-v, is in
Archimedes, Diophantus and Eutocius L", in Heron C or a sign
similar to a capital S*.

A favourite way of expressing fractions with numerators greater
than unity was to separate them into component fractions with
numerator unity, when juxtaposition as usual meant addition. Thus
was written L"S" = + i; if was CS'VV" = \ + J + |+iV ;
Eutocius writes L"fS" or ^ + ^ for J, and so on. Sometimes the
same fraction was separated into several different sums ; thus in
Heron (p. 119, ed. Hultsch) M is variously expressed as

(a) i + i + ^ + i + ^r,

( b ) i + i + TF + TiV + Tra

and (c) + + A' + Tl-r + ^-r-

Sexagesimal fractions. This system has to be mentioned because
the only instances of the working out of some arithmetical operations
which have been handed down to us are calculations expressed in
terms of such fractions ; and moreover they are of special interest
as having much in common with the modern system of decimal
fractions, with the difference of course that the submultiple is 60
instead of 10. The scheme of sexagesimal fractions was used by the
Greeks in astronomical calculations and appears fully developed in
the O-WTCII$ of Ptolemy. The circumference of a circle, and along
with it the four right angles subtended by it at the centre, are
divided into 360 parts (r/^/xaTa or fjiolpai) or as we should say degrees,
each fjLolpa into 60 parts called (first) sixtieths, (717x0701) efi/Koora,
or minutes (AcTrra), each of these again into Scvrepa c^Koara (seconds),
and so on. A similar division of the radius of the circle into 60

* Diophantus has a general method of expressing fractions which is the
exact reverse of modern practice; the denominator is written above the

7 K a. ww

numerator, thus 7=5/3, KO. = 21/25, and puf . $& = 1,270,568/10,816. Some-
times he writes down the numerator and then introduces the denominator

with to noply or wptov, e.g. r* . /juop . X^. a^or = 3,069,000/331,776.

ARITHMETIC IN ARCHIMEDES. Ixxi

parts (rp.rip.arcL) was also made, and these were each subdivided into
sixtieths, and so on. Thus a convenient fractional system was
available for general arithmetical calculations, expressed in units of
any magnitude or character, so many of the fractions which we
should represent by -fa, so many of those which we should write
(ffV) 2 ' (sV) 1 * an d so on to an y extent. It is therefore not surprising
that Ptolemy should say in one place " In general we shall use the
method of numbers according to the sexagesimal manner because of
the inconvenience of the [ordinary] fractions." For it is clear that
the successive submultiples by 60 formed a sort of frame with fixed
compartments into which any fractions whatever could be located,
and it is easy to see that e.g. in additions and subtractions the
sexagesimal fractions were almost as easy to work with as decimals
are now, 60 units of one denomination being equal to one unit of
the next higher denomination, and "carrying' 1 and "borrowing"
being no less simple than it is when the number of units of one
denomination necessary to make one of the next higher is 10 instead
of 60. In expressing the units of the circumference, degrees, /uupai
or the symbol jl was generally used along with the ordinary numeral
which had a stroke above it ; minutes, seconds, etc. were expressed
by one, two, etc. accents affixed to the numerals. Thus /t/J = 2,
/xotpon/ p. fji/3' p." 47 42' 40". Also where there was no unit in any
particular denomination O was used, signifying ovSe/xia /xoipa, ovSev
cf-QKocrrov and the like ; thus O a' /3" O'" = 1' 2" 0"'. Similarly, for
the units representing the divisions of the radius the word rp.rip.ara.
or some equivalent was used, and the fractions were represented as
before ; thus rp.rjp.u.ruv 8' vt" - 67 (units) 4' 55".

2. Addition and Subtraction.

There is no doubt that, in writing down numbers for these
purposes, the several powers of 10 were kept separate in a manner
corresponding practically to our system of numerals, and the
hundreds, thousands, etc., were written in separate vertical rows.
The following would therefore be a typical form of a sum in addition ;

avK8'=: 1424
p y 103

MjScnra' 12281

M A' 30030

_ ^

r{ 43838

Ixxii INTRODUCTION.

and the mental part of the work would be the same for the Greek as
for us.

Similarly a subtraction would be represented as follows :

'=93636
fl " ' %

M,yv & 23409
M CTK' 70227

3. Multiplication.

A number of instances are given in Eutocius' commentary on
the Measurement of a circle, and the similarity to our procedure is
just as marked as in the above cases of addition and subtraction.
The multiplicand is written first, and below it the multiplier preceded
by iri'(="into"). Then the highest power of 10 in the multiplier
is taken and multiplied into the terms containing the separate
multiples of the successive powers of 10, beginning with the highest
and descending to the lowest ; after which the next highest power
of 10 in the multiplier is multiplied into the various denominations
in the multiplicand in the same order. The same procedure is
followed where either or both of the numbers to be multiplied
contain fractions. Two instances from Eutocius are appended from
which the whole procedure will be understood.

(1) I/or' 780

CTTI I/-' x 780

MMr' 490000 56000
M,<r,rt/ 56000 6400

bfjiov M irjv sum 608400

(2)

/yiy L"S"

CTT! vr/ U'o"

9,000,000 30,000 9,000 1500 750

30,000 100 30 5 2

o' L" L"8" 9,000 30 9 1J J + J

,a^Ya' L"8V 1,500 5 1J i I

L" L"8Yir" 750 2J J + J $ T V

-" [9,041,250 + 30,137J + 9,041J + 1506-
= 9,082,689 T V.

ARITHMETIC IN ARCHIMEDES. Ixxiii

One instance of a similar multiplication of numbers involving
fractions may be given from Heron (pp. 80, 81). It is only one of
many, and, for brevity, the Greek notation will be omitted. Heron
has to find the product of 4 and 7f , and proceeds as follows :

4 . 7 = 28,

33 7 _ 2
FT ' "

<T4T '

The result is accordingly

33 2 _ V 4 6 1 _ 3 1 , 6 2 1

<T4T ' T4T " ~r FT ~ FT + FT FT-

The multiplication of 37 4' 55" (in the sexagesimal system) by
itself is performed by Theon of Alexandria in his commentary on
Ptolemy's vvvrafa in an exactly similar manner.

4. Division.

The operation of dividing by a number of one digit only was
easy for the Greeks as for us, and what we call "long division" was
with them performed, mutatis mutandis, in the same way as now
with the help of multiplication and subtraction. Suppose, for
instance, that the operation in the first case of multiplication given

above had to be reversed and that Mrjv (608,400) had to be divided
by \l/w (780). The terms involving the different powers of 10 would
be mentally kept separate as in addition and subtraction, and the
first question would be, how many times will 7 hundreds go into 60
myriads, due allowance being made for the fact that the 7 hundreds
have 80 behind them and that 780 is not far short of 8 hundreds ?
The answer is 7 hundreds or i//, and this multiplied by the divisor

rf $

J/>TT' (780) would give M,r' (546,000) which, subtracted from

(608,400), leaves the remainder M,/ft/ (62,400). This remainder has
then to be divided by 780 or a number approaching 8 hundreds, and
8 tens or it would have to be tried. In the particular case the
result would then be complete, the quotient being I^TT' (780), and
there being no remainder, since v (80) multiplied by \l/ir f (780) gives

s-
the exact figure MJ&/ (62,400).

Ixxiv

INTRODUCTION.

An actual case of long division where the dividend and divisor
contain sexagesimal fractions is described by Theon. The problem
is to divide 1515 20' 15" by 25 12' 10", and Theon's account of the
process comes to this.

Quotient
15" First term 60

Divisor
25 12' 10"

Dividend
1515 20'

25 . 60 = 1 500

Remainder 15-900'

Sum

12'. 60 =

Remainder

10". 60 =

Remainder

25 . 7' -

Sum

12'. 7'

Remainder

10". 7'

Remainder

25 . 33"

Remainder

12'. 33"

920'

720'

"20(7

10'

'190'
175^
15 7 =900'

Second term 7'

84"

831"

r_i<r

~829^50 7 "
_825^'__

4" /r 50'" = 290"'

_396'"

(too great by) 106'"

Third term 33"

Thus the quotient is something less than 60 7' 33". It will be
observed that the difference between this operation of Theon's and

that followed in dividing M^v' (608,400) by ITT' (780) as above is
that Theon makes three subtractions for one term of the quotient,
whereas the remainder was arrived at in the other case after one
subtraction. The result is that, though Theon's method is quite
clear, it is longer, and moreover makes it less easy to foresee what
will be the proper figure to try in the quotient, so that more time
would be apt to be lost in making unsuccessful trials.

5. Extraction of the square root.

We are now in a position to see how the operation of extracting
the square root would be likely to be attacked. First, as in the case
of division, the given whole number whose square root is required
would be separated, so to speak, into compartments each containing

ARITHMETIC IN ARCHIMEDES. IxXV

such and such a number of units and of the separate powers of 10.
Thus there would be so many units, so many tens, so many hundreds,
etc., and it would have to be borne in mind that the squares of
numbers *from 1 to 9 would lie between 1 and 99, the squares of
numbers from 10 to 90 between 100 and 9900, and so on. Then the
first term of the square root would be some number of tens or
hundreds or thousands, and so on, and would have to be found in
much the same way as the first term of a quotient in a "long
division," by trial if necessary. If A is the number whose square
root is required, while a represents the first term or denomination of
the square root and x the next term or denomination still to be
found, it would be necessary to use the identity (a + xf - a 2 4- 2ax + x*
and to find x so that 2ax + x 2 might be somewhat less than the
remainder A a 2 . Thus by trial the highest possible value of x
satisfying the condition would be easily found. If that value were
6, the further quantity 2ab + 6 2 would have to be subtracted from
the first remainder A a 2 , and from the second remainder thus left
a third term or denomination of the square root would have to be
derived, and so on. That this was the actual procedure adopted is
clear from a simple case given by Theon in his commentary on the
crvi/Tai5. Here the square root of 144 is in question, and it is
obtained by means of Eucl. n. 4. The highest possible denomina-
tion (i.e. power of 10) in the square root is 10 ; 10 2 subtracted from
144 leaves 44, and this must contain not only twice the product of
10 and the next term of the square root but also the square of that
next term itself. Now, since 2 . 10 itself produces 20, the division
of 44 by 20 suggests 2 as the next term of the square root ; and
this turns out to be the exact figure required, since

2 . 20 + 2 2 - 44.

The same procedure is illustrated by Theon's explanation of
Ptolemy's method of extracting square roots according to the
sexagesimal system of fractions. The problem is to find approxi-
mately the square root of 4500 /xoipcu or degrees, and a geometrical
figure is used which makes clear the essentially Euclidean basis of
the whole method. Nesselmann gives a complete reproduction of
the passage of Theon, but the following purely arithmetical represen-
tation of its purport will probably be Jpuod clearer, when looked at
side by side with the figure.

Ptolemy has first found the irtegral part of \/4500 to be 67.

Ixxvi

INTRODUCTION.

Now 67 2 = 4489, so that the remainder is 11. Suppose now that
the rest of the square root is expressed by means of the usual
sexagesimal fractions, and that we may therefore put

N/4500 =

2. 67#
where x, y are yet to be found. Thus x must be such that ^ -

is somewhat less than 11, or x must be somewhat less than ~ an

2i . D /

330
or ^ , which is at the same time greater than 4. On trial, it

turns out that 4 will satisfy the conditions of the problem, namely

/ 4\ 2

that f 67 + ^ J must be less than 4500, so that a remainder will

be left by means of which y may be found.

67 U

55"

4489

268'

-rfl

4' 268'

16"

55" 3688" 40'"

Now 1 1 ' : f 7^. J is the remainder, and this is equal to

11.60 2 -2.67.4.60-16 7424

60 2

GO 2 "'

, civ' 4\ y . A 7424

Thus we must suppose that 21 , -T^J ) /./-va approximates to -^QT >

or that 8048?/ is approximately equal to 7424 . 60.

ARITHMETIC IN ARCHIMEDES. Ixxvii

Therefore y is approximately equal to 55. We have then to
subtract

' *-\L f^\* 4*2640 302 ^
+ 60/ 60 8 + \60 V ' r 60 3 + 60 4 '

from the remainder -^ above found.

T , , , ,. . 442640 , 7424 . 2800 46 40
The subtraction of 6Q3 from -^ gives -g^ T , or - 2 + 3 ;

but Theon does not go further and subtract the remaining -^y 4 - ,

instead of which he merely remarks that the square of

approximates to g- 2 + - . As a matter of fact, if we deduct the

3025 2800 ^ . . . . . . .

-^TTT- trom -pjprj- , so as to obtain the correct remainder, it is
ulr oU

. , . . 164975
found to be TT^T
GO 4

To show the power of this method of extracting square roots by
means of sexagesimal fractions, it is only necessary to mention that

103 55 23

Ptolemy gives --- - + -^ + ^ as an approximation to v3, which

approximation is equivalent to 1*7320509 in the ordinary decimal
notation and is therefore correct to 6 places.

But it is now time to pass to the question how Archimedes
obtained the two approximations to the value of \/3 which he
assumes in the Measurement of a circle. In dealing with this
subject I shall follow the historical method of explanation adopted
by Hultsch, in preference to any of the mostly a priori theories
which the ingenuity of a multitude of writers has devised at
different times.

6. Early investigations of surds or incommensurables.

From a passage in Proclus' commentary on Eucl. I.* we learn
that it was Pythagoras who discovered the theory of irrationals
(ij TWV a\dyo)v ?r/)ay/AaTta). Further P lj ito says (Tlteaetetus 147D),
"On square roots this Theodoi J P Li Gyrene] wrote a work in

* p. 65 (ed. Friedlein).

Ixxviii INTRODUCTION.

which he proved to us, with reference to those of 3 or 5 [square] feet
that they are incommensurable in length with the side of one square
foot, and proceeded similarty to select, one by one, each [of the other
incommensurable roots] as far as the root of 17 square feet, beyond
which for some reason he did not go." The reason why \/2 is not
mentioned as an incommensurable square root must be, as Cantor
says, that it was before known to be such. We may therefore
conclude that it was the square root of 2 which was geometrically
constructed by Pythagoras and proved to be incommensurable with
the side of a square in which it represented the diagonal. A clue
to the method by which Pythagoras investigated the value of \/2
is found by Cantor and Hultsch in the famous passage of Plato
(Rep. vui. 546 B, c) about the 'geometrical' or * nuptial* number.
Thus, when Plato contrasts the farr) and apprjros Sia/mcrpos r^$
Tre/ATraSos, he is referring to the diagonal of a square whose side
contains five units of length ; the d/a/oTyros Staj^cr/aos, or the irrational
diagonal, is then \/50 itself, and the nearest rational number is
\/50-l, which is the prjrrj Sia/AT/>os. We have herein the
explanation of the way in which Pythagoras must have made the
first and most readily comprehensible approximation to >/2 ; he
must have taken, instead of 2, an improper fraction equal to it but
such that the denominator was a square in any case, while the
numerator was as near as possible to a complete square. Thus

Pythagoras chose ^ , and the first approximation to \/2 was

7 7

accordingly ^ , it being moreover obvious that \/2 > - . Again,

Pythagoras cannot have been unaware of the truth of the
proposition, proved in Eucl. n. 4, that (a + 6) 2 = a 2 + 2ab + b 3 , where
a, b are any two straight lines, for this proposition depends solely
upon propositions in Book i. which precede the Pythagorean
proposition I. 47 and which, as the basis of I. 47, must necessarily
have been in substance known to its author. A slightly different
geometrical proof would give the formula (a - b) 2 = a 2 2a6 + 6 8 ,
which must have been equally well known to Pythagoras. It could
not therefore have escaped the discoverer of the first approximation
v50 1 for \/50 that the use . -/ 3 formula with the positive sign

would give a much nearer approximation, viz. 7 + ^-j, which is only

ARITHMETIC IN ARCHIMEDES.

Ixxix

greater than \/50 to the extent of ( rr ) Thus we may properly
assign to Pythagoras the discovery of the fact represented by

H :

s 1

The consequential result that \/2 > ^ \/50 1 is used by

Aristarchus of Samos in the 7th proposition of his work On the
size and distances of the sun and moon*.

With reference to the investigations of the values of \/3, V5,

\/6, \l\l by Theodorus, it is pretty certain that \/3 was

geometrically represented by him, in the same way as it appears

* Part of the proof of this proposition was a sort of foretaste of the first part
of Prop. 3 of Archimedes' Measurement of a
circle, and the substance of it is accordingly
appended as reproduced by Hultsch.

ABEK is a square, KB a diagonal, L HBE
-\L KBE, L FBE = 3, and AC is perpendicu-
lar to BF so that the triangles ACB, BEF are
similar.

Aristarchus seeks to prove that
AB : BC > 18 : 1.

If R denote a right angle, the angles KBE,
HBE, FBE are respectively $J.R, jR, ^R.

Then HE : FE > L HBE : L FBE.

[This is assumed as a known lemma by Aristarchus as well as Archimedes.]

Therefore HE : FE > 15 : 2 (a).

Now, by construction, BK*=2BE*.

Also [Eucl. vi. 3] BK : BE=KH : HE ;

whence

And, since

KH : HE > 7 : 5,

so that KE :EH>12 :5

From (a) and (), ex aequali,

KE : FE > 18 : 1.
Therefore, since BF > BE (or KE),

BF : FE > 18 : 1,
so that, by similar triangles,

AB : BC > 18 : 1.

1XXX INTRODUCTION.

afterwards in Archimedes, as the perpendicular from an angular
point of an equilateral triangle on the opposite side. It would
thus be readily comparable with the side of the " 1 square foot "
mentioned by Plato. The fact also that it is the side of three
square feet (rpnrovs 8wa/us) which was proved to be incommensurable
suggests that there was some special reason in Theodorus' proof for
specifying feet, instead of units of length simply; and the ex-
planation is probably that Theodorus subdivided the sides of his
triangles in the same way as the Greek foot was divided into
halves, fourths, eighths and sixteenths. Presumably therefore,

exactly as Pythagoras had approximated to \/2 by putting ^

for 2, Theodorus started from the identity 3 = ^ - I* would then
be clear that

48 + 1 . 7
16 ' *'*' 4"

To investigate \/48 further, Theodorus would put it in the form
\/49-l, as Pythagoras put \/50 into the form N/49 + 1, and the
result would be

-,, ^ 1

We know of no further investigations into incommensurable
square roots until we come to Archimedes.

7. Archimedes' approximations to V3.

Seeing that Aristarchus of Samos was still content to use the
first and very rough approximation to \/2 discovered by Pythagoras,
it is all the more astounding that Aristarchus' younger contemporary
Archimedes should all at once, without a word of explanation, give
out that

1351 ,- 265
~780 >N/3> 153'

as he does in the Measurement of a circle.

In order to lead up to the explanation of the probable steps by
which Archimedes obtained these approximations, Hultsch adopts
the same method of analysis as was used by the Greek geometers in
solving problems, the method, that is, of supposing the problem
solved and following out the necessary consequences. To compare

ARITHMETIC IN ARCHIMEDES. Ixxxi

the two fractions TKO and _ Qr . , we first divide both denominators
loo loO

into their smallest factors, and we obtain

780 = 2.2.3.5.13,
153 = 3.3.17.

We observe also that 2 . 2 . 13 = 52, while 3.17^ 51, and we may
therefore show the relations between the numbers thus,

780 - 3 . 5 . 52,
153 = 3.51.

For convenience of comparison we multiply the numerator and

265
denominator of r-^ by 5 ; the two original fractions are then

1351 , 1325
-^s and

15.52 " 15.51'
so that we can put Archimedes' assumption in the form
1351 ,- 1325

and this is seen to be equivalent to

Now 26-_^= */ 26 2 ~ 1 + (KO) , and the latter expression
is an approximation to \/26 2 1.
We have then 26 - >

As 26 - r = was compared with 15^/3, and we want an ap-

OJj

proximation to /^/3 itself, we divide by 15 and so obtain

But ^'-l---^; d it follows

that

The lower limit for \/3 was given by

H. A. /

Ixxxii INTRODUCTION.

and a glance at this suggests that it may have been arrived at by
simply substituting (52 1) for 52.

Now as a matter of fact the following proposition is true. If
a*b is a w/tole number which is not a square, while a 2 is the nearest
square number (above or below the first number, as the case may be),
then

a ~- > N/ S b > a ----- .
2a 2al

Hultsch proves this pair of inequalities in a series of propositions
formulated after the Greek manner, and there can be little doubt
that Archimedes had discovered and proved the same results in
substance, if not in the same form. The following circumstances
confirm the probability of this assumption.

(1) Certain approximations given by Heron show that he
knew and frequently used the formula

*Ja* 6 co a Y- ,
'la

(where the sign co denotes "is approximately equal to").
Thus he gives \/50 co 7 + -,-j ,

,
lo

(2) The formula Va 2 +6eoa + - - - is used by the Arabian

Alkarkhl (llth century) who drew from Greek sources (Cantor,
p. 719 sq.).

It can therefore hardly be accidental that the formula

~ > ^ a * t a

gives us what we want in order to obtain the two Archimedean
approximations to V3, and that in direct connexion with one
another*.

* Most of the a priori theories as to the origin of the approximations are
open to the serious objection that, as a rule, they give series of approximate
values in which the two now in question do not follow consecutively, but are
separated by others which do not appear in Archimedes. Hultsch's explanation
is much preferable as being free from this objection. But it is fair to say that
the actual formula used by Hultsch appears in Hunrath's solution of the puzzle

ARITHMETIC IN ARCHIMEDES. Ixxxiii

We are now in a position to work out the synthesis as follows.
From the geometrical representation of V3 as the perpendicular
from an angle of an equilateral triangle on the opposite side we
obtain \/2* I = /3 and, as a first approximation,

2-l>V3.
Using our formula we can transform this at once into

V3>2- l 1 ,or 8-J.

(1\ 5
2 )> r o> an d would obtain

25 27

^r , which he would compare with 3, or ^- ; i.e. he would put

" ^ and would obtain

=./

To obtain a still nearer approximation, he would proceed in the

/26\ a 676 . . _ 675 ,

same manner and compare ( TF ) > or o^ > with 3, or ^ , whence it

\1D/ 2*2d 225

would appear that /s/3

and therefore that ^5 ( 26 ~ 59) >

1 Qr>l

that is, ~ > \/3.

The application of the formula would then give the result

.... /T 1326-1 265

that is, V3>- IO - 1 -,or 1T3 .

The complete result would therefore be

1351 265

(Die Berechnung irratiotialer Quadratwurzeln vor der Herrschaft der Decimal-
briiche, Kiel, 1884, p. 21 ; of. Ueber das Attsziehen der Quadratiourzel bei
Griechen und Indern, Hadersleben, 1883), and the same formula is implicitly
used in one of the solutions suggested by Tannery (Sur la mesure du cercle
d'Archimede in M&moires de la 8ocit6 des sciences physiques et naturelles de
Bordeaux, 2 e serie, iv. (1882), p. 313-337).

lxxxi\ INTRODUCTION.

Thus Archimedes probably passed from the first approximation
7 to ? from o to 15* and from 75 directly to 7go , the closest

approximation of all, from which again he derived the less close

^65

approximation !* . The reason why he did riot proceed to a still
153

1351
nearer approximation than 7 7r is probably that the squaring of

this fraction would have brought in numbers much too large to be
conveniently used in the rest of his calculations. A similar reason

5 7

will account for hr having started from ^ instead of -\ if he had

O T

used the latter, he would first have obtained, by the same method,
Tv- and thence -^ TT > \/3, or ^> \/3; the squaring
of -7, would have given \/3 == & > and the corresponding

x- U U - 1881 ? 1 - xl_ U

approximation would have given -^~ ok wnere a g aln the numbers

t)0 J. i/T

are inconveniently large for his purpose.

8. Approximations to the square roots of large
numbers.

Archimedes gives in the Measurement of a circle the following
approximate values :

(1) 3013J >V9082321,

(2) 1838^- > V3380929,

(3) 1009> ViOi8405,
(4)

(5) 591i<N/349450,

(6) 1172|<Vl373943|f,

(7) 2339i<N/5472132 T V

There is no doubt that in obtaining the integral portion
of the square root of these numbers Archimedes used the method
based on the Euclidean theorem (a + b)* = a 9 + 2ab + 6 s which has

ARITHMETIC IN ARCHIMEDES.

already been exemplified in the instance given above from Theon,
where an approximation to V4500 is found in sexagesimal fractions.
The method does not substantially differ from that now followed; but
whereas, to take the first case, \/9082321, we can at once see what
will be the number of digits in the square root by marking off pairs
of digits in the given number, beginning from the end, the absence
of a sign for in Greek made the number of digits in the square
root less easy to ascertain because, as written in Greek, the number

a only contains six signs representing digits instead of seven.
Even in the Greek notation however it would not be difficult to see
that, of the denominations, units, tens, hundreds, etc. in the square
root, the units would correspond to KO! in the original number, the

* *

tens to ,/3r, the hundreds to M, and the thousands to M. Thus it

would be clear that the square root of 9082321 must be of the form

where x, y t z, w can only have one or other of the values 0, 1, 2, ... 9.
Supposing then that x is found, the remainder ^-(1000#) 2 , where
N is the given number, must next contain 2. 1000#. lOOy and
(1(%) 2 , then 2(1000aj+100y).10;s and (10s)*, after which the
remainder must contain two more numbers similarly formed.

In the particular case (1) clearly x = 3. The subtraction of
(3000) 2 leaves 82321, which must contain 2 . 3000 . lOOy. But, even
if y is as small as 1, this product would be 600,000, which is greater
than 82321. Hence there is no digit representing hundreds in the
square root. To find z, we know that 82321 must contain

and has to be obtained by dividing 82321 by 60,000. Therefore
s=l. Again, to find w, we know that the remainder

(82321 -2. 3000. 10 - 10),

or 22221, must contain 2 . 3010w + wr, and dividing 22221 by
2.3010 we see that w = 3. Thus 3013 is the integral portion of
the square root, and the remainder is 22221 -(2. 3010. 3 + 3 2 ), or
4152.

The conditions of the proposition now require that the approxi-
mate value to be taken for the square root must not be less than

Ixxxvi INTRODUCTION.

the real value, and therefore the fractional part to be added to 3013
must be if anything too great. Now it is easy to see that the

fraction to be added is greater than ^ because 2. 3013. g+ ( ^) * s
less than the remainder 4152. Suppose then that the number
required (which is nearer to 3014 than to 3013) is 3014--,

and - has to be if anything too small.

Now (3014)' = (3013) 8 + 2 . 3013 + 1 = (3013) 2 + 6027

= 9082321-4152 + 6027,
whence 9082321 = (3014) 2 - 1875.

By applying Archimedes' formula *Ja a b < a ^ , we obtain

ACL

The required value - has therefore to be not greater than xpo

P 1

It remains to be explained why Archimedes put for - the value j

which is equal to * . In the first place, he evidently preferred

fractions with unity for numerator and some power of 2 for
denominator because they contributed to ease in working, e.g. when
two such fractions, being equal to each other, had to be added.

9 i

(The exceptions, the fractions - and ^, are to be explained by

exceptional circumstances presently to be mentioned.) Further, in
the particular case, it must be remembered that in the subsequent

work 2911 had to be added to 3014-^ and the sum divided by 780,

or 2 . 2 . 3 . 5 . 13. It would obviously lead to simplification if a
factor could be divided out, e.g. the best for the purpose, 13. Now,
dividing 2911 +3014, or 5925, by 13, we obtain the quotient 455,

and a remainder 10, so that 10- - remains to be divided by 13.
Therefore ^ has to be so chosen that I0q p is divisible by 13, while
~ approximates to, but is not greater than, 7 . The solution

would therefore be natural and easy.

ARITHMETIC IN ARCHIMEDES. Ixxxvii

(2) x/3380929.

The usual process for extraction of the square root gave as the
integral Rart of it 1838, and as the remainder 2685. As before, it
was easy to see that the exact root was nearer to 1839 than to 1838,
and that

x/3380929 = 1838 s + 2685 = 1839' - 2 . 1838 - 1 + 2685

= 1839 -992.
The Archimedean formula then gave

It could not have escaped Archimedes that -7 was a near approxima-

.. 4 992 1984 . 1 1839 ,1 . , , .. ., ,

tl0n t0 36f8 r 7356' Smce 4= 7356 ; and 4 WOuld have 8atlsfied

the necessary condition that the fraction to be taken must be less

2
than the real value. Thus it is clear that, in taking -_-=- as the

approximate value of the fraction, Archimedes had in view the
simplification of the subsequent work by the elimination of a factor.

f) f)

If the fraction be denoted by -, the sum of 1839-- and 1823, or

3662 - P -, had to be divided by 240, i.e. by 6 . 40. Division of 3662
by 40 gave 22 as remainder, and then p, q had to be so chosen that

22-- was conveniently divisible by 40, while - was less than but
q J J ' q

992
approximately equal to 0^70 The solution p 2, q\\ was easily

OUlO

seen to satisfy the conditions.

(3) N/l 018405.

The usual procedure gave 1018405 = 1009* + 324 and the ap-
proximation

324
It was here necessary that the fraction to replace 577^7* should be

greater but approximately equal to it, and ^ satisfied the conditions,
while the subsequent work did not require any change in it.

Ixxxviil INTRODUCTION.

(4) 74069284^.

The usual process gave 4069284^- = 2017 s + 995^; it followed
that

36 .995 + 1

2017 36. LMIT

and 201 7 J was an obvious value to take as an approximation
somewhat greater than the left side of the inequality.

(5) V349450.

In the case of this and the two following roots an approximation
had to be obtained which was less, instead of greater, than the true
value. Thus Archimedes had to use the second part of the formula

In the particular case of N /34945(T the integral part of the root is
591, and the remainder is 169. This gave the result

and since 169=13*, while 2. 591 + 1 =7 . 13 8 , it resulted without
further calculation that

N /349450>591|.

Why then did Archimedes take, instead of this approximation,
another which was not so close, viz. 591^? The answer which the
subsequent working and the other approximations in the first part of
the proof suggest is that he preferred, for convenience of calculation,

to use for his approximations fractions of the form ^ only. But he

could not have failed to see that to take the nearest fraction of this
form, , instead of -= might conceivably affect his final result and

make it less near the truth than it need be. As a matter of fact,
as Hultsch shows, it does not affect the result to take 59 \\ and to
work onwards from that figure. Hence we must suppose that
Archimedes had satisfied himself, by taking 59 1| and proceeding on
that basis for some distance, that he would not be introducing any
appreciable error in taking the more convenient though less accurate
approximation 59 1J.

ARITHMETIC IN ARCHIMEDES. Ixxxix

(6) 7137394311.

In this case the integral portion of the root is 1172, and the
remainder 359 J. Thus, if denote the root,

359
1 172 + '

359
Now 2.11724-1 = 2345; the fraction accordingly becomes

1 / 359 \
and =r ( = O^TQ) satisfies the necessary conditions, viz. that it must

I \ &DLO/

be approximately equal to, but not greater than, the given fraction.
Here again Archimedes would have taken 1172| as the approximate
value but that, for the same reason as in the last case, 1172^ was
more convenient.

(7) V5472132 r V

The integral portion of the root is here 2339, and the remainder
, so that, if R is the exact root,

> 2339J, a fortiori.

A few words may be added concerning Archimedes' ultimate
reduction of the inequalities

to the simpler result 3 - > TT > 3 ^ .

As a matter of fact = = 7.^r , so that in the first fraction it was

only necessary to make the small change of diminishing the de-
nominator by 1 in order to obtain the simple 3 = .

As regards the lower limit for TT, we see that ^n^-, - QACn ; and

JUi t oOo"

Hultsch ingeniously suggests the method of trying the effect of
increasing the denominator of the latter fraction by 1. This

XC INTRODUCTION.

1137 379
produces ^A^ or 9 /,q n ; and, if we divide 2690 by 379, the quotient

is between 7 and 8, so that

1 379^ 1
7 > 2690 > 8'

Now it is a known proposition (proved in Pappus vn. p. 689)

,,,., c .. a a + c

that, if r > -, , then -,- > , , .

b d o b+d

Similarly it may be proved that

a + c c
b~+~d^ d'

It follows in the above case that

379 379 +L 1

which exactly gives ^ > ^ ,

71 o

, 10 . , A 379 ^ 1 .

and == is very much nearer to TA than - is.
71 UoyU o

Note on alternative hypotheses with regard to the
approximations to \/3.

For a description and examination of all the various theories put
forward, up to the year 1882, for the purpose of explaining Archimedes'
approximations to \/3 the reader is referred to the exhaustive paper by
Dr Siegmund GUnther, entitled Die yuadratiscfan Irratwrwlitaten der Alten
und deren Entwickelungsmethoden (Leipzig, 1882). The same author gives
further references in his Abriss der Ceschichte der Mathematik und der Natur-
tvis&enschaften im Altertum forming an Appendix to Vol. v. Pt. 1 of I wan von
Miiller's Handbuch der klassiscJien Altertums-wissenschaft (Miinchen, 1894).

GUnther groups the different hypotheses under three general heads :

(1) those which amount to a more or less disguised use of the
method of continued fractions and under which are included the solutions
of De Lagny, Mollweide, Hauber, Buzengeiger, Zeuthen, P. Tannery (first
solution), Heilermami ;

(2) those which give the approximations in the form of a series
of fractions such as cH --- 1 ------ 1 ----- \- ... ; under this class come the

solutions of Eadicke, v. Pessl, Eodet (with reference to the (Julvasutras),
Tannery (second solution) ;

ARITHMETIC IN ARCHIMEDES. xci

(3) those which locate the incommensurable surd between a greater
and lesser limit and then proceed to draw the limits closer and closer.
This class includes the solutions of Oppermann, Alexejeff, Schonborn,
Hunrath, though the first two are also connected by GUnther with the
method of continued fractions.

Of the methods so distinguished by Gunther only those need be here
referred to which can, more or less, claim to rest on a historical basis
in the sense of representing applications or extensions of principles laid
down in the works of Greek mathematicians other than Archimedes which
have come down to us. Most of these quasi-historical solutions connect
themselves with the system of side- and diagonal-numbers (ir\vpiKo\ and
faancTpiKol apifyioi) explained by Theon of Smyrna (c. 130 A.D.) in a work
which was intended to give so much of the principles of mathematics as
was necessary for the study of the works of Plato.

The side- and diagonal-numbers are formed as follows. We start with
two units, and (a) from the sum of them, (6) from the sum of twice
the first unit and once the second, we form two new numbers ; thus

1.1 + 1=2, 2.1 + 1 = 3.

Of these numbers the first is a side- and the second a diagonal-number
respectively, or (as we may say)

2 =2, d 2 =3.

In the same way as these numbers were formed from ! = !, c?i=l, suc-
cessive pairs of numbers are formed from </ 2 , ^> an< ^ so on > m accordance
with the formula

whence we have

3 =1. 2+3 = 5, rf 3 = 2. 2 + 3 = 7,

4 =1. 5 + 7 = 12, d 4 =2. 5 + 7 = 17,
and so on.

Theon states, with reference to these numbers, the general proposition
which we should express by the equation

The proof (no doubt omitted because it was well-known) is simple. For
we have

n -2 2 2a n -2 2 ) anc * so on,
while d^ 2a x 2 = - 1 ; whence the proposition is established.

Cantor has pointed out that any one familiar with the truth of this
proposition could not have failed to observe that, as the numbers were
successively formed, the value of d n 2 /a n 2 would approach more and more
nearly to 2, and consequently the successive fractions d^a* would give

XC11 INTRODUCTION.

nearer and nearer approximations to the value of V2, or in other words that

1 3 7 17 41

1* 2' 5' 12' 29'

are successive approximations to \/2. It is to be observed that the third

of these approximations, -, is the Pythagorean approximation which
o

appears to be hinted at by Plato, while the above scheme of Theon,
amounting to a method of finding all the solutions in positive integers of
the indeterminate equation

2.r 2 -y*=l,

and given in a work designedly introductory to the study of Plato,
distinctly suggests, as Tannery has pointed out, the probability that even
in Plato's lifetime the systematic investigation of the said equation had
already begun in the Academy. In this connexion Proclus' commentary
on Eucl. i. 47 is interesting. It is there explained that in isosceles
right-angled triangles "it is not possible to find numbers corresponding to
the sides ; for there is no square number which is double of a square
except in the sense of approximately double, e.g. 7 2 is double of 5 2 less 1."
When it is remembered that Theon's process h;is for its object the finding
of any number of squares differing only by unity from double the squares
of another series of numbers respectively, and that the sides of the two
sets of squares are called diagonal- and .nV&-numbers respectively, the
conclusion becomes almost irresistible that Plato had such a system in
mind when he spoke of fary dtdpcrpor (rational diagonal) as compared
with apprjros dtafjLTpof (irrational diagonal) TTJS nt^nabos (cf. p. Ixxviii above).
One supposition then is that, following a similar line to that by which
successive approximations to \/2 could be obtained from the successive
solutions, in rational numbers, of the indeterminate equations 2# 2 -# 2 = 1,
Archimedes set himself the task of finding all the solutions, in rational
numbers, of the two indeterminate equations bearing a similar relation

to V3, viz.

A- 2 -3^ = 1,

A' 2 -3y 2 =-2.

Zeuthen appears to have been the first to connect, eo nomine, the ancient
approximations to V3 with the solution of these equations, which are also
made by Tannery the bams of his first method. But, in substance, the
same method had been used as early as 1723 by De Lagny, whose
hypothesis will be, for purposes of comparison, described after Tannery's
which it so exactly anticipated.

Zeuthen' * solution.

After recalling the fact that, even l>efore Euclid's time, the solution
of the indeterminate equation # 2 -fy 2 =z 2 by means of the substitutions

ARITHMETIC IN ARCHIMEDES. XClii

is well known, Zeuthen concludes that there could have been no
fficulty in deducing from Eucl. n. 5 the identity

>m which, by multiplying up, it was easy to obtain the formula
3 (2mtt) 2 + (wi 2 - 3tt 2 ) 2 = (m 2 + 3n 2 ) 2 .

If therefore one solution w 2 - 3n 2 = 1 was known, a second could at once
5 found by putting

x = w 2 + 3/& 2 , y = 2mn.

Now obviously the equation

7/i 2 -3w 2 =l

satisfied by the values m = 2, n 1 ; hence the next solution of the
[uation

=7, ^ = 2.2.1=4;
id, proceeding in like manner, we have any number of solutions as
. 4 2 =97, y 2 = 2 - 7.4 = 56,

817, y 3 = 2. 97. 56 = 10864,
id so on.

Next, addressing himself to the other equation

# 2 -3y 2 = -2,
euthen uses the identity

bus, if we know one solution of the equation in 1 3>i 2 = 1, we can proceed
> substitute

Suppose 7?i = 2, 7? = 1, as before ; we then have

'*'i = 5 #i = 3 -
If we put # 2 =#j + 3^ = 14, y 2 =.>' 1 +y i = 8, we obtain

uid m = 7, w = 4 is seen to be a solution of wi 2 -3n 2 =l).
Starting again from o: 2 , y 2 , we have

^=38, ^ = 22,

nd ft "

#* n

w=19, ?i=ll lx)ing a solution of the equation w 2 -3>i 2 =-2);
.r 4 =104, ^4=60,

whence ^ 4 = ?

XC1V INTRODUCTION.

(and ?>i=26, ?i=15 satisfies w 2 -3w 2 =l),

o; 6 =284, # fi = 164

. .. . >r fi 97 A\ 265 ,
Similarly - = , -^ = -- , and so on.

#6 56 y? 153

This method gives all the successive approximations to \/3, taking
account as it does of both the equations

Tannery's first solution.

Tannery asks himself the question how Diophantus would have set
about solving the two indeterminate equations. He takes the first equation
in the generalised form

-r 2 oy 2 = l,

and then, assuming one solution (p, q) of the equation to be known, he
supposes

p^mx-p, q^x+q.

Then p* - aqf =s w 2 *; 2 - 2mpx +p 2 - ax 1 Zaqx - cup = 1 ,

whence, since p 2 aq 2 =l, by hypothesis,

so that

ri m-a m a

arid Pi 2 i 2 =l-

The valuas of jo^ ^ so found are rational but not necessarily integral ;
if integral solutions are wanted, we have only to put

where (w, v) is another integral solution of x 2 - ay 2 =l.
Generally, if ( JE?, q) be a known solution of the equation

suppose ^?! =q/>+^, qi=yp + 8q, and " il sufl&t pour determiner a, 0, y, 8 de
connattre les trois groupes de solutions les plus simples et de rosoudre
deux couples d'dquations du premier degr<$ & deux inconnues." Thus
(1) for the equation

*-^-i,

the first three solutions are

(^ = 1,^=0), (^ = 2,^ = 1), (p = 7, ? =4),

so that a = 2, 0=3, y=l, d=2,

ARITHMETIC IN ARCHIMEDES. XCV

and it follows that the fourth solution is given by

p = 2. 7 + 3. 4 = 26,

= 1.7 + 2.4=15;

(2) for the equation a? - 3y 2 = - 2,

the first three solutions being (1, 1), (5, 3), (19, 11), we have

and '

whence a = 2, j3=3, y=l, 5=2, and the next solution is given by
^=2.19 + 3.11 = 71,

and so on.

Therefore, by using the two indeterminate equations and proceeding as
shown, all the successive approximations to V3 can be found.

Of the two methods of dealing with the equations it will be seen that
Tannery's has the advantage, as compared with Zeuthen's, that it can be
applied to the solution of any equation of the form x* ay L =^r.

De Lagntfs method.

The argument is this. If </3 could be exactly expressed by an im-
proper fraction, that fraction would fall between 1 and 2, and the square of
its numerator would be three times the square of its denominator. Since
this is impossible, two numbers have to be sought such that the square of
the greater differs as little as possible from 3 times the square of the
smaller, though it may be either greater or less. De Lagny then evolved
the following successive relations,

2 2 =3.1 2 + 1, 5 2 = 3.3 2 -2, 7 2 = 3.4 2 + l, 19 2 =3.11 2 -2,

26*=3.15*+1, 71 2 =3.41 2 -2, etc.
From these relations were derived a series of fractions greater than \'3,

viz. Y , - , i e ^ c -> an< l another series of fractions less than \^3, viz.
-,-,-, etc. The law of formation was found in each case to be that, if

- was one fraction in the series and -- the next, then
2 '/

This led to the results

2 7 26 97 362 1351 ,

5 19 71 265 989 3691

XCV1 INTRODUCTION.

while the law of formation of the successive approximations in each series
is precisely that obtained by Tannery as the result of treating the two
indeterminate equations by the Diophantine method.

Heilernianvts method.

This method needs to be mentioned because it also depends upon a
generalisation of the system of side- and diagonal-numbers given by Thcon
of Smyrna.

Theon's rule of formation was

and Heilermann simply substitutes for 2 in the second relation any
arbitrary number a, developing the following scheme,

It follows that

! Z) n -i

By subtraction, D* - aS* = (1 - a) (/> w -i 2 - ^n-f)

= (1 - a)* (/> n -2 2 - >W) similarly,

This corresix>nds to the most general form of the " Pellian" equation

x*-ay 2 = (const.).
If now we put /> =>V = 1, we have

/> , (l-)" + l

>V" W~ ''

from which it appears that, where the fraction on the right-hand side

approaches zero as n increases, ,~ is an approximate value for Va.

Clearly in the case where a=3, /7 =2, # = 1 we have

Z>o_2 A_5 A_14_7 A_19 /> 4 _52_26
So"""!' ^""3' ^ 8 4' S z ~ir t>t 30"" 15'

^5 = 71 /^^m^D? /> 7== 265
S 6 41' ^"112 56' S 7 153'
and so on.

ARITHMETIC IN ARCHIMEDES.

But the method is, as shown by Heilennann, more rapid if it is used to
find, not \/a, but 6\/a, where b is so chosen as to make b'*a (which takes

the place of a) somewhat near to unity. Thus suppose a = , so that

Va = ^V3, and we then have (putting /) =,S' =1)
o

c. -, n 5 2 l/o 5 26 2f >

^ = 2 > A = 25 ""I V3~-. 26 , or -,

c , 102 .. 54+52 106 /- 5 106 265

and

208 102.27 106 5404

25 ' 3 ~ 25 . 25 + 25 ~ 25 . 25 J

5404 5 1351

780-

This is one of the very few instances of success in bringing out the two
Archimedean approximations in immediate sequence without any foreign
values intervening. No other methods appear to connect the two values
in this direct way except those of Hunrath and Hultsch depending on the
formula

aT->V*~&> .,-
-2a 2al

\\ r e now pass to the second class of solutions which develops the
approximations in the form of the sum of a series of fractions, and under
this head comes

Tantivry's second method.

This may be exhibited by means of its application (1) to the case of the
square root of a large number, e.g. \/349450 or V^"l" + 23409, the first of
the kind appearing in Archimedes, (2) to the case of \/3.

(1) Using the formula

we try the effect of putting for \/571--f 23409 the expression

r _ 23409
571 + 1142'

It turns out that this gives correctly the integral part of the root, and we
now suppose the root to be

571-f20-f-.
m

{Squaring and regarding ^ as negligible, we have

H. A.

571 2 + 400 + 22840 ++- = 571 2 + 23409,
m m

XCV111 INTRODUCTION.

whence - =169,

, 1 169 1

and

so that \/349450>59li.

(2) Bearing in mind that

_ 2

we have v3 = \/l 2 + 2ev>H-_ -, ^

2. 1 + 1

1 - -

+3' r 3'

Assuming then that \/3 = f - 4- - J , squaring and neglecting ^ , we obtain

^^a

whence w&=15, and we get as the second approximation

5 1 26

3 + F5' r 15*

We have now 26 2 - 3 . 15 2 = 1,

and can proceed to find other approximations by means of Tannery's first

method.

(2 1
1+- -f +

and, neglecting - 2 , we get

26 2 52

whence = 15 . 52= 780, and

V3^fi+? + l- L. 18B1 V

v \ ^3^15 780 780 /

1351
It is however to be observed that this method only connects - - with

/oU

26 265

and not with the intermediate approximation ' , to obtain which

15 15*3

Tannery implicitly uses a particular case of the formula of Hunrath and
Hultsch.

Rodet's method was apparently invented to explain the approximation
in the (Julvastttras*

^-~ AT 3 T 3. 4 374734'

* See Cantor, Vorlesungen fiber Gesch. d. Math. p. 600 sq.

ARITHMETIC IN ARCHIMEDES. xcix

4
but, given the approximation - , the other two successive approximations

indicated by the formula can be obtained by the method of squaring just
described* without such elaborate work as that of Rodet, which, when
applied to \/3 only gives the same results as the simpler method.

Lastly, with reference to the third class of solutions, it may be
mentioned

(1) that Oppermann used the formula
a + b
2

2 / 3

which gave successively y > v 3 > '= ,

but only led to one of the Archimedean approximations, and that by
combining the last two ratios, thus

97 + 1 68 _ 265

56 + 97 ~153 J

(2) that Schonborn came somewhat near to the formula successfully used
by Huurath and Hultsch when he proved t that

* Cantor had already pointed this out in his first edition of 1880.
t Xeitochrift fiir Math. u. Physik (Hist. litt. Altheilung) xxvin. (1883),
p. 169 sq.

CHAPTER V.

ON THE PROBLEMS KNOWN AS NET2EIS.

THE word vcvcns, commonly inclinatio in Latin, is difficult to
translate satisfactorily, but its meaning will be gathered from some
general remarks by Pappus having reference to the two Books of
Apollonius entitled veixrets (now lost). Pappus says*, "A line is
said to verge (veveiv) towards a point if, being produced, it reach the
point," and he gives, among particular eases of the general form of
the problem, the following.

"Two lines being given in position, to place between them a
straight line given in length and verging towards a given point."

"If there be given in position (1) a semicircle and a straight
line at right angles to the base, or (2) two semicircles with their
bases in a straight line, to place between the two lines a straight
line given in length and verging towards a corner (ywviav) of a
semicircle."

Thus a straight line has to be laid across two lin.es or curves so
that it passes through a given point and the intercept on it between
the lines or curves is equal to a given length t.

1. The following allusions to particular vcv'tms are found in
Archimedes. The proofs of Props. 5, 6, 7 of the book On Sj)irals
use respectively three particular cases of the general theorem that,

* Pappus (ed. Hultsch) vn. p. 670.

t In the German translation of Zeuthen's work, Die Lehre von den
KegeUchnitten im Altertum, i>eD<m is translated by " Einschiebung," or as we
might say " insertion," but this fails to express the condition that the required
line must pass through a given point, just as inclinatio (and for that matter the
Greek term itself) fails to express the other requirement that the intercept on
the line must be of given length.

ON THE PROBLEMS KNOWN AS NET2EI2.

if A be any point on a circle and BC any diameter, it is possible to
draw through A a straight line, meeting the circle again in P and
EG produced in R, such that the intercept PR is equal to any given

length. In each particular case the fact is merely stated as true
without any explanation or proof, and

(1) Prop. 5 assumes the case where the tangent at A is parallel
to EC,

(2) Prop. 6 the case where the points A, P in the figure are
interchanged,

(3) Prop. 7 the case where A, P are in the relative positions
shown in the figure.

Again, (4) Props. 8 and 9 each assume (as before, without proof,
and without giving any solution of the
implied problem) that, if AE, BC be two
chords of a circle intersecting at right
angles in a point D such tJtat ED > DC,
tfien it is possible to draw through A
another line ARP, meeting BC in R and
the circle again in P, such tlmt PR DE.

Lastly, with the assumptions in Props.
5, 6, 7 should be compared Prop. 8 of the
Liber Assumptorum, which may well be
due to Archimedes, whatever may be said of the composition of the
whole book. This proposition proves that, if in the first figure
APR is so drawn that PR is equal to the radius OP, tJien the arc
AB is three times the arc PC. In other words, if an arc AB of a
circle be taken subtending any angle at the centre 0, an arc equal
to one- third of the given arc can be found, i.e. t/ie given angle can be
trisected, if only APR can be drawn through A in such a manner

Cll INTRODUCTION.

that the intercept PR between the circle and BO produced is equal to
the radius of the circle. Thus the trisection of an angle is reduced to
a vcvcrts exactly similar to those assumed as possible in Props. 6, 7
of the book On Spirals.

The vcvWs so referred to by Archimedes are not, in general,
capable of solution by means of the straight line and circle alone,
as may be easily shown. Suppose in the first figure that x
represents the unknown length OR, where is the middle point
of JSC, and that k is the given length to which PR is to be equal ;
also let OD = a, AD = 6, BC = 2c. Then, whether EC be a diameter
or (more generally) any chord of the circle, we have

and therefore k Jb" + (x - a)' 2 = x 2 - r 2 .

The resulting equation, after rationalisation, is an equation of the
fourth degree in #; or, if we denote the length of AR by y, we have,
for the determination of x and ?/, the two equations

In other words, if we have a rectangular system of coordinate
axes, the values of x and y satisfying the conditions of the problem
can be determined as the coordinates of the points of intersection of
a certain rectangular hyperbola and a certain parabola.

In one particular case, that namely in which D coincides with
the middle point of JtC, or in which A is one extremity of the
diameter bisecting BC at right angles, a = 0, and the equations
reduce to the single equation

y*-ky = b* + c*i

which is a quadratic and can be geometrically solved by the
traditional method of application of areas ; for, if u be substituted
for y k, so that u = AP, the equation becomes

u (k + u) = b* + c 8 ,

and we have simply "to apply to a straight line of length k a
rectangle exceeding by a square figure and equal to a given
area (& 2 4-c 2 )."

The other vcOo-is referred to in Props. 8 and 9 can be solved in
the more general form where k, the given length to which PR
is to be equal, has any value within a certain maximum and is not

ON THE PROBLEMS KNOWN AS NET2EI2. ciii

necessarily equal to DE, in exactly the same manner ; and the two
equations corresponding to (a) will be for the second figure

Here, again, the problem can be solved by the ordinary method
of application of areas in the particular case where AE is the
diameter bisecting EC at right angles ; and it is interesting to note
that this particular case appears to be assumed in a fragment
of Hippocrates' Quadrature of lune* preserved in a quotation
by Simplicius* from Eudemus' History of Geometry, while Hippo-
crates flourished probably as early as 450 B.C.

Accordingly we find that Pappus distinguishes different classes
of vvcrts corresponding to his classification of geometrical problems
in general. According to him, the Greeks distinguished three kinds
of problems, some being plane, others solid, and others linear. He
proceeds thusf : " Those which can be solved by means of a straight
line and a circumference of a circle may properly be called plane
(cTrnrcSa) ; 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 (eupcni>) one or more of the
sections of the cone have been called solid (o-Tcpea) ; for the
construction requires the use of surfaces of solid figures, namely,
those of cones. There remains a third kind of problem, that
which is called linear (ypa/A/ufcdv) ; for other lines [curves] besides
those mentioned are assumed for the construction whose origin
is more complicated and less natural, as they are generated from
more irregular surfaces and intricate movements." Among other
instances of the linear class of curves Pappus mentions spirals, the
curves known as quadratrices, conchoids and cissoids. He adds
that "it seems to be a grave error which geometers fall into
whenever any one discovers the solution of a plane problem by
means of conies or linear curves, or generally solves it by means of
a foreign kind, as is the case, for example, (1) with the problem in
the fifth Book of the Conies of Apollonius relating to the parabola J,

* Simplicius, Comment, in Aristot. Phys. pp. 6168 (ed. Diels). The whole
quotation is reproduced by Bretschueider, Die Geometric und die Geometer vor
Euklides, pp. 109121. As regards the assumed construction see particularly
p. 64 and p. xxiv of Diels 1 edition; cf. Bretschneider, pp. 114, 115, and Zeuthen,
DiV? Lehrc von den Keijelschnitten im Altertum, pp. 269, 270.

t Pappus iv. pp. 270272.

Cf. Apollonius of Perya, pp. cxxviii. cxxix.

CIV INTRODUCTION.

and (2) when Archimedes assumes in his work on the spiral a
vevo-i? of a solid character with reference to a circle; for it is
possible without calling in the aid of anything solid to lind the
[proof of the] theorem given by the latter [Archimedes], that is, to
prove that the circumference of the circle arrived at in the tirst
revolution is equal to the straight line drawn at right angles to the
initial line to meet the tangent to the spiral."

The " solid veCo-is " referred to in this passage is that assumed to
be possible in Props. 8 and 9 of the book On Spirals, and is mentioned
again by Pappus in another place where he shows how to solve the
problem by means of conies*. This solution will be given later, but,
when Pappus objects to the procedure of Archimedes as unorthodox,
the objection appears strained if we consider what precisely it is that
Archimedes assumes. It is not the actual solution which is assumed,
but only its possibility j and its possibility can be perceived without
any use of conies. For in the particular case it is only necessary,
as a condition of possibility, that DE in the second figure above
should not be the maximum length which the intercept PR could
have as APR revolves about A from the position ADK in the
direction of the centre of the circle ; and that DE is not the
maximum length which PR can have is almost self-evident. In
fact, if P, instead of moving along the circle, moved along the
straight line through E parallel to /?(7, and if ARP moved from the
position ADE in the direction of the centre, the length of PR would
continually increase, and a fortiori, so long as P is on the arc of the
circle cut off by the parallel through E to BC, PR must be greater
in length than DE ; and on the other hand, as ARP moves further
in the direction of <#, it must sometime intercept a length PR
equal to DE before P reaches /?, when PR vanishes. Since, then,
Archimedes* method merely depends upon the theoretical possibility
of a solution of the vevo-is, and this possibility could be inferred
from quite elementary considerations, he had no occasion to use
conic sections for the purpose immediately in view, and he cannot
fairly be said to have solved a plane problem by the use of conies.

At the same time we may safely assume that Archimedes
was in possession of a solution of the ycvcrts referred to. But there
is no evidence to show how he solved it, whether by means of conies,
or otherwise. That he would have been able to effect the solution,

* Pappus iv. p. 298 sq.

ON THE PROBLEMS KNOWN AS XET2EIS. CV

as Pappus does, by the use of conies cannot be doubted. A precedent
for the introduction of conies where a " solid problem " had to be
solved was at hand in the determination of two mean proportionals
between two unequal straight lines by Menaechmus, the inventor of
the conic sections, who used for the purpose the intersections of a
parabola and a rectangular hyperbola. The solution of the cubic
equation on which the proposition On tlie Sphere and Cylinder u. 4:
depends is also effected by means of the intersections of a parabola
with a rectangular hyperbola in the fragment given by Eutocius
and by him assumed to be the work of Archimedes himself*.

Whenever a problem did not admit of solution by means of the
straight line and circle, its solution, where possible, by means of
conies was of the greatest theoretical importance. First, the
possibility of such a solution enabled the problem to be classified
as a "solid problem"; hence the importance attached by Pappus
to solution by means of conies. But, secondly, the method had
other great advantages, particularly in view of the requirement that
the solution of a problem should be accompanied by a Stoptcr/xo?
giving the criterion for the possibility of a real solution. Often too
the StopioyAos involved (as frequently in Apollonius) the determination
of the number of solutions as well as the limits for their possibility.
Thus, in any case where the solution of a problem depended on the
intersections of two conies, the theory of conies afforded an effective
means of investigating

2. But though the solution of " solid problems " by means of
conies had such advantages, it was not the only method open to
Archimedes. An alternative would be the use of some mechanical
construction such as was often used by the Greek geometers and is
recognised by Pappus himself as a legitimate substitute for conies,
which are not easy to draw in a planet. Thus in Apollonius'
solution of the problem of the two mean proportionals as given by
Eutocius a ruler is supposed to be moved about a point until the
points at which the ruler crosses two given straight lines at right
angles are equidistant from a certain other fixed point; and the
same construction is also given under Heron's name. Another
version of Apollonius' solution is that given by loannes Philoponus,
which assumes that, given a circle with diameter OC and two

* See note to On the Sphere and Cylinder, n. 4.
t Pappus in. p. 54.

CV1 INTRODUCTION.

straight lines OD, OE through and at right angles to one
another, a line can be drawn through (7, meeting the circle again
in F and the two lines in D, E respectively, such that the in-
tercepts CD, FE are equal. This solution was no doubt discovered
by means of the intersection of the circle with a rectangular hyper-
bola drawn with OD, OE as asymptotes and passing through C ;
and this supposition accords with Pappus* statement that Apollonius
solved the problem by means of the sections of the cone*. The
equivalent mechanical construction is given by Eutocius as that
of Philo Byzantinus, who turns a ruler about C until CD, FE are
equal f.

Now clearly a similar method could be used for the purpose of
effecting a vewris. We have only to suppose a ruler (or any object
with a straight edge) with two marks made on it at a distance
equal to the given length which the problem requires to be
intercepted between two curves by a line passing through the
fixed point ; then, if the ruler be so moved that it always passes
through the fixed point, while one of the marked points on it follows
the course of one of the curves, it is only necessary to move the
ruler until the second marked point falls on the other curve. Some
such operation as this may have led Nicomedes to the discovery of
his curve, the conchoid, which he introduced (according to Pappus)
into his doubling of the cube, and by which he also trisected an
angle (according to the same authority). From the fact that
Nicomedes is said to have spoken disrespectfully of Eratosthenes'
mechanical solution of the duplication problem, and therefore must
have lived later than Eratosthenes, it is concluded that his date
must have been subsequent to 200 B.C., while on the other hand
he must have written earlier than 70 B.C., since Geminus knew the
name of the curve about that date ; Tannery places him between
Archimedes and Apollonius J. While therefore there appears to
be no evidence of the use, before the time of Nicomedes, of such
a mechanical method of solving a i/v<rts, the interval between
Archimedes and the discovery of the conchoid can hardly have
been very long. As a matter of fact, the conchoid of Nicomedes
can be used to solve not only all the vcwrcis mentioned in Archimedes
but any case of such a problem where one of the curves is a straight

* Pappus in. p. 56.

t For fuller details see Apollonius of Perga, pp. cxxvcxxvii.

Bulletin des Sciences Mathtmatiques, 2 s6rie vn. p. 284.

ON THE PROBLEMS KNOWN AS NET2EIS.

CV11

line. Both Pappus and Eutocius attribute to Nicomedes the inven-
tion of a machine for drawing his conchoid. AB is supposed to be

a ruler with a slot in it parallel to its length, FE a second ruler at
right angles to the first with a fixed peg in it, C. This peg moves
in a slot made in a third ruler parallel to its length, while this
ruler has a fixed peg on it, />, in a straight line with the slot in
which C moves ; and the peg D can move along the slot in AB. If
then the ruler PD moves so that the peg D describes the length of
the slot in AB on each side of F, the extremity of the ruler, P,
describes the curve which is called a conchoid. Nicomedes called
the straight line AB the ruler (KOLVU>V\ the fixed point C the pole
(TTO'AOS), and the length PD the distance (StaorTy/xa) ; and the
fundamental property of the curve, which in polar coordinates
would now be denoted by the equation r-a + b sec 0, is that, if
any radius vector be drawn from C to the curve, as CP, the length
intercepted on the radius vector between the curve and the straight
line AB is constant. Thus any vvo-is in which one of the two
given lines is a straight line can be solved by means of the
intersection of the other line with a certain conchoid whose pole
is the fixed point to which the required straight line must verge
(vvtv). In practice Pappus tells us that the conchoid was not
always actually drawn, but that "some," for greater convenience,
moved the ruler about the fixed point until by trial the intercept
was made equal to the given length*.

3. The following is the way in which Pappus applies
conic sections to the solution of the vv<rts referred to in Props. 8, 9
of the book On Spirals. He begins with two lemmas.

* Pappus iv. p. 246.

CV111

INTRODUCTION.

(1) If from a given point A any straight line be drawn meeting
a straight line EC given in position in R, and if RQ be drawn
perpendicular to J3C and bearing a given ratio to A ft, the locus of
Q is a hyperbola.

For draw AD perpendicular to 71C, and on AD produced take A'

such that

QR : RA = A'D >. DA = (the given ratio).

Measure DA" along DA equal to DA'.
Then, if QNbe perpendicular to AN,

(AR* - AD 2 ) : (QR 3 - A'D*) = (const.),
or QN 2 : A'N . A"N - (const.)

(2) If EC be given in length, and if RQ, a straight line drawn
at right angles to EC from any point R on it, be such that

]}R . RC = k . RQ,

where k is a straight line of given length, then the locus of Q is a
parabola.

Let be the middle point of EG, and let OK be drawn at right
angles to it and of such length that

OC* -- k . KO.

Draw QN' perpendicular to OK.
Then QN' 2 = OR 2 - OC* - BR . RC

= k . (KO - RQ), by hypothesis,

- k . KN'.

ON THE PROBLEMS KNOWN AS NET2EI2. cix

In the particular case referred to by Archimedes (with the slight
generalisation that the given length k to which PR is to be equal is
not necessarily equal to DE) we have

(1) the given ratio RQ : AR is unity, or RQ = AR, whence A"
coincides with J, and, by the first lemma,

QN* = AN . A'N,
so that Q lies on a rectangular hyperbola.

(2) BR . RC = AR . RP = k . AR = k . RQ, and, by the second
lemma, Q lies on a certain parabola.

If now we take as origin, OC as axis of x and OK as axis of y,
and if we put OD a, AD = /;, JIC 2c, the hyperbola and parabola
determining the position of Q are respectively denoted by the
equations

(a-*)' = y*-6 a ,

c 2 -x 2 = A 1 //,

which correspond exactly to the equations (/?) above obtained by
purely algebraical methods.

Pappus says nothing of the Sio/ncr/xo's which is necessary to the
complete solution of the generalised problem, the Sioptoyxo's namely
which determines the nutxittium value of k for which the solution is
possible. This maximum value would of course correspond to the
case in which the rectangular hyperbola and the parabola touch one
another. Zeuthen has shown* that the corresponding value of k can
be determined by means of the intersection of two other hyperbolas or
of a hyperbola and a parabola, and there is no doubt that Apollonius,
with his knowledge of conies, and in accordance with his avowed
object in giving the properties useful and necessary for 8toptcr/xot,
would have been able to work out this particular Siopicr/Ao's by means
of conies ; but there is no evidence to show that Archimedes investi-
gated it by the aid of conies, or indeed at all, it being clear, as shown
above, that it was not necessary for his immediate purpose.

This chapter may fitly conclude with a description of (1) some
important applications of vevcrets given by Pappus, and (2) certain
particular cases of the same class of problems which are plane, that
is, can be solved by the aid of the straight line and circle only, and
which were (according to Pappus) shown by the Greek geometers to
be of that character.

* Zeuthen, Die Lehre von den Kegelachnitten im Altertum, pp. 273 5.

INTRODUCTION.

4. One of the two important applications of 'solid' vcvo-ets was
discovered by Nicomedes, the inventor of the conchoid, who intro-
duced that curve for solving a vcvo-is to which he reduced the problem
of doubling the cube* or (what amounts to the same thing) the finding
of two mean proportionals between two given unequal straight lines.

Let the given unequal straight lines be placed at right angles as
CL, LA. Complete the parallelogram ABCL, and bisect AB at D,
and EC at E. Join LD and produce it to meet CB produced in //.
From E draw EF at right angles to BC, and take a point F on EF
such that CF is equal to AD. Join 11 F, and through C draw CG
parallel to IIP. If we produce BC to K, the straight lines CG, CK

form an angle, and we now draw from the given point F a straight
line FGK, meeting CG, CK in G, K respectively, such that the
intercept GK is equal to AD or FC. (This is the i/cuo-ts to which
the problem is reduced, and it can be solved by means of a conchoid
with ^as pole.)

Join KL and produce it to meet BA produced in M.

Then shall CK, AM be the required mean proportionals between
CL, LA, or

CL : CK^CK : AM=AM : AL.

We have, by Eucl. n. 6,

BK . KC +
If we add EF 2 to each side,

BK . KC
Now, by parallels,

MA :AB=ML:LK

* Pappus iv. p. 242 sq. and in. p. 58 sq. ; Eutocius on Archimedes, On the
Sphere and Cylinder, u. 1 (Vol. in. p. 114 sq.)

ON THE PROBLEMS KNOWN AS NET2EI2. Cxi

and, since A B = 2AD, and BC = \HC,

MA :AD = IIC:CK

= FG : GK, by parallels,
whence, componendo,

MD:AD = FK;GK.

But GK= AD ; therefore MD = FK, and MD* = FK\

Again, MD * = ^J/ . MA + ^Z) 2 ,

and ^A" = 5 A r . A"(7 + CF\ from above,

while MD * = FK\ and A D* -

therefore BM . MA = BK . KG.

Hence CK : MA = 2?Jf : BK

:AL

that is, LC : CK = CK : MA = MA : AL.

5. The second important problem which can be reduced to
a ' solid ' i/cvo-is is the trisection of any angle. One method of
reducing it to a i/evcris has been mentioned above as following from
Prop. 8 of the Liber Assumptorum. This method is not mentioned
by Pappus, who describes (iv. p. 272 sq.) another way of effecting
the reduction, introducing it with the words, "The earlier
geometers, when they sought to solve the aforesaid problem about
the [trisection of the] angle, a problem by nature 'solid/ by
'plane' methods, were unable to discover the solution; for they
were not yet accustomed to the use of the sections of the cone,
and were for that reason at a loss. Later, however, they trisected
an angle by means of conies, having used for the discovery of it
the following vevo-ts."

The vi><ris is thus enunciated : Given a rectangle A BCD, let it
be required to draw through A a straight line AQR, meeting CD in
Q and BC produced in /?, such that the intercept QR is equal to a
given length, k suppose.

Suppose the problem solved, QR being equal to k. Draw DP
parallel to QR and RP parallel to CD, meeting in P. Then, in the
parallelogram DR, DP = QR = k.

Hence P lies on a circle with centre D and radius k.

Again, by Eucl. I. 43 relating to the complements of the
parallelograms about the diagonal of the complete parallelogram,
BC . CD=BR. QD

cxii

INTRODUCTION.

and, since BC . CD is given, it follows that P lies on a rectangular
hyperbola with BR, BA as asymptotes and passing through D.

Therefore, to effect the construction, we have only to draw this
rectangular hyperbola and the circle with centre D and radius equal
to k. The intersection of the two curves gives the point P, and R
is determined by drawing PR parallel to DC. Thus AQR is found.

[Though Pappus makes A BCD a rectangle, the construction
applies equally if ABCD is any parallelogram.]

Now suppose ABC to be any acute angle which it is required to
trisect. Let AC be perpendicular to BC. Complete the parallelo-
gram ADBC, and produce DA.

Suppose the problem solved, and let the angle CBE be one-third
of the angle ABC. Let BE meet AC in E arid DA produced in F.
Bisect EF in If, and join AH.

Then, since the angle ABE is equal to twice the angle EBC and,
by parallels, the angles EBC, EFA are equal,

_ ABE= 2 L AFff = L AHB.
Therefore Al

and

Hence, in order to trisect the angle ABC, we have only to solve
the following vewis: Given the rectangle ADBC whose diagonal

ON THE PROBLEMS KNOWN AS NETZEI2. cxiii

is AJB, to draiv through B a straight line BEF, meeting AC in E and
DA produced in F, such that EF may be equal to twice AB ; and this
i/cvcris is solved in the manner just shown.

These methods of doubling the cube and trisecting any acute
angle are seen to depend upon the application of one and the same
i/v(ris, which may be stated in its most general form thus. Given
any tivo straight lines forming an angle and any fixed point
which is not on either line, it is required to draw through the
fixed point a straight line such that the portion of it intercepted
"between the fixed lines is equal to a given length. If AE, AC be

the fixed lines and B the fixed point, let the parallelogram ACBD
be completed, and suppose that BQR, meeting CA in Q and AE in
A', satisfies the conditions of the problem, so that QR is equal to
the given length. If then the parallelogram CQRP is completed,
we may regard P as an auxiliary point to be determined in order
that the problem may be solved ; and we have seen that P can be
found as one of the points of intersection of (1) a circle with centre
C and radius equal to k, the given length, and (2) the hyperbola
which passes through C and has DE, DB for its asymptotes.

It remains only to consider some particular cases of the problem
which do not require conies for their solution, but are ' plane 7
problems requiring only the use of the straight line and circle.

6. We know from Pappus that Apollonius occupied him-
self, in his two Books of i/cvcm?, with problems of that type
which were capable of solution by 'plane 9 methods. As a matter
of fact, the above vewis reduces to a 'plane' problem in the
particular case where B lies on one of the bisectors of the angle
between the two given straight lines, or (in other words) where the
parallelogram ACBD is a rhombus or a square. Accordingly we
find Pappus enunciating, as one of the 'plane' cases which had

cxiv INTRODUCTION.

been singled out for proof on account of their greater utility for
many purposes, the following*: Given a rhombus with one side
produced, to fit into the exterior angle a straight line given in
length and verging to the opposite angle ; and he gives later on, in
his lemmas to Apollonius' work, a theorem bearing on the problem
with regard to the rhombus, and (after a preliminary lemma)
a solution of the vewris with reference to a square.

The question therefore arises, how did the Greek geometers
discover these and other particular cases, where a problem which
is in general 'solid,' and therefore requires the use of conies (or a
mechanical equivalent), becomes t plane' ? Zeuthen is of opinion that
they were probably discovered as the result of a study of the general
solution by means of conies f. I do not feel convinced of this, for
the following reasons.

(1) The authenticated instances appear to be very rare in
which we should be justified in assuming that the Greeks used
the properties of conies, in the same way as we should combine
and transform two Cartesian equations of the second degree, for
the purpose of proving that the intersections of two conies also
lie on certain circles or straight lines. Tt is true that we may
reasonably infer that Apollonius discovered by a method of this sort
his solution of the problem of doubling the cube where, in place
of the parabola and rectangular hyperbola used by Menaechmus,
he employs the same hyperbola along with the circle which passes
through the points common to the hyperbola and parabola J ; but
in the only propositions contained in his conies which offer an
opportunity for making a similar reduction , Apollonius does not
make it, and is blamed by Pappus for not doing so. In the pro-
positions referred to the feet of the normals to a parabola drawn
from a given point are determined as the intersections of the
parabola with a certain rectangular hyperbola, and Pappus objects

* Pappus vii. p. 670.

t "Mit dieser selben Aufgabe ist namlich ein wichtiges Beispiel dafur
verknupft, dass man bemiiht war solche Falle zu entdecken, in denen Aufgaben,
zu deren Losung im allgemeinen Kegelschnitte erforderlich sind, sich mittels
Zirkel und Lineal losen las sen. Da nun das Studium der allgemeinen Losung
durch Kegelschnitte das beste Mittel gewahrt solche Falle zu entdecken, so ist
es ziemlich wahrscheinlich, dass man wirklich diesen Weg eingeschlagen hat.' 1
Zeuthen, op. cit. p. 280.

Apollonius of Perga, p. cxxv, cxxvi.

Ibid. p. cxxviii and pp. 182, 186 (Conies, v. 58, 62

ON THE PROBLEMS KNOWN AS NETSEIS. CXV

to this method as an instance of discovering the solution of a
'plane' problem by means of conies*, the objection having reference
to the use of a hyperbola where the same points could be obtained
as the intersections of the parabola with a certain circle. Now the
proof of this latter fact would present no difficulty to Apollonius,
and Pappus must have been aware that it would not ; if therefore
he objects in the circumstances to the use of the hyperbola, it is at
least arguable that he would equally have objected had Apollonius
brought in the hyperbola and used its properties for the purpose
of proving the problem to be c plane ' in the particular case.

(2) The solution of the general problem by means of conies
brings in the auxiliary point P and the straight line CP. We
should therefore naturally expect to find some trace of these in the
particular solutions of the vcwris for a rhombus and square; but
they do not appear in the corresponding demonstrations and figures
given by Pappus.

Zeuthen considers that the vevo-is with reference to a square was
probably shown to be ' plane ' by means of the same investigation
which showed that the more general case of the rhombus was also
capable of solution with the help of the straight line and circle
only, i.e. by a systematic study of the general solution by means of
conies. This supposition seems to him more probable than the view
that the discovery of the plane construction for the square may have
been accidental ; for (he says) if the same problem is treated solely
by the aid of elementary geometrical expedients, the discovery that
it is ' plane ' is by no means a simple matter f. Here, again, I am
not convinced by Zeuthen's argument, as it seems to me that a
simpler explanation is possible of the way in which the Greeks were
led to the discovery that the particular vcvarei? were plane. They
knew in the first place that the trisection of a right angle was a
'plane' problem, and therefore that half a right angle could be
trisected by means of the straight line and circle. It followed

* Pappus iv. p. 270. Cf. p. ciii above.

t "Die Ausfuhrbarkeit kann dann auf die zuerst angedeutete Weise gefunden
sein, die den allgemeinen Fall, wo der Winkel zwischen den gegebenen Geraden
beliebig 1st, in sich begreift. Dies scheint mir viel wahrscheinlicher als die
Annahme, dass die Entdeckung dieser ebenen Konstruction zufallig sein sollte ;
denn wenn man dieselbe Aufgabe nur mittels rein elementar-geometrischer
Htilfsmittel behandelt, so liegt die Entdeckung, dass sie eben 1st, ziemlich fern."
Zeuthen, op. eft. p. 282.

CXV1 INTRODUCTION.

therefore that the corresponding vcvcns, i.e. that for a square, was
a * plane' problem in the particular case where the given length
to which the required intercept was to be equal was double of
the diagonal of the square. This fact would naturally suggest
the question whether the problem was still plane if k had
any other value; and, when once this question was thoroughly
investigated, the proof that the problem was 'plane, 1 and the
solution of it, could hardly have evaded for long the pursuit of
geometers so ingenious as the Greeks. This will, T think, be
clear when the solution given by Pappus and reproduced below
is examined. Again, after it had been proved that the vevcns with
reference to a square was 'plane,' what more natural than the further
inquiry as to whether the intermediate case between that of the
square and parallelogram, that of the rhombus, might perhaps be a
s plane ' problem ?

As regards the actual solution of the plane i>v<reis with respect
to the rhombus and square, i.e. the cases in general where the fixed
point B lies on one of the bisectors of the angles between the two
given straight lines, Zeuthen says that only in one of the cases have
we a positive statement that the Greeks solved the i/cveris by means
of the circle and ruler, the case, namely, where ACtiD is a square*.
This appears to be a misapprehension, for not only does Pappus
mention the case of the rhombus as one of the plane vcvvtts which
the Greeks had solved, but it is clear, from a proposition given by
him later, how it was actually solved. The proposition is stated
by Pappus to be " involved " (Trapatfewpov/Acvov, meaning presumably
"the subject of concurrent investigation") in the 8th problem of
Apollonius' first Book of vevoreis, and is enunciated in the following
form f. Given a rhombus AD with diameter BC produced to E, if EF
be a mean pi'Oportional between BE^ JEC, and if a circle be described
with centre E and radius EF cuttiny CD in K and AC produced in
H, BKH shall be a straight line. The proof is as follows.

Let the circle cut AC in L, and join HE, KE, LE. Let LK
meet BC in M.

* "Indessen besitzen wir doch nur in einem einzelnen hierher gehorigen
Falle eine positive Angabe dariiber, dass die Griechen die Einschiebung mittels
Zirkel und Lineal ausgefiihrt haben, wenn namlich die gegebenen Geraden
zugleich rechte Winkel bilden, AIBC also ein Quadrat wird." Zeuthen, 02?. cit.
p. 281.

t Pappus vn. p. 778.

ON THE PROBLEMS KNOWN AS NET2EI2.

CXV11

Since, from the property of the rhombus, the angles LCM, KCM
are equal, and therefore CL, CK make equal angles with the diameter
FG of the circle, it follows that CL = CK.

Also EK EL) and CE is common to the triangles ECK, ECL.
Therefore the said triangles are equal in all respects, and

Now, by hypothesis,

or EB : EK = EK : EC (since EF = EK),

and the angle CEK is common to the triangles BEK, KEC ; there-
fore the triangles BEK, KEC are similar, and

= ^ CHE, from above.
Again, _ HCE =_ACB=- BCK.

Thus in the triangles CBK, CHE two angles are equal re-
spectively ;
therefore ^ CEH = L CKB.

But, since ^ CKE = i C//^, from above, the points K, C\ E, II
are concyclic.

Hence _ CEH + _ CA'/^ .= (two right angles).

Accordingly, since .1 CEH ' _ CA"^,

L CKH + ~ CKB = (two right angles),
and BKH is a straight line.

CXV111 INTRODUCTION.

Now the form of the proposition at once suggests that, in the
8th problem referred to, Apollonius had simply given a construction
involving the drawing of a circle cutting CD and AC produced in
the points K, H respectively, and Pappus' proof that BKH is a
straight line is intended to prove that HK verges towards fi, or (in
other words) to verify that the construction given by Apollonius
solves a certain vcvo-ts requiring BKH to be drawn so that KH is
equal to a given length.

The analysis leading to the construction must have been worked
out somewhat as follows.

Suppose BKH drawn so that KH is equal to the given length k.
Bisect KH at N, and draw NE at right angles to KH meeting KC
produced in E.

Draw KM perpendicular to BC and produce it to meet CA in L.
Then, from the property of the rhombus, the triangles KCM, LCM
are equal in all respects.

Therefore KM=ML- 9 and accordingly, if MN be joined, MN,
LH are parallel.

Now, since the angles at J/, A" are right, a circle can be described
about EMKN.

Therefore L. CEK = L. MNK, in the same segment,

- L CHK, by parallels.

Hence a circle can be described about CEHK. It follows that
L BCD = L CEK + L CKE

Therefore the triangles EKH, DEC are similar.
Lastly, L CKN = L CBK + L. BCK ;

and, subtracting from these equals the equal angles EKN, BCK
respectively, we have

L EKC = L. EBK.

Hence the triangles EBK, EKC are similar, and

or BE.EC^EK*.

But, by similar triangles, EK : KH = DC : CB,

and the ratio DC : CB is given, while KH is also given (~ k).

ON THE PROBLEMS KNOWN AS NET2EI2. cxix

Therefore EK is given, and, in order to find E, we have only, in
the Greek phrase, to "apply to BG a rectangle exceeding by a square
figure and equal to the given area EK*"

Thus the construction given by Apollonius was clearly the
following*.

Ifk be the given length, take a straight line p such that

p:k = AB:C.

Apply to BO a rectangle exceeding by a square figure and equal to
the area p 2 . Let BE . EC be this rectangle, and with E as centre and
radius equal to p describe a circle cutting AC produced in fl and
CD in K.

11 K is then equal to k, and verges towards B, as proved by
Pappus; the problem is therefore solved.

The construction used by Apollonius for the ' plane ' vevo-is with
reference to the rhombus having been thus restored by means of the
theorem given by Pappus, we are enabled to understand the purpose

* This construction was suggested to me by a careful examination of
Pappus' proposition without other aid ; but it is no new discovery.
Samuel Horsley gives the same construction in his restoration of Apollonii
Pergaei Inclinatiomim libri duo (Oxford, 1770); he explains, however, that
he went astray in consequence of a mistake in the figure given in the MSS.,
and was unable to deduce the construction from Pappus's proposition until he
was recalled to the right track by a solution of the same problem by Hugo
d'Omerique. This solution appears in a work entitled, Analysis yeoinetrica, sive
nova et vcra metlwdus resolvendi tarn problemata geometrica quam arithmeticas
quaext tones, published at Cadiz in 1698. D'Omerique's construction, which is
practically identical with that of Apollonius, appears to have been evolved by
means of an independent analysis of his own, since he makes no reference to
Pappus, as he does in other cases where Pappus is drawn upon (e.g. when giving
the construction for the case of the square attributed by Pappus to one
Heraclitus). The construction differs from that given above only in the fact
that the circle is merely used to determine the point K, after which BK is joined
and produced to meet AC in If. Of other solutions of the same problem two
may here be mentioned. (1) The solution contained in Marino Ghetaldi's
posthumous work DC Resolutione et Compositione Mathematica Libri quinque
(Borne, 1630), and included among the solutions of other problems all purporting
to be solved " methodo qua antiqui utebantur," is, though geometrical, entirely
different from that above given, being effected by means of a reduction of the
problem to a simpler plane peG<rt* of the same character as that assumed by
Hippocrates in his Quadrature of lunes. (2) Christian Huygens (De circuit
mag ni tad i tie inventa; accedunt problettuitum quorundam illustrium coiixtructiones,
Lugduni Batavorum, 1654) gave a rather complicated solution, which may be
described as a generalisation of Heraclitus' solution in the case of a square.

CXX INTRODUCTION.

for which Pappus, while still on the subject of the " 8th problem "
of Apollonius, adds a solution for the particular case of the square
(which he calls a "problem after Heraclitus") with an introductory
lemma. It seems clear that Apollonius did not treat the case of the
square separately from the rhombus because the solution for the
rhombus was equally applicable to the square, and this supposition
is confirmed by the fact that, in setting out the main problems
discussed in the rcwms, Pappus only mentions the rhombus and not
the square. Being however acquainted with a solution by one
Heraclitus of the vewrts relating to a square which was not on the
same lines as that of Apollonius, while it was not applicable to the
case of the rhombus, Pappus adds it as an alternative method for
the square which is worth noting*. This is no doubt the explanation
of the heading to the lemma prefixed to Heraclitus' problem which
Hultsch found so much difficulty in explaining and put in brackets
as an interpolation by a writer who misunderstood the figure
and the object of the theorem. The words mean "Lemma useful
for the [problem] with reference to squares taking the place
of the rhombus" (literally "having the same property as the
rhombus"), i.e. a lemma useful for Heraclitus' solution of the

* This view of the matter receives strong support from the following
facts. In Pappus* summary (p. 670) of the contents of the i>ei/<reis of Apollonius
"two cases" of the feucm with reference to the rhombus are mentioned last
among the particular problems given in the first of the two Books. As we have
seen, one case (that given above) was the subject of the "8th problem" of
Apollonius, and it is equally clear that the other case was dealt with in the
" 9th problem." The other case is clearly that in which
the line to be drawn through B, instead of crossing the H A c

exterior angle of the rhombus at C, lies across the angle
C itself, i.e. meets CA, CD both produced. In the former
case the solution of the problem is always possible what-
ever be the length of k; but in the second case clearly
the problem is not capable of solution if k, the given
length, is less than a certain minimum. Hence the
problem requires a diopurfjiAs to determine the minimum
length of k. Accordingly we find Pappus giving, after

the interposition of the case of the square, a " lemma useful for the Sio/woy-tos of
the 9th problem," which proves that, if CH = CK and 7f be the middle point of
HK, then HK is the least straight line which can be drawn through B to meet
CH, CK. Pappus adds that the hopurfMs for the rhombus IB then evident ; if
HK be the line drawn through B perpendicular to CB and meeting CA, CD
produced in H, K, then, in order that the problem may admit of solution, the
given length k must be not less than HK.

ON THE PROBLEMS KNOWN AS NET2EI2. CXXl

vvo-is in the particular case of a square*. The lemma is as
follows.

A BCD being a square, suppose BHE drawn so as to meet CD in
H and AD produced in E, and let EF be drawn perpendicular to BE
meeting BC produced in F. To prove that
CF 2 = BC* + HE*.

Suppose EG drawn parallel to DC meeting CF in G. Then
since BEF is a right angle, the angles HBC, FEG are equal.

Therefore the triangles BCH, EGF are equal in all respects, and

EF ~BH.

Now BF* = BE* + EF*,

or BC . BF + BF. FC = BII . BE + BE . EH + EF 2 .

But, the angles HCF, HEF being right, the points (7, //, E, F
are concyclic, and therefore

BC.BF=BI/. BE.
Subtracting these equals, we have

---BE.EH+BU*

= EB . BU + EH*
-FB.BC +

* Hultsch translates the words \^/Lt/Aa xpfaww *<* r *> M Terpaywvwv TOIOIJV
rd ai)rd T$ frippv (p. 780) thus, " Lemma utile ad problema de quadratis quorum
summa rhombo aequalis eat," and has a note in his Appendix (p. 1260) explaining
what he supposes to be meant. The ' squares' he takes to be the given square
and the square on the given length of the intercept, and the rhombus to be one
for which he indicates a construction but which is not shown in Pappus* figure.
Thus he is obliged to translate r<f ^o^V as " rhombus," which is one objec-
tion to his interpretation, while "whose squares are equal" scarcely seems a
possible rendering of iroio^vruv rd atfrd.

CXX11 INTRODUCTION.

Take away the common part EC . CF, and
OF 3 = BC* + EH\

Heraclitus* analysis and construction are now as follows.

Suppose that we have drawn ERE so that HE has a given
length k.

Since CF* = BC* + EH\ or EC* + Jc a ,

and EC and k are both given,

CF is given, and therefore EF is given.

Thus the semicircle on EF as diameter is given, and therefore
also E, its intersection with the given line ADE hence EE is
given.

To effect the construction, we first find a square equal to the
sum of the given square and the square on k. We then produce
EC to F so that CF is equal to the side of the square so found. If
a semicircle be now described on BF as diameter, it will pass above
D (since OF> CD, and therefore EC . CF> CD*), and will therefore
meet AD produced in some point E.

Join EE meeting CD in //.

Then HE = k, and the problem is solved.

CHAPTER VI.

CUBIC EQUATIONS.

IT has often been explained how the Greek geometers were able
to solve geometrically all forms of the quadratic equation which give
positive roots ; while they could take no account of others because
the conception of a negative quantity was unknown to them. The
quadratic equation was regarded as a simple equation connecting
areas, and its geometrical expression was facilitated by the methods
which they possessed of transforming any rectilineal areas whatever
into parallelograms, rectangles, and ultimately squares, of equal
area ; its solution then depended on the principle of application, of
areas, the discovery of which is attributed to the Pythagoreans.
Thus any plane problem which could be reduced to the geometrical
equivalent of a quadratic equation with a positive root was at once
solved. A particular form of the equation was the pure quadratic,
which meant for the Greeks the problem of finding a square equal
to a given rectilineal area. This area could be transformed into a
rectangle, arid the general form of the equation thus became or = ab,
so that it was only necessary to find a mean proportional between a
and b. In the particular case where the area was given as the
sum of two or more squares, or as the difference of two squares,
an alternative method depended on the Pythagorean theorem of
Eucl. I. 47 (applied, if necessary, any number of times successively).
The connexion between the two methods is seen by comparing
Eucl. vi. 13, where the mean proportional between a and b is
found, and Eucl. n. 14, where the same problem is solved without
the use of proportions by means of i. 47, and where in fact the
formula used is

CXX1V INTRODUCTION.

The choice between the two methods was equally patent when the
equation to be solved was ,<r = pa*, where p is any integer ; hence
the * multiplication* of squares was seen to be dependent on the
finding of a mean proportional. The equation x 2 = 2a* was the
simplest equation of the kind, and the discovery of a geometrical
construction for the side of a square equal to twice a given square
was specially important, as it was the beginning of the theory of
incommensurables or 'irrationals' (aXoywr 7iy)ay/LiaT6ia) which was
invented by Pythagoras. There is every reason to believe that this
successful doubling of the square was what suggested the question
whether a construction could not be found for the doubling of the
cube, and the stories of the tomb erected by Minos for his son and
of the oracle bidding the Delians to double a cubical altar were no
doubt intended to invest the purely mathematical problem with an
element of romance. It may then have been the connexion between
the doubling of the square and the finding of one mean proportional
which suggested the reduction of the doubling of the cube to the
problem of finding two mean proportionals between two unequal
straight lines. This reduction, attributed to Hippocrates of Chios,
showed at the same time the possibility of multiplying the cube
by any ratio. Thus, if .r, y are two mean proportionals between

a, 6, we have

a : x = x : y - y : b,
and we derive at once

a : b - a 3 : x- 3 ,

whence a cube (x j ) is obtained which bears to a 3 the ratio b : a,

?}
while any fraction - can be transformed into a ratio between lines

of which one (the consequent) is equal to the side a of the given
cube. Thus the finding of two mean proportionals gives the solution
of any pure cubic equation, or the equivalent of extracting the cube
root, just as the single mean proportional is equivalent to extracting
the square root. For suppose the given equation to be tf = bcd.
We have then only to find a mean proportional a between c and <l,

and the equation becomes a; 3 = a 3 . b = a? . - which is exactly the

multiplication of a cube by a ratio between lines which the two
mean proportionals enable us to effect.

As a matter of fact, we do not find that the great geometers
were in the habit of reducing problems to the multiplication of the

CUBIC EQUATIONS. CXXV

cube eo nomine, but to the equivalent problem of the two mean
proportionals ; and the cubic equation x* = cPb is not usually stated
in that form but as a proportion. Thus in the two propositions On
the Sphere and Cylinder n. 1, 5, where Archimedes uses the two
mean proportionals, it is required to find x where

<? : x 2 = x : b ;

he does not speak of finding the side of a cube equal to a certain
parallelepiped, as the analogy of finding a square equal to a given
rectangle might have suggested. So far therefore we do not find
any evidence of a general system of adding and subtracting solids
by transforming parallelepipeds into cubes and cubes into parallel-
epipeds which we should have expected to see in operation if the
Greeks had systematically investigated the solution of the general
form of the cubic equation by a method analogous to that of the
application of areas employed in dealing with quadratic equations.

The question then arises, did the Greek geometers deal thus
generally with the cubic equation

which, on the supposition that it was regarded as an independent
problem in solid geometry, would be for them a simple equation
between solid figures, x and a both representing linear magnitudes,
B an area (a rectangle), and F a volume (a parallelepiped) 1 And
was the reduction of a problem of an order higher than that which
could be solved by means of a quadratic equation to the solution of
a cubic equation in the form shown above a regular and recognised
method of dealing with such a problem ? The only direct evidence
pointing to such a supposition is found in Archimedes, who reduces
the problem of dividing a sphere by a plane into two segments
whose volumes are in a given ratio (On the Sphere and Cylinder n. 4)
to the solution of a cubic equation which he states in a form
equivalent to

4a 8 :x* = (3a-x) : -*- a (1),

where a is the radius of the sphere, m : n the given ratio (being a
ratio between straight lines of which m > ?t), and x the height of the
greater of the required segments. Archimedes explains that this is
a particular case of a more general problem, to divide a straight
line (a) into two parts (#, a - x) such that one part (a - x) is to an-
other given straight line (c) as a given area (which for convenience'

CXXV1 INTRODUCTION.

sake we suppose transformed into a square, b' 2 ) is to the square on
the other part (as 2 ), i.e. so that

(a-x) :c-^b* :ar. ........................... (2).

He further explains that the equation (2) stated thus generally
requires a 810/3107x05, i.e. that the limits for the possibility of a real
solution, etc., require to be investigated, but that the particular case
(with the conditions obtaining in the particular proposition) requires
no Siopioyio?, i.e. the equation (1) will always give a real solution.
He adds that " the analysis and synthesis of both these problems
will be given at the end." That is, he promises to give separately a
complete investigation of the equation (2), which is equivalent to the
cubic equation

x* (a-x) = b*c ........................... (3)

and to apply it to the particular case (1).

Wherever the solution was given, it was temporarily lost, having
apparently disappeared even before the time of Dionysodorus and
Diocles (the latter of whom lived, according to Cantor, not later
than about 100 B.C.) ; but Eutocius describes how he found an
old fragment which appeared to contain the original solution of
Archimedes, and gives it in full. It will be seen on reference to
Eutocius' note (which I have reproduced immediately after the
proposition to which it relates, On the Sphere and Cylinder n. 4)
that the solution (the genuineness of which there seems to be no
reason to doubt) was effected by means of the intersection of a
parabola and a rectangular hyperbola whose equations may re-
spectively be written thus,

(a x) y ac.

The 810/3107x05 takes the form of investigating the maximum
possible value of x 2 (a x), and it is proved that this maximum

value for a real solution is that corresponding to the value a; = - a.

This is established by showing that, if b*c -^_ a 8 , the curves touch

at the point for which x = ^ a. If on the other hand b*c < ^ a 2 , it

is proved that there are two real solutions. In the particular case
(1) it is clear that the condition for a real solution is satisfied, for

CUBIC EQUATIONS. CXXVii

the expression in (1) corresponding to b*c in (2) is - 4a 3 , and it

is only necessary that

which is obviously true.

Hence it is clear that not only did Archimedes solve the cubic
equation (3) by means of the intersections of two conies, but he also
discussed completely the conditions under which there are 0, 1 or 2
roots lying between and a. It is to be noted further that the
Stopioyxo's is similar in character to that by which Apollonius
investigates the number of possible normals that can be drawn
to a conic from a given point*. Lastly, Archimedes' method is
seen to be an extension of that used by Menaechmus for the solution
of the pure cubic equation. This can be put in the form

a 3 : a? = a : b,
which can again be put in Archimedes' form thus,

a 2 : x 2 - x : b,
and the conies used by Menaechmus are respectively

x 2 = ay, xy = ab,

which were of course suggested by the two mean proportionals
satisfying the equations

a : x = x : y - y : 6.

The case above described is not the only one where we may
assume Archimedes to have solved a problem by first reducing it
to a cubic equation and then solving that. At the end of the
preface to the book On Conoids and Spheroids he says that the
results therein obtained may be used for discovering many theorems
and problems, and, as instances of the latter, he mentions the
following, "from a given spheroidal figure or conoid to cut off,
by a plane drawn parallel to a given plane, a segment which shall
be equal to a given cone or cylinder, or to a given sphere." Though
Archimedes does not give the solutions, the following considerations
may satisfy us as to his method.

(1) The case of the 'right-angled conoid* (the paraboloid of
revolution) is a ' plane ' problem and therefore does not concern us
here.

* Cf. Apollonius ofPerga, p. 168 sqq.

CXXViii INTRODUCTION.

(2) In the case of the spheroid, the volume of the whole
spheroid could be easily ascertained, and, by means of that, the
ratio between the required segment and the remaining segment;
after which the problem could be solved in exactly the same way
as the similar one in the case of the sphere above described,
since the results in On Conoids and Spheroids, Props. 29 32,
correspond to those of On the Sphere and Cylinder II. 2. Or
Archimedes may have proceeded in this case by a more direct
method, which we may represent thus. Let a plane be drawn
through the axis of the spheroid perpendicular to the given
plane (and therefore to the base of the required segment). This
plane will cut the elliptical base of the segment in one of its
axes, which we will call 2?/. Let x be the length of the axis
of the segment (or the length intercepted within the segment
of the diameter of the spheroid passing through the centre of the
base of the segment). Then the area of the base of the segment will
vary as y 2 (since all sections of the spheroid parallel to the given
plane must be similar), and therefore the volume of the cone which
has the same vertex and base as the required segment will vary as
y*x. And the ratio of the volume of the segment to that of the
cone is (On Conoids and Spheroids, Props. 29 32) the ratio
(3a - x) : (2a a;), where 2a is the length of the diameter of the
spheroid which passes through the vertex of the segment. There-
fore

where C is a known volume. Further, since x, y are the coordinates
of a point on the elliptical section of the spheroid made by the plane
through the axis perpendicular to the cutting plane, referred to a
diameter of that ellipse and the tangent at the extremity of the
diameter, the ratio y* : x ( 2a - x) is given. Hence the equation
can be put in the form

and this again is the same equation as that solved in the fragment
given by Eutocius. A Stopioyxo's is formally necessary in this case,
though it only requires the constants to be such that the volume
to which the segment is to be equal must be less than that of the
whole spheroid.

(3) For the ' obtuse-angled conoid ' (hyperboloid of revolution)
it would be necessary to use the direct method just described for

CUBIC EQUATIONS. CXxix

the spheroid, and, if the notation be the same, the corresponding
equations will be found, with the help of On Conoids and SpJwroida,

Props. 25, 26, to be

2 3a + x

y x . ^ = C ,
J 2a + x

and, since the ratio y 1 : x (2a + x) is constant,
x 3 (3a + x) = b*c.

If this equation is written in the form of a proportion like the
similar one above, it becomes

6* : x 2 = (3a + x):c.

There can be no doubt that Archimedes solved this equation as
well as the similar one with a negative sign, i.e. he solved the two
equations

x 3 ax 2 + b*c = 0,

obtaining all their positive real roots. In other words, he solved
completely, so far as the real roots are concerned, a cubic equation
in which the term in x is absent, although the determination of the
positive and negative roots of one and the same equation meant for
him two separate problems. And it is clear that all cubic equations
can be easily reduced to the type which Archimedes solved.

We possess one other solution of the cubic equation to which
the division of a sphere into segments bearing a given ratio to one
another is reduced by Archimedes. This solution is by Dionysodorus,
and is given in the same note of Eutocius*. Dionysodorus does not
generalise the equation, however, as is done in the fragment quoted
above ; he merely addresses himself to the particular case,

a* :x*=(3a-x) : a,
v ' m + n '

thereby avoiding the necessity for a Stopwr/Aos. The curves which he
uses are the parabola

m+n
and the rectangular hyperbola

a (3a x) = y*
v ' y

When we turn to Apollonius, we find him emphasising in his
* On the Sphere and Cylinder n. 4 (note at end).

CXXX INTRODUCTION.

preface to Book iv. of the Conies* the usefulness of investigations
of the possible number of points in which conies may intersect one
another or circles, because " they at all events afford a more ready
means of observing some things, e.g. that several solutions are
possible, or that they are so many in number, and again that no
solution is possible"; and he shows his mastery of this method
of investigation in Book v., where he determines the number of
normals that can be drawn to a conic through any given point, the
condition that two normals through it coincide, or (in other words)
that the point lies on the evolute of the conic, and so on. For these
purposes he uses the points of intersection of a certain rectangular
hyperbola with the conic in question, and among the cases we find
(v. 51, 58, 62) some which can be reduced to cubic equations, those
namely in which the conic is a parabola and the axis of the parabola
is parallel to one of the asymptotes of the hyperbola. Apollonius
however does not bring in the cubic equation ; he addresses himself
to the direct geometrical solution of the problem in hand without
reducing it to another. This is after all only natural, because the
solution necessitated the drawing of the rectangular hyperbola in
the actual figure containing the conic in question ; thus, e.g. in the
case of the problem leading to a cubic equation, Apollonius can, so
to speak, compress two steps into one, and the introduction of the
cubic as such would be mere surplusage. The case was different
with Archimedes, when he had no conic in his original figure ; and
the fact that he set himself to solve a cubic somewhat more general
than that actually involved in the problem made separate treatment
with a number of new figures necessary. Moreover Apollonius was
at the same time dealing, in other propositions, with cases which did
not reduce to cubics, but would, if put in an algebraical form, lead
to biquadratic equations, and these, expressed as such, would have
had no meaning for the Greeks ; there was therefore the less reason
in the simpler case to introduce a subsidiary problem.

As already indicated, the cubic equation, as a subject of syste-
matic and independent study, appears to have been lost sight of
within a century or so after the death of Archimedes. Thus Diocles,
the discoverer of the cissoid, speaks of the problem of the division of
the sphere into segments in a given ratio as having been reduced
by Archimedes "to another problem, which he does not solve in
his work on the sphere and cylinder"; and he then proceeds to
* Apollonius of Perga, p. Ixxiii.

CUBIC EQUATIONS. CXXxi

solve the original problem directly, without in any way bringing
in the cubic. This circumstance does not argue any want of
geometrical ability in Diocles ; on the contrary, his solution of the
original problem is a remarkable instance of dexterity in the use of
conies for the solution of a somewhat complicated problem, and it
proceeds on independent lines in that it depends on the intersection
of an ellipse and a rectangular hyperbola, whereas the solutions of
the cubic equation have accustomed us to the use of the parabola
and the rectangular hyperbola. I have reproduced Diocles 7 solution
in its proper place as part of the note of Eutocius on Archimedes'
proposition; but it will, I think, be convenient to give here its
equivalent in the ordinary notation of analytical geometry, in
accordance with the plan of this chapter. Archimedes had proved
[On the Sphere and Cylinder n. 2] that, if k be the height of a
segment cut off by a plane from* a sphere of radius a, and if h be
the height of the cone standing on the same base as that of the
segment and equal in volume to the segment, then

(3a-k) : (2a-k) = h : k.

Also, if h' be the height of the cone similarly related to the
remaining segment of the sphere,

( + *) :k = h':(2a-k).
From these equations we derive

(h - k) : k - a : (2a - k),
and (h' -2a + k) : (2a - k) = a : k.

Slightly generalising these equations by substituting for a in the
third term of each proportion another length 6, and adding the
condition that the segments (and therefore the cones) are to bear to
each other the ratio m : n, Diocles sets himself to solve the three

equations

(h-k) : k = b : (2a-k)\

(h'-2a + k):(2a-k)=:b:k [ ( A )-

and h : h' = m : n )

Suppose m>n, so that &>. The problem then is to divide a
straight line of length 2a into two parts k and (2a k) of which k is
the greater, and which are such that the three given equations are
all simultaneously satisfied.

Imagine two coordinate axes such that the origin is the middle
point of the given straight line, the axis of y is at right angles to it,

CXXX11 INTRODUCTION.

and x is positive when measured along that half of the given straight
line which is to contain the required point of division. Then the
conies drawn by Diocles are

(1) the ellipse represented by the equation

and (2) the rectangular hyperbola

(x + a)(y + b) = 2ab.

One intersection between these conies gives a value of x between
and a, and leads to the solution required. Treating the equations
algebraically, and eliminating y by means of the second equation
which gives

a-x ,

y -* . ,
J a + x

we obtain from the first equation

a + x,
that is, (a + x)*(a + b - x) = - (a - x? (a + b + x) (B).

In other words Diodes' method is the equivalent of solving a
complete cubic equation containing all the three powers of x and a
constant, though no mention is made of such an equation.

To verify the correctness of the result we have only to remember
that, x being the distance of the point of division from the middle
point of the given straight line,

Thus, from the first two of the given equations (A) we obtain
respectively

7 i ,

h = a + x + . 0,
a x

A. a x j

~ax+- - .0,

whence, by means of the third equation, we derive

(a + x) z (a + b - x) = (a - x) 2 (a + b + x),

which is the same equation as that found by elimination above (B).

CUBIC EQUATIONS. CXXxiii

I have purposely postponed, until the evidence respecting the
Greek treatment of the cubic equation was complete, any allusion
to an interesting hypothesis of Zeuthen's* which, if it could be
accepted as proved, would explain some difficulties involved in
Pappus' account of the orthodox classification of problems and loci.
I have already quoted the passage in which Pappus distinguishes
the problems which are jtlane (cTrtVcSa), those which are solid (crrepca)
and those which are linear (ypa^iKa) f. Parallel to this division of
problems into three orders or classes is the distinction between three
classes of loci%. The first class consists of plane loci (TOTTOI eTrwrcSoi)
which are exclusively straight lines and circles, the second of solid
loci (TOTTOI orcpcoi) which are conic sections , and the third of
linear loci (TOTTOI ypa/x/xt/cot). It is at the same time clearly implied
by Pappus that problems were originally called plane, solid or linear
respectively for the specific reason, that they required for their
solution the geometrical loci which bore the corresponding names.
But there are some logical defects in the classification both as
regards the problems and the loci.

(1) Pappus speaks of its being a serious error on the part of
geometers to solve a plane problem by means of conies (i.e. 'solid
loci ') or * linear ' curves, and generally to solve a problem " by means
of a foreign kind " (e ai/oiKciou ycVovs). If this principle were
applied strictly, the objection would surely apply equally to the
solution of a * solid ' problem by means of a ' linear ' curve. Yet,
though e.g. Pappus mentions the conchoid and the cissoid as being
' linear ' curves, he does not object to their employment in the
solution of the problem of the two mean proportionals, which is a
1 solid ' problem.

(2) The application of the term 'solid loci' to the three conic
sections must have reference simply to the definition of the curves
as sections of a solid figure, viz. the cone, and it was no doubt in
contrast to the ' solid locus ' that the ' plane locus ' was so called.
This agrees with the statement of Pappus that ' plane' problems may

* Die Lehre von den KegehchmUen, p. 226 sqq.

t p. chi.

Pappus vn. pp. 652, 662.

It is true that Proclus (p. 394, ed. Friedlein) gives a wider definition of
' solid lines" as those which arise " from some section of a solid figure, as the
cylindrical helix and the conic curves''; but the reference to the cylindrical
helix would seem to be due to some confusion.

CXXX1V INTRODUCTION.

properly be so called because the lines by means of which they are
solved "have their origin in a plane." But, though this may be
regarded as a satisfactory distinction when * plane * and * solid ' loci
are merely considered in relation to one another, it becomes at once
logically defective when the third or ' linear ' class is also brought
in. For, on the one hand, Pappus shows how the ' quadratrix ' (a
'linear* curve) can be produced by a construction in three
dimensions (" by means of surface-loci," Sta TO>I> TT^OS C7rt<^ai/tat?
TOTTOJV) ; and, on the other hand, other ' linear ' loci, the conchoid
and cissoid, have their origin in a plane. If then Pappus' account
of the origin of the terms * plane ' and 'solid 7 as applied to problems
and loci is literally correct, it would seem necessary to assume that
the third name of ' linear ' problems and loci was not invented until
a period when the terms * plane ' and ' solid loci ' had been so long
recognised and used that their origin was forgotten.

To get rid of these difficulties, Zeuthen suggests that the terms
* plane ' and ' solid ' were first applied to problems, and that they
came afterwards to be applied to the geometrical loci which were
used for the purpose of solving them. On this interpretation, when
problems which could be solved by means of the straight line and
circle were called 'plane/ the term is supposed to have had reference,
not to any particular property of the straight line or circle, but to
the fact that the problems were such as depend on an equation of a
degree not higher than the second. The solution of a quadratic
equation took the geometrical form of application of areas, and the
term ' plane ' became a natural one to apply to the class of problems
so soon as the Greeks found themselves confronted with a new class
of problems to which, in contrast, the term ' solid ' could be applied.
This would happen when the operations by which problems were
reduced to applications of areas were tried upon problems which
depend on the solution of a cubic equation. Zeuthen, then,
supposes that the Greeks sought to give this equation a similar
shape to that which the reduced ' plane ' problem took, that is, to
form a simple equation between solids corresponding to the cubic
equation

the term ' solid ' or ' plane ' being then applied according as it had
been reduced, in the manner indicated, to the geometrical equivalent
of a cubic or a quadratic equation.

Zeuthen further explains the term 'linear problem* as having

CUBIC EQUATIONS. CXXXV

been invented afterwards to describe the cases which, being
equivalent to algebraical equations of an order higher than the
third, would not admit of reduction to a simple relation between
lengths, areas and volumes, and either could not be reduced to an
equation at all or could only be represented as such by the use of
compound ratios. The term 'linear* may perhaps have been applied
because, in such cases, recourse was had to new classes of curves,
directly and without any intermediate step in the shape of an
equation. Or, possibly, the term may not have been used at all
until a time when the original source of the names * plane' and
' solid ' problems had been forgotten.

On these assumptions, it would still be necessary to explain how
Pappus came to give a more extended meaning to the term ' solid
problem,' which according to him equally includes those problems
which, though solved by the same method of conies as was used to
solve the equivalent of cubics, do not reduce to cubic equations but
to biquadratics. This is explained by the supposition that, the
cubic equation having by the time of Apollonius been obscured
from view owing to the attention given to the method of solution
by means of conies and the discovery that the latter method was
one admitting of wider application, the possibility of solution by
means of conies came itself to be regarded as the criterion deter-
mining the class of problem, and the name ' solid problem ' came
to be used in the sense given to it by Pappus through a natural
misapprehension. A similar supposition would account, in Zeuthen's
view, for a circumstance which would otherwise seem strange, viz.
that Apollonius does not use the expression ' solid problem/ though
it might have been looked for in the preface to the fourth Book
of the Conies. The term may have been avoided by Apollonius
because it then had the more restricted meaning attributed to it by
Zeuthen and therefore would not have been applicable to all the
problems which Apollonius had in view.

It must be admitted that Zeuthen's hypothesis is in several
respects attractive. I cannot however feel satisfied that the
positive evidence in favour of it is sufficiently strong to outweigh
the authority of Pappus where his statements tell the other way.
To make the position clear, we have to remember that Menaechmus,
the discoverer of the conic sections, was a pupil of Eudoxus who
flourished about 365 B.C. ; probably therefore we may place the
discovery of conies at about 350 B.C. Now Aristaeus 'the elder*

CXXXV1 INTRODUCTION.

wrote a book on solid loci (crrcpeot rorrot) the date of which Cantor
concludes to have been about 320 B.C. Thus, on Zeuthen's hypo-
thesis, the * solid problems ' the solution of which by means of conies
caused the latter to be called 'solid loci' must have been such as
had been already investigated and recognised as solid problems
before 320 B.C., while the definite appropriation, so to speak, of the
newly discovered curves to the service of the class of problems must
have come about in the short period between their discovery and
the date of Aristaeus' work. It is therefore important to consider
what particular problems leading to cubic equations appear to have
been the subject of speculation before 320 B.C. We have certainly
no ground for assuming that the cubic equation used by Archimedes
(On t/ie Sphere and Cylinder n. 4) was one of these problems ; for
the problem of cutting a sphere into segments bearing a given ratio
to one another could not have been investigated by geometers who
had not succeeded in finding the volume of a sphere and a segment
of a sphere, and we know that Archimedes was the first to discover
this. On the other hand there was the duplication of the cube, or
the solution of a pure cubic equation, which was a problem dating
from very early times. Also it is certain that the trisection of an
angle had long exercised the minds of the Greek geometers. Pappus
says that " the ancient geometers " considered this problem and first
tried to solve it, though it was by nature a solid problem (Trpo'/SA/ij/jia
77J <vo-i orrcpcbv vTrap^ov), by means of plane considerations (Sta TWI/
cTTtTrcoW) but failed ; and we know that Hippias of Elis invented,
about 420 B.C., a transcendental curve which was capable of being
used for two purposes, the trisection of an angle, and the quadrature
of a circle*. This curve came to be called the Quadratrix t, but, as
Deinostratus, a brother of Menaechmus, was apparently the first to
apply the curve to the quadrature of the circle J, we may no doubt
conclude that it was originally intended for the purpose of trisecting

* Proems (ed. Friedlein), p. 272.

t The character of the curve may be described as follows. Suppose there
are two rectangular axes Oy, Ox and that a straight line OP of a certain length
(a) revolves uniformly from a position along Oy to a position along Ox, while a
straight line remaining always parallel to Ox and passing through P in its
original position also moves uniformly and reaches Ox in the same time as the
moving radius OP. The point of intersection of this line and OP describes the
Quadratrix, which may therefore be represented by the equation

y la =28 1 IT.
J Pappus iv. pp. 250 2.

CUBIC EQUATIONS. CXXXV11

an angle. Seeing therefore that the Greek geometers had used their
best efforts to solve this problem before the invention of conies, it
may easily be that they had succeeded in reducing it to the geo-
metrical equivalent of a cubic equation. They would not have been
unequal to effecting this reduction by means of the figure of the
i/v<rts given above on p. cxii. with a few lines added. The proof
would of course be the equivalent of eliminating x between the two
equations

where x=DF, y = FP--EC, a^DA, 1>
The second equation gives

(x + a) (x - 3a) = (y + b) (36 - y).
From the first equation it is easily seen that
(x + a) : (y + b) = a : y,

and that (x 3a) y a (b -- 3//) ;

we have therefore a 2 (b - 3y) = y 1 (36 - y) ..................... ((3)

[or 2/ 3 - Zby- ~ 3a s y + orb = 0].

If then the trisection of an angle had been reduced to the geo-
metrical equivalent of this cubic equation, it would be natural for
the Greeks to speak of it as a solid problem. In this respect it
would be seen to be similar in character to the simpler problem of
the duplication of the cube or the equivalent of a pure cubic
equation ; and it would be natural to see whether the transformation
of volumes would enable the mixed cubic to be reduced to the form
of the pure cubic, in the same way as the transformation of areas
enabled the mixed quadratic to be reduced to the pure quadratic.
The reduction to the pure cubic would soon be seen to be impossible,
and the stereometric line of investigation would prove unfruitful
and be abandoned accordingly.

The two problems of the duplication of the cube and the
trisection of an angle, leading in one case to a pure cubic equation
and in the other to a mixed cubic, are then the only problems
leading to cubic equations which we can be certain that the Greeks
had occupied themselves with up to the time of the discovery of the
conic sections. Menaechmus, who discovered these, showed that
they could be successfully used for finding the two mean propor-
tionals and therefore for solving the pure cubic equation, and the

CXXXVlll INTRODUCTION.

next question is whether it had been proved before the date of
Aristaeus' Solid Loci that the trisection of an angle could be
effected by means of the same conies, either in the form of the
vcvcris above described directly and without the reduction to a cubic
equation, or in the form of the subsidiary cubic (/3). Now (1) the
solution of the cubic would be somewhat difficult in the days when
conies were still a new thing. The solution of the equation (ft) as
such would involve the drawing of the conies which we should
represent by the equations

xy = a\

bx = 3a* + 3by - y*,

and the construction would be decidedly more difficult than that
used by Archimedes in connexion with his cubic, which only requires
the construction of the conies

hence we can hardly assume that the trisection of an angle in the
form of the subsidiary cubic equation was solved by means of conies
before 320 B.C. (2) The angle may have been trisected by means
of conies in the sense that the vi;<ris referred to was effected by
drawing the curves (a), i.e. a rectangular hyperbola and a circle.
This could easily have been done before the date of Aristaeus ; but
if the assignment of the name 'solid loci' to conies had in view their
applicability to the direct solution of the problem in this manner
without any reference to the cubic equation, or simply because
the problem had been before proved to be ' solid ' by means of the
reduction to that cubic, then there does not appear to be any
reason why the Quadratrix, which had been used for the same
purpose, should not at the time have been also regarded as a * solid
locus,' in which case Aristaeus could hardly have appropriated the
latter term, in his work, to conies alone. (3) The only remaining
alternative consistent with Zeuthen's view of the origin of the
name 'solid locus' appears to be to suppose that conies were so
called simply because they gave a means of solving one ' solid
problem/ viz. the doubling of the cube, and not a problem of the
more general character corresponding to a mixed cubic equation, in
which case the justification for the general name ' solid locus ' could
only be admitted on the assumption that it was adopted at a time

CUBIC EQUATIONS. CXXxix

when the Greeks were still hoping to be able to reduce the general
cubic equation to the pure form. I think however that the
traditional explanation of the term is more natural than this
would be. Conies were the first curves of general interest for
the description of which recourse to solid figures was necessary as
distinct from the ordinary construction of plane figures in a plane*;
hence the use of the term 'solid locus' for conies on the mere ground
of their solid origin would be a natural way of describing the new
class of curves in the first instance, and the term would be likely
to remain in use, even when the solid origin was no longer thought
of, just as the individual conies continued to be called " sections of
a right-angled, obtuse-angled, and acute-angled " cone respectively.

While therefore, as I have said, the two problems mentioned
might naturally have been called ' solid problems ' before the dis-
covery of * solid loci/ I do not think there is sufficient evidence
to show that * solid problem' was then or later a technical term
for a problem capable of reduction to a cubic equation in the sense
of implying that the geometrical equivalent of the general cubic
equation was investigated for its own sake, independently of its
applications, and that it ever occupied such a recognised position
in Greek geometry that a problem would be considered solved so
soon as it was reduced to a cubic equation. If this had been so,
and if the technical term for such a cubic was 'solid problem/ I
find it hard to see how Archimedes could have failed to imply some-
thing of the kind when arriving at his cubic equation. Instead of
this, his words rather suggest that he had attacked it as res integra.
Again, if the general cubic had been regarded over any length of
time as a problem of independent interest which was solved by
means of the intersections of conies, the fact could hardly have been
unknown to Nicoteles who is mentioned in the preface to Book IV.
of the Conies of Apollonius as having had a controversy with Conon
respecting the investigations in which the latter discussed the maxi-
mum number of points of intersection between two conies. Now
Nicoteles is stated by Apollonius to have maintained that no use

* It is true that Archytas' solution of the problem of the two mean propor-
tionals used a curve of double curvature drawn on a cylinder ; but this was not
such a curve as was likely to be investigated for itself or even to be regarded as
a locus, strictly speaking; hence the solid origin of this isolated curve would
not be likely to suggest objections to the appropriation of the term ' solid locus'
to conies.

CXI INTRODUCTION.

could be made of the discoveries of Conon for 3toptoy*ot ; but it seeins
incredible that Nicoteles could have made such a statement, even for
controversial purposes, if cubic equations then formed a recognised
class of problems for the discussion of which the intersections of
conies were necessarily all-important.

I think therefore that the positive evidence available will not
justify us in accepting the conclusions of Zeuthen except to the
following extent.

1. Pappus' explanation of the meaning of the term 'plane
problem' (cTriircSov Trpd/JA^/xa) as used by the ancients can hardly
be right. Pappus says, namely, that "problems which can be
solved by means of the straight line and circle may properly be
called plane (Ac-yon-' av CIKOTCOS cTrnrcSa) ; for the lines by means of
which such problems are solved have their origin in a plane." The
words "may properly be called" suggest that, so far as plane
problems were concerned, Pappus was not giving the ancient
definition of them, but his own inference as to why they were
called 'plane.' The true significance of the term is no doubt, as
Zeuthen says, not that straight lines and circles have their origin
in a plane (which would be equally true of some other curves), but
that the problems in question admitted of solution by the ordinary
plane methods of transformation of 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.

2. When further problems were attacked which proved to be
beyond the scope of the plane methods referred to, it would be
found that some of such problems, in particular the duplication
of the cube and the trisection of an angle, were reducible to simple
equations between volumes instead of equations between areas ; and
it is quite possible that, following the analogy of the distinction
existing in nature between plane figures and solid figures (an analogy
which was also followed in the distinction between numbers as 'plane'
and 'solid' expressly drawn by Euclid), the Greeks applied the term
'solid problem' to such a problem as they could reduce to an
equation between volumes, as distinct from a ' plane problem '
reducible to a simple equation between areas.

3. The first 'solid problem* in this sense which they succeeded

CUBIC EQUATIONS. Cxli

in solving was the multiplication of the cube, corresponding to the
solution of a pure cubic equation in algebra, and it was found that
this could be effected by means of curves obtained by making plane
sections of a solid figure, namely the cone. Thus curves having a
solid origin were found to solve one particular solid problem, which
could riot but seem an appropriate result ; and hence the conic, as
being the simplest curve so connected with a solid problem, was
considered to be properly termed a ' solid locus/ whether because of
its application or (more probably) because of its origin.

4. Further investigation showed that the general cubic equation
could not be reduced, by means of stereometric methods, to the
simpler form, the pure cubic ; and it was found necessary to try
the method of conies directly either (1) upon the derivative cubic
equation or (2) upon the original problem which led to it. In
practice, as e.g. in the case of the trisection of an angle, it was
found that the cubic was often more difficult to solve in that
manner than the original problem was. Hence the reduction of
it to a cubic was dropped as an unnecessary complication, and
the geometrical equivalent of a cubic equation stated as an in-
dependent problem never obtained a permanent footing as the
'solid problem* par excellence.

5. It followed that solution by conies came to be regarded as
the criterion for distinguishing a certain class of problem, and, as
conies had retained their old name of * solid loci/ the corresponding
term ' solid problem ' came to be used in the wider sense in which
Pappus interprets it, according to which it includes a problem
depending on a biquadratic as well as a problem reducible to a
cubic equation.

6. The terms ' linear problem ' and ' linear locus ' were then
invented on the analogy of the other terms to describe respectively
a problem which could not be solved by means of straight lines,
circles, or conies, and a curve which could be used for solving such
a problem, as explained by Pappus.

CHAPTER VII.

ANTICIPATIONS BY ARCHIMEDES OF THE INTEGRAL CALCULUS.

IT has been often remarked that, though the method of exhaustion
exemplified in Euclid xn. 2 really brought the Greek geometers face
to face with the infinitely great and the infinitely small, they
never allowed themselves to use such conceptions. It is true that
Antiphon, a sophist who is said to have often had disputes with
Socrates, had stated* that, if one inscribed any regular polygon,
say a square, in a circle, then inscribed an octagon by constructing
isosceles triangles in the four segments, then inscribed isosceles
triangles in the remaining eight segments, and so on, "until the
whole area of the circle was by this means exhausted, a polygon
would thus be inscribed whose sides, in consequence of their small -
ness, would coincide with the circumference of the circle." But as
against this Simplicius remarks, and quotes Eudemus to the same
effect, that the inscribed polygon will never coincide with the
circumference of the circle, even though it be possible to carry
the division of the area to infinity, and to suppose that it would
is to set aside a geometrical principle which lays down that magni-
tudes are divisible ad infinitum^. The time had, in fact, riot come
for the acceptance of Antiphon's idea, and, perhaps as the result of
the dialectic disputes to which the notion of the infinite gave rise,
the Greek geometers shrank from the use of such expressions as
infinitely great and infinitely small and substituted the idea of things
greater or less than any assigned magnitude. Thus, as Hankel says J,
they never said that a circle is a polygon with an ir finite number of

* Bretschneider, p. 101.
t Bretschneider, p. 102.

Hankel, Zur Geschichte der Mathematik im Alter thum und Mittelalter,
p. 123.

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cxliii

infinitely small sides ; they always stood still before the abyss of the
infinite and never ventured to overstep the bounds of clear con-
ceptions. They never spoke of an infinitely close approximation or
a limiting value of the sum of a series extending to an infinite
number of terms. Yet they must have arrived practically at such
a conception, e.g., in the case of the proposition that circles are to
one another as the squares on their diameters, they must have been
in the first instance led to infer the truth of the proposition by the
idea that the circle could be regarded as the limit of an inscribed
regular polygon with an indefinitely increased number of corre-
spondingly small sides. They did not, however, rest satisfied with
such an inference ; they strove after an irrefragable proof, and this,
from the nature of the case, could only be an indirect one. Ac-
cordingly we always find, in proofs by the method of exhaustion,
a demonstration that an impossibility is involved by any other
assumption than that which the proposition maintains. Moreover
this stringent verification, by means of a double reductio ad ab-
surdum, is repeated in every individual instance of the use of the
method of exhaustion; there is no attempt to establish, in lieu of
this part of the proof, any general propositions which could be
simply quoted in any particular case.

The above general characteristics of the Greek method of
exhaustion are equally present in the extensions of the method
found in Archimedes. To illustrate this, it will be convenient,
before passing to the cases where he performs genuine integrations,
to mention his geometrical proof of the property that the area of a
parabolic segment is four-thirds of the triangle with the same base
and vertex. Here Archimedes exhausts the parabola by continually
drawing, in each segment left over, a triangle with the same base
and vertex as the segment. If A be the area of the triangle so
inscribed in the original segment, the process gives a series of areas

A, {A, (D'A,...
and the area of the segment is really the sum of the infinite series

But Archimedes does not express it in this way. He first proves
that, if A 19 A 2 ,...A n be any number of terms of such a series, so that
A l = 44 2 , AZ- 4tA SJ ... , then

A l + At + A., + . .. + A n + $A n = 4 lf

or A {i + i + a) 2 + ... + ay-' + jar 1 } = $A.

Cxliv INTRODUCTION.

Having obtained this result, we should nowadays suppose n to
increase indefinitely and should infer at once that (j-)*" 1 becomes
indefinitely small, and that the limit of the sum on the left-hand side
is the area of the parabolic segment, which must therefore be equal
to $A. Archimedes does not avow that he inferred the result in
this way ; he merely states that the area of the segment is equal
to A 9 and then verifies it in the orthodox manner by proving that
it cannot be either greater or less than A.

I pass now to the extensions by Archimedes of the method
of exhaustion which are the immediate subject of this chapter. It
will be noticed, as an essential feature of all of them, that
Archimedes takes both .an inscribed figure and a circumscribed
figure in relation to the curve or surface of which he is investigating
the area or the solid content, and then, as it were, compresses the
two figures into one so that they coincide with one another and
with the curvilinear figure to be measured ; but again it must
be understood that he does not describe his method in this way or
say at any time that the given curve or surface is the limiting form
of the circumscribed or inscribed figure. I will take the cases
in the order in which they come in the text of this book.

1 . Surface of a sphere or spherical segment.

The first step is to prove (On the Sphere and Cylinder I. 21, 22)
that, if in a circle or a segment of a circle there be inscribed
polygons, whose sides AB, BC, CD, ... are all equal, as shown
in the respective figures, then

(a) for the circle

(BB 1 + CC' + . ..) : AA' = A'B : BA,

(b) for the segment

(BB' + CC'+...+KK'+LM) :AM=A'B:BA.

Next it is proved that, if the polygons revolve about the
diameter AA', the surface described by the equal sides of the
polygon in a complete revolution is [i. 24, 35]

(a) equal to a circle with radius <J AB (BB f + CC 1 + ... + YY')
or (b) equal to a circle with radius J~ATT(B]f r +C~C 7 + ... + LAf~).

Therefore, by means of the above proportions, the surfaces
described by the equal sides are seen to be equal to

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cxlv

(a) a circle with radius J AA' . A'JJ,
and (b) a circle with radius JAM. All
they are therefore respectively [i. 25, 37] less tJian

(a) a circle with radius AA\

(b) a circle with radius AL.

Archimedes now proceeds to take polygons circumscribed to the
circle or segment of a circle (supposed in this case to be less than a
semicircle) so that their sides are parallel to those of the inscribed
polygons before mentioned (cf. the figures on pp. 38, 51); and he
proves by like steps [i. 30, 40] that, if the polygons revolve about the
diameter as before, the surfaces described by the equal sides during
a complete revolution are greater than the same circles respectively.

Lastly, having proved these results for the inscribed and
circumscribed figures respectively, Archimedes concludes and proves
[i. 33, 42, 43] that the surface of the sphere or the segment of the
sphere is equal to the first or the second of the circles respectively.

In order to see the effect of the successive steps, let us express
the several results by means of trigonometry. If, in the figures on
pp. 33, 47 respectively, we suppose 4n to be the number of sides in
the polygon inscribed in the circle and 2>i the number of the equal
sides in the polygon inscribed in the segment, while in the latter
case the angle AOL is denoted by a, the proportions given above
are respectively equivalent to the formulae*

IT . 2?T _ 7T 7T

sin + sin - - 4- . . . + sin (2n. 1) = cot ,
2n '2n ^ ' "2n 4?i

- f . a . 2a . , , .a)

2 { sin + sin + ... + sin (n 1 ) - [ + sin a
. I n n n] a

and , = cot -^ .

1 cos a 2?i

Thus the two proportions give in fact a summation of the series
sin 6 + sin 20 + . . . + sin (n 1 )

both generally where nO is equal to any angle a less than TT, and in
the particular case where n is even and = TT/H.

Again, the areas of the circles which are equal to the surfaces
described by the revolution of the equal sides of the inscribed

* These formulae are taken, with a slight modification, from Loria, II periodo
aureo della geometria greca, p. 108.

H. A. k

Cxlvi INTRODUCTION.

polygons are respectively (if a be the radius of the great circle
of the sphere)

and

If ( T 2?T . -iv 71 ") A 2 V

sin - x sm _ - + sm + . . . + sin (2n - 1 ) h k or bira cos - - ,
4n [ 2n 2n v ' 2n) 4?i

o ~ . a I", ( . a . 2a . . nx a) "1

a 2 . 2 sm-^- 2 ^sin- + sm + ... 4- sin(n-l) - x +sma ,
2n \_ [ n n ^ ' n) J

or Tra 2 . 2 cos T - ( 1 cos a).

2n v '

The areas of the circles which are equal to the surfaces described
by the equal sides of the circumscribed polygons are obtained from
the areas of the circles just given by dividing them by cos 2 7r/4?i and
cos 8 a/2n respectively.

Thus the results obtained by Archimedes are the same as would
be obtained by taking the limiting value of the above trigonometri-
cal expressions when n is indefinitely increased, and when therefore
cos 7r/4?i and cos a/2n are both unity.

But the first expressions for the areas of the circles are (when n
is indefinitely increased) exactly what we represent by the
integrals

Cir

ira 2 . i I sin dO, or 47ra 2 ,

- Jo

and no, 2 . / 2 sin dO, or 2ira 2 (I - cos a).

Thus Archimedes' procedure is the equivalent of a genuine
integration in each case.

2. Volume of a sphere or a sector of a sphere.

The method does not need to be separately set out in detail here,
because it depends directly 011 the preceding case. The investiga-
tion proceeds concurrently with that of the surface of a sphere or a
segment of a sphere. The same inscribed and circumscribed figures
are used, the sector of a sphere being of course compared with the
solid figure made up of the figure inscribed or circumscribed to the
segment and of the cone which has the same base as that figure and
has its vertex at the centre of the sphere. It is then proved,

(1) for the figure inscribed or circumscribed to the sphere, that its
volume is equal to that of a cone with base equal to the surface of
the figure and height equal to the perpendicular from the centre of
the sphere on any one of the equal sides of the revolving polygon,

(2) for the figure inscribed or circumscribed to the sector, that the

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cxlvii

volume is equal to that of a cone with base equal to the surface of
the portion of the figure which is inscribed or circumscribed to the
segment of the sphere included in the sector and whose height is the
perpendicular from the centre on one of the equal sides of the
polygon.

Thus, when the inscribed and circumscribed figures are, so
to speak, compressed into one, the taking of the limit is practically
the same thing in this case as in the case of the surfaces, the
resulting volumes being simply the before-mentioned surfaces
multiplied in each case by \a.

3. Area of an ellipse.

This case again is not strictly in point here, because it does
not exhibit any of the peculiarities of Archimedes' extensions of
the method of exhaustion. That method is, in fact, applied in
the same manner, mutatis mutandis, as in Eucl. XH. 2. There
is no simultaneous use of inscribed and circumscribed figures, but
only the simple exhaustion of the ellipse and auxiliary circle by
increasing to any desired extent the number of sides in polygons
inscribed to each (On Conoids and Spheroids, Prop. 4).

4. Volume of a segment of a paraboloid of revolution.
Archimedes first states, as a Lemma, a result proved incidentally

in a proposition of another treatise (On Spirals, Prop. 11), viz. that,
if there be n terms of an arithmetical progression h, '2h, 37*, ..., then

h + 2k + 3h + . . . -f nh > \tfh\
and h + 2h + '3h + ... + (w- 1)A < \ri-h} (a ''

Next he inscribes and circumscribes to the segment of the
paraboloid figures made up of small cylinders (as shown in the figure
of On Conoids and Spheroids, Props. 21, 22) whose axes lie along
the axis of the segment and divide it into any number of equal
parts. If c is the length of the axis AD of the segment, and if
there are n cylinders in the circumscribed figure and their axes are
each of length h, so that c = nh, Archimedes proves that

cylinder CE _ u-h

(1)

inscribed fig. h -4- 2h + 3h+ ... 4- (/t 1)7*
> 2, by the Lemma,

cylinder (7 #_ _ n' 2 h

^' circumscribed fig. ~ h + 2h + 37* + . .. + nh

Cxlviii INTRODUCTION.

Meantime it has been proved [Props. 19, 20] that, by increasing
n sufficiently, the inscribed and circumscribed figure can be made
to differ by less than any assignable volume. It is accordingly
concluded and proved by the usual rigorous method that

(cylinder CE) = '2 (segment),
so that (segment AJ2C) = (cone AJ3C).

The proof is therefore equivalent to the assertion, that if h is
indefinitely diminished and n indefinitely increased, while nh remains
equal to c,

limit of h \h + 2A + 37* + ... + (n- 1) h] - ic 2 ;

that is, in our notation,

i'C

I xdx - ic 2 .
Jo

Thus the method is essentially the same as ours when we
express the volume of the segment of the paraboloid in the form

.'0

where K is a constant, which does not appear in Archimedes' result
for the reason that he does not give the actual content of the
segment of the paraboloid but only the ratio which it bears to the
circumscribed cylinder.

5. Volume of a segment of a hyperbola id of revolution.

The first step in this case is to prove [On Conoids and Spheroids,
Prop. 2] that, if there be a series of n terms,

ah + h\ a.2h + (2/<) 2 , a . 37* + (37*) 2 , ... a. nh + (w7*) 2 ,
and if (ah + tf) + \a . 2h + (27t) 2 J + ... + !. nh + (nlif] = S M ,

then n {a . nh + (nh)*}/M n < (a + nh) i ( + )

/V2 3;

and n [a . nh + (n/ t )*J /.?_, > ( + nh) ? + -

Next [Props. 25, 26] Archimedes draws inscribed and circum-
scribed figures made up of cylinders as before (figure on p. 137), and

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cxlix

proves that, if AD is divided into n equal parts of length h, so that
nh -AD, and if A A' = a, then

cylinder EB' _ n [a . nh + (nhf\
inscribed tigure ~~ _!

fa nh\

. cylinder KB' n {a . nh + (nhf\

and .- J - ------ --7 nr- - - - c.

circumscribed hg. o n

a nh

The conclusion, arrived at in the same manner as before, is that
cylinder EB' . ..//a nh\

A !>/ = ( a + n '0 / ( ^ + "^~ ) '

segment ABB ^ ' \2 3/

This is the same as saying that, if nk = b, and if h be indefinitely
diminished while )i is indefinitely increased,

limit of n (ab + b*)/S H = (a + b)/ (^ -t- |) ,

r -4. f & v if a 'A
or limit of - ^ = 6- (^- + - ) .

Now >S' H -= re (// -f '2/i + . . . + nh) + J/r -f (2/*) 2 + . . . + (nh)-\,
so that A,^ - a/*, (A + 2A -f . . . + nh ) + h \h 2 + ( 2h) 2 + . . . + (nhf\ .
The limit of the last expression is what we should write as

f h

I (a.v + ar) rf.r,

JQ
which is equal to 6 2 ( - + j ;

and Archimedes has given the equivalent of this integration.

6. Volume of a segment of a spheroid.

Archimedes does not here give the equivalent of the integration

f b

(*-rf),

.'o

presumably because, with his method, it would have required yet
another lemma corresponding to that in which the results (/?) above
are established.

cl INTRODUCTION.

Suppose that, in the case of a segment less than half the spheroid
(figure on p. 142), A A' = a, 07) = ic, AD = 6; and let AD be divided
into n equal parts of length h.

The gnomons mentioned in Props. 29, 30 are then the differences
between the rectangle cb + lr and the successive rectangles

and in this case we have the conclusions that (if S n be the sum of
n terms of the series representing the latter rectangles)

cylinder EB' n(cb + b 2 )

inscribed figure n (cb + b 2 ) - *V /t

. I /c -2b

cy Under EB' n(cb + b*

circumscribed fig. n (cb + b s )

i- .1 T ^ cylinder EB' . , x //c 2/A
and in the limit J .inn/ = -" ( c + ^)/ ( -, + -,- ) .
segment J/?J5 x / \2 3/

Accordingly \ve have the limit taken of the expression

and the integration performed is the same as that in the case of the
hyperboloid above, with c substituted for a.

Archimedes discusses, as a separate case, the volume of half a
spheroid [Props. 27, 28]. It differs from that just given in that c
vanishes and b = <&, so that it is necessary to find the limit of

(3/ 4 )^f_. . . + (nil f

and this is done by means of a corollary to the lemma given on
pp. 107 9 [On Spirals, Prop. 10] which proves that

A 9 + (2h)* + ... + (nhf > \n (nh)\
and h 2 + (2A) 2 + ... + \(n~ 1) 7/j 2 < \n (nh)\

The limit of course corresponds to the integral

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cli

7. Area of a spiral.

(1) Archimedes finds the area bounded by the first complete
turn of a spiral and the initial line by means of the proposition just
quoted, viz.

A 2 + (2h) 2 + . . . + (nltf > \n (nhf,

hr + (2A) 2 + . . . + {(w - 1 ) A} 8 < Jn (nh)*.

He proves [Props. 21, 22, 23] that a figure consisting of similar
sectors of circles can be circumscribed about any arc of a spiral such
that the area of the circumscribed figure exceeds that of the spiral
by less than any assigned area, and also that a figure of the same
kind can be inscribed such that the area of the spiral exceeds that
of the inscribed figure by less than any assigned area. Then, lastly,
he circumscribes and inscribes figures of this kind [Prop. 24] ; thus
e.g. in the circumscribed figure, if there are n similar sectors, the
radii will be n lines forming an arithmetical progression, as A, 27i,
3//, ... nh, and nh will be equal to a, where a is the length inter-
cepted 011 the initial line by the spiral at the end of the first turn.
Since, then, similar sectors are to one another as the square of their
radii, and n times the sector of radius nh or a is equal to the circle
with the same radius, the first of the above formulae proves that

(circumscribed fig.) > J-?ra 2 .

A similar procedure for the inscribed figure leads, by the use of the
second formula, to the result that

(inscribed fig.) < J^rrr.
The conclusion, arrived at in the usual manner, is that

(area of spiral) $7ra~ \
and the proof is equivalent to taking the limit of

or of -* [A 3 + (-IKf + ... + [( - 1) A} 2 ],

which last limit we should express as

Clii INTRODUCTION.

[It is clear that this method of proof equally gives the area
bounded by the spiral and any radius vector of length b not being
greater than a ; for we have only to substitute trb/a for TT, and to
remember that in this case nh - b. We thus obtain for the area

fit

^ P x*dx, or

(2) To find the area bounded by an arc on any turn of the
spiral (not being greater than a complete turn) and the radii
vectores to its extremities, of lengths b and c say, where c > /;,
Archimedes uses the proposition that, if there be an arithmetic
progression consisting of the terms

6, b + h, b+'2h, ... 6 + (M -!)/,
and if /? - 6 2 + (6 + h)~+ (b + 2h)' 2 + ... + [b + (n - 1) 7*J a ,

and

[0/i Spirals, Prop. 1 1 and note.]

Then in Prop. 26 he circumscribes and inscribes figures consisting
of similar sectors of circles, as before. There are n\ sectors in
each figure and therefore n radii altogether, including both b and c,
so that we can take them to be the terms of the arithmetic progres-
sion given above, where [b + (n 1) k\ = c. It is thus proved, by
means of the above inequalities, that

sector OB'C \b + (n - 1) A| 2 ^ sector OJfC

circumscribed fig. {b + (n-l)h\b + \(n-l)/i^ inscr. fig.
and it is concluded after the usual manner that

{b + (n-})h}' 2

_
spiral OUC

Remembering that n - 1 = (c - &)//*, we see that the result is the

ARCHIMEDES' ANTICIPATIONS OF THE INTEGRAL CALCULUS, cliii

same thing as proving that, in the limit, when n becomes indefinitely
great and h indefinitely small, while b + (n - 1) h = c,

limit of h[b* + (b + h) 3 + ... 4- [b + (n - 2)h\*]

that is, with our notation,

= i (c 3 -

(3) Archimedes works out separately [Prop. 25], by exactly
the same method, the particular case where the area is that described
in any one complete turn of the spiral beginning from the initial
line. This is equivalent to substituting (n I) a for b and na for c,
where a is the radius vector to the end of the first complete turn of
the spiral.

It will be observed that Archimedes does not use the result
corresponding to

f x 2 dx- {' x 2 dx^ C x"d.c.

Jo Jb JO

8. Area of a parabolic segment.

Of the two solutions which Archimedes gives of the problem of
squaring a parabolic segment, it is the mechanical solution which
gives the equivalent of a genuine integration. In Props. 14, 15 of
the Quadrature of the Parabola it is proved that, of two figures
inscribed and circumscribed to the segment and consisting in each
case of trapezia whose parallel sides are diameters of the parabola,
the inscribed figure is less, and the circumscribed figure greater,
than one-third of a certain triangle (EqQ in the figure on p. 242).
Then in Prop. 1C we have the usual process whicli is equivalent to
taking the limit when the trapezia become infinite in number and
their breadth infinitely small, and it is proved that

(area of segment) = \ A EqQ.

The result is the equivalent of using the equation of the parabola
referred to Qq as axis of x and the diameter through Q as axis of
2/, viz.

p\j = x (2a - o?) f

which can, as shown on p. 236, be obtained from Prop. 4, and finding

ydx,
o

Cliv INTRODUCTION.

where y has the value in terms of x given by the equation ; and of
course

a (2ax - x*) dx = ^ .

The equivalence of the method to an integration can also be
seen thus. It is proved in Prop. 16 (see figure on p. 244) that, if
qE be divided into n equal parts and the construction of the
proposition be made, Qq is divided at 19 2 , ... into the same
number of equal parts. The area of the circumscribed figure is then
easily seen to be the sum of the areas of the triangles

(~}n 7?* (~\ 7? TF ^~) 7? IT
that is, of the areas of the triangles

QqF, QO&, QO,D>, ...

Suppose now that the area of the triangle QqF is denoted by A, and
it follows that

(circumscribed fig.) = A \ 1 + ., + ,"" + ... + [

I tt 71 1L I

Similarly we obtain

(inscribed fig.) = -, . A JA 2 + 2 2 A 2 +... + (n- 1) 2 A L> |.

Taking the limit we have, if A denote the area of the triangle EqQ,
so that A = nA,

1 r A
(area of segment) -- / A 2 c/A

A." JQ

If the conclusion be regarded in this manner, the integration is
the same as that which corresponds to Archimedes' squaring of the
spiral.

CHAPTER VIII.

THE TERMINOLOGY OF ARCHIMEDES.

So far as the language of Archimedes is that of Greek geometry
in general, it must necessarily have much in common with that of
Euclid and Apollonius, and it is therefore inevitable that the
present chapter should repeat many of the explanations of terms of
general application which I have already given in the corresponding
chapter of my edition of Apollonius' Conic**. But I think it will
be best to make this chapter so far as possible complete and self-
contained, even at the cost of some slight repetition, which will
however be relieved (1) by the fact that all the particular phrases
quoted by way of illustration will be taken from the text of
Archimedes instead of Apollonius, and (2) by the addition of a large
amount of entirely different matter corresponding to the great
variety of subjects dealt with by Archimedes as compared with the
limitation of the work of Apollonius to the one subject of conies.

One element of difficulty in the present case arises out of the
circumstance that, whereas Archimedes wrote in the Doric dialect,
the original language has been in some books completely, and in others
partially, transformed into the ordinary dialect of Greek. Uni-
formity of dialect cannot therefore be preserved in the quotations
about to be made ; but I have thought it best, when explaining
single words, to use the ordinary form, and, when illustrating their
use by quoting phrases or sentences, to give the latter as they appear
in Heiberg's text, whether in Doric or Attic in the particular case.
Lest the casual reader should imagine the paroxytone words

7T(T6T(U, 7T(TOlWttl, <TOrlTai, 8wttl/TCU, a7TTT<U,

and the like to be misprints, I add that the quotations in
Doric from Heiberg's text have the unfamiliar Doric accents.

I shall again follow the plan of grouping the various technical
* Apollonius of Perga, pp. clvii clxx.

clvi INTRODUCTION.

terms under certain general headings, which will enable the Greek
term corresponding to each expression in the ordinary mathematical
phraseology of the present day to be readily traced wherever such
a Greek equivalent exists.

Points and lines.

A point is orTy/xeTov, the point B TO B O-^/ACIOV or TO B simply ; a
point on (a line or curve) a-yjaelov CTTI (with gen.) or lv ; a point
raised above (a plane) o-jy/xeioi/ /XCTCW/OOI/ ; any two points whatever
beiny taken Svo <rr)p.iwv Xa^fBavo^tv^v oVotwvovv.

At a point (e.g. of an angle) Trpo's (with dat.), having its vertex at
the centre of the sphere Kopvtfr-rjv <.\wv Trpos TO) KcWpu) T^S <r<ai'pas ; of
lines meeting in a point, touching or dividing at a point, etc., Kara
(with ace.), thus AE is bisected at Z is d AE Si^a r^virai Kara TO Z ;
of a point falling on or being placed on another CTTI or Kara (with
ace.), thus Z will fall on F, TO /xev Z im TO F TrcorctVat, so tfto E fo'es

On A, 0>OT TO /XV E KdTtt TO A KLO"OaL.

Particular points are extremity Wpa?, vertex Kopv<f>rj, centre
KcVrpov, poitttf 0/ 1 division oWpco-is, ^outl o/ meeting O-VHTTTWO-L?, point
of section TO/XT/, point of bisection 8t^oTo/xta, ^/te middle point TO
fjico'ov ] the points of division H, I, K, Ta TWV Statpecrtwv o~a/*ta TCI H,
I, K ; let B be its middle point /xcVov 8c avTas IO-TO) TO B ; the point of
section in which (a circle) cuts a To/xa, /ca^' av Tcpvci.

A /i?ie is ypait/xT/, a curved line Ka^irvXt) ypappy, a straight line
cvOtia with or without ypappy. The straight line IKA, a TKA
eutfcia ', but sometimes the older expression is used, the straight line
on which (cm with gen. or dat. of the pronoun) are placed certain
letters, thus let it be the straight line M, COTW <' a TO M, other
straight lines K, A, aAAcu ypa/x/xat, c<' av ret K, A. The straight
lines between the points at [j.Tav T&V o-TyticiW tv^etat, o/* ^/te ^tev
which have the same extremities the straight line is the least TWI> Ta
auTct TrepaTa c^ovcrcSv ypa/x/x,(3i/ eXa^arTTyv eivat Tiyv v0ctav, straight lines
cutting one another tvfaiat Te/mvovcrai aXAaAas.

For points in relation to lines we have such expressions as the
following : the points F, , M are on a straight line eV cvflcta? cart
Ta F, , M o*a/ita, ^e ^?oin^ o/" bisection of the straight line containing
the centres of the middle magnitudes a S^oTo/xia Ta? i36cta5 Tas
xov<ras ra KcvTpa T(3v /xeo-wv /xcyc^ea)!/. A very characteristic phrase
for a a /?oin^ which divides the straight line in such a proportion
that.., is CTT! Tas cutfci'as Siaipc^cta-a? wore...; similarly CTTI Tas XE

THE TERMINOLOGY OF ARCHIMEDES. clvii

T/xa0io~a OVTWS, wore. A certain point will be on the straight line...
dividing it so that... ccrcrctrai CTT! ras v0tas...8iatpeoi/ OVTWS rav
tlprjfjitvav v0tav, OKTTC....

The middle point of a line is often elegantly denoted by an
adjective in agreement ; thus at the middle point of the segment lirl
/Ao-ov TOV TfJid/JLOLTos, (a line) drawn from F to the middle point of
EB, drro TOV F CTTI ptcrav TO.V EB d^^ctcra, drawn to the middle point of
the base lirl /xctrav rav fid&iv ayo/xcVa.

A straight line produced is the (straight line) in the same straight
line with it 77 ITT ev0t'as avrfj. In the same straight line with the
axis eVt ras avTas cvdct'as TU> dovt. Of a straight line falling on
another line Kara (with gen.) is used, e.g. TTITTTOUO-I KO.T avnjs; CTTI
(with ace.) is also used of a straight line placed on another, thus if
EH be placed on BA, T0iVas ras EH ITTL rav BA.

For lines passing through points we find the following ex-
pressions : will pass through N, ^ct 8ta TOV N ; will pass through the
centre Sta TOV /ceWpou Tropevrrcrat, ^(;i^ y*rtZ/ through Trco-ctVat 8ta rov
, verging toward* B vcvovcra CTTI TO B, pas# through tl& same point
7Tt TO avro o-a/xciov p\6vTai ; ^/w diagonals of the parallelogram fall
(i.e. meet) at (*), KaTtt 8c TO (*D at 8ta/jiTpoi TOV TrapaXXTyXoypa/x/xov
t ; EZ (pf isses) through the points bisecting AB, FA, ?rt 8c Tav
Tai/ AB, FA a EZ. The verb ct/u is also used of passing
through, thus Icro'tiTai. 8r) auTa 8ta. TOU 0.

For lines in relation to other lines we have perpendicular to
Koi0Tos 7rt (with ace.), parallel to TrapdXA^Xo? with dat. or -rrapd
(with ace.); /e KA be (drawn) from K parallel to FA, aTro TOV K
Trapct TCLV FA o~Ta> a KA.

Lines meeting one another o-v/xTriTnrovo-ai dAA^Xats; </ie point in
which ZH, MN produced meet one another and AF, TO o-T^xetoi/, /ca^' o
o-v/x)Sa\Xovo-ii/ K/3a\\op.cvai at ZH, MN aXXT/Xats TC Kat T>J AF ; o as
to meet the tangent uxrre ^7ro > tv Ta 7rt^avovora, ^e?^ straight lines be
drawn parallel to AF o 7>iee^ ^//,e section of the cone a^^wv cv^ctat
Trapa TOLI/ AF CO-TC TTOT! TO.V TOV Kaivov TO/ACIV, draw a straight line to
meet its circumference TTOT! TOIV 7rpt</)pciav avTOv 7TOTt/?aXctv cvtfctdv,
/Ae ^i?ie drawn to meet a TroTtTrea-ovo-a, ^e^ AE, AA be drawn from the
point A to meet the spiral and produced to meet the circumference of
the circle TroTiTriTTToVnov diro TOV A cra/xctov TTOT! TO.V iXtKa at AE, AA
Kal cKTrtTTTovTwv 7ron TCLV TOV KVK\ov 7rpi0epciav ; until it meets 0A in
O, co-Tc ica (TVfjLTrtar] Ta 0A KaTa TO O (of a circle).

clviii INTRODUCTION.

(TJie straight line) will fall outside (i.e. will extend beyond) P,
CKTOS TOV P Trco-ciTcu ; will fall within the section of the figure cvro9

TTCOTOVVTat TO*? TOV O-XyP-O-TOS TOfJLCLS.

The (perpendicular) distance between (two parallel lines) AZ, BH,
TO oWoTTy/xa Tav AZ, BH. Other ways of expressing distances are the
following : the magnitudes equidistant from the middle one TO. LVOV
aTTtxpvTa awo TOV fjico-ov //.ey0a, are at equal distances from one
another ra air aXXaAwv Sieora/ccv ; the segments (lengths) on AH
equal to N, TO, iv ra AH radfjiaTa to*o/xy0a Ta N ; greater by one
segment Ivl T/xa/xaTt /xt<ov.

The word cvfleta itself is also often used in the sense of distance ;
cf. the terms Trporrry ev0ta etc. in the book On Spirals, also a evflcta
a utTav TOV Ktvrpov TOV dAt'ov Kat TOL Ktvrpov ras yas ^/ie distance
between the centre of the sun and the centre of the earth.

The word for ^'o^i is cTri&uyvva) or cTrt&v'yvv/u ; the straight Hue
joining the points of contact a TGLS a<f>as iTri&vyvvovcra cv^cta, BA when
joined a BA cTri^cv^^ctcra ; let EZ Jom ^Ae jtoints of bisection of AA,
BF, a oe EZ 7rt^vyvvTO) TU? OIXOTOJAIOLS rav AA, BF. In one case
the word seems to be used in the sense of drawing simply, et *a
cv0ta cin&vxOrj ypa/x/xa ci/

Angles.

An a?i<7^ is yaw'a, the three kinds of angles are right opOrf, acute
ota, obtuse d/xj8Xta ; right-angled etc. optfoyawos, o^vywvtos, cl/x^A.v-
yojvios ; equiangular to-oywvtos ; ?t*i^A an eue/i number of angles
apTioywvos or dpTtoyaivto?.

ul< ri<7/4< angles to opOos Trpo? (with ace.) or Trpos op^as (with dat.
following); thus if a line be erected at right angles to the plane, ypa^i/xa?
aveo-TaKouVas opOas TTOTI TO cTriVcSoF, <A planes are at right angles to
one another opOa TTOT* aAAaAa IVTL ra cTrtTrcSa, being at right angles
to ABF, ?rpos 6p6as iljv TW ABF ; KF, HA are at right angles to one
another iror opOds VTL aXXaXats at KF, HA, to cut at right angles
rcfjLVw Trpos opOds. The expression making right angles with is also
used, e.g. op0as Troiovo-a yaw'as TTOTI Tav AB.

The complete expression for the angle contained by the lines AH,
AF is a yaWa a Trepte^o/xci/a VTTO rav AH, AF ; but there are a great
variety of shorter expressions, yowa itself being often understood ;
thus the angles A, E, A, B, at A, E, A, B yoivtat ; the angle at , a TTOTI
TO ; the angle contained by A A, AZ, a yawa a VTTO Tav A A, AZ ; the
angle AHF, 17 VTTO TWV AHF ywvta, >J VTTO AHF (with or without yaw'a).

THE TERMINOLOGY OF ARCHIMEDES. clix

Making the angle K equal to the angle , ywiav Troiovcra rav K
to-av ra ; the aw/fe into which the sun fits and which has its vertex
at the eye yawa, cts av 6 aXios cvap/xd^et rav Kopvtfrav l^ovo'av TTOTI ra
oi/rci j of the sides subtending the right angle (hypotenuses) rav VTTO
rav opOav yawai/ VTroreti/ouo-av, they subtend the same angle Ivrl vwo
TOLV avrav yawai/.

If a line through an angular point of a polygon divides it
exactly symmetrically, the opposite angles of the polygon, al dircvavTiov
ywvi'cu TOV TToAirywi/ov, are those answering to each other on each side
of the bisecting line.

Planes and plane figures.

A plane cTrtTrcSoi/ ; the plane through BA, TO cTrtVeSov TO Kara
TTJV BA, or TO OLO. rrjs BA, plane of the base eTrtVcSov T^S /?aa-o>5, plane
(i.e. base) of the cylinder eTrtVeSov TOV Kv\tvopov ; cutting plane CTTL-
TrcSov TCJULVOV, tangent plane 7rt7rc8ov 7ru/fai}ov ; the intersection of
planes is their common section Kowr) TO^JLYJ.

In the same plan?, as the circle cv TO> avT<5 cTrtTrcSa) TW KVK\U.

Let a plane be, erected on HZ at right angles to the plane iti which
AB, FA are 0,77-0 ras HZ 7rt7rc8oi' cli/crrTaKCTu) opObv TTOTI TO cTrtTrcSov TO,
v <S CVTL at AB, TA.

The plane surface rj cTrtTrcSos (cTrtc^aveta), a plane segment cTrarcSov
rp.rjfJLa y a plane Jig ure o^T/fia ITTLTT&QV.

A rectilineal figure evOvypa^jjiov (a-\rjfjLa) 9 a side TrAcvpa, perimeter
ij 7Tpt)uLTpos, similar O/AOIOS, similarly situated o/xotcos KI/ACVOS.

To coincide with (when one figure is applied to another),
tyapfjio&iv followed by the dative or cVt (with ace.) ; o?&e />ar2
coincides with t/ie other <^>ap//o^t TO Tpov /ncpos CTTI TO cTcpov; ^/ie
plane through NZ coincides ivith the plane through AF, TO cTrnreSov TO
KaTa TOLV NZ l<f>apfj.6^L T<5 eTTiTreSa) TO) KUTO, Tav AF. The passive is
also used; if equal and similar plane figures coincide with one another
l 6/xotcov or^r//xttTo)v cTrtTrcSwi/ </>apyu,oo/x,Va>v CTT* aXXaAa.

Triangles.

A triangle is Tptytovoi/, the triangles bounded by (their three
sides) TO, 7rept^d)u.va rpiywva UTTO T<OV. ... A riffht-angled triangle
Tpiytovov op^oycji/tov, o/K? o/" <A6 sirf^v about the right angle fjua rv 7Tpl
rrjv opOyv. The triangle through the a.cis (of a cone) TO Stot rov am

Clx INTRODUCTION.

Quadrilaterals.

A quadrilateral is a four-sided figure (TCTpaTrXeupoi/) as dis-
tinguished from a four-angled figure, TCTpaywvov, which means a
square. A trapezium, rpa-rr^iov, is in one place more precisely
described as a trapezium having its two sides parallel TpaTrc'fiov ras
Svo TrXcupas e^oi/ TrapaXXa'Xous aXXaXats.

A parallelogram TrapaXX^Xoypa/x/xov ; for a parallelogram on a
straight line as base Im (with gen.) is used, thus the parallelograms
on tJiem are of equal height lo-Tlv lo-ofyrj ra TrapaXX^Xoypa/A/xa ra iir
avTwv. A diagonal of a parallelogram is Sia/ncrpos, the opposite sides
of the parallelogram at KCLT cvavrtoi/ TOV TrapaXXi/Xoypa/x^ou TrXcupai.

Rectangles.

The word generally used for a rectangle is \upiov (space or area)
without any further description. As in the case of angles, the
rectangles contained by straight lines are generally expressed more
shortly than by the phrase ra 7repicxo/x.i/a x w p t/a VTTO ; either xwpt'ov
may be omitted or both xupiov smd Trcptc^o/xci/oi/, thus the rectangle
AF, FE may be any of the following, TO VTTO TWI/ AF, FE, TO virb
AF, FE, TO vVo AFE, and the rectangle under (s)K, AH is TO VTTO rrjs
OK KOL rfjs AH. Rectangles , I, K, A, ^wpta cV ols Ta (or c<* <!v

KaO*TOV TCOl/) 0, I, K, A.

To apply a rectangle to a straight line (in the technical sense) is
7rapa/3aA\iVy and TrapaTriTTTco is generally used in place of the passive;
the participle 7rapaKet)u.cvo9 is also used in the sense of applied to. In
each case applying to a straight line is expressed by Trapa (with ace.).
Examples are, areas which we can apply to a given straight line (i.e.
which we can transform into a rectangle of the same area) x^P^t <*
Swa//.c0a Trapa TOLV 8o0io-ai/ tvOelav Trapa/JaXctv, let a rectangle be
applied to each of them TrapaTrcTrroiKCTW Trap* Kacrrav avrav \^piov ;
if there be applied to each of them a rectangle exceeding by a square
figure, and the sides of the excesses exceed each other by an equal
amount (i.e. form an arithmetical progression) et /ca Trap* *ao-Tav
avraV TrapaTrccrr) TL xwpi'oi/ virepfiaXXov ctSct TCTpaywVw, IWI/TI 8c ai
TrXevpat TO>V V7Tp^X>;/xttTO)v TO> to-u) dXXaXav VTrfpe^ovcrat.

The rectangle applied is Trapa/SX^/xa.

Squares.

A square is TCTpa'ycoi/ov, a square on a straight line is a square
(erected) from it (oVo'). The square on F3, TO aTro Ta? FH

THE TERMINOLOGY OF ARCHIMEDES. clxi

is shortened into TO airb ras TH, or TO OLTTO PS simply. The square
next in order to it (when there are a number of squares in a row) is
TO Trap 9 avTw rcrpaytDVOV or TO \6fivov TCTpaycovop.

With reference to squares, a most important part is played by
the word ovvapts and the various parts of the verb Swa/xcu. vi/afu?
expresses a square (literally a power) ; thus in Diophantus it is used
throughout as the technical term for the square of the unknown
quantity in an algebraical equation, i.e. for x*. In geometrical
language it is the dative singular Swa/xci which is mostly used ;
thus a straight line is said to be potentially equal, SiW/xei wra, to a
certain rectangle where the meaning is that the square oil the straight
line is equal to the rectangle ; similarly for the square on BA is less
than double the square on AK we have 77 BA cXao-o-o^v cartv rj oWAao-tW
8wa/xt Tifc AK. The verb 8v'vaa0ai (with or without io~ov) has the
sense of being &W/xi io-a, and, when ovvacrOai is used alone, it is
followed by the accusative ; thus the square (on a straight line) is
equal to the rectangle contained by... is (cvtfcta) icrov Swarcu TU>
TrcpLxojjLv<o vTTo . . . ] let the square on the radius be equal to the
rectangle BA, AZ, 77 IK TOV KcWpo-u 8uvcu70a> TO VTTO TO>V BAZ, (the
difference) by which the square on ZT is greater than the square on
half the other diameter <S /xaoi/ Swdrai a ZF Tas ^/xtcrctas TCI?

A gnomon is yi/a>ju.u>v, and its breadth (TrAotTos) is the breadth of
each end; a gnomon of bread fh equal to BI, -yvcJ/xwv TrActTos C^CDV to-ov
TO. BI, (a gnomon) whose breadth is greater by one segment than the
breadth of the gnomon last taken away ov TrAaTos evl T/ia/xaTi
TOV TrAaTcos TO? Trpo avrov a^atpou/xcVou

Polygons.

A polygon is TroAirywvoi', an equilateral polygon is
a polygon of an even number of sides or angles dpTioVAcvpov or
a polygon with all its sides equal except BA, AA, ?o-as
TOLS TrAcupa? xwpk TOJI/ BAA ; a pol} r gon with its sides, excluding
the base, equal and even in number TOLS TrAcvpas l^ov \<apl<s rf)<; ^ao-cws
10*05 KOI apTtov? ; an equilateral polygon the number of whose sides is
measured by four 7roAu'yu>vov tcroTrAevpov, ov at TrAevpat VTTO T
wwmfcer o/' its sides be measured by four TO
VTTO TTpa8os. A chiliagon ^tAiaya>vo^.

straight lines subtending two sides of the polygon (i.e. joining
angles next but one to each other) at vrro Bvo rrAevpas TOV TroAvyaJvov

H. A. /

Clxii INTRODUCTION.

the straight line subtending one less than half the
number of the sides 17 vTroretvovcra ras tu eXacrcrova? TO>V

Circles.

A circle is /cv*Aos, 2/ie circle ^ is 6 ^ KVK\OS or 6 Kv/cXos lv <S TO ^,

the given circle be tfiat drawn below CCTTW 6 Sonets KVK\OS 6

The centre is KeVrpov, the circumference Trcpi^cpeta, the former
word having doubtless been suggested by something sfacA m and
the latter by something, e.g. a cord stretched tight, earned round
the centre as a fixed point and describing a circle with its other
extremity. Accordingly Trepi^cpeia is used for a circular arc as well
as for the whole circumference ; thus the arc BA is 77 BA TrcpK^epeta,
the (part of the) circumference of the circle cut off by the same
(straight line) 77 rov KVK\OV Trcpt^epcta 77 VTTO rrjs aurijs aTroTc/xvotui^.
Though the circumference of a circle is also sometimes called its
perimeter (77 TrcpiiuTpos) in the treatises On the Sphere and Cylinder
and on the Measurement of a Circle, the word does not seem to have
been used by Archimedes himself in this sense ; he speaks, however,
in the Sand-reckoner of the perimeter of the earth (Trcpt/urpos ras yas).

The radius is 77 IK rov Ktvrpov simply, and this expression
without the article is used as a predicate as if it were one word ;
thus the circle whose radius is E is 6 KVK\OS of CK rov /ccVrpov a
0E; BE is a radius of the circle 17 ot BE CK rov Kci/rpov cort rov KVK\OV.

A diameter is Sict/xerpos, the circle on AE as diameter 6 Trcpl
Sta/ticrpov rrjv AE KVK\OS.

For drawing a chord of a circle there is no special technical
term, but we find such phrases as the following : cap efc rov KVK\OV
tvOtLo. ypa/x/x?; e/xircoTy if in a circle a straight line be placed, and the
chord is then tfie straight line so placed y c/xTreaovo-a, or quite
commonly >J cv TW KVK\U (cvOda) simply. For the chord subtending
one 656th part of the circumference of a circle we have the following
interesting phrase, d v7roTiVovo~a cv r/^a/xa Staipc^ctVa? ras rov ABF

A segment of a circle is rtdj/xa KUK\OU ; sometimes, to distinguish
it from a segment of a splwre, it is called a jo&we segment
Tprjfjia cirtVcSov. A semicircle is i?/u/cwcAiov; a segment less tlian a
semicircle cut off by AB, rtt^/xa cXaoro-ov TJP,IKVK\{OV o aTrorc/xvci
17 AB. 7% segments on AE, EB (as bases) are ra CTTI rc3i/
AE, EB rfU7/xara; but </*6 semicircle on ZH a diameter is TO

THE TERMINOLOGY OF ARCHIMEDES. clxiii

TO ircpl 3ta/u,eTpov rav ZH or TO THJUKVK\IOV TO wepi rav ZH
simply. The expressiort the angle of tlw semicircle, a TOV ty/ujcvjcXtov
(yaw'a), is used of the (right) angle contained by the diameter and
the arc (or tangent) at one extremity of it.

A sector of a circle is TO/ULCUS or, when it is necessary to
distinguish it from what Archimedes calls a * solid sector, 7 cir/WcSo?
To/tews KVK\OV a plane sector of a circle. The sector including the
right angle (at the centre) is 6 TO/ACT)? 6 rav opOdv yow'av
Either of the radii bounding a sector is called a side of it,
each of the sectors (is) equal to the sector which has a side common

(With it) CKCWTTOS TWV TO/i(i)l/ U7OS TO) KOlvdv I^OVTt wXcvpOLV TOfJLCl > B,

sector is sometimes regarded as described on one of the bounding
radii as a side, thus similar sectors have been described on all (the
straight lines) ai/ayeypa^aVai diro vao'av d/xotot TOftecs.

Of polygons inscribed in or circumscribed about a circle yypa<^iv
cis or cv and 7rpiypa<> Trcpi (with ace.) are used ; we also find
Treptycypcyi/xcVos used with the simple dative, thus TO Trepiye-
ypap.fivov crxfjua T<O To/it is the figure circumscribed to the sector.
A polygon is said to be inscribed in a segment of a circle when
the base of the segment is one side and the other sides subtend
arcs making up the circumference ; thus let a polygon be inscribed
on AF in the segment ABF, CTT! T^S AT? iroXvywov cyycypa^w
t? TO ABF rp.rjp.a. A regular polygon is said to be inscribed in
a sector when the two radii are two of the sides and the other sides
are all equal to one another, and a similar polygon is said to be
circumscribed about a sector when the equal sides are formed by the
tangents to the arc which are respectively parallel to the equal
.sides of the inscribed polygon and the remaining two sides are the
bounding radii produced to meet the adjacent tangents. Of a
circle circumscribed to a polygon rrcptXa/x^ai/civ is also used ; thus
iro\vy<t)vov KVK\O$ TTtptycypa/x/xcVo? TrepiXa/AjSavcVo) Trept TO avro Ktvrpov
ytvd/xcvos, as we might say let a circumscribed circle be drawn with
the same centre going round the polygon. Similarly the circle ABFA
containing the polygon 6 ABFA KVK\OS c^cov TO TroAuycoi/ov.

When a polygon is inscribed in a circle, the segments left over
between the sides of the polygon and the subtended arcs are
TTcptXctTTOficva TfiiJfiaTa; when a polygon is circumscribed to the
circle, the spaces between the two are variously called TCI Trcpt-
XctTrd/Aeva -rijs 7Tptypa^iys T/xiy/utaTa, TCI TrepiXeiTrd^ieva o^fwrra, TO,
or TO, aTroXei/x/xara.

INTRODUCTION.

Spheres, etc.

In connexion with a sphere (<r<t>aipa) a number of terms are
used on the analogy of the older and similar terms connected with
the circle. Thus the centre is xeWpov, the radius ij IK TOV KcVrpov,
the diameter 17 Sid/xTpos. Two segments, rp^fjiara cr^cupas or
TfuJiAaTa cr^>atptxa, are formed when a sphere is cut by a plane;
a hemisphere is rj^a-^aipiov ; the segment of the sphere at F, TO Kara TO
F T/xry/xa T^S cr<aipa$ ; the segment on the side of ABF, TO aVo ABF
Tfwy/xa; the segment including the circumference BAA, TO Kara -n)i/ BAA
Trcpi^epctav T/x>//xa. The curved surface of a sphere or segment
is e7ri<a'vea ; thus of spherical segments bounded by equal surfaces th?
hemisphere is greatest is T<UV TjJ to-ry cTrt^ai/eca Trepcc^o/xcvov
T/xiy/xaTcav /xciov COTI TO Tf/uo-^cuptov. The terms &nse
(Kopv<f>jj) and height (ityos) are also used with reference to a segment
of a sphere.

Another term borrowed from the geometry of the circle is the
word sector (TO/LUVS) qualified with the adjective orepfo's (solid).
A solid sector (TO/XCVS orepeo's) is defined by Archimedes as the
figure bounded by a cone which has its vertex at the centre of
a sphere and the part of the surface of the sphere within the cone.
The segment of the sphere included in the sector is TO T/xV/xia Tijs
cr^atpas TO Iv TO> TO/XCI or TO KdTa TOV TO/XC'CU

A great circle of a sphere is 6 /xe'yioros KVK\OS TWI> iv rrj u-<aipa
and often 6 ttytorTos KV*AOS alone.

Let a sphere be cut by a frtane not through the centre TT/X^O-^U>
cr<f>alpa /XT) Sta TOV Ktvrpov cTriTrcSw ; a sphere cut by a plane through
the centre in tJte circle EZHG), o-^aTpa cTrtTreSw TCT/x^/xe'n; Sia TOV
Kara TOV EZH0 KVK\OV.

Prisms and pyramids.

A prism is Trpur/xa, a pyramid Trvpaiu's. As usual, dvaypdfaiv diro
is used of describing a prism or pyramid on a rectilineal figure
as base; thus let a prism be described on tJte rectilineal figure
(as base) dvaycypa^Sw airo TOV cv^vypd/xttov Trpto'/xa, on the polygon
circumscribed about the circle A let a pyramid be set up awo TOV Trept
fov A KVicAov Trepiyeypatt/xevov TroXvyoJvov Trvpaiu? avco-ToiTco di/ayeypax*-
fjuevrj. A pyramid with an equilateral base ABF is irvpaius iVoVXevpov
cxovaa jSdcrtv TO ABF.

The surface is, as usual, eVt^aveta and, when any particular face
or a base is excluded, some qualifying phrase has to be used.

THE TERMINOLOGY OP ARCHIMEDES. clxv

Thus the surface of the prism consisting of the parallelograms
(i.e. excluding the bases) y rudi/ta TOV TrptayxaTos 17 CK TWV
TrapaAA^Xoypa/Afuov o"vyKifjivrj \ the surface (of a pyramid) excluding
the base or the triangle AEF, 17 cTrt^avcta x<opt? '"J 5 /Jewrews or TOV
AEF rpiywov.

The triangles bounding tfie pyramid ra Trepte^ovTa rpiycova rijv
(as distinct from the base, which may be polygonal).

Cones and solid rhombi.

The Elements of Euclid only introduce right cones, which are
simply called cones without the qualifying adjective. A cone is
there denned as the surface described by the revolution of a right-
angled triangle about one of the sides containing the right angle.
Archimedes does not define a cone, but generally describes a right
cone as an isosceles cone (KWVOS to-oo-KcA^s), though once he calls it
right (o/j0os). J. H. T. Miiller rightly observes that the term
isosceles applied to a cone was suggested by the analogy of the
isosceles triangle, but T doubt whether such a cone was thought of
(as he supposes) as one which could be described by making an
isosceles triangle revolve about the perpendicular from the vertex
on the base ; it seems more natural to connect it with the use of
the word side (irXevpa) by which Archimedes designates a generator
of the cone, a right cone being thus directly regarded as a cone having
all its legs equal. The latter supposition would also accord better
with the term scalene cone (KWI/OS o-KaXryvds) by which Apollonius
denotes an oblique circular cone; such a cone could not of course
be described by the revolution of a scalene triangle. An oblique
circular cone is simply a cone for Archimedes, and he does not
define it ; but, while he speaks of finding a cone with a given
vertex and passing through every point on a given ' section of an
acute-angled cone ' [ellipse], he regards the finding of the cone as
being equivalent to finding the circular sections, and we may
therefore conclude that lie would have defined the cone in
practically the same way as Apollonius does, namely as the surface
described by a straight line always passing through a fixed point
and moving round the circumference of any circle not in the same
plane with the point.

The vertex of a cone is, as usual, Kopv(f>rj, the base /?cum, the axis
ao>v and the height vif/os ; the cones are of the same Iwight ctorlv oi
VTTO TO avTo tyo?. A generator is called a side (irAcvpa) ; if a

INTRODUCTION.

cone be cut by a plane meeting all the generators of the cone ei K<X
jccovo? cTTHrcSo) TfJLa&fi avfjimirTOVTi munu? rais TOV Kuvav irXcvpais.

The surface of the cone excluding the base y eTrt^aVcia TOV KWOV
^S /?a<ra>9 ; the conical surface between (tivo generators) AA, AB,
cVi<^avcta 17 fj.Tav TWV AAB.
There is no special name for what we call a frustum of a cone
or the portion intercepted between two planes parallel to the base ;
the surface of such a frustum is simply the surface of the cone
betiveen the parallel planes TJ Vt(/>aVcia TOV ican/ov p.Tav run/

A curious term is segment of a cone (cwro'r/ia/ia KWI/OV), which is
used of the portion of any circular cone, right or oblique, cut off
towards the vertex by any plane which makes an elliptic and not a
circular section. With reference to a segment of a cone the axis
(<x<ov) is defined as the straight line drawn from the vertex of the
cone to the centre of the elliptic base.

As usual, avaypdfaw a?ro is used of describing a cone on a circle
as base. Similarly, a very common phrase is dirb TOV KVK\OV KUJVOS
IOTW let there be a cone on the circle (as base).

A solid rhombus (pd/A/3os orcpco's) is the figure made up of two
cones having their base common, their vertices on opposite sides of
it, and their axes in one straight line. A rhombus made up of
isosceles cones pd/A/}o? c to-oo-KeXcov KWI/COI/ <nryKt/Ai/o5, and the two
cones are spoken of as the cones bounding the rhombus ot KUJVOL ol
TOV pd/x/tov.

Cylinders.

A right cylinder is jcvXivSpos op0ds, and the following terms
apply to the cylinder as to the cone : base /JaVts, one base or the
other 77 ercpa pdcris, of which the circle AB is a base and FA opposite
to it ov /?a<rts /xcv 6 AB /cu/cXo?, aTrcravrtov 8c 6 FA ; axis a<Df, height
VJ/TOS, generator TrXcvpa. The cylindrical surface cut off by (two
generators) AF, BA, rj a7roTfAvofAvrj KvAtvSptKT) e?rt<^aVeia VTTO TWV AF,
BA ; the surface of the cylinder adjacent to the circumference ABF, 77
firi<f>dvia TOV KvXivSpov rj Kara TTJV ABF 7rcpi<f>pciav denotes the
surface of the cylinder between the two generators drawn through
the extremities of the arc.

A frustum of a cylinder TO/HOS nvXtvopov is a portion of a
cylinder intercepted between two parallel sections which are elliptic
and not circular, and the axis (ao>v) of it . is the straight line

THE TERMINOLOGY OF ARCHIMEDES. clxvii

joining the centres of the two sections, which is in the same straight
line with the axis of the cylinder.

Conic Sections.

General 'terms are KWIKCL oroi^cta, elements of conies, ra jcuvucd
(the theory of) conies. Any conic section KWVOV roxtiy OTTOKZOW.
Chords are simply cv^et'cu iv ra rov KVVOV ropf ay/xewu. Archimedes
never uses the word axis (da>v) with reference to a conic ; the axes
are with him diameters (8id/xTpoi), and Std/xcrpo?, when it has
reference to a complete conic, is used in this sense exclusively. A
tangent is Tn\l/avovo-a or c^aTrrofxc'i/T; (with gen.).

The separate conic sections are still denoted by the old names ;
a parabola is a section of a right-angled cone opfloyow'ov KWVOV TO/AT/,
a hyperbola a section of an obtuse-angled cone ci/x/?A.vya>viov KOJVOV
TO/X?;, and an ellipse a section of an acute-angled cone o&yow'ov KCWOV
TO/XT?.

The parabola.

Only the axis of a complete parabola is called a diameter, and
the other diameters are simply lines parallel to the diameter. Thus
parallel to the diameter or itself the diameter is Trapa rav Sid/xcrpov %
avra 8id/x6Tpos; AZ 15 parallel to the diameter a AZ Trapa rav
&a'/xTpoV corn. Once the term principal or original (diameter) is
used, applied (sc. Sia'/xcrpos).

A segment of a parabola is r/x^/xa, which is more fully described
as the segment bounded by a straight line and a section of a right-
angled cone T/xa/xa TO Treptc^o/utcyov VTTO T V0t'as KCU opOoytavtov KOJVOV
To/xa?. The word 8td/xerpos is again used with reference to a
segment of a parabola in the sense of our word axis ; Archimedes
defines the diameter of any segment as the line bisecting all the
straight lines (chords) drawn parallel to its base rav St^a rc/xvovo-av
ras c0ct'a? TrdVas ras Trapa. rav jSdaiv avrou ayottcvas.

The part of a parabola included between two parallel chords is
called a frustum rd/xos (oVo optfoyuvt'ov icoivov ro/xas ac^atp
the two chords are its lesser and greater base (c'AdWcoi/ and
)8aVis) respectively, and the line joining the middle points of the
two chords is the diameter (Sta/xcrpos) of the frustum.

What we call the latus rectum of a parabola is in Archimedes
the line which is double of the line drawn as far as the axis a SiirXaala
ras /xe'xpt TOV dfovos. In this expression the axis (a&ov) is the axis

Clxviii INTRODUCTION.

of the right-angled cone from which the curve was originally derived
by means of a section perpendicular to a generator*. Or, again, the
equivalent of our word parameter (irap* oV ovvdvrai, at aVo ras ro/xas)
is used by Archimedes as by Apollonius, meaning the straight line
to which the rectangle which has its breadth equal to the abscissa
of a point and is equal to the square of the ordinate must be
applied as base. The full phrase states that the ordinates have
their squares equal to the rectangles applied to the line equal to N (or
the parameter) which have as their breadth the lines which they (tJie
ordinates) cut off from AZ (the diameter) towards the extremity A,
Svvdvrai ra irapa rav t<rav ra N TrapaTriirrovra TrAdVos c^ovra, as aural
d7To\a(jL/3dvovri aVo Tas AZ TTOTI TO A irepas.

Ordinates are the lines drawn from the section to the diameter
(of the segment) parallel to the base (of the segment) al OLTTO Tas TOGO'S
trrl rav AZ dyo/xcvat Trapa rav AE, or simply at aVo Tas TO/xas. Once
also the regular phrase drawn ordinate-ivise TTay/xe'vo>s Karr/y/xcVry is
used to describe an ordinate, as in Apollonius.

The hyperbola.

What we call the asymptotes (at a<//.7rTa>Tot in Apollonius) are
in Archimedes the lines (apjtroaching) nearest to the section of the
obtuse-angled cone at eyyiara ras rov d/xj8Xvyo>vtov KOJVOV TO/UUS.

The centre is not described as such, but it is the point at which
the lines nearest (to tJie curve) meet TO ora/uietoi/, Ka0 f o at eyyto-Ta

This is a property of tlie sections of obtuse-angled cones TOVTO yap
ev Tats rov dfjL/3Xvy<ovLov KCOVOV To/xats

The ellipse.

The major and minor axes are the greater and lesser diameters
yu,to>v and cAao-o-wv Sta/icrpo?. Let the greater diameter be AF,
Scattcrpo? Se (awas) a /xcv ttct^cov O*T<O <^>* as Ta A, I\ The rectangle
contained by the diameters (axes) TO Trcpte^d/Ltevoi/ VTTO rav Sta/xeVpwv.
One axis is called conjugate (o-v&yys) to the other: thus let the
straight line N be equal to half of the other diameter which is
conjugate to AB, d 6t N wOtta ura CCTTW rq. ^tcrcta ras crcpas 8taju,Tpov,
a ccTTt crvfcvyr}? rq, AB.

The centre is here /cevrpov.

* Cf. Apollonius of Perga, pp. xxiv, xxv.

THE TERMINOLOGY OF ARCHIMEDES. clxix

Conoids and Spheroids.

There is a remarkable similarity between the language in which
Archimedes describes the genesis of his solids of revolution and that
used by Euclid in defining the sphere. Thus Euclid says: when, the
diameter of a semicircle remaining fixed, the semicircle revolves and
returns to the same position from which it began to move, the included
figure is a sphere cr<j>aipd COTIV, orav rjp.iKVK\iov p.vov<rrjs rrjs 8tapierpov
ircpt.VxOev TO ly/ufcvjcAiov ts TO avro traXiv aTTOKaraaraOfj, oOtv rjpfaro
</>epeo-0cu, TO 7rcpiA>;</>0i> o-xfjjjLa', and he proceeds to state that the
axis of the sphere is the fixed straight line about which the semicircle
turns a<uv 6* -rijs o-<aipas eoriv -q /xcvovcra iEv0ia, Trcpl r)v TO TJ^LKVK\IOV
<rTp<t>Tai. Compare with this e.g. Archimedes' definition of the
right-angled conoid (paraboloid of revolution) : if a section of a
right-angled cone, with its diameter (axis) remaining fixed, revolves
and returns to the position from which it started, the figure included
by the section of the right-angled cone is called a right-angled conoid,
and its axis is defined as the diameter which has remained fixed,

1 Ktt 6p6oyO)VLOV KWVOV TO/XCl /tX,VOi;0-aS TttS ia/ATpOU 7rptV^lCra

uTTOKaTaorafl]} 7raA.ii/, o#ev a>p/xa<rev, TO Trcpi\a<t>0V a^J/xa VTTO TOIS TOV
opQoywiov KCUVOV To/xas opOoyoivLov KcovoetSc? KaXticrOai, KOL aova
p.v avTov TOLV fie/xcvaKovcrav Sta/xeTpov KaAeurdat, and it will be seen
that the several phrases used are practically identical with those of
Euclid, except that <2p/txao-v takes the place of f/pa.To <f>cpo-0ai ; and
even the latter phrase occurs in Archimedes' description of the
genesis of the spiral later on.

The words conoid KtovoctSc? (o"x^xa) and spfieroid o-<^atpoctSc9
(orx^fta) are simply adapted from KWVOS and cr^aipa, meaning that
the respective figures have the appearance (cKos) of, or resemble,
cones and spheres ; and in this respect the names are perhaps more
satisfactory thaji paraboloid, hyperboloid and ellipsoid, which can
only be said to resemble the respective conies in a different sense.
But when KCDVOCI&'S is qualified by the adjective right-angled
opOoywviov to denote the paraboloid of revolution, and by a/u./3Av-
ywVtoi/ obtuse-angled to denote the hyperboloid of revolution, the
expressions are less logical, as the solids do not resemble right-
angled and obtuse-angled cones respectively ; in fact, since the
angle between the asymptotes of the generating hyperbola may be
acute, a hyperboloid of revolution would in that case more resemble
an acwte-angled cone. The terms right-angled and obtuse-angled

clxx INTRODUCTION.

were merely transferred to the conoids from the names for the
respective conies without any more thought of their meaning.

It is unnecessary to give separately the definition of each
conoid and spheroid; the phraseology is in all cases the same
as that given above for the paraboloid. But it may be remarked
that Archimedes does not mention the conjugate axis of a hyperbola
or the figure obtained by causing a hyperbola to revolve about that
axis ; the conjugate axis of a hyperbola first appears in Apollonius,
who was apparently the first to conceive of the two branches of a
hyperbola as one curve. Thus there is only one obtuse- angled
conoid in Archimedes, whereas there are two kinds of spheroids
according as the revolution takes place about the greater diameter
(axis) or lesser diameter of the generating section of an acute-
angled cone (ellipse) ; the spheroid is in the former case oblong
or^xupoetSe?) and in the latter case fiat

A special feature is, however, to be observed in the description
of the obtuse-angled conoid (hyperboloid of revolution), namely that
the asymptotes of the hyperbola are supposed to revolve about the
axis at the same time as the curve, and Archimedes explains that
they will include an isosceles cone (KWVOV tcroo-KeXca TreptXa^ovvrai),
which he thereupon defines as the cone enveloping the conoid
(Treptcxcov TO jcuwoetSc'?). Also in a spheroid the term diameter
(&afiTpos) is appropriated to the straight line drawn through
the centre at right angles to the axis (a oia rov Ktvrpov TTOT' 6p0as
dyo/xcca TO> aoi/i). The centre of a spheroid is the middle point of
the axis TO /XCTOV TOV aovos.

The following terms are used of all the conoids and spheroids.
The vertex (Kopv<f>ij) is the point at which the axis meets the surface TO
cra/Actov, Ka0* o aTrreTcu 6 aa>v Tas 7n.<^>avxs, the spheroid having of
course two vertices. A segment (T/xa/xa) is a part cut off by a plane,
and the base (/Sdo-is) of the segment is defined as the plane (figure)
included by the section of the conoid (or spheroid) in the cutting
plane TO cirittSov TO 7rcpiXa</>#v vwb Tas TOU KcovoctSco? (or a^atpoctSco?)
TO/MX? cv TW a7TOTftvovTt OTMrc'Su). The vertex of a segment is the point
at which the tangent plane parallel to the base of the segment meets
the surface, TO cra/uiW, Koff o dnreTai TO cTrtVeSov TO iri\l/avov (rov
KcovoctScos). The axis (dfav) of a segment is differently defined for
the three surfaces ; (a) in the paraboloid it is the straight line cut off
within t/ie segment from the line drawn through the vertex of the

THE TERMINOLOGY OF ARCHIMEDES. clxxi

segment parallel to the axis of the conoid d ci/a7roXa<^0i<ra cv0ta cv TO>
T/xa/xaTi awo ras dxOtfcas &a ras /copv^as TOV T/xa/xaTos Trapa TOV
aova TO /ccoi/oeiSeos, (6) in the hyperboloid it is the straight line cut
off within the segment from the line drawn through the vertex of the
segment and the vertex of the cone enveloping the conoid cwro ra$
Sta ras KOpv<j>a<; rov r/ia/xaro? Kat ra? /copulas TOT) KOJVOV TOV
TO jcwi/ociSes, (c) in the spheroid it is the part similarly
cut off from the straight line joining the vertices of tJie two segments
into which the base divides the spheroid, airb ras cv0eta? ras ras
KopiK^a? auToSi/ (TCOV r/Aa/xarcov) CTTI^ cvyvvovera?.

Archimedes does not use the word centre with respect to the
hyperboloid of revolution, but calls the centre the vertex of t/w
enveloping cone. Also the axis of a hyperboloid or a segment is
only that part of it which is within the surface. The distance
between the vertex of the hyperboloid or segment and the vertex
of the enveloping cone is the line adjacent to the axis a Troreovcra
TO) afovi.

The following are miscellaneous expressions. The part inter-
cepted ivithm the conoid of the intersection of the planes d eraTro-
Aa^>0etcra v rw /ccwoeiSci ras ycvo/^cvas ro/xas TO>V cTriTreSwi/, (the plane)
uill have cut the spheroid through its axis TCT/UUXKO? eo-o-eiVai TO
o-<cupoi85 Sia TOU aovos, so that the section it makes ivill be a
conic section OKTTC rav rop.ov Trot^Vct KWVOV To/u.ai>, let two segments be
cut off in any manner diroTTp.da-0(j> Svo TfwifLaTa cos CTV^CV or by
planes drawn in any manner cTrtWSois OTTWO-OW ay/xcvois.

Half the spheroid TO d/xtVeoi/ TOV <T</>atpoct8c'os, Aa//* <Ae /i?w
joining the vertices of the segments (of a spheroid), i.e. what we should
call a semi-diameter, d ^/xtcrca arras TCXS eTrt^cuyvvovtras Tas

The spiral.

We have already had, in the conoids and spheroids, instances of
the evolution of figures by the motion of curves about an axis. The
same sort of motion is used for the construction of solid figures
inscribed in and circumscribed about a sphere, a circle and an
inscribed or circumscribed polygon being made to revolve about
a diameter passing through an angular point of the polygon and
dividing it and the circle symmetrically. In this case, in Archimedes'
phrase, the angular points of the polygon will move along tJie circum-
ferences of circles, at ywvtat KaTa KVK\MV Trepi^cpcuov fax/fajo-ovTat (or

clxxii INTRODUCTION.

oio-ftjo-ovrcu) and the sides will move on certain cones, or on the surface
of a cone Kara TWW moi/cov cv\6ij(rovTat, or Kar 67rt^avtas KOJI/OV ; and
sometimes the angular points or the points of contact of the sides of
a circumscribed polygon are said to describe circles ypd<ovo-t KVK\OVS.
The solid figure so formed is TO ycvrjOw crrpov cr^^a, an ^ ^ ^ ie
sphere by its revolution make a figure Trepicvex^ura >/ o-<aipa 7roitTa>

For the construction of the spiral, however, we have a new
element introduced, that of time, and we have two different uniform
motions combined ; if a straight Hue in a plane turn uniformly
about one extremity which remains fixed, and return to the position
from which it started and if, at the same time as the line is revolving,
a point move at a uniform rate along the line starting from the fixed
extremity, the point will des&ribe a spiral in the plane, et Ka v0ta...ev

7Tl7T'Sa>.../>ieVoi/TOS TOU CTCpOV 7TpaTOS (ZUTttS l<TOTa\<JS TTtpltVtxPtivOL

dTroicaTaoTaflfl 7rdA.iv, o0v <opju.ao-V, a/xa S ra ypa/x/xa Treptayo/AcVa
<f>pTJTai TI <ra.fJLiov itroTa^ecDS avro cavra! Kara ras u0tas apdp.vov CLTTO
rov /LICVOVTOS 7Tparo5, TO <rafjLiov cA.t/ca ypai/^ft iv TO) cTrtWSo).

The spiral (described) in the first, second, or any turn is a c\i a v
Ta TrpoJra, SeuTcpa, or oTroiaovv Trcpt^opa yeypa/Lt/xeVa, and the turns
other than any particular ones are 'the other spirals at aAAai cXtKc?.

The distance traversed by the point along the line in any time is
a tvOtia a Stawo-^etcra, and the times in which the point moved over the
distances 01 xpovoi, iv ois TO aa^tov TCLS ypa/A^utu,? iiropevBvj ', in the titne
in which the revolving line reaches AT from AB, iv & \povw a Trepiayo/xeVa
ypa/x/xa aTro Tas AB CTT! TOLV AF d<^tKVtTai.

The origin of the spiral is ap^a Tas eAtKo?. <A<j initial line ap\a Tas
7Tpi</)opa5. The distance described by the point along the line in
the first complete revolution is v0ta TrpwTa (/wvtf distance), that
described during the second revolution the second distance cv0ia
Sfvrepa, and so on, the distances being called by tfw number of the
revolutions 6/AuW/zo>s Tats 7rept^>opats. The first area y ^wptov TrpwToi/,
is <Ae ara bounded by the spiral described in the first revolution and
by tJte f first distance ' TO ^wptov TO TrcptXa^^ev VTTO T Tas IXtKos Tas ev
TCI TrpwTtt 7Tpt<f>opa ypa<^tVas Kat Tas eu^ta?, a CO-TII/ TrpcoTa ; the second
area is that bounded by the spiral in the second turn and the * second
distance,' and so on. Tlw area added by the spiral in any turn is TO

X<l>plbv TO 7TOTtA.a<$V V7TO TUS cAtKOf V TtVt TTCpt^Op^L.

The ^/ir^ circfe, KVKXOS TrpwTo?, is the circle described with the
* first distance 7 as radius and the origin as centre, the second circle

THE TERMINOLOGY OF ARCHIMEDES. clxxiii

that with the origin as centre and twice the 'first distance* as
radius, and so on.

Together with as many times the whole of the circumference of the
circle as (is represented by) t/ie number less by one titan (that of)
the revolutions ptQ* oXas ras TOV Kv/cXov Trept^epcta? TocravraKis AO./A-
/Javo/uVas, ocro? c'orlv 6 ivl cXao-o-wv apitfftos rdv 7Tpopav, the circle
called by the number corresponding to that of the revolutions 6 KVK\OS
6 Kara TOV avrbv dpiOfJibv Xcyo'/xei/os TCUS 7Tpt<^opat?.

With reference to any radius vector, the side which is in the
direction of the revolution is forward ra rrpoayou/^va, the other
backward ra c7ro/Ai/a.

Tangents, etc.

Though the word aTrro/xat is sometimes used in Archimedes of a
line touching a curve, its general meaning is not to touch but simply
to meet] e.g. the axis of a conoid or spheroid meets (cwrTCTcu) the
surface in the vertex. (The word is also often used elsewhere than
in Archimedes of points lying on a locus ; e.g. in Pappus, p. 664, tlve
point will lie on a straight line given in position a^crai TO <rr}fjiclov
00-i ScSo/utcVr/s tvSci'as.)

To touch a curve or surface is generally <a7rro-0 ABF KVK\OV <^a7rTo/xi/at ^tfaxrai/; if straight lines be drawn
touching the circles lav a^<3o-tV rives 7rii^avov<rai TWV KVK\CDV. The
full phrase of touching without cutting is sometimes found in
Archimedes ; if a plane touch (any of) the conoidal figures
without cutting the conoid ci KCL T<OV KovoctScW (r^/iarcov cTrwrcSov
c^aTTT^Tat fjiYf TC/XVOV TO Kcoi/oftSe?. The simple word i/ravciv is
occasionally used (participially), the tangent planes ra iirtire&a ra
\l/avovra.

To touch at a point is expressed by Kara (with ace.) ; the points
at which the sides... touch (or meet) the circle <n?/mcca, *ca0' a aTrrovrat
TOV KVK\OV at irXcvpat.... Ze^ </iem touch the circle at tJie middle
points of the circumferences cut off by the sides of the inscribed
polygon CTrii^averwo-av TOV KVK\OV Kara ftcicra TO>V 7rept<pci<3> TO>K
aTTOTe/xvo/xeVwv wo TOV cyycypa^i/xei/ov TroXvyaivov 7rA.rup(ov.

The distinction between eTrt^avctv and aTrro/xat is well brought
out in the following sentence; &u lAal Md planes touching the
spheroid meet its surface at one point only we shall prove on Sk

TOL im\l/avovra CTrnreSa rov ox^aipoctSe'o? *ca0* cv p,6vov airroWai cra/Xtov
ras cTTt^avcfa? avrov

jTAe point of contact

Tangents dratvn from (a point) dy/u'vat CXTTO; we find also the
elliptical expression euro TOU H </>a7rT<r0<o 17 OHT, e OHII be the
tangent from H, where, in the particular case, H is on the circle.

Constructions.

The richness of the Greek language in expressions for con-
structions is forcibly illustrated by the variety of words which
may be used (with different shades of meaning) for draining a
line. Thus we have in the first place ayo> and the compounds
Sia'yw (of drawing a line through a figure, with ets or V following,
of producing a plane beyond a figure, or of drawing a line in a
plane), Karayo) (used of drawing an ordinate doivn from a point on
a conic), 7rpocrayo> (of drawing a line to meet another). As an
alternative to Trpoorayw, 7rpo<r/?aAX is also used ; and irpocrmirTu
may take the place of the passive of either verb. To produce is
K/?a'XXo>, and the same word is also used of a plane drawn through a
point or through a straight line ; an alternative for the passive is
supplied by cKTrnrra). Moreover TrpoorKcifwu is an alternative word
for being produced (literally being added).

In the vast majority of cases constructions are expressed by the
elegant use of the perfect imperative passive (with which may be
classed such forms as yeyoveVo) from yiyvo/xcu, coro) from CI/AI, and
Kur0<o from Ki/buu), or occasionally the aorist imperative passive.
The great variety of the forms used will be understood from the
following specimens. Let BF be made (or supposed) equal to A,
Ktcr0o> TO) A LVOV TO BF ; let it be drawn faOu, let a straight line be
drawn in it (a chord of a circle) So/'xfloi TIS cts avrbv cu0ta, let KM be
drawn equal to... i<nj KanJxOv ^ KM, let it be joined cire&vxdco, let
KA be drawn to meet Trpocr/Jc/JA^o-fla) T; KA, let them be produced
'K/?/?\i70-0axrav, suppose them found cvpijcrflaxrav, let a circle be set out
KKur0a> KVKA.OS, let it be taken cfAr^fla), let K, H be taken coraxrav
ciAij/x/uicVai at K, H, let a circk ty be taken \\a<^0u> KVK\OS cV <S TO ^, let
it be cut TCTpqcr6<a t let it be divided Statp^o-da) (o^pi/'o-dw) ; let one cone be
cut by a plane parallel to the base and produce the section EZ, rp^O^r^ 6

CTCpO? K<5vOS CTTtTTcSa) TTttpaXX^Xo) Tj} /^CUTCl Kttt TTOiCtTO) TO/X7/V T7/V EZ, /0

TZ 6c cw< q/P aTToXcXa^^o a TZ ; fee (wc/t an a^r^) 6e fo# anc? ^6 i
6e NHF, XcXct</>^a) Kal corai 17 wo NHF, fe< a figure be made

THE TERMINOLOGY OF ARCHIMEDES. clxXV

let the sector be made COTOJ ycycviy/xcyos 6 TO/XCTJS, let canes be
described on t/te circles (as bases) avaycypa^cocrav oVo TWV KVK\UXV
KuJi/oi, CXTTO TOV KvVcXov Kwvos OTa>, Ze it be inscribed or circumscribed
yyypa<0a> (or eyycypa/xfiei/ov COTCD), ?rcptyeypa00a> ; let aw area (equal
to that) of AB 6e applied to AH,- 7rapa/3j3AiJ(70a> irapct rav AH TO x<*>pi'ov
TOV AB ; e a segment of a circle be described on K, CTT! -nys 0K
KvVcXov r/x^/xa <^OTao > 0a>, Ze tfAe circle be completed avaTrcTrXrjpucrOo) 6
NH (a parallelogram) be completed o~v/A7T7rX>7pwV0(i> TO NS,
roirfcrB^ let the rest of the construction be the same as
before TCL aAAa *caT07Cvcur0a> TOV avVoi/ rpoirov TOIS TrpoVepoi/. Suppose
it done ycyoverto.

Another method is to use the passive imperative of voc'cu (e i< be
conceived). Let straight lines be conceived to be drawn

i, let the sphere be conceived to be cut vour0a> 17
let a figure (generated) from tfw inscribed polygon be
conceived as inscribed in tlw sj)here (XTTO TOV TroXvyalvov TOV eyypa^o-
/icvov voi<rOw TL cts Tirjv &<l>cupav cyypac^ci/ (rxfjfjia. Sometimes the
participle for drawn is left out ; thus air aurov votiaOw eTr^avcia let
a surface be conceived (generated) from it.

The active is much more rarely used ; but we find ( 1 ) cav with
subjunctive, if we cut lav TC/XCJ/XCV, if we draw lav ayayw/xcv, if you
produce lav lK/3a\fjs; (2) the participle, it is possible to inscribe... and
(ultimately) to leave Swarov earn/ 6yypa^>ovTa...Xt7rtv, if ice con-
tinually circumscribe polygons, bisecting the remaining circumferences
and drauring tangents, we shall (ultimately) leave act 81; 7repiypa'<ovres
7roXvya)i/a St^a T/x/o/xVu)V TCOJ/ TreptXetTro/xci/o)!/ Trcpt^cpciwi' Kat ayoftcvwi/
e^aTTTo/jLcvcov Xcu/^o/xcv, i< w possible, if we take the area..., to inscribe
Xa/?o'i/Ta (or Xa/x/JaVovra) TO x a >P t ' l/ ---8 VI/aTO/ / <rrv...yypai/rai; (3) the
first person singular, / a&e fafo straight lines \afjL/3dvtt> 8uo cv^cta?,
/ ^ooX; a straight line cXajSov Ttva cv^eiav ; / rfraw M/?'om parallel
to AZ, aya> aTro TOV Tav 0M TrapaXX^Xoi/ Ta AZ, having drawn FK
perpendicular, I cut off AK e^ta^ <o FK ayaywv KCL#CTOV TCLV FK TGL
FK to-ar aTT&afiov rav AK, / inscribed a solid figure... and circum-
scribed another cveypai^a ax>J/uia o"Tpov.,.Kat aXXo 7Tpicypai//a.

The genitive of the passive participle is used absolutely,
cvpc0cVro? 817 t oei?^ supposed found, yypa</>eVros 817 (^Ae figure)
being inscribed.

To make a figure similar to one (and equal to another) 6/xotakrat,
to find experimentally opyavticcfc \aptlv, to cut into unequal parts cis
avicra

Clxxvi INTRODUCTION.

Operations (addition, subtraction, etc.).
1. Addition, and sums, of magnitudes.

To add is Trpoort'&y/u, for the passive of which 7rpo<nca//.at is often
used; thus one segment being added evos r/Aajnaros TroriTcfoWos, the
added (straight line) a TroTtKct/^cra, let the common HA, ZF be added
jcotvat fl-pooWo-flawrav at HA, ZF ; the words are generally followed
by Trpo's (with ace. of the thing added to), but sometimes by the
dative, that to which the addition was made <S TTOTCTC'^.

For being added together we have trvvriO co-Oat, ; thus being added
to itself o*wTi0/jivov avro laura), added together cs TO avro cruvr0VTa,
itself (continually) imcrvvTiBifjiwov eavrw.

are commonly expressed for two magnitudes by owa/A<d-
Tcpos used in the following different ways ; the sum of BA, AA
os 17' BAA, lA0 sum of AF, FB o-vi/a/x^orcpos 77 AF, FB, ^e
of the area and the circle TO <rvva.fji<l>6Tcpov o TC KVK\OS Kal TO
\<apiov. Again for sums in general we have such expressions as ^e
line which is equal to both the radii 97 on? a/x<oTpcus TCUS CK TOV
Kcvrpov, the line equal to (the sum of) all the lines joining y iarrj
TrdVcus rats 7ri^vyvvovo-at9. Also all the circles ot TTCIVTCS KVK\OL
means the sum of all the circles ; and <TvyKira.i IK is used for is
equal to the sum of (two other magnitudes).

To denote plus /XCTGL (with gen.) and ow are used ; toy ether with
the bases /ACTOI ruv ^Sao-ewv, together with half the base of the segment
<nw rrj ^/uo-cia -ny? TOV T/xif/xaTos ^ao-ccos ; T and Kat also express the
same thing, and the participle of Trpoo-Aa/A/JdVo) gives another way of
describing having something added to it', thus the squares on (all)
the lines equal to the greatest together with the square on the greatest...
is TCI TTpdywva TO. oVo TO> laav ra /xcyibra TroTtXa/ijSavovra TO TC aVo
Tas fieyarTas TTpaya)i/ov....

2. Subtraction and differences.

To subtract from is d^atpetv aVo ; i/ 1 (<Ae rhombus) be conceived as
taken away lav vo^Brj a^T/pTy/xcVos, let the segments be subtracted
a<(up0Wa>v Ta Tfujfjuara. Terms common to each side in an
equation are Koiva ; the squares are common to both (sides) KOIVO. cvn
eKaTcpow Ta TTpay<ova. Then let the common area be subtracted
is KOIVOV d<t>i)pr}(r6(D TO \<apiov, and so on ; the remainder is denoted
by the adjective Xowro's, e.g. the conical surface remaining \ourrj 17

The difference or excess is vTrcpoxv, or more fully the excess by

THE TERMINOLOGY OF ARCHIMEDES. clxxvii

which (one magnitude) exceeds (another) VTTC/DO^T;, jj vTrc/ae^c... or
vTrepoxa, a /iiv cori.... The esecess is also expressed by means of
the verb vVepexeiy alone ; let the difference by which the said triangles
exceed the triangle AAF be 0, w 817 vircp\t, TO, dprjfAcva rpiywva TOV
AAF rpiyuvov lorco TO 0, to exceed by less than the excess of the cone
ty over the fialf of the spheroid v-n-cpcxctv cAcurcrovt r} <S (or dAt'icw)
vTTp*xi 6 ^ KOJJ/OS rov ^/xtVcos TOV o^tupociSco? (where <5 v7Tp\t m ay
also be omitted). Again the eojces* may be <5 /jttW com' The
opposite to vTrepex^ is AeizrcTai (with gen.).

/fyua to foaice a certain excess tou 8vo-iv vVcpoxais, with which
equal to one excess, ura ftta vTrcpo^ot, is contrasted.

The following sentence practically states the equivalent of an
algebraical equation; the rectangle under ZH, HA exceeds the rect-
angle under ZE, EA by the (sum of) the rectangle contained by EA,
EH and the rectangle under ZE, HE, wrcpcxct TO VTTO rav ZH, HA TOV
VTTO rav ZE, EA TU) T VTTO rclv HA, EH Trepi^o/xeVo) /cat TW VTTO Tar ZE,
HE. Similarly fowce PH together with IIS is (equal to) the sum of
2P, PII, 8vo p.V al PH jxcTa T<S IIS o*vva/x<^>OTpos co-Tti/ a SPH.

3. Multiplication.

To multiply'is 7ro\\a7rXao-ia^<o ; multiply one another (of numbers)
TroAAaTrAao-ta'&ii' aAAaAovs ; to multiply % a number is expressed by
the dative ; let A be multiplied by 7re7roXAa7rAao-iao*0o> 6 A TO) .

Multiplied in^o is sometimes cm (with ace.) ; thus the rectangle
H, A into 0A (i.e. a solid figure) is TO VTTO TO>V H0, A irt

4. Division.

To divide Sicupeiv ; /etf i< 6e divided into three equal parts at the
points K, 0, SiT/p^crflo) et? Tpta tcra Kara TGI K, cra/xcta ; ^o 6e divisible
VTTO.

Proportions.

A ra^io is Xoyos, proportional is expressed by the phrase tn
proportion avaXoyov, and a proportion is aVoXoyto, We find in
Archimedes some uses of the verb Acyco which seem to throw light
on the definition found in Euclid of the relation or ratio between
two magnitudes. One passage (On Conoids and Spheroids, Prop. 1 )
says if tJie terms similarly placed have, two and two, the same ratio
and the first magnitudes are taken in relation to some other mag-
nitudes in any ratios whatever et *a KOTOI 8vo TOP avrov Aoyov

clxxviii INTRODUCTION.

TO. 6/Ot'a>s TCTay/Acva, Xcyifrai 8e TO, ?rpa>Ta /u,cyc0ea TTOTI nva aXXa
fiye'0ea...ev Xo'yois oirotowrow, i/* A, B... 6e wz. relation to N, B... &tt$
Z 6e wo m relation to anything (i.e. has no term corresponding to
it) ? *ca... TOI ftev A, B,... Xeywvrai irorl ra N, B,... TO o* Z
iroff ev Xcyifrai.

A mean proportional between is /xeo-r/ a'vaXoyov T<OV..., ts a
proportional between /ACO-OV Xoyov e^ci -n^s...Kat T^S..., tfwo ?nee
portionals Svo /Liccrai avaXoyov with or without Kara TO o~vv;(S in
continued proportion.

If three straight lines be proportioned eav Tpeis cv^etat avaXoyov
cSo-t, a fourth proportional rerdpra ava'Xoyov, if four straight lines be
proportional in continued proportion * KOL Tco-crapcs ypa/x/xai avaXoyov
lo)VT4 ev TCX o-vvXt avaXoyia, a^ <7ie jooin^ dividing (tfie line) in the
said proportion Kara rav avaXoyov To/xdv Ta ctp^eva.

The ra^o o/ 1 one straight line to another is e.g. o* T^S PA Trpos AX
Xo'yos or o* (Xo'yos), ov lxt ^ PA TT/JOS T^V AX ; the ratio of tlie bases 6
T<OV ^ao"io)v Xo'yos ; has the ratio of 5 to 2 Xoyov Ixet, ov TTC'VTC Trpos
Svo.

For having the same ratio as we find the following constructions.
Have the same ratio to one anotlier as the bases TOV avrov cfxovTt Xoyov
TTOT' aXXaXous Tat? /3d<rc<riv, as the squares on the radii ov at IK TWV
KvTpa>v 8vva/Lit ; TA has to PZ /*6 (linear) ratio which the square on
TA has to the square on H, ov l\ti Xoyov 17 TA ?rpos T^V H 6Wafit,
TOVTOV c^et TOV Xoyov 17 TA Trpos PZ /A^KCI. / divided in the same
ratio ts TOV avrov Xoyov TCT/xijTat, or simply o^oicos; wt7 divide the
diameter in t/ie proportion of the successive odd numbers, unity
corresponding to the (part) adjacent to the vertex of the segrnent rav

Sia/ACTpOV T/JlOl5vTl CIS TOVS T<OV ^S TTCptO'O'WV aplOfJLWV XoyOVS, VOS

Xcyo/xevov TTOTI TOI K0pv(f>a TOV Tftd/xaros.

To Aat?e a ^es (or greater) ratio than is l^etv Xoyov eXao-crova (or
fict^ova) with the genitive of the second ratio or a phrase introduced
by ij ; to have a less ratio than the greater magnitude has to the less,
I^civ Xoyov cXdWova rj TO /xtov /teyc^os ?rpos TO IXaacrov.

For duplicate, triplicate etc. ratios we have the following
expressions : lias the triplicate ratio of the same ratio TpiTrXao-tova
Xoyov Jxi TOT) avTov Xoyov, Aas tlw duplicate ratio of EA to AK
StTrXao-tbva Xo'yov \i ^Trcp y EA -jrpos AK, are in the triplicate ratio
of the diameters in the bases ev TptTrXao-tovt Xoya> ctcrt ro>v cv Tats
jSacrco-i &a/xTpa>v, sesquialterate ratio jJ/uoXtos Xoyos. With these
expressions must be contrasted the use of double, quadruple etc.

THE TERMINOLOGY OF ARCHIMEDES. clxxix

ratio in the sense of a simple multiple by 2, 4 etc., e.g. if any
number of areas be placed in order, each being four times the next ct
Ka xupta Te0a>m $Js OTTOOUOVV w T<5 TCTpairXacrion Xoyw.

The ordinary expression for a proportion is as A is to B so is P
to A, <Js ?? A wpos rijv B, ovro>? 17 P ?rpos TT)V A. Ze AE be made so
that AE is to PE as the sum of A, AE is to AE, 7r7roM?V0a>, ws
(rwajn^oTepos 17 A, AE Trpos TI/V AE, OVTCDS 17 AE Trpos PE. The
antecedents are ret ijyovficva, tfAe cofwegwewte ra CTo/xcva.

For reciprocally proportional the parts of avrurtirovOa are used ;
the 6ase are reciprocally proportional to the heights avTiireirovOao-iv
at /3aVcts rat? v^(rtv, to 6e reciprocally in the same proportion
avTwr7rov0//,V Kara TOV avrov Xoyov.

A ra^o compounded of is \oyos crvi^/A/Aei^os (or <nryKt/Avos) c TC
Toi5...KaJ rov...; ^Ae ratio of PA to AX is equal to that compounded of
6 rrjs PA Trpos AX Xoyos crvvrj-rrra.!. /c.... Two other expressions for
compounded ratios are 6 TOV oVo A TT/DOS TO CITTO B Kat 6 (or
Trpoo-Aa/Swi'Toi/) iT75 A TT/OOS B, ^Ae ra^'o q/" the square on A
the square on B multiplied by the ratio of A to B.

The technical terms for transforming such a proportion as
a : b = c : d are as follows :

1. va\\d alternately (usually called permutando or alternando)
means transforming the proportion into a : c = b : d.

2. oVcwraXiv reversely (usually invertendo), b : a = d : c.

3. <n5i/0(ris Xoyov is composition of a ratio by which the ratio
a : b becomes a + b : b. The corresponding Greek term to com-
ponendo is owtfcm, which means no doubt literally "to one who
has compounded," i.e. "if we compound," the ratios. Thus <rvv0cVn
denotes the inference that a + b : b = c + d : d. Kara <rvv0ccrw is also
used in the same sense by Archimedes.

4. 8iatpeo*is Xoyov signifies the division of a ratio in the sense of
separation or subtraction by which a : b becomes a b : b. Similarly
SieXoVn (or Kara Statpco^tv) denotes the inference that a - 6 : 6 =
c - d : d. The translation dividendo is therefore somewhat mis-
leading.

5. aVaorpo(/>77 Xoyov conversion of a ratio and aVaoTpfyai/Tt
correspond respectively to the ratio a : a 6 and to the inference
that a:a 6 c : c d.

clxxx INTRODUCTION.

6. 81* to-ov ex aequali (sc. distantia) is applied e.g. to the
inference from the proportions

a : b : c : d etc. = A : B : C : D etc.
that a : d = A : D.

When this dividing-out of ratios takes place between proportions
with corresponding terms placed crosswise, it is described as 8e,' to-ov
v r{j rcTapayfjLevy oVoXoyia, ex- aequali in disturbed proportion or
dvo/AotW Tc!>i/ Xoya)i> TTay/xi>G)v the ratios being dissimilarly placed ;
this is the case e.g. when we have two proportions

a:b=: C,

b :c = A :7?,

and we infer that a : c = A : C.

Arithmetical terms.

Whole multiples of any magnitude are generally described as the
double of, the triple of etc., 6 SwrXdVios, 6 rpwrXao-tos K.T.X., following
the gender of the particular magnitude ; thus the (surface which is)
four times tJw greatest circle in the sphere y TTpa7rXao~ia TOV /Acyterrov
KVK\OV TWV v rfj ox^atpa ; five times the sum of AB, BE together with
ten times the sum of TB, B A, a ircvTaTrXao-i'a o-vva/x^orcpov ras AB, BE
fiera ras ScxaTrXao-tas oT)vap^orpov ras FB, BA. The same multiple
as Too-avTa7r\ao'tW...6o-a7rXao-t<ov O*T6, or iVaVts 7ro\A.a7rXao'ta)i/...Kat.
^he general word for a multiple of is TroAAaTrAao-ios or iroAAaTrAao'iW,
which may be qualified by any expression denoting the number of
times multiplied ; thus multiplied by the same number TroAAairXacrto?
ra> avra> apt0fuj>, multiples according to the successive numbers
TroAAaTrAcwria Kara TOUS ^75 dpiOpovs.

Another method is to use the adverbial forms twice oYs, thrice
TP&, etc., which are either followed by the nominative, e.g. twice EA
815 $ EA, or constructed with a participle, e.g. twice taken 8ts Xa/x-
/3av6jjLvos or 8ts ctp^ei/os ; together with twice the whole circumference
of the circle /A60' oAas ras rov KVK\OV Treptc^epetas 8U Xa/xj8avo/xi/as.
Similarly the same number of times (the said circumference) as is
expressed by the number one less than (tJtat of) the revolutions
TOcravTOLKis Xa/x/favo/ACfa?, 6Vos COTIV 6 Ivl cXao-o-cuc dpt^os rav
7rcpi0opav. An interesting phrase is the following, as many times as
the line FA is contained (literally added' toyetJwr) in AA, so many times
let the time ZH be contained in the time AH, oo-a'/a? o-vyjcet'rat a PA

THE TERMINOLOGY OF ARCHIMEDES. clxxxi

cv ra A A, rocravTa/as <rvyKt'cr0a> 6 \povos 6 ZH ev TO> XP vt 9 T< ?

AH.

Submultiples are denoted by the ordinal number, followed by
; one-seventh is /88o/*ov /xcpo? and so on, one-half being however
When the denominator is a large number, a circumlocutory
phrase is used ; thus less than ^^th part of a rig/ti angle cXdrrvv rj
Statpefctcra? ra? opOas ets po" rovraii/ ti/ /uepo?.

When the numerator of a fraction is not unity, it is expressed
by the ordinal number, and the denominator by a compound
substantive denoting such and such a submultiple ; e.g. tivo-thirds
Svo TptTa/topta, three-JiftJis rpla Trefwrra/Aopta.

There are two improper fractions which have special names,
thus one-and-a-lwdf of is ^/zioAtos, one-and-a-third of eTrirpiTo?.
Where a number is partly integral and partly fractional, the integer
is first stated and the fraction follows introduced by K<U In or K<U
and besides. The phrases used to express the fact that ilie cir-
cumference of a circle is less tJian 3f but greater tfian 3$$ times its
diameter deserve special notice; (1) Trarros KVK\OV 17 Trcpt/xeTpos rf}s
Sta/xerpov TptTrXacrtW ccrri, KOL In V7Tpc\t. \a<r<rovi jj.lv irj e/5Sd/xa> ficpct
rfjs 8ia/xrpov, /xet^ovt 8e ^ SeVa /58o/xr;KocrTO/xovot9, and (2) rptTrXacrtcuv
ccrri Kai eXacrcrovt /xtv ^ cj33o/ma> /uiepct, /xctfovt Se ^ i' oa x/ /j,ciu>v. We
also have the phrase for the first part eAdWtov >/ rpiTrAao-ttov Kal

To measure fterpetj/, common measure KOLVOV jucVpov, commensurable,
incommensurable o-v/Afterpos, dav/x/^crpos.

Mechanical terms.

Mecfuinics ra /xr;xtviKa, weight /?apo? ; centre q/* gravity iccvrpov
TOU ^apeos with another genitive of the body or magnitude ; in the
plural we have either ra iccvrpa avr<5v TOU ySaptos or ra Kevrpa r<3v
fiapeuv. Ktvrpov is also used alone.

A Zever ^vyos or fvytov, <Ae horizon 6 6pio>i> ; in a vertical line is
represented by perpendicularly Kara KaOerov, thus ^e jt?oiw^ o/*
suspension and the centre of gravity of tlie body suspended are in a
vertical line Kara KaOcrov ccrri TO TC cra/xcioi/ TOV Kpepaarov KOI TO
Kevrpov rov /3apos rov /cpcyaa/xcVov. Of suspension from or a^ CK or
Kara (with ace.) is used. Ze^ ^e triangle be suspended from tlie
points B, F, Kpc/xa(r^a> TO Tptywvov CK TCOI' B, F (ra/xeiW; t/" ^Ae
suspension of the triangle BAF a< B, F 60 set free, and it be suspended
at E, &6 triangle remains in its position ct *a TOV BAF Tpiywvou a

clxxxii INTRODUCTION.

p,V Kara TO B, F fcpe/uio-t? Xv0#, *cara 8e TO E Kpe//,a<r0fl, fie'w TO*
Tptycovov, cos vw l^ei.

To incline towards PCTTCIV CTTI (ace.) ; to 50 in equilibrium
toroppoTTcti', $A0y twW 00 in equilibrium with A Ae< fast KaTc^ofiei/ov
TOV A MToppoTnfo-ei, they mil be in equilibrium at A (i.e. will balance
about A) Kara TO A lo-oppoTr^o-oviri ; AB is too great to balance F
/utcioV OTI TO AB 17 <*xrT Mroppo7rtv T<3 F. The adjective for in
equilibrium is icroppeTnys; ta ft 00 w equilibrium with tlw triangle
FAH, uroppcTre? IOTCO T<3 FAH TptyoJ^o). To balance a certain
distances (from the point of support or the centre of gravity of a
system) is GOTO TII/OW ftcuceW tcroppoTrctv.

Theorems, problems, etc.

A theorem dccop^a (from dccopciv to investigate) ; a problem
irpo/?A.i?fia, with which the following expressions may be compared,
^/w (questions) propounded concerning the figures TO. Trpojfo/SAry/xeVa
rapt TWV o^/xaTO)v, <Aese things are propounded for investigation
irpo/faAAc'TCH TaSc tf ccup^crat ; also Trpo/cci/xai takes the place of the
passive, which it was proposed (or required) to find oircp TrpocWro
cvpeiv.

Another similar word is cTrtray/xa, direction or requirement ;
thus tffo theorems and directions necessary for the proofs of them ra
0t}pTJfjLaTa Kal ra imTa.yp.aTa Ta \ptiav fyovTa ets TOLS ctTroScifi'as avrwv,
in orcfer ^Aai &e re^mremen^ may 6e fulfilled OTTOS ycvyTat. TO CTTI-
raxQcv (or cTrtray/xa). To satisfy the requirement is Trotetv TO eTrtray/xa
(either e.g. of lines in a figure, or of the person solving the
problem).

After the setting out (cK0o-is) in any proposition there follows
the short statement of what it is required to prove or to do. In
the former case (that of a theorem) Archimedes uses one of three
expressions SCIKTCOV it is required to prove, Xeyw or <a/u Sij I assert
or say; and in the second case (that of a problem) Set &/ it is
required (to do so and so).

In a problem the analysis dvaXvo-ts and syntltesis o-vvOco-is are
distinguished, the latter being generally introduced with the words
the synthesis of the problem will be as follows <rvvTcOrj<rcTai TO
irpofiXrjfjia ovro>5. The parts of the verb avaXvtiv are similarly
used ; thus the analysis and synthesis of each of these (problems) will
be given at the end IjcaTcpa 8c ravra CTT! TeXct dvaXvOijcreTai TC /cat

THE TERMINOLOGY OF ARCHIMEDES. clxxxiii

A notable term in connexion with problems is the Stoptoyio?
(determination), which means the determination of the limits within
which a solution is possible*. If a solution is always possible, the
problem does not involve a Stopt<r/j,<>9, OVK c^ct SiopioyxoV ; otherwise
it does involve it, exi Stopta/iov.

Data and hypotheses.

For given some part of the verb St'S<o/ut is used, generally the
participle Sodct?, but sometimes ScSo/Aevo? and once or twice StSo/ucvos.
Zel a ctrcfe 6e 0we?& ScSoo-fla) KVK\O?, given two unequal magnitudes
Svo /w.ey0a>v <mcra>v So0i/Ta>i', eacA o/* Ae w;o Ziwe* FA, EZ is given
ccrrtv So0Mra cKarcpa TOJV PA, EZ, &<? same raio as /ie given one
Ao'yos 6 avros ra> Sodcm. Similar expressions are the assigned ratio
o resets Aoyos, the given area TO irpoT*.0\v (or TrpoKct/mcvov) \<apiov.

Given in position 0c<ri simply (sc. SeSo/xcny).

Of hypotheses the parts of the verb vV<m0/*eu and (for the
passive) VTTOKCI/LKU are used ; with the same suppositions row avrcov
^Ae saic^ suppositions be made uTro/ccto-^a) rot ctp^/xeva,
suppositions viroTiOfycOa raSe.

Where in a reductio ad absurdum the original hypothesis is
referred to, and generally where 'an earlier step is quoted, the past
tense of the verb is used ; but it was not (so) OVK fy 8, for it tuas less
rjv yap cAaVirwv, they were proved equal aTrc&tx&jerav tcroi,/0r this has
been proved to be possible SeSeticrat yap TOVTO Swarov eov. Where a
hypothesis is thus quoted, the past tense of v7roKt/juu has various
constructions after it, (1) an adjective or participle, AZ, BH were
supposed equal torn wrcjccivro at AZ, BH, it is by hypothesis a tangent
VTTCKCITO cTri^avovo-a, (2) an infinitive, for by hypothesis it does not
cut vTreKetTo yap firj TfjLVw, the axis is by hypothesis not at right
angles to the parallel planes vVcKetro 6 ao>y p; et/xcv opOos TTOTI ra
TrapaAAaXa cTriVcSa, (3) the plane is supposed to have been drawn
through the centre TO eiriTreSov vTroKetrai 8ta TOV xcvrpov d\0ai.

Supposing it found cvpcden-os absolutely. Suppose it done

The usual idiomatic use of ct Sc /x?/ after a negative statement
may be mentioned ; it will not meet the surface in another point,
otherwise... ov yap d^crat /car* aAAo o-a/Actov ras CTri^avcias* ci Sc
/;....

* Cf. Apollonius of Perga, p. Ixx, note.

Clxxxiv INTRODUCTION.

Inferences, and adaptation to different cases.

The usual equivalent for therefore is apa; o?v and rolwv are
generally used in a somewhat weaker sense to mark the starting-
point of an argument, thus 7m ovv may be translated as since, then.
Since is CTTCI, because Sidri.

woXXoJ fjia\X.ov much more then is apparently not used in Archi-
medes, who has TroXXo) alone ; thus much kss then is the ratio of the
circumscribed figure to the inscribed than that of K to H 7roXX<3
dpa TO ircptypa^cv Trpds TO eyypox^cv \ao*o"ova Xdyov cxct TOT), ov e^ei 17
K 7rp6s H.

Sia with the accusative is a common way of expressing the
reason why; because the cone is isosceles &a TO urocriccXiJ ctvat TOV
K<ovov,for the same reason 8ta Tavrd.

8ia with the genitive expresses the means by which a proposition
is proved ; by means of tlie construction 8ta T^S Karao-Kvrj^, by the
same means Sia TWV avrwv, by the same method &a TOV avrov TpoVov.

Whenever this is the case, the surface is greater oVav TOVTO ,
/xct^cov yivcTat 17 cVi^xiveia..., if this is the case, the angle BA is
equal... , et o\ TOVTO, l<ra tvrtv a VTTO BA ywi/ta..., which is the same
thing as slwwing tliat... o ravrov cort TO Sct^at, oTi....

Similarly for the sector 6/xoiW /cat ?ri TOV TO/JLCCOS, <Ae proof
is the same as (tliat used to show) that a ama aTrdSct^ts a7Tp ^at on,
the proof that. ..is the same a avra a7rooik Ivn nal SIOTL..., the same
argument holds for all rectilineal figures inscribed in the segments in
the recognised manner (see p. 204) cm TTCLVTWI/ cvfluypa/x/^wv TWV
yypa<^o/xva>v 9 TOL T/xa/xara yvwpt/xws 6 auros Xdyos ; i^ ?^7Z 6e possible,
fiaving proved it for a circle, to transfer the same argument in
the case of the sector co-Tat cVi KvicXov SeifavTa /xeTayayctv TOI/ o/xotov
Xdyoi/ ical eVt TOV TO/LICCJS ; lAi r^ t^7Z 5s ^A<? same, %l ^ ?^i7/ ^6 the
lesser of the diameters which will be intercepted within the spheroid
(instead of the greater) TO, /xcv aXXa ra avra co-o-etrai, TOLV 8e 8ia/ATpa>v
a cXao'O'aiv co'O'ctTat a cva?roXa<^0eto < a cv T<3 o'^aipoctSct ; i^ t&tZZ maA;^
rw difference whet}ier...or...oioi<rtt, 8e ovScV, IT...ITC....

Conclusions.

J%6 proposition is therefore obvious, or is proved S^Xov ovv eon
(or ScSctKTcu) TO Trporc^eV; similarly <f>avpov ovv CCTTIV, o ISci Sct^at,
and ISei Sc TOVTO Sct'ae. IFAicA i* absurd, or impossible oVep UTOTTOV,
or dSvvarov.

A curious use of two negatives is contained in the following :

THE TERMINOLOGY OF ARCHIMEDES. clxxxv

OVK Spa OVK ecrrt Ktvrpov TOV /?apcos TOV AEZ rpiycovou TO N <ra/mcTov.
IVTW apa, therefore it is not possible that the point N should not be the
centre of gravity of the triangle AEZ. It must therefore be so.

Thus a rhombus will have been formed <rrai 817 yeyovcis po/*/?os ;
two unequal straight lines have been found satisfying the requirement-
tvprj/Acvat, i(Tiv apa Svo i>$iat avtcroi Troiovcrai TO cTrtray/xa.

Direction, concavity, convexity.

In the same direction irl ra avrd, in the other direction Ivl
TO. erepa, concave in the same direction ITTI TO. aura KOL\T) ; in the same
direction as CTTI TO. avVa with the dative or e$' a, thus in the same
direction as the vertex of tJie cone errl ra avra ra TOV KUVOV /<opv</>a,
drawn in the same direction as (that of) the convex side of it ri TO,
avra dyo/Acvat, cfi a Ivn ra Kvpra a^rov. For on the same side of CTTI
TO. avTa is followed by the genitive, they fall on the same side of the
line cin ra avTa TrtTTTOvcrt T^s ypaft^?.

On each side of *<f> cKarepa (with gen.) ; on each side of the plane
of the base <' c/cctTcpa TOV firiireoov -njs ^acrews.

Miscellaneous.

Property O-VJUTTTCD//^ Proceeding thus continually, acl TOVTO
7rotovyT5, act TovTov ycvojAtvov, or TovTov c^s yivo/xvou. In the
elements ev ry (rTotxctwo-ct.

One special difference between our terminology and the Greek is
that whereas we speak of any circle, any straight line and the like,
the Greeks say every circle, every straight line, etc. Thus any
pyramid is one third part of the prism with the same base as the
pyramid and equal height iraa-a Trvpa/xts rptrov ficpos TT! TOV Trpwr/xaTos
TOV raV avrav j8a<rtv IXOVTOS ra irvpa/xtftt KOI v^os taov. / define the
diameter of any segment as Sia'/xerpoi/ /caXca) Travros T/xd7iaTos. To
exceed any assigned (magnitude) of those which are comparable tvith
one another V7rcp\iv TravTos rov irpvrtQivros TWV Trpos a\\rj\a

Another noteworthy difference is illustrated in the last sentence.
The Greeks did not speak as we do of a given area, a given ratio
etc., but of the given area, the given ratio, and the like. Thus It is
possible... to leave certain segments less than a given area Swarov

<TTLV...\l7Tll> TiVtt TfJilj/JLaLTa, GLTTCp COTat .\aCT(TOVa TOV 7TpOKtfLVOV

^(optov ; to divide a given sphere by a plane so that the segments have
to one another an assigned ratio rav 8o0io-av er^aipav cTrtirc'Sw
cS<rre ra rp.aip.ara avra? TTOT* aAAaAa TOV ra^Oivra \6yov e^ctv.

H. A. n

INTRODUCTION.

Magnitudes in arithmetical progression are said to exceed each
other by an equal (amount) ; if there be any number of magnitudes in
arithmetical progression et KCL cuvrt /ueyc^ca OTTOO-GLOW ra> tcra) dXXaXa>v
vTrepexovTa. The common difference is the eascess virepoxa,, and the
terms collectively are spoken of as the magnitudes exceeding by the
equal (difference) rot TW or vTrcpe^ovTa.

Terms of a geometrical progression are simply m (Continued)
proportion ai/aAoyoi/, the series is then 77 avaXoyia, the proportion^
and a term of the series is TIS TUP ev T^I avra avaXoyta. Numbers in
geometrical progression beginning from unity are apifytol avaAoyov
a?ro fiovaSos. Ze ^Ae term A o/* <7ie progression be taken which
is distant the same number of terms from as A is distant from
unity \e\d<t>6<D e*c ras avaXoytas 6 A a.TT\u>v cwro rov TOCTOVTOVS, ocrovs
o A CXTTO fAOvdSos aTre^ct.

THE WORKS OF
ARCHIMEDES.

ON THE SPHERE AND CYLINDEE.

BOOK I.

" ARCHIMEDES to Dositheus greeting.

On a former occasion I sent you the investigations which
I had up to that time completed, including the proofs, showing
that any segment bounded by a straight line and a section of a
right-angled cone [a parabola] is four-thirds of the triangle
which has the same base with the segment and equal height.
Since then certain theorems not hitherto demonstrated (ai/e-
\eyicT(ov) have occurred to me, and I have worked out the proofs
of them. They are these : first, that the surface of any sphere
is four times its greatest circle (TOV p&yio-rov KVK\OV)\ next,
that the surface of any segment of a sphere is equal to a circle
whose radius (rj etc TOV icevrpov) is equal to the straight line
drawn from the vertex (tcopv<f>ij) of the segment to the circum-
ference of the circle which is the base of the segment ; and,
further, that any cylinder having its base equal to the greatest
circle of those in the sphere, and height equal to the diameter
of the sphere, is itself [i.e. in content] half as large again as the
sphere, and its surface also [including its bases] is half as large
again as the surface of the sphere. Now these properties were
all along naturally inherent in the figures referred to (avry ry
<t>v<ri, TrpovTrfipxGv 7Tpl TO, elprj/jbeva o-^/xara), but remained
unknown to those who were before my time engaged in the
study of geometry. Having, however, now discovered that the
properties are true of these figures, I cannot feel any hesitation

H. A. 1

2 ARCHIMEDES

in setting them side by side both with my former investiga-
tions and with those of the theorems of Eudoxus on solids
which are held to be most irrefragably established, namely,
that any pyramid is one third part of the prism which has the
same base with the pyramid and equal height, and that any
cone is one third part of the cylinder which has the same
base with the cone and equal height. For, though these
properties also were naturally inherent in the figures all along,
yet they were in fact unknown to all the many able geometers
who lived before Eudoxus, and had not been observed by any
one. Now, however, it will be open to those who possess the
requisite ability to examine these discoveries of mine. They
ought to have been published while Conon was still alive,
for I should conceive that he would best have been able to
grasp them and to pronounce upon them the appropriate
verdict ; but, as I judge it well to communicate them to those
who are conversant with mathematics, I send them to you with
the proofs written out, which it will be open to mathematicians
to examine. Farewell.

I first set out the axioms* and the assumptions which I
have used for the proofs of my propositions.

DEFINITIONS.

1. There are in a plane certain terminated bent lines
(icafji7rv\ai, ypa/ji/j,al TreTrepcuT/jLevai)^, which either lie wholly on
the same side of the straight lines joining their extremities, or
have no part of them on the other side.

2. I apply the term concave in the same direction

to a line such that, if any two points on it are taken, either
all the straight lines connecting the points fall on the same
side of the line, or some fall on one and the same side while
others fall on the line itself, but none on the other side.

* Though the word used is dtu/uara, the " axioms" are more of the nature
of definitions ; and in fact Eutocius in his notes speaks of them as such (tipoi).

f Under the term bent line Archimedes includes not only curved lines of
continuous curvature, but lines made up of any number of lines which may be
either straight or curved.

ON THE SPHERE AND CYLINDER I. 3

3. Similarly also there are certain terminated surfaces, not
themselves being in a plane but having their extremities in a
plane, and such that they will either be wholly on the same
side of the plane containing their extremities, or have no part
of them on the other side.

4. I apply the term concave in the same direction

to surfaces such that, if any two points on them are taken, the
straight lines connecting the points either all fall on the same
side of the surface, or some fall on one and the same side of
it while some fall upon it, but none on the other side.

5. I use the term solid sector, when a cone cuts a sphere,
and has its apex at the centre of the sphere, to denote the
figure comprehended by the surface of the cone and the surface
of the sphere included within the cone.

6. I apply the term solid rhombus, when two cones with
the same base have their apices on opposite sides of the plane
of the base in such a position that their axes lie in a straight
line, to denote the solid figure made up of both the cones.

ASSUMPTIONS.

1. Of all lines which have the same extremities the straight
line is the least*.

* This well-known Archimedean assumption is scarcely, as it stands, a
definition of a straight line, though Proclus says [p. 110 ed. Friedlein] "Archi-
medes defined (upla-a.ro} the straight line as the least of those [lines] which have
the same extremities. For because, as Euclid's definition says, { ftroi; KTCU rots
t<p (O.VTTJS <n?/xelots, it is in consequence the least of those which have the same
extremities." Proclus had just before [p. 109] explained Euclid's definition,
which, as will be seen, is different from the ordinary version given in our text-
books; a straight line is not " that which lies evenly between its extreme points,"
but "that which e tvov rots i<p eavrfy ffweiot* KCITCU." The words of Proclus
are, " He [Euclid] shows by means of this that the straight line alone [of all
lines] occupies a distance (icarlxeci' &d0Ti?/a) equal to that between the points
on it. For, as far as one of its points is removed from another, so great is the
length (fjuiytBot) of the straight line of which the points are the extremities;
and this is the meaning of r6 <? foov jteto-ftu roc; e'0* tavrrjs (rwteloi?. But, if you
take two points on a circumference or any other line, the distance cut off
between them along the line is greater than the interval separating them ; and
this is the case with every line except the straight line." It appears then from
this that Euclid's definition should be understood in a sense very like that of

4 ARCHIMEDES

2. Of other lines in a plane and having the same extremi-
ties, [any two] such are unequal whenever both are concave in
the same direction and one of them is either wholly included
between the other and the straight line which has the same
extremities with it, or is partly included by, and is partly
common with, the other; and that [line] which is included is
the lesser [of the two].

3. Similarly, of surfaces which have the same extremities,
if those extremities are in a plane, the plane is the least [in
area].

4. Of other surfaces with the same extremities, the ex-
tremities being in a plane, [any two] such are unequal when-
ever both are concave in the same direction and one surface
is either wholly included between the other and the plane which
has the same extremities with it, or is partly included by, and
partly common with, the other; and that [surface] which is
included is the lesser [of the two in area].

5. Further, of unequal lines, unequal surfaces, and unequal
solids, the greater exceeds the less by such a magnitude as,
when added to itself, can be made to exceed any assigned
magnitude among those which are comparable with [it and
with] one another*.

These things being premised, if a polygon be inscribed in a
circle, it is plain that the perimeter of the inscribed polygon is
less than the circumference of the circle ; for each of the sides
of the polygon is less than that part of the circumference of the
circle which is cut off by it."

Archimedes* assumption, and we might perhaps translate as follows, (( A straight
line is that which extends equally (<? t<rov Ketrat) with the points on it," or, to
follow Proclus' interpretation more closely, "A straight line is that which
represents equal extension with [the distances separating] the points on it. 1 '
* With regard to this assumption compare the Introduction, chapter in. 2.

ON THE SPHERE AND CYLINDER I.

Proposition 1.

// a polygon be circumscribed about a circle, the perimeter
of the circumscribed polygon is greater
than the perimeter of the circle.

Let any two adjacent sides, meet-
ing in A, touch the circle at P, Q
respectively.

Then [Assumptions, 2]

PA+AQ>(arcPQ).
A similar inequality holds for each
angle of the polygon; and, by ad-
dition, the required result follows.

Proposition 2.

Given two unequal magnitudes, it is possible to find two un-
equal straight lines such that the greater straight line has to the
less a ratio less than the greater magnitude has to the less.

Let AB, D represent the two unequal magnitudes, AB being
the greater.

Suppose BO measured along BA equal to D, and let GH be
any straight line.

Then, if CA be added to itself a sufficient
number of times, the sum will exceed D. Let
AF be this sum, and take E on GH produced
such that GH is the same multiple of HE that
AF is of AC.

Thus EH:HG = AC:AF.
But, since AF>D (or CB),

AC :AF<AC :CB.
Therefore, componendo,

EG:GH<AB:D.
Hence EG, GH are two lines satisfying the given condition.

ARCHIMEDES

Proposition 3.

Given two unequal magnitudes and a circle, it is possible to
inscribe a polygon in the circle and to describe another about it
so that the side of the circumscribed polygon may have to the side
of the inscribed polygon a ratio less than that of the greater
magnitude to the less.

Let A, B represent the given magnitudes, A being the
greater.

Find [Prop. 2] two straight lines F, KL, of which F is the
greater, such that

F:KL<A :B (1).

Draw LM perpendicular to LK and of such length that

In the given circle let CE, DG be two diameters at right
angles. Then, bisecting the angle DOC, bisecting the half
again, and so on, we shall arrive ultimately at an angle (as
NOG) less than twice the angle LKM.

Join NC, which (by the construction) will be the side of a
regular polygon inscribed in the circle. Let OP be the radius
of the circle bisecting the angle NOG (and therefore bisecting
NC at right angles, in H, say), and let the tangent at P meet
0(7, ON produced in S 9 T respectively.

Now, since CO N < 2 Z LKM,

ON THE SPHERE AND CYLINDER I.

and the angles at H, L are right ;

therefore MK:LK>OG: OH
> OP : OH.
Hence ST:CN<MK: LK

therefore, a fortiori, by (1),

8T:CN<A :B.
Thus two polygons are found satisfying the given condition.

Proposition 4.

Again, given two unequal magnitudes and a sector, it is
possible to describe a polygon about the sector and to inscribe
another in it so that the side of the circumscribed polygon may
have to the side of the inscribed polygon a ratio less than the
greater magnitude has to the less.

[The " inscribed polygon " found in this proposition is one
which has for two sides the two radii bounding the sector, while
the remaining sides (the number of which is, by construction,
some power of 2) subtend equal parts of the arc of the sector ;
the "circumscribed polygon" is formed by the tangents parallel
to the sides of the inscribed polygon and by the two bounding
radii produced.]

In this case we make the same construction as in the last
proposition except that we bisect the angle COD of the sector,
instead of the right angle between two diameters, then bisect
the half again, and so on. The proof is exactly similar to the
preceding one.

ARCHIMEDES

Proposition 5.

Given a circle and two unequal magnitudes, to describe a
polygon about the circle and inscribe another in it, so that the
circumscribed polygon may have to the inscribed a ratio less than
the greater magnitude has to the less.

Let A be the given circle and B, C the given magnitudes, B
being the greater.

E-
F-

Take two unequal straight lines D t E, of which D is the
greater, such that D : E < B : G [Prop. 2], and let F be a mean
proportional between D, E, so that D is also greater than F.

Describe (in the manner of Prop. 3) one polygon about the
circle, and inscribe another in it, so that the side of the former
has to the side of the latter a ratio less than the ratio D : F.

Thus the duplicate ratio of the side of the former polygon
to the side of the latter is less than the ratio D* : F*.

But the said duplicate ratio of the sides is equal to the
ratio of the areas of the polygons, since they are similar ;

therefore the area of the circumscribed polygon has to the
area of the inscribed polygon a ratio less than the ratio /)* : F 9 ,
or D : E, and a fortiori less than the ratio B : C.

ON THE SPHERE AND CYLINDER I.

Proposition 6.

" Similarly we can show that, given two unequal magnitude?
and a sector, it is possible to circumscribe a polygon about the
sector and inscribe in it another similar one so that the circum-
scribed may have to the inscribed a ratio less than the greater
magnitude has to the less.

And it is likewise clear that, if a circle or a sector, as well
as a certain area, be given, it is possible, by inscribing regular
polygons in the circle or sector, and by continually inscribing
such in the remaining segments, to leave segments of the circle or
sector which are [together] less than the given area. For this is
proved in the Elements [Eucl. xu. 2].

But it is yet to be proved that, given a circle or sector and
an area, it is possible to describe a polygon about the circle or
sector, siich that the area remaining between the circumference
and the circumscribed figure is less than the given area"

The proof for the circle (which, as Archimedes says, can be
equally applied to a sector) is as follows.

Let A be the given circle and B the given area.

Now, there being two unequal magnitudes A + B and A, let
a polygon (C) be circumscribed about the circle and a polygon
(/) inscribed in it [as in Prop. 5], so that

C:I<A+B:A (1).

The circumscribed polygon (C) shall be that required.

10 ARCHIMEDES

For the circle (A) is greater than the inscribed polygon (/).

Therefore, from (1), a fortiori,

C:A<A+B :A,
whence C < A + J5,

or C-A<B.

Proposition 7.

If in an isosceles cone [i.e. a right circular cone] a pyramid
be inscribed having an equilateral base, the surface of the
pyramid excluding the base is equal to a triangle having its
base equal to the perimeter of the base of the pyramid and its
height equal to the perpendicular drawn from the apex on one
side of the base.

Since the sides of the base of the pyramid are equal, it
follows that the perpendiculars from the apex to all the sides
of the base are equal ; and the proof of the proposition is
obvious.

Proposition 8.

If a pyramid be circumscribed about an isosceles cone t the
surface of the pyramid excluding its base is equal to a triangle
having its base equal to the perimeter of the base of the pyramid
and its height equal to the side [i.e. a generator] of the cone.

The base of the pyramid is a polygon circumscribed about
the circular base of the cone, and the line joining the apex of
the cone or pyramid to the point of contact of any side of the
polygon is perpendicular to that side. Also all these perpen-
diculars, being generators of the cone, are equal ; whence the
proposition follows immediately.

ON THE SPHERE AND CYLINDER I. 11

Proposition 9.

If in the circular base of an isosceles cone a chord be placed,
and from its extremities straight lines be drawn to the apex of
the cone, the triangle so formed will be less than the portion of
the surface of the cone intercepted between the lines drawn to the
apex.

Let ABC be the circular base of the cone, and its apex.

Draw a chord AB in the circle, and join OA, OB. Bisect
the arc ACB in C, and join AC, BC, OC.

Then A OAC + A OBC > A OAB.

Let the excess of the sum of the first two triangles over the
third be equal to the area D.

Then D is either less than the sum of the segments AEG,
CFB, or not less.

I. Let D be not less than the sum of the segments referred
to.

We have now two surfaces

(1) that consisting of the portion OAEC of the surface
of the cone together with the segment AEG, and

(2) the triangle OAC]

and, since the two surfaces have the same extremities (the
perimeter of the triangle OAC), the former surface is greater
than the latter, which is included by it [Assumptions, 3 or 4].

12 ARCHIMEDES

Hence (surface OAEC) + (segment AEG) > A OAC.
Similarly (surface OCFB) + (segment CFB) > A OBC.

Therefore, since D is not less than the sum of the segments,
we have, by addition,

(surface OAECFB) + D > A OA C + A OBC

> A GAB + D, by hypothesis.

Taking away the common part D, we have the required
result.

II. Let D be less than the sum of the segments A EC,
CFB.

If now we bisect the arcs AC, CB, then bisect the halves,
and so on, we shall ultimately leave segments which are
together less than D. [Prop. 6]

Let AGE, EEC, CKF, FLB be those segments, and join
OE, OF.

Then, as before,

(surface OA GE) + (segment AGE) > A OAE
and (surface OEHG) + (segment EHG) > A OEC.
Therefore (surface OAGHC) + (segments AGE, EEC)

> A OA C, a fortiori.

Similarly for the part of the surface of the cone bounded by
OC, OB and the arc CFB.

Hence, by addition,
(surface OA GEHCKFLB)+ (segments AGE, EEC, CKF, FLB)

> A OAB + D, by hypothesis.

But the sum of the segments is less than D, and the re-
quired result follows.

ON THE SPHERE AND CYLINDER I.

Proposition 1O.

If in the plane of the circular base of an isosceles cone two
tangents be drawn to the circle meeting in a point, and the points
of contact and the point of concourse of the tangents be respectively
joined to the apex of the cone, the sum of the two triangles
formed by the joining lines and the two tangents are together
greater than the included portion of the surface of the cone.

Let ABC be the circular base of the cone, its apex, AD,
BD the two tangents to the circle meeting in D. Join OA,
OB, OD.

Let EOF be drawn touching the circle at C, the middle
point of the arc ACB, and therefore parallel to AB. Join
OE, OF.

Then ED + DF>EF,

and, adding AE + FB to each side,

AD + DB > AE + EF+FB.

Now OA, OC, OB, being generators of the cone, are equal,
and they are respectively perpendicular to the tangents at A,
C,B.

14 ARCHIMEDES

It follows that

A OAD+ A ODB > A OAE+ A OEF+ A OFB.

Let the area G be equal to the excess of the first sum over
the second.

G is then either less, or not less, than the sum of the spaces
EAHC, FCKB remaining between the circle and the tangents,
which sum we will call L.

I. Let G be not less than L.
We have now two surfaces

(1) that of the pyramid with apex and base AEFB t
excluding the face OAB,

(2) that consisting of the part OACB of the surface of the
cone together with the segment ACB.

These two surfaces have the same extremities, viz. the
perimeter of the triangle OAB, and, since the former includes
the latter, the former is the greater [Assumptions, 4].

That is, the surface of the pyramid exclusive of the face
OAB is greater than the sum of the surface OACB and the
segment ACB.

Taking away the segment from each sum, we have
A OAE + A OEF+ A OFB + L > the surface OAHCKB.
And G is not less than L.
It follows that

A OAE + A OEF+ A OFB + G,

which is by hypothesis equal to A OAD + A ODB, is greater
than the same surface.

II. Let G be less than L.

If we bisect the arcs AC, CB and draw tangents at their
middle points, then bisect the halves and draw tangents, and
so on, we shall lastly arrive at a polygon such that the sum
of the parts remaining between the sides of the polygon and
the circumference of the segment is less than G.

ON THE SPHERE AND CYLINDER I. 15

Let the remainders be those between the segment and the
polygon APQRSB, and let their sum be M. Join OP, OQ,
etc.

Then, as before,

Also, as before,

(surface of pyramid OAPQRSB excluding the face OA B)

>the part OACB of the surface of the

cone together with the segment ACB.

Taking away the segment from each sum,
A GAP + A OPQ + ... -f M> the part OACB of the

surface of the cone.
Hence, a fortiori,

A OAE + A OEF+ A OFB + G,
which is by hypothesis equal to

&OAD + &ODB,
is greater than the part OACB of the surface of the cone.

Proposition 11.

If a plane parallel to the axis of a right cylinder cut the
cylinder, the part of the surface of the cylinder cut off by the
plane is greater than the area of the parallelogram in which the
plane cuts it.

Proposition 12.

If at the extremities of two generators of any right cylinder
tangents be drown to the circular bases in the planes of those
bases respectively, and if the pairs of tangents meet, the
parallelograms formed by each generator and the two corre-
sponding tangents respectively are together greater than the
included portion of the surface of the cylinder between the two
generators.

[The proofs of these two propositions follow exactly the
methods of Props. 9, 10 respectively, and it is therefore un-
necessary to reproduce them.]

ARCHIMEDES

" From the properties thus proved it is clear (1) that, if a
pyramid be inscribed in an isosceles cone, the surface of the
pyramid excluding the base is less than the surface of the cone
[excluding the base], and (2) that, if a pyramid be circumscribed
about an isosceles cone, the surface of the pyramid excluding the
base is greater than the surface of the cone excluding the base.

" It is also clear from what has been proved both (1) that,
if a prism be inscribed in a right cylinder, the surface of the
prism made up of its parallelograms [i.e. excluding its bases] is
less than the surface of the cylinder excluding its bases, and
(2) that, if a prism be circumscribed about a right cylinder, the
surface of the prism made up of its parallelograms is greater
than the surface of the cylinder excluding its bases.' 9

Proposition 13.

The surface of any right cylinder excluding the bases is equal
to a circle whose radius is a mean proportional between the side
[i.e. a generator] of the cylinder and the diameter of its base.

Let the base of the cylinder be the circle A, and make CD
equal to the diameter of this circle, and EF equal to the height
of the cylinder.

ON THE SPHERE AND CYLINDER I. 17

Let JET be a mean proportional between CD, EF, and B
a circle with radius equal to H.

Then the circle -B shall be equal to the surface of the
cylinder (excluding the bases), which we will call 8.

For, if not, B must be either greater or less than 8.

I. Suppose B<8.

Then it is possible to circumscribe a regular polygon about
JB, and to inscribe another in it, such that the ratio of the
former to the latter is less than the ratio S : B.

Suppose this done, and circumscribe about A a polygon
similar to that described about B\ then erect on the polygon
about A a prism of the same height as the cylinder. The
prism will therefore be circumscribed to the cylinder,

Let KD, perpendicular to CD, and FL, perpendicular to
EF, be each equal to the perimeter of the polygon about A.
Bisect CD in Jf, and join MK.

Then A KDM = the polygon about A.

Also O EL = surface of prism (excluding bases).

Produce FE to N so that FE = EN, and join NL.

Now the polygons about A, B, being similar, are in the
duplicate ratio of the radii of A, B.

Thus

A KDM : (polygon about B) = MD* : H*

= MD:NF
= &KDM:

(since DK = FL).

Therefore (polygon about 5) = A LFN

= EL
~ (surface of prism about A\

from above.

But (polygon about B) : (polygon in B) < S : B.
11. A. 2

18 ARCHIMEDES

Therefore

(surface of prism about A) : (polygon in B) < S : B,
and, alternately,

(surface of prism about A) : 8 < (polygon in B) : S ;

which is impossible, since the surface of the prism is greater
than S, while the polygon inscribed in B is less than B.

Therefore B^S.

II. Suppose B>S.

Let a regular polygon be circumscribed about B and another
inscribed in it so that

(polygon about jB) : (polygon in B) < B : 8.

Inscribe in A a polygon similar to that inscribed in B, and
erect a prism on the polygon inscribed in A of the same height
as the cylinder.

Again, let DK, FL, drawn as before, be each equal to the
perimeter of the polygon inscribed in A.

Then, in this case,

A RDM > (polygon inscribed in A)

(since the perpendicular from the centre on a side of the
polygon is less than the radius of A).

Also A LFN = O EL = surface of prism (excluding bases).
Now
(polygon in A) : (polygon in B) = MD* : H*,

= &KDM : &LFN, as before.
And A KDM > (polygon in A ).

Therefore

A LFN, or (surface of prism) > (polygon in B).
But this is impossible, because

(polygon about JB) : (polygon in B) < B : S,

< (polygon about B) : 8, a fortiori,
so that (polygon in B) > S,

> (surface of prism), a fortiori.
Hence B is neither greater nor less than S, and therefore

ON THE SPHERE AND CYLINDER I.

Proposition 14.

The surface of any isosceles cone excluding the base is equal
to a circle whose radius is a mean proportional between the side
of the cone [a generator] and the radius of the circle which is the
base of the cone.

Let the circle A be the base of the cone ; draw C equal to
the radius of the circle, and D equal to the side of the cone, and
let E be a mean proportional between C, D.

Draw a circle B with radius equal to E.

Then shall B be equal to the surface of the cone (excluding
the base), which we will call 8.

If not, B must be either greater or less than S.

I. Suppose B < 8.

Let a regular polygon be described about B and a similar
one inscribed in it such that the former has to the latter a ratio
less than the ratio 8 : B.

Describe about A another similar polygon, and on it set up
a pyramid with apex the same as that of the cone.

Then (polygon about A) : (polygon about B)

= (7:1)

= (polygon about A) : (surface of pyramid excluding base).

20 ARCHIMEDES

Therefore

(surface of pyramid) = (polygon about B).
Now (polygon about B) : (polygon in B) < 8 : B.
Therefore

(surface of pyramid) : (polygon in E) < 8 : B,
which is impossible, (because the surface of the pyramid is
greater than S, while the polygon in B is less than B).
Hence B { S.

II. Suppose B > S.

Take regular polygons circumscribed and inscribed to B such
that the ratio of the former to the latter is less than the ratio
B:S.

Inscribe in A a similar polygon to that inscribed in B, and
erect a pyramid on the polygon inscribed in A with apex the
same as that of the cone.
In this case

(polygon in A) : (polygon in B) = (7 s : E*

= 0:J5
> (polygon in A) : (surface of pyramid excluding base).

This is clear because the ratio of G to D is greater than the
ratio of the perpendicular from the centre of A on a side of the
polygon to the perpendicular from the apex of the cone on the
same side*.

Therefore

(surface of pyramid) > (polygon in B).
But (polygon about B) : (polygon in B) < B : S.
Therefore, a fortiori,

(polygon about B) : (surface of pyramid) < B : 8 ;
which is impossible.

Since therefore B is neither greater nor less than 8,

* This is of course the geometrical equivalent of saying that, if a, p be two
angles each less than a right angle, and a>0, then sin a>sin /3.

ON THE SPHERE AND CYLINDER I. 21

Proposition 15.

The surface of any isosceles cone has the same ratio to its
base as the side of the cone has to the radium of the base.

By Prop. 14, the surface of the cone is equal to a circle
whose radius is a mean proportional between the side of the
cone and the radius of the base.

Hence, since circles are to one another as the squares of
their radii, the proposition follows.

Proposition 16.

If an isosceles cone be cut by a plane parallel to the base, the
portion of the surface of the cone between the parallel planes is
equal to a circle whose radius is a mean proportional between (1)
the portion of the side of the cone intercepted by the parallel
planes and (2) the line which is equal to the sum of the radii oj
the circles in the parallel planes.

Let OA B be a triangle through the axis of a cone, DE its
intersection with the plane cutting off the
frustum, and OFC the axis of the cone.

Then the surface of the cone OAB is
equal to a circle whose radius is equal to
V(M .AC. [Prop. 14.]

Similarly the surface of the cone ODE
is equal to a circle whose radius is equal
to *JOlTDF.

And the surface of the frustum is
equal to the difference between the two circles.

Now

OA.AC-OD.DF = DA.AC+OD.AC-OD.DF.
But OD.AC=OA.DF,

since OA : AC = OD : DF.

22 ARCHIMEDES

Hence OA . AC - OD . DF = DA . AC + DA . DF

And, since circles are to one another as the squares of their
radii, it follows that the difference between the circles whose
radii are \IOA.AC, ^OD.DF respectively is equal to a circle
whose radius is ^DA.(AC + DF).

Therefore the surface of the frustum is equal to this circle.

Lemmas.

" 1. Cones having equal height have the same ratio as their
bases; and those having equal bases have the same ratio as their
heights*.

2. If a cylinder be cut by a plane parallel to the base, then,
as the cylinder is to the cylinder t so is the aods to the emsf.

3. The cones which have the same bases as the cylinders [and
equal height] are in the same ratio as the cylinders.

4. Also the bases of equal cones are reciprocally proportional
to their heights; and those cones whose bases are reciprocally
proportional to their heights are equal J.

5. Also the cones y the diameters of whose bases have the same
ratio as their axes, are to one another in the triplicate ratio of the
diameters of the bases .

And all these propositions have been proved by earlier
geometers."

* Euclid xn. 11. " Cones and cylinders of equal height are to one another
as their bases."

Euclid xn. 14. " Cones and cylinders on equal bases are to one another as
their heights."

t Euclid xn. 13. " If a cylinder be cut by a plane parallel to the opposite
planes [the bases], then, as the cylinder is to the cylinder, so will the axis be
to the axis."

J Euclid xii. 15. " The bases of equal cones and cylinders are reciprocally
proportional to their heights ; and those cones and cylinders whose bases are
reciprocally proportional to their heights are equal."

Euclid xii. 12. " Similar cones and cylinders are to one another in the
triplicate ratio of the diameters of their bases."

ON THE SPHERE AND CYLINDER I.

Proposition 17.

If there be two isosceles cones, and the surface of one cone be
equal to the base of the other, while the perpendicular from the
centre of the base [of the first cone] on the side of that cone is
equal to the height [of the second}, the cones mil be equal

Let OAB, DEF be triangles through the axes of two cones
respectively, G 9 G the centres of the respective bases, OH the

perpendicular from G on FD ; and suppose that the base of the
cone OAB is equal to the surface of the cone DEF, and
that OC =

Then, since the base of OAB is equal to the surface of
DEF,

(base of cone OAB) : (base of cone DEF)

= (surface of DEF) : (base of DEF)
= DF:FG [Prop. 15]

= DG : GH t by similar triangles,
= DG:OC.

Therefore the bases of the cones are reciprocally propor-
tional to their heights ; whence the cones are equal. [Lemma
4]

ARCHIMEDES

Proposition 18.

Any solid rhombus consisting of isosceles cones is equal to
the cone which has its base equal to the surface of one of the
cones composing the rhombus and its height equal to the perpen-
dicular drawn from the apex of the second cone to one side of
the first cone.

Let the rhombus be OABD consisting of two cones with
apices 0, D and with a common base (the circle about AB as
diameter).

Let FHK be another cone with base equal to the surface of
the cone OAB and height FG equal to DE, the perpendicular
from D on OB.

Then shall the cone FHK be equal to the rhombus.

Construct a third cone LMN with base (the circle about
MN) equal to the base of OAB and height LP equal to OD.

Then, since LP=OD,

LP:CD=OD:CD.

But [Lemma 1] OD:CD = (rhombus OADB) : (cone DAB\
and LP : CD - (cone LMN) : (cone DAB).

It follows that

(rhombus 04IXB) = (cone LMN) (1).

ON THE SPHERE AND CYLINDER I. 25

Again, since AB*=MN t and

(surface of OAB) = (base of FHK\
(base of FHK) : (base of LMN)

= (surface of OAB) : (base of OAB)
= OB:BC [Prop. 15]

= OD : DE, by similar triangles,
= LP : FG, by hypothesis.

Thus, in the cones FHK, LMN, the bases are reciprocally
proportional to the heights.

Therefore the cones FHK, LMN are equal,

and hence, by (1), the cone FHK is equal to the given
solid rhombus.

Proposition 19.

If an isosceles cone be cut by a plane parallel to the base,
and on the resulting circular section a cone be described having
as its apex the centre of the base [of the first cone], and if the
rhombus so formed be taken away from the whole cone t the part
remaining will be equal to the cone with base equal to the mrface
of the portion of the first cone between the parallel planes and
with height equal to the perpendicular drawn from the centre of
the base of the first cone on one side of that cone.

Let the cone OAB be cut by a plane parallel to the base in
the circle on DE as diameter. Let C be the centre of the base
of the cone, and with C as apex and the circle about DE as base
describe a cone, making with the cone ODE the rhombus
ODCE.

Take a cone FGH with base equal to the surface of the
frustum DABE and height equal to the perpendicular (CK)
from C on AO.

Then shall the cone FGH be equal to the difference between
the cone OAB and the rhombus ODCE.

Take (1) a cone LMN with base equal to the surface of the
cone OAB, and height equal to CK,

ARCHIMEDES

(2) a cone PQR with base equal to the surface of the cone
ODE and height equal to CK.

M N Q R

Now, since the surface of the cone OAB is equal to the
surface of the cone ODE together with that of the frustum
DABE, we have, by the construction,

(base of LM N) = (base of FGH) + (base of PQR)
and, since the heights of the three cones are equal,
(cone LMN) = (cone FGH) + (cone PQR).

But the cone LMN is equal to the cone OAB [Prop. 17],
and the cone PQR is equal to the rhombus ODGE [Prop. 18].

Therefore (cone OA 5) = (cone FGH) + (rhombus ODCE),
and the proposition is proved.

Proposition 2O.

If one of the two isosceles cones forming a rhombus be cut
by a plane parallel to the base and on the resulting circular
section a cone be described having the same apex as the second
cone, and if the resulting rhombus be taken from the whole
rhombus, the remainder will be equal to the cone with base equal
to the surface of the portion of the cone between the parallel
planes and with height equal to the perpendicular drawn from
the apex of the second * cone to the Me of the first cone.

* There is a slight error in Heiberg's translation "prioris ooni " and in the
corresponding note, p. 93. The perpendicular is not drawn from the apex of
the cone which is cut by the plane hut from the apex of the other.

ON THE SPHERE AND CYLINDER I.

Let the rhombus be OACB, and let the cone OAB be cut
by a plane parallel to its base in the circle about DE as diameter.
With this circle as base and C as apex describe a cone, which
therefore with ODE forms the rhombus ODCE.

Take a cone FGH with base equal to the surface of the
frustum DABE and height equal to the perpendicular (CK)
from C on OA.

The cone FGH shall be equal to the difference between the
rhombi OACB, ODCE.

For take (1) a cone LMN with base equal to the surface of
OAB and height equal to CK,

(2) a cone PQR, with base equal to the surface of ODE,
and height equal to CK.

Then, since the surface of OAB is equal to the surface of
ODE together with that of the frustum DABE, we have, by
construction,

(base of LMN) = (base of PQR) + (base of FGH),
and the three cones are of equal height ;
therefore (cone LMN) = (cone PQR) + (cone FGH).

But the cone LMN is equal to the rhombus OACB, and the
cone PQR is equal to the rhombus ODCE [Prop. 18].

Hence the cone FGH is equal to the difference between the
two rhombi OACB f ODCE.

ARCHIMEDES

Proposition 21.

A regular polygon of an even number of sides being inscribed
in a circle, as ABC...A'...C'B'A, so that A A' is a diameter,
if two angular points next but one to each other, as B 9 B f , be
joined, and the other lines parallel to BB f and joining pairs
of angular points be drawn, as CC', DD'..., then
(BB r + CO' +...): AA' = A'B: BA.

Let BB', CC', DD',... meet AA' in F, 0, H,... ; and let
CB', DC',... be joined meeting A A' in K, L,... respectively.

AB.

Then clearly GB', DC',... are parallel to one another and to

Hence, by similar triangles,

BF:FA=B'F:FK
= CG:GK
=C'G : GL

E'liIA';

ON THE SPHERE AND CYLINDER I.

and, summing the antecedents and consequents respectively, we
have

(BB' + GG' + ...): A A' = BF:FA
= A'B:BA.

Proposition 222.

If a polygon be inscribed in a segment of a circle LAL' so
that all its sides excluding the base are equal and their number
even, as LK. ..A.. .K'L', A being the middle point of the segment,
and if the lines BB', (7(7',... parallel to the baseLL' and joining
pairs of angular points be drawn, then

where M is the middle point of LL' and A A' is the diameter
through M.

Joining CB', DC',...LK', as in the last proposition, and
supposing that they meet AM in P, Q,...R 9 while BB', CO',...,
KK' meet AM in F, (?,... H, we have, by similar triangles,

= CG:PG

ARCHIMEDES

and, summing the antecedents and consequents, we obtain

:AM=BF:FA
= A'B:BA.

Proposition 23.

Take a great circle ABC... of a sphere, and inscribe in it
a regular polygon whose sides are a multiple of four in number.
Let AA' t MM' be diameters at right angles and joining
opposite angular points of the polygon.

Then, if the polygon and great circle revolve together about
the diameter AA' t the angular points of the polygon, except A,
A', will describe circles on the surface of the sphere at right
angles to the diameter A A'. Also the sides of the polygon
will describe portions of conical surfaces, e.g. BC will describe
a surface forming part of a cone whose base is a circle about
CC' as diameter and whose apex is the point in which CB,
G'B' produced meet each other and the diameter AA'.

Comparing the hemisphere MAM' and that half of the
figure described by the revolution of the polygon which is
included in the hemisphere, we see that the surface of the
hemisphere and the surface of the inscribed figure have the
same boundaries in one plane (viz. the circle on MM' as

ON THE SPHERE AND CYLINDER I.

diameter), the former surface entirely includes the latter, and
they are both concave in the same direction.

Therefore [Assumptions, 4] the surface of the hemisphere
is greater than that of the inscribed figure ; and the same is
true of the other halves of the figures.

Hence the surface of the sphere is greater than the surface
described by the revolution of the polygon inscribed in the great
circle about the diameter of the great circle.

Proposition 24.

If a regular polygon AB...A'...B'A, the number of whose
sides is a multiple of four, be inscribed in a great circle of a
sphere, and if BB' subtending two sides be joined, and all the
other lines parallel to BB' and joining pairs of angular points
be drawn, then the surface of the figure inscribed in the sphere
by the revolution of the polygon about the diameter A A' is equal
to a circle the square of whose radius is equal to the rectangle
B A (BB' + CC' + ...).

The surface of the figure is made up of the surfaces of parts
of different cones.

Now the surface of the cone ABB' is equal to a circle whose
radius is VSZTffiF. [Prop. 14]

ARCHIMEDES

The surface of the frustum BB'C'C is equal to a circle of
radius V f BC . (BB' ' + CC'\ [Prop. 16]

and so on.

It follows, since BA = BC = . . ., that the whole surface is
equal to a circle whose radius is equal to

Proposition 25.

The surface of the figure inscribed in a sphere as in the last
propositions, consisting of portions of conical surfaces, is less than
four times the greatest circle in the sphere.

Let AB...A'...B'A be a regular polygon inscribed in a
great circle, the number of its sides being a multiple of four.

As before, let BB' be drawn subtending two sides, and
C f C7 / ,...FF parallel to BB'.

Let R be a circle such that the square of its radius is equal
to

so that the surface of the figure inscribed in the sphere is equal
to R. [Prop. 24]

ON THE SPHERE AND CYLINDER I.

Now

(BB f + 00'+... + FF) : A A' = A'B: AB, [Prop. 21]
whence A B (BB f + CC' + ...+ YY') = A A' . A'B.
Hence (radius of Kf = A A' . A'B

<AA*.

Therefore the surface of the inscribed figure, or the circle JR,
is less than four times the circle AMA'M'.

Proposition 26.

The figure inscribed as above in a sphere is equal [in volume]
to a cone whose base is a circle equal to the surface of the figure
inscribed in the sphere and whose height is equal to the
perpendicular drawn from the centre of the sphere to one side of
the polygon.

Suppose, as before, that AB...A' ...B'A is the regular
polygon inscribed in a great circle, and let BB' 9 CG\ ... be
joined.

With apex construct cones whose bases are the circles
on BB' y CC', ... as diameters in planes perpendicular to A A'.

H. A.

34 ARCHIMEDES

Then OBAB' is a solid rhombus, and its volume is equal to
a cone whose base is equal to the surface of the cone ABB' and
whose height is equal to the perpendicular from on AB
[Prop. 18]. Let the length of the perpendicular be p.

Again, if CB, C'B' produced meet in T, the portion of the
solid figure which is described by the revolution of the triangle
BOG about AA' is equal to the difference between the rhombi
OCTC' and OBTB' t i.e. to a cone whose base is equal to the
surface of the frustum BB'G'C and whose height is p [Prop. 20].

Proceeding in this manner, and adding, we prove that, since
cones of equal height are to one another as their bases, the
volume of the solid of revolution is equal to a cone with height
p and base equal to the sum of the surfaces of the cone BAB',
the frustum BB'C'C, etc., i.e. a cone with height p and base
equal to the surface of the solid.

Proposition 27.

The figure inscribed in the sphere as before is less than
four times the cone whose base is equal to a great circle of
the sphere and whose height is equal to the radius of the
sphere.

By Prop. 26 the volume of the solid figure is equal to a cone
whose base is equal to the surface of the solid and whose height
isp, the perpendicular from on any side of the polygon. Let
It be such a cone.

Take also a cone S with base equal to the great circle, and
height equal to the radius, of the sphere.

Now, since the surface of the inscribed solid is less than four
times the great circle [Prop. 25], the base of the cone R is less
than four times the base of the cone S.

Also the height (p) of R is less than the height of S.
Therefore the volume of R is less than four times that of S',
and the proposition is proved.

ON THE SPHERE AND CYLINDER I.

Proposition 28.

Let a regular polygon, whose sides are a multiple of four in
number, be circumscribed about a great circle of a given
sphere, as AB...A'...B'A ; and about the polygon describe
another circle, which will therefore have the same centre as the
great circle of the sphere. Let AA' bisect the polygon and

cut the sphere in a, of.

If the great circle and the circumscribed polygon revolve
together about AA' y the great circle will describe the surface
of a sphere, the angular points of the polygon except A, A will
move round the surface of a larger sphere, the points of contact
of the sides of the polygon with the great circle of the inner
sphere will describe circles on that sphere in planes perpen-
dicular to AA' y and the sides of the polygon themselves will
describe portions of conical surfaces. The circumscribed figure
will thus be greater than the sphere itself.

Let any side, as BM, touch the inner circle in K, and let K'
be the point of contact of the circle with B'M'.

Then the circle described by the revolution of KK ' about
A A' is the boundary in one plane of two surfaces

(1) the surface formed by the revolution of the circular
segment KaK', and

ARCHIMEDES

(2) the surface formed by the revolution of the part
KB...A...B'K' of the polygon.

Now the second surface entirely includes the first, and they
are both concave in the same direction ;

therefore [Assumptions, 4] the second surface is greater
than the first.

The same is true of the portion of the surface on the opposite
side of the circle on KK' as diameter.

Hence, adding, we see that the surface of the figure
circumscribed to the given sphere is greater than that of the

Proposition 29.

In a figure circumscribed to a sphere in the manner shown
in the previous proposition the surface is equal to a circle the
square on whose radius is equal to AB(BB' + CC' + ...).

For the figure circumscribed to the sphere is inscribed in a
larger sphere, and the proof of Prop. 24 applies.

Proposition 3O.

The surface of a figure circumscribed as before about a sphere
is greater than four times the great circle of the sphere.

ON THE SPHERE AND CYLINDER I. 37

Let AB...A'...B'A be the regular polygon of 4en sides
which by its revolution about AA' describes the figure circum-
scribing the sphere of which amam! is a great circle. Suppose
aa', A A' to be in one straight line.

Let R be a circle equal to the surface of the circumscribed
solid.

Now (BB' + CC' + ...):AA' = A'B:BA, [as in Prop. 21]
so that AB(BB' + CC'+ ...) = AA'.A'B.

Hence (radius of R) = */AA .A'B [Prop. 29]

>A'B.

But A 'E = 20P, where P is the point in which AB touches
the circle ama'm'.

Therefore (radius of R) > (diameter of circle amam') ;
whence R> and therefore the surface of the circumscribed solid,
is greater than four times the great circle of the given sphere.

Proposition 31.

The solid of revolution circumscribed as before about a sphere
is equal to a cone whose base is equal to the surface of the solid
and whose height is equal to the radius of the sphere.

The solid is, as before, a solid inscribed in a larger sphere ;
and, since the perpendicular on any side of the revolving polygon
is equal to the radius of the inner sphere, the proposition is
identical with Prop. 26.

COR. The solid circumscribed about the smaller sphere is
greater than four times the cone whose base is a great circle
of the sphere and whose height is equal to the radius of the
sphere.

For, since the surface of the solid is greater than four times
the great circle of the inner sphere [Prop. 30], the cone whose
base is equal to the surface of the solid and whose height is the
radius of the sphere is greater than four times the cone of
the same height which has the great circle for base. [Lemma 1.]

Hence, by the proposition, the volume of the solid is greater
than four times the latter cone.

ARCHIMEDES

Proposition 32.

If a regular polygon with n sides be inscribed in a great
circle of a sphere, as ab...a'...b'a, and a similar polygon
AB...A'...B'A be described about the great circle, and if the
polygons revolve with the great circle about the diameters aa,
A A! respectively, so that they describe the surfaces of solid
figures inscribed in and circumscribed to the sphere respectively,
then

(1) the surfaces of the circumscribed and inscribed figures
are to one another in the duplicate ratio of their sides, and

(2) the figures themselves [i.e. their volumes] are in the
triplicate ratio of their sides.

(1) Let AA', aa' be in the same straight line, and let
MmOrn'M' be a diameter at right angles to them.

Join BB', CO',... and W,cc,... which will all be parallel
to one another and MM'.

Suppose R, S to be circles such that

R = (surface of circumscribed solid),
S = (surface of inscribed solid).

ON THE SPHERE AND CYLINDER I. 39

Then (radius of R)* = AB (BB f + CC' + . . .) [Prop. 29]
(radius of 8)* = ab (W + cc' + ...). [Prop. 24]

And, since the polygons are similar, the rectangles in these
two equations are similar, and are therefore in the ratio of

AB 2 : ab*.
Hence

(surface of circumscribed solid) : (surface of inscribed solid)

(2) Take a cone V whose base is the circle R and whose
height is equal to Oft, and a cone W whose base is the circle 3
and whose height is equal to the perpendicular from on ab,
which we will call p.

Then V, W are respectively equal to the volumes of the
circumscribed and inscribed figures. [Props. 31, 26]

Now, since the polygons are similar,
AB : ab = 0a : p

= (height of cone V) : (height of cone W) ;

and, as shown above, the bases of the cones (the circles R t 8)
are in the ratio of AB* to ab*.

Therefore V : W = AB 9 : ab 3 .

Proposition 33.

The surface of any sphere is equal to four times the greatest
circle in it.

Let C be a circle equal to four times the great circle.

Then, if C is not equal to the surface of the sphere, it must
either be less or greater.

I. Suppose C less than the surface of the sphere.

It is then possible to find two lines , 7, of which $ is the
greater, such that

ft : 7 < (surface of sphere) : C. [Prop. 2]

Take such lines, and let 8 be a mean proportional between
them.

ARCHIMEDES

Suppose similar regular polygons with 4sn sides circum-
scribed about and inscribed in a great circle such that the ratio
of their sides is less than the ratio ft : S. [Prop. 3]

Let the polygons with the circle revolve together about
a diameter common to all, describing solids of revolution as
before.

Then (surface of outer solid) : (surface of inner solid)

= (side of outer) 8 : (side of inner) 8 [Prop. 32]

<':$', or : 7

< (surface of sphere) : (7, a fortiori.

But this is impossible, since the surface of the circum-
scribed solid is greater than that of the sphere [Prop. 28], while
the surface of the inscribed solid is less than C [Prop. 25],

Therefore C is not less than the surface of the sphere.
II. Suppose C greater than the surface of the sphere.
Take lines /3 9 7, of which ft is the greater, such that
ft : y < G : (surface of sphere).

Circumscribe and inscribe to the great circle similar regular
polygons, as before, such that their sides are in a ratio less than
that of ft to S, and suppose solids of revolution generated in the
usual manner.

ON THE SPHERE AND CYLINDER I. 41

Then, in this case,

(surface of circumscribed solid) : (surface of inscribed solid)

< : (surface of sphere).

But this is impossible, because the surface of the circum-
scribed solid is greater than C [Prop. 30], while the surface of
the inscribed solid is less than that of the sphere [Prop. 23].

Thus C is not greater than the surface of the sphere.

Therefore, since it is neither greater nor less, C is equal to
the surface of the sphere.

Proposition 34.

Any sphere is equal to four times the cone which has its base
equal to the greatest circle in the sphere and its height equal
to the radius of the sphere.

Let the sphere be that of which ama'm' is a great circle.

If now the sphere is not equal to four times the cone
described, it is either greater or less.

I. If possible, let the sphere be greater than four times the
cone.

Suppose V to be a cone whose base is equal to four times
the great circle and whose height is equal to the radius of the
sphere.

Then, by hypothesis, the sphere is greater than F; and two
lines ft, 7 can be found (of which fj is the greater) such that

ft ' 7 < (volume of sphere) : F.
Between and y place two arithmetic means 8, e.

As before, let similar regular polygons with sides 4;i in
number be circumscribed, about and inscribed in the great
circle, such that their sides are in a ratio less than /8 : S.

Imagine the diameter aa of the circle to be in the same
straight line with a diameter of both polygons, and imagine
the latter to revolve with the circle about aa', describing the

ARCHIMEDES

surfaces of two solids of revolution. The volumes of these solids
are therefore in the triplicate ratio of their sides. [Prop. 32]
Thus (vol. of outer solid) : (vol. of inscribed solid)

< /9 s : S s , by hypothesis,

< 13 : 7, a fortiori (since : y > /3 s : 8 8 )*,

< (volume of sphere) : V, a fortiori.

But this is impossible, since the volume of the circumscribed

* That ft : y>fP : 5 :j is assumed by Archimedes,
property in his commentary as follows.

Take x such that /3 : 5 = 5 : .r.

Thus 8 8 i 8=8 x '. d

and, since /9>5, (3 8>8-x.

But, by hypothesis, - 8 = 3 - e.

Eutocius proves the

Therefore

8 - > 5 - x,

Again, suppose d : x=x : y t

and, as before, we have 8 - x > x - y,

so that, a fortiori, i

Therefore e
and, since g>e, y>y.

Now, by hypothesis, ft 5, or, y are in continued proportion ;

therefore s : &=p : y

ON THE SPHERE AND CYLINDER I. 43

jolid is greater than that of the sphere [Prop. 28], while the
volume of the inscribed solid is less than V [Prop. 27].

Hence the sphere is not greater than F, or four times the
sone described in the enunciation.

II. If possible, let the sphere be less than V.
In this case we take 0, y (@ being the greater) such that
ft : 7 < F: (volume of sphere).

The rest of the construction and proof proceeding as before,
ve have finally

(volume of outer solid) : (volume of inscribed solid)
< F : (volume of sphere).

But this is impossible, because the volume of the outer
solid is greater than F [Prop. 31, Cor.], and the volume of the
nscribed solid is less than the volume of the sphere.

Hence the sphere is not less than F.

Since then the sphere is neither less nor greater than F, it
s equal to F, or to four times the cone described in the enun-
ciation.

COR. From what has been proved it follows that every
cylinder whose base is the greatest circle in a sphere and whose
height is equal to the diameter of the sphere is f of the sphere,
ind its surface together with its bases is f of the surface of the
sphere.

For the cylinder is three times the cone with the same
mse and height [Eucl. XII. 10], i.e. six times the cone with
he same base and with height equal to the radius of the
jphere.

But the sphere is four times the latter cone [Prop. 34].
Therefore the cylinder is f of the sphere.

Again, the surface of a cylinder (excluding the bases) is
jqual to a circle whose radius is a mean proportional between
;he height of the cylinder and the diameter of its base
Prop. 13].

ARCHIMEDES

In this case the height is equal to the diameter of the base
and therefore the circle is that whose radius is the diameter of
the sphere, or a circle equal to four times the great circle of
the sphere.

Therefore the surface of the cylinder with the bases is equal
to six times the great circle.

And the surface of the sphere is four times the great circle
[Prop. 33] ; whence

(surface of cylinder with bases) = f . (surface of sphere).

Proposition 35.

If in a segment of a circle LAL' (where A is the middle
point of the arc) a polygon LK...A...K'L' be inscribed of which
LL f is one side, while the other sides are 2?i in number and all
equal, and if the polygon revolve with the segment about the
diameter AM, generating a solid figure inscribed in a segment of
a sphere, then the surface of the inscribed solid is equal to a
circle the square on whose radius is equal to the rectangle

LL'\
h 2

The surface of the inscribed figure is made up of portions of
surfaces of cones.

ON THE SPHERE AND CYLINDER I. 45

If we take these successively, the surface of the cone BAB'
is equal to a circle whose radius is

[Prop. 14]

The surface of the frustum of a cone BCC'B' is equal to
a circle whose radius is

A T* BB -\- CG rrfc -.,

AB. g- ; [Prop. 16]

and so on.

Proceeding in this way and adding, we find, since circles
are to one another as the squares of their radii, that the
surface of the inscribed figure is equal to a circle whose radius

'AB(BB' + CC'+... + KK'

Proposition 36.

The surface of the figure inscribed as before in the segment
of a sphere is less than that of the segment of the sphere.

This is clear, because the circular base of the segment is a
common boundary of each of two surfaces, of which one, the
segment, includes the other, the solid, while both are concave
in the same direction [Assumptions, 4],

Proposition 37.

The surface of the solid figure inscribed in the segment of the
sphere by the revolution of LK. ..A.. .K'L' about AM is less than
a circle with radius equal to AL.

Let the diameter AM meet the circle of which LAL' is a
segment again in A'. Join A'B.

As in Prop. 35, the surface of the inscribed solid is equal to
a circle the square on whose radius is

AB (BB' + CC' + . . . + KK' + LM).

ARCHIMEDES

But this rectangle

[Prop. 22]

Hence the surface of the inscribed solid is less than the
circle whose radius is AL.

Proposition 38.

The solid figure described as before in a segment of a sphere
less than a hemisphere, together with the cone whose base is the
base of the segment and whose apex is the centre of the sphere,
is equal to a cone whose base is equal to the surface of the
inscribed solid and whose height is equal to the perpendicular
from the centre of the sphere on any side of the polygon.

Let be the centre of the sphere, and p the length of the
perpendicular from on AB.

Suppose cones described with as apex, and with the
circles on BB', CC',... as diameters as bases.

Then the rhombus OBAB' is equal to a cone whose base is
equal to the surface of the cone BAB', and whose height is p.

[Prop. 18]

Again, if CB, C'B' meet in T, the solid described by the
triangle BOG as the polygon revolves about AO is the difference

ON THE SPHERE AND CYLINDER I.

between the rhombi OCTC' and OBTB', and is therefore equal
to a cone whose base is equal to the surface of the frustum
BCC'B' and whose height is p. [Prop. 20]

Similarly for the part of the solid described by the triangle
COD as the polygon revolves ; and so on.

Hence, by addition, the solid figure inscribed in the segment
together with the cone OLL' is equal to a cone whose base is
the surface of the inscribed solid and whose height is p.

COR. The cone whose base is a circle with radius equal to
AL and whose height is equal to the radius of the sphere is
greater than the sum of the inscribed solid and the cone OLL'.

For, by the proposition, the inscribed solid together with
the cone OLL' is equal to a cone with base equal to the surface
of the solid and with height p.

This latter cone is less than a cone with height equal to OA
and with base equal to the circle whose radius is AL y because
the height p is less than OA, while the surface of the solid is
less than a circle with radius AL. [Prop. 37]

Proposition 39.

Let lal' be a segment of a great circle of a sphere, being less
than a semicircle. Let be the centre of the sphere, and join
Ol t 01'. Suppose a polygon circumscribed about the sector Olal'
such that its sides, excluding the two radii, are 2/t in number

ARCHIMEDES

and all equal, as LK, ... BA, AB' t ... K'L' ; and let OA be that
radius of the great circle which bisects the segment lal'.

The circle circumscribing the polygon will then have the
same centre as the given great circle.

Now suppose the polygon and the two circles to revolve
together about OA. The two circles will describe spheres, the

angular points except A will describe circles on the outer
sphere, with diameters BB' etc., the points of contact of the
sides with the inner segment will describe circles on the inner
sphere, the sides themselves will describe the surfaces of cones
or frusta of cones, and the whole figure circumscribed to the
segment of the inner sphere by the revolution of the equal
sides of the polygon will have for its base the circle on LL'
as diameter.

The surface of the solid figure so circumscribed about the
sector of the sphere [excluding its base] will be greater than that
of the segment of the sphere whose base is the circle on IV as
diameter.

For draw the tangents IT, I'T' to the inner segment at I, I'.
These with the sides of the polygon will describe by their
revolution a solid whose surface is greater than that of the
segment [Assumptions, 4].

But the surface described by the revolution of IT is less
than that described by the revolution of LT, since the angle TIL
is a right angle, and therefore LT>IT.

Hence, a fortiori, the surface described by LK...A...K'L'
is greater than that of the segment.

ON THE SPHERE AND CYLINDER I.

COR. The surface of the figure so described about the sector
of the sphere is equal to a circle the square on whose radius
is equal to the rectangle

For the circumscribed figure is inscribed in the outer sphere,
and the proof of Prop. 35 therefore applies.

Proposition 4O.

The surface of the figure circumscribed to the sector as before
is greater than a circle whose radius is equal to al.

Let the diameter AaO meet the great circle and the circle
circumscribing the revolving polygon again in a', A'. Join
A'B, arid let ON be drawn to N, the point of contact of AB
with the inner circle.

Now, by Prop. 39, Cor., the surface of the solid figure
circumscribed to the sector 01 AV is equal to a circle the square
on whose radius is equal to the rectangle

But this rectangle is equal to A'B.AM [as in Prop. 22],
H. A. 4

50 ARCHIMEDES

Next, since AL', al' are parallel, the triangles AML', ami'
are similar. And AL' > al' ; therefore A M > am.

Also A'B = 20N=aa'.

Therefore A 'B . AM > am . ao!

> al'\

Hence the surface of the solid figure circumscribed to the
sector is greater than a circle whose radius is equal to al', or al.

COR. 1. The volume of the figure circumscribed about the
sector together with the cone whose apex is and base the circle
on LL' as diameter, is equal to the volume of a cone whose base
is equal to the surface of the circumscribed figure and whose
height is ON.

For the figure is inscribed in the outer sphere which has the
same centre as the inner. Hence the proof of Prop. 38 applies.

COR. 2. The volume of the circumscribed figure with the cone
OLL' is greater than the cone whose base is a circle with radius
equal to al and whose height is equal to the radius (Oa) of the
inner sphere.

For the volume of the figure with the cone OLL' is equal to
a cone whose base is equal to the surface of the figure and
whose height is equal to ON.

And the surface of the figure is greater than a circle with
radius equal to al [Prop. 40], while the heights Oa, ON are
equal.

Proposition 41.

Let laV be a segment of a great circle of a sphere which is
less than a semicircle.

Suppose a polygon inscribed in the sector Olal' such that
the sides Ik,... ba, ai', ...&T are 2n in number and all equal.
Let a similar polygon be circumscribed about the sector so that
its sides are parallel to those of the first polygon ; and draw
the circle circumscribing the outer polygon.

Now let the polygons and circles revolve together about
OaA, the radius bisecting the segment lal*.

ON THE SPHERE AND CYLINDER I.

Then (1) the surfaces of the outer and inner solids of revolution
so described are in the ratio ofAB* to a& 8 , and (2) their volumes
together with the corresponding cones with the same base and
with apex in each case are as AB 3 to ab 9 .

[Prop. 39, Cor.]
ab (bV + cc' + . . . + kk' + -} . [Prop. 35]

(1) For the surfaces are equal to circles the squares on
whose radii are equal respectively to

and

But these rectangles are in the ratio of AB 9 to ab 9 . Therefore
so are the surfaces.

(2) Let OnNbe drawn perpendicular to ab and AB\ and
suppose the circles which are equal to the surfaces of the outer
and inner solids of revolution to be denoted by 8, s respectively.

Now the volume of the circumscribed solid together with
the cone OLL' is equal to a cone whose base is S and whose
height is ON [Prop. 40, Cor. 1].

And the volume of the inscribed figure with the cone Oil' is
equal to a cone with base s and height On [Prop. 38].

But S:s = AB*:ab*,

and ON : On = AB:ab.

Therefore the volume of the circumscribed solid together with
the cone OLL' is to the volume of the inscribed solid together
with the cone 01V as AB 9 is to 06* [Lemma 5],

ARCHIMEDES

Proposition 42.

If lal' be a segment of a sphere less than a hemisphere and
Oa the radium perpendicular to the base of the segment, the
surface of the segment is equal to a circle whose radius is equal
to al.

Let R be a circle whose radius is equal to al. Then the
surface of the segment, which we will call S, must, if it be not
equal to R, be either greater or less than R.

I. Suppose, if possible, S > R.

Let lal' be a segment of a great circle which is less than a
semicircle. Join 01, 01 ', and let similar polygons with 2n equal
sides be circumscribed and inscribed to the sector, as in the
previous propositions, but such that

(circumscribed polygon) : (inscribed polygon) < S : R.

[Prop. 6]

Let the polygons now revolve with the segment about OaA,
generating solids of revolution circumscribed and inscribed to
the segment of the sphere.

Then

(surface of outer solid) : (surface of inner solid)
= AB* : ab* [Prop. 41]

= (circumscribed polygon) : (inscribed polygon)
< S : R, by hypothesis.
But the surface of the outer solid is greater than S [Prop. 39].

ON THE SPHERE AND CYLINDER I.

Therefore the surface of the inner solid is greater than R ;
which is impossible, by Prop. 37.

II. Suppose, if possible, 8 < R.

In this case we circumscribe and inscribe polygons such that
their ratio is less than R : S ; and we arrive at the result that

(surface of outer solid) : (surface of inner solid)
<R:S.

But the surface of the outer solid is greater than R [Prop. 40].
Therefore the surface of the inner solid is greater than 8 : which
is impossible [Prop. 36].

Hence, since S is neither greater nor less than jR,

Proposition 43.

Even if the segment of the sphere is greater than a hemisphere,
its surface is still equal to a circle whose radium is equal to al.

For let led' a' be a great circle of the sphere, aa being the
diameter perpendicular to IV ; and let
la'l' be a segment less than a semi-
circle.

Then, by Prop. 42, the surface of
the segment la'l' of the sphere is
equal to a circle with radius equal to
al

Also the surface of the whole
sphere is equal to a circle with radius
equal to aa' [Prop. 33].

But aa' 2 a7 2 = a*, and circles are to one another as the
squares on their radii.

Therefore the surface of the segment lal', being the difference
between the surfaces of the sphere and of la'l', is equal to a
circle with radius equal to al.

ARCHIMEDES

Proposition 44.

The volume of any sector of a sphere is equal to a cone whose
base is equal to the surface of the segment of the sphere included
in the sector, and whose height is equal to the radius of the

Let R be a cone whose base is equal to the surface of the
segment laV of a sphere and whose height is equal to the radius
of the sphere ; and let S be the volume of the sector Olal'.

Then, if S is not equal to R, it must be either greater or

less.

I. Suppose, if possible, that 8 > R.

Find two straight lines ft, 7, of which ft is the greater, such
that

ft : 7 < S : R ;

and let 8, be two arithmetic means between ft, 7.

Let lal' be a segment of a great circle of the sphere.
Join 01, 01', and let similar polygons with Zn equal sides be
circumscribed and inscribed to the sector of the circle as before,
but such that their sides are in a ratio less than ft : S.
[Prop. 4].

ON THE SPHERE AND CYLINDER I. 55

Then let the two polygons revolve with the segment about
OaA, generating two solids of revolution.

Denoting the volumes of these solids by F, v respectively,
we have

( F + cone OLL') : (v 4- cone OIF) = A B* : ab* [Prop. 41]

</3 3 :5 8

< $ : 7, a fortiori*,

< S : R t by hypothesis.
Now ( F 4- cone OLL') > S.

Therefore also (v + cone Oil') > R.

But this is impossible, by Prop. 38, Cor. combined with Props.
42, 43.

Hence 8 } R.

II. Suppose, if possible, that S < R.
In this case we take /8, 7 such that
0:y<R:S t

and the rest of the construction proceeds as before.
We thus obtain the relation

(F+ cone OLL) : (v + cone OH') < R : S.
Now (v + cone Oil') < S.

Therefore ( F 4- cone OLL) < R ;

which is impossible, by Prop. 40, Cor. 2 combined with Props.
42, 43.

Since then S is neither greater nor less than -R,

S = R.
* Cf. note on Prop. 34, p. 4*2..

ON THE SPHEKE AND CYLINDEK.

BOOK II.

" ARCHIMEDES to Dositheus greeting.

On a former occasion you asked me to write out the proofs of
the problems the enunciations of which I had myself sent to
Conon. In point of fact they depend for the most part on the
theorems of which I have already sent you the demonstrations,
namely (1) that the surface of any sphere is four times the
greatest circle in the sphere, (2) that the surface of any
segment of a sphere is equal to a circle whose radius is equal
to the straight line drawn from the vertex of the segment to
the circumference of its base, (3) that the cylinder whose base
is the greatest circle in any sphere and whose height is equal
to the diameter of the sphere is itself in magnitude half as
large again as the sphere, while its surface [including the two
bases] is half as large again as the surface of the sphere, and
(4) that any solid sector is equal to a cone whose base is the
circle which is equal to the surface of the segment of the sphere
included in the sector, and whose height is equal to the radius
of the sphere. Such then of the theorems and problems as
depend on these theorems I have written out in the book
which I send herewith ; those which are discovered by means
of a different sort of investigation, those namely which relate
to spirals and the conoids, I will endeavour to send you soon.

OX THE SPHERE AND CYLINDER II.

The first of the problems was as follows : Given a sphere, to
find a plane area equal to the surface of the sphere.

The solution of this is obvious from the theorems aforesaid.
For four times the greatest circle in the sphere is both a plane
area and equal to the surface of the sphere.

The second problem was the following."

Proposition 1. (Problem.)

Given a cone or a cylinder, to find a sphere equal to the cone
or to the cylinder.

If V be the given cone or cylinder, we can make a cylinder
equal to f F. Let this cylinder be the cylinder whose base
is the circle on AB as diameter and whose height is OD.

Now, if we could make another cylinder, equal to the
cylinder (OD) but such that its height is equal to the diameter
of its base, the problem would be solved, because this latter
cylinder would be equal to f F, and the sphere whose diameter
is equal to the height (or to the diameter of the base) of the
same cylinder would then be the sphere required [I. 34, Cor.].

Suppose the problem solved, and let the cylinder (CO) be
equal to the cylinder (OD), while EF, the diameter of the base,
is equal to the height CG.

58 ARCHIMEDES

Then, since in equal cylinders the heights and bases are
reciprocally proportional,

= EF: OD .................. (1).

Suppose MN to be such a line that

EF* = AB.MN ....................... (2).

Hence AB : EF=EF:MN,

and, combining (1) and (2), we have

AB:MN=EF:OD,

or AB:EF=MN:OD.

Therefore AB:EF = EF : MN - MN : OD,
and EF, MN are two mean proportionals between AB, OD.

The synthesis of the problem is therefore as follows. Take
two mean proportionals EF, MN between AB and OD> and
describe a cylinder whose base is a circle on EF as diameter
and whose height CG is equal to EF.

Then, since

AB : EF = EF: MN=MN :

and therefore AB* : EF* = AB : MN

= EF:OD

whence the bases of the two cylinders (OJD), (CG) are recipro-
cally proportional to their heights.

Therefore the cylinders are equal, and it follows that
cylinder (Cff) = f7.

The sphere on EF as diameter is therefore the sphere
required, being equal to V.

ON THE SPHERE AND CYLINDER II.

Proposition 2.

// BAB' be a segment of a sphere, BB' a diameter of the
base of the segment, and the centre of the sphere, and if AA'
be the diameter of the sphere bisecting BB 1 in M, then the volume
of the segment is equal to that of a cone whose base is the same
as that of the segment and whose height is h, where

h:AM=OA' + A'M:A'M.

Measure MH along MA equal to A, and MH' along MA
equal to h', where

h':A'M=OA+AM:AM.

Suppose the three cones constructed which have 0, H
H' for their apices and the base (BB') of the segment for their
common base. Join AB, A'B.

Let C be a cone whose base is equal to the surface of the
segment BAB' of the sphere, i.e. to a circle with radius equal
to AB [I. 42], and whose height is equal to OA.

Then the cone Cis equal to the solid sector OBAR [I. 44].
Now, since HM : MA = OA 1 + AM : A'M,
dividendo, HA:AM=OA: A'M,

and, alternately, H A : AO = AM : MA\
so that

HO : OA = A A' : A'M
= AB* : BM*
= (base of cone C) : (circle on BB' as diameter).

60 ARCHIMEDES

But OA is equal to the height of the cone C ; therefore, since
cones are equal if their bases and heights are reciprocally
proportional, it follows that the cone C (or the solid sector
OBABf) is equal to a cone whose base is the circle on BR as
diameter and whose height is equal to OH.

And this latter cone is equal to the sum of two others
having the same base and with heights OM, MH, i.e. to the
solid rhombus OBHB'.

Hence the sector OBAB' is equal to the rhombus OBHB'.
Taking away the common part, the cone OB&,

the segment BAB' = the cone HBB 1 .
Similarly, by the same method, we can prove that

the segment BA 'B' = the cone H 'BB'.
Alternative proof of the latter property.

Suppose D to be a cone whose base is equal to the surface
of the whole sphere and whose height is equal to OA.

Thus D is equal to the volume of the sphere. [I. 33, 34]
Now, since OA' + A'M : A'M = EM : MA,
dividendo and alternando, as before,

OA : AH = A'M: MA.

Again, since H'M : MA' = OA + A M : AM,
H'A' : OA = A'M : MA

= OA : AH, from above.

Componendo, H'O : OA = OH : HA (1).

Alternately, H'0:OH=OA : AH (2),

and, componendo, EH' : HO = OH : HA,

= H'0 : OA, from (1),

whence HH ' . OA = H'O . OH (3).

Next, since H'O : OH = OA: AH, by (2),

-A'M :MA 9
(H'O + OH? : H'O . OH = (A'M + MA? : A'M . MA,

ON THE SPHERE AND CYLINDER II.

whence, by means of (3),

HH " : HE' . OA = AA'* : A'M. MA,
or HH' : OA = A A 9 : BM\

Now the cone D, which is equal to the sphere, has for its base
a circle whose radius is equal to A A', and for its height a line
equal to OA.

Hence this cone D is equal to a cone whose base is the circle
on BB' as diameter and whose height is equal to HH' ;

therefore the cone D = the rhombus HBH'B',
or the rhombus HBH'B' = the sphere.

But the segment BAB' = the cone EBB' ;

therefore the remaining segment BA'B' the cone H'BB'.

COR. The segment BAB' is to a cone with the same base and
equal height in the ratio of OA' + A'M to A'M.

Proposition 3. (Problem.)

To cut a given sphere by a plane so that the surfaces of the
segments may have to one another a given ratio.

Suppose the problem solved. Let A A' be a diameter of a
great circle of the sphere, and suppose that a plane perpendicular
to A A' cuts the plane of the great circle in the straight

line BB' t and AA' in M, and that it divides the sphere so that
the surface of the segment BAB' has to the surface of the
segment BA'B' the given ratio.

ARCHIMEDES

Now these surfaces are respectively equal to circles with
radii equal to AB, A'B [I. 42, 43].

Hence the ratio AB* : A'B 9 is equal to the given ratio, i.e.
A M is to MA' in the given ratio.

Accordingly the synthesis proceeds as follows.

If H : K be the given ratio, divide A A' in M so that

Then AM : MA' = AB* : A'B*

= (circle with radius AB) : (circle with radius A'B)

= (surface of segment BAB') : (surface of segment BA'B').

Thus the ratio of the surfaces of the segments is equal to
the ratio H : K.

Proposition 4. (Problem.)

To cut a given sphere by a plane so that the volumes of the
segments are to one another in a given ratio.

Suppose the problem solved, and let the required plane cut
the great circle ABA' at right angles in the line BE'. Let
AA' be that diameter of the great circle which bisects BB' at
right angles (in M ), and let be the centre of the sphere.

that
and

Take H on OA produced, and H' on OA' produced, such

OA' + A'M : A'M = HM : MA, (1),

OA+AM:AM=H'M:MA' (2).

Join BH, B'H, BH', B'H'.

ON THE SPHERE AND CYLINDER II. 63

Then the cones HBB', H'BB' are respectively equal to the
segments BAB', BA'B' of the sphere [Prop. 2].

Hence the ratio of the cones, and therefore of their altitudes,
is given, i.e.

HM : H'M = the given ratio (3).

We have now three equations (1), (2), (3), in which there
appear three as yet undetermined points M, H, H' ; and it is
first necessary to find, by means of them, another equation in
which only one of these points (M ) appears, i.e. we have, so to
speak, to eliminate H, H'.

Now, from (3), it is clear that HH' : H'M is also a given
ratio ; and Archimedes' method of elimination is, first, to find
values for each of the ratios A'H' : H'M and HH' : H'A which
are alike independent of H, H', and then, secondly, to equate
the ratio compounded of these two ratios to the known value
of the ratio HH 1 : H'M.

(a) To find such a value for A'H' : H'M.

It is at once clear from equation (2) above that

A'H':H'M = OA : OA + AM (4).

(b) To find such a value for HH' : A'H'.
From (1) we derive

AM : MA = OA + AM : HM

= OA':AH (5);

and, from (2), AM : MA = H'M : OA + AM

= A'H': OA (6).

Thus HA:AO = OA':A'H',

whence OH : OA f = OH' : A'H',

or OH:OH'=OA':A'H'.

It follows that

HH': OH'^OH' :A'H',
or HH'.H'A'=OH".

Therefore HH' : H'A' - OH" : H'A"

= AA'* : A'M*, by means of (6)

64 ARCHIMEDES

(c) To express the ratios A'H' : H'M and HH' : H'M more
simply we make the following construction. Produce OA to D
so that OA = AD. (D will lie beyond H, for A'M> MA, and
therefore, by (5), OA >AH.)

Then A'H' : H'M = OA:OA+AM

=AD:DM ..................... (7).

Now divide AD at E so that

HH':H'M = AD : DE .................. (8).

Thus, using equations (8), (7) and the value of HH 1 : H'A'
above found, we have

AD:DE = HH':H'M

- (HH' : H'A') . (A'H' : H'M)
= (AA'*:A'M*).(AD:DM).
But AD : DE = (DM : DE) . (AD : DM).

Therefore MD : DE = AA* : A'M* ................. (9).

And D is given, since -4J9 = OA. Also AD : DE (being equal
to HH' : H'M) is a given ratio. Therefore DE is given.

Hence the problem reduces itself to the problem of dividing
A'D into two parts at M so that

MD : (a given length) = (a given area) : A'M\
Archimedes adds : " If the problem is propounded in this
general form, it requires a 40/3407409 [i.e. it is necessary to
investigate the limits of possibility], but, if there be added the
conditions subsisting in the present case, it does not require a
810/0107105."

In the present case the problem is :

Given a straight line A' A produced to D so that A' A = 2AD,
and given a point E on AD, to cut AA' in a point M so that

" And the analysis and synthesis of both problems will be
given at the end*."

The synthesis of the main problem will be as follows. Let
12 : 8 be the given ratio, R being less than 8. AA' being a

* See the note following this proposition.

ON THE SPHERE AND CYLINDER II. 65

diameter of a great circle, and the centre, produce OA to D
so that OA=AD, and divide AD in E so that

AE : ED = R : 8.
Then cut A A' in M so that

MD : DE = AA" : A'M*.

Through M erect a plane perpendicular to AA' ; this plane
will then divide the sphere into segments which will be to one
another as R to S.

Take H on A' A produced, and H' on A A' produced, so that
OA' + A'M :A'M = HM:MA, .............. (1),

OA + AM : AM = H'M : MA' ............ (2).

We have then to show that

HM : MH' = R : S, or AE : ED.
(a) We first find the value of HE' : H'A' as follows.
As was shown in the analysis (6),

HH'.H'A'=OH",

or HE' : H'A' = OH'* : H'A'*

= AA'*:A'M*
= MD : DE y by construction.
(/8) Next we have

H'A' : H'M = OA : OA + AM

= AD:DM.

Therefore HH ' :H'M = (HH' : H'A') . (H'A : H'M)

= (MD:DE).(AD:DM)

whence HM : MH' = AE : ED

R:S. Q. E. D.

Note. The solution of the subsidiary problem to which the
original problem of Prop. 4 is reduced, and of which Archimedes
promises a discussion, is given in a highly interesting and
important note by Eutocius, who introduces the subject with
the following explanation.

H. A. 5

66 ARCHIMEDES

"He [Archimedes] promised to give a solution of this
problem at the end, but we do not find the promise kept in any
of the copies. Hence we find that Dionysodorus too failed to
light upon the promised discussion and, being unable to grapple
with the omitted lemma, approached the original problem in a
different way, which I shall describe later. Diocles also ex-
pressed in his work irepl Trvpicov the opinion that Archimedes
made the promise but did not perform it, and tried to supply
the omission himself. His attempt I shall also give in its
order. It will however be seen to have no relation to the
omitted discussion but to give, like Dionysodorus, a construction
arrived at by a different method of proof. On the other hand,
as the result of unremitting and extensive research, I found in
a certain old book some theorems discussed which, although the
reverse of clear owing to errors and in many ways faulty as
regards the figures, nevertheless gave the substance of what I
sought, and moreover to some extent kept to the Doric dialect
affected by Archimedes, while they retained the names familiar in
old usage, the parabola being called a section of a right-angled
cone, and the hyperbola a section of an obtuse-angled cone;
whence I was led to consider whether these theorems might
not in fact be what he promised he would give at the end. For
this reason I paid them the closer attention, and, after finding
great difficulty with the actual text owing to the multitude of
the mistakes above referred to, I made out the sense gradually
and now proceed to set it out, as well as I can, in more familiar
and clearer language. And first the theorem will be treated
generally, in order that what Archimedes says about the limits
of possibility may be made clear ; after which there will follow
the special application to the conditions stated in his analysis
of the problem."

The investigation which follows may be thus reproduced.
The general problem is:

Given two straight lines AB, AC and an area D, to divide
ABatMsothat

ON THE SPHERE AND CYLINDER II.

Analysis.

Suppose M found, and suppose AC placed at right angles to
AB. Join CM and produce it. Draw EBN through B parallel
to AC meeting CM in N, and through C draw CHE parallel to
AB meeting EBN in E. Complete the parallelogram CENF,
and through M draw PMH parallel to AC meeting FN in P.

Measure EL along EN so that

CE. EL (or A B . EL) = D.
Then, by hypothesis,
AM :AC=CE.EL: MB\
And

AM: AC=CE :EN,

by similar triangles,

= CE.EL:EL.EN. c H

It follows that PJV 2 = Jf J? 2 = EL . EN.

Hence, if a parabola be described with vertex E t axis EN> and
parameter equal to EL, it will pass through P ; and it will be
given in position, since EL is given.

Therefore P lies on a given parabola.
Next, since the rectangles FH, AE are equal,
FP.PH=AB.BE.

Hence, if a rectangular hyperbola be described with CE, CF
as asymptotes and passing through B, it will pass through P.
And the hyperbola is given in position.

Therefore P lies on a given hyperbola.

Thus P is determined as the intersection of the parabola
and hyperbola. And since P is thus given, M is also given.

AM : AC = D : M B*,

Now, since

But AC. D is given, and it will be proved later that the mcmmum
value of AM. MB* is that which it assumes when BM = 2AM.

ARCHIMEDES

Hence it is a necessary condition of the possibility of a
solution that AG.D must not be greater than $AB.($AB)*> o?*

Synthesis.

If be such a point on AB that BO = 2AO, we have seen
that, in order that the solution may be possible,

AC.D^AO.OB*.
Thus AC. D is either equal to, or less than, AO . OB*.

(1 ) If A C . D = 4 . OB\ then the point itself solves the
problem.

(2) Let AC. D be less than AO. OB 9 .

Place AC at right angles to AB. Join CO, and produce it
to E. Draw EBR through B parallel to A G meeting CO in JR,
and through C draw CE parallel
to AB meeting EBR in E. Com-
plete the parallelogram CERF,
and through draw QOK parallel
to AC meeting FR in Q and CE
iuK.

Then, since

AC.D<AO.OB\
measure RQ' along RQ so that

Q Q'

or AO:AC=D:Q'R\

Measure EL along ER so that

D = CE.EL(orAB.EL).
Now, since AO : AC = D : Q'R*, by hypothesis,

= CE.EL:Q'R* t
and AO:AC = CE: ER, by similar triangles,

= CE . EL : EL . J?jf2,
it follows that

ON THE SPHERE AND CYLINDER II. 69

Describe a parabola with vertex E, axis ER, and parameter
equal to EL. This parabola will then pass through Q'.

Again, rect. FK = rect. A E,

or FQ.QK = AB.BE;

and, if we describe a rectangular hyperbola with asymptotes
CE, OF and passing through B, it will also pass through Q.

Let the parabola and hyperbola intersect at P, and through
P draw PMH parallel to AC meeting AB in M and CE
in H, and GPN parallel to AB meeting CF in G and ER
inN.

Then shall M be the required point of division.
Since PG.PH=AB.BE,

rect. GM=rect.ME,
and therefore CMN is a straight line.

Thus AB.BE = PO.PH = AM.EN (1).

Again, by the property of the parabola,
PN* = EL.EN,
or MB* = EL. EN (2).

From (1) and (2)

or AM.AB:AB.EL = AB.AC:MB\

Alternately,

AM.AB : AB.AC = AB.EL : MB\
or AM :AG = D:MB\

Proof of 8 1 o/j to-/* o 9.

It remains to be proved that, t/ -45 be divided at so that
BO = 2AO, then AO . OB" is the maximum value of AM. MB\

or AO.OB*>AM.MB\

where M is any point on AB other than 0.

ARCHIMEDES

Suppose that AO : AC
so that A O.OB*

Join CO, and produce it to N;
draw EBN through B parallel w
to AC, and complete the paral-
lelogram CENF.

Through draw POH
parallel to AC meeting FN
in P and CE in H.

With vertex E, axis EN,
and parameter EL', describe
a parabola. This will pass
through P, as shown in the
analysis above, and beyond P
will meet the diameter CF of
the parabola in some point.

Next draw a rectangular
hyperbola with asymptotes CE,
CF and passing through B.
This hyperbola will also pass
through P, as shown in the
analysis.

Produce NE to T so that
TE = EN. Join TP meeting
CE in F, and produce it to
meet CF in W. Thus TP will
touch the parabola at P.

Then, since BO =

CE. EL' : OB % ,
CE.EL'.AC.

2AO,

And TP = 2PY.

Therefore PW=PY.

Since, then, WY bet ween the asymptotes is bisected at P, the
point where it meets the hyperbola,

WY is a tangent to the hyperbola.

Hence the hyperbola and parabola, having a common tangent
at P, touch one another at P.

ON THE SPHERE AND CYLINDER II, 71

Now take any point M on AB, and through M draw QM K
parallel to AC meeting the hyperbola in Q and CE in K.
Lastly, draw GqQR through Q parallel to A B meeting OF in G,
the parabola in q, and EN in R.

Then, since, by the property of the hyperbola, the rectangles
GK, AE are equal, CMR is a straight line.

By the property of the parabola,

qR* = EL'.ER,

so that QR*<EL' .ER.

Suppose QR* = EL . ER,

and we have AM : AC = CE : ER

= CE.EL:EL.ER

= CE.EL:QR*

= CE.EL:MB*,

or AM.MB*=CE.EL.AC.

Therefore AM. MB 9 < CE.EL' . AC

<AO.OB\

If AC. D < AO . OB*, there are two solutions because there
will be two points of intersection between the parabola and the
hyperbola.

For, if we draw with vertex E and axis EN a parabola
whose parameter is equal to EL, the parabola will pass through
the point Q (see the last figure) ; and, since the parabola meets
the diameter CF beyond Q, it must meet the hyperbola again
(which has CF for its asymptote).

[If we put AB = a, BM = x, AC = c, and D = b\ the pro-
portion

AM: AC = D:Mff

is seen to be equivalent to the equation

a? 2 (a x) b*c,
being a cubic equation with the term containing x omitted.

Now suppose EN, EC to be axes of coordinates, EN being
the axis of y*

ARCHIMEDES

Then the parabola used in the above solution is the
parabola

, V
--.*

and the rectangular hyperbola is

y (a x) SB ac.

Thus the solution of the cubic equation and the conditions
under which there are no positive solutions, or one, or two
positive solutions are obtained by the use of the two conies.]

[For the sake of completeness, and for their intrinsic interest,
the solutions of the original problem in Prop. 4 given by
Dionysodorus and Diocles are here appended.

Dionysodorus' solution.

Let AA! be a diameter of the given sphere. It is required
to find a plane cutting A A' at right angles (in a point M,
suppose) so that the segments into which the sphere is divided
are in a given ratio, as CD : DE.

Produce A 'A to F so that AF OA, where is the centre
of the sphere.

c ' E

Draw AH perpendicular to AA' and of such length that

ON THE SPHERE AND CYLINDER II. 73

and produce AH to K so that

AK* = FA.AH ...................... (a).

With vertex F, axis FA, and parameter equal to AH
describe a parabola. This will pass through K, by the equa-
tion (a).

Draw A'K' parallel to AK and meeting the parabola in K' ;
and with A'F, A'K' as asymptotes describe a rectangular
hyperbola passing through H. This hyperbola will meet the
parabola at some point, as P, between K and K'.

Draw PM perpendicular to A A' meeting the great circle in
B, B', and from H, P draw HL, PR both parallel to AA' and
meeting A'K' in L, R respectively.

Then, by the property of the hyperbola,

PR.PM = AH.HL,
i.e. PM.MA' = HA.AA' t

or PM:AH = AA':A'M,

and PM* : AH* = AA" : A'M\

Also, by the property of the parabola,

FM.AH,

or FM : AH = PM* : AH*

= AA'* : A'M*, from above.

Thus, since circles are to one another as the squares of their
radii, the cone whose base is the circle with A'M as radius and
whose height is equal to FM t and the cone whose base is the
circle with AA' as radius and whose height is equal to AH,
have their bases and heights reciprocally proportional.

Hence the cones are equal ; i.e., if we denote the first cone
by the symbol c (A'M), FM, and so on,

c (A'M), FM = c (A A'), AH.
Now c(AA'), FA : c(AA'), AH = FA : AH

= CE : ED, by construction.

74 ARCHIMEDES

Therefore

c (A A), FA : c (AM), FM = CE:ED (0).

But (1) c (AA'\ FA = the sphere. [I. 34]

(2) c (AM), FM can be proved equal to the segment of
the sphere whose vertex is A' and height AM.

For take 6 on A A' produced such that
GM : MA = FM : MA

= OA + AM : AM.
Then the cone GEE' is equal to the segment A'EE' [Prop. 2].

And FM:MG = AM : MA, by hypothesis,

= BM* : A'M\
Therefore

(circle with rad. BM) : (circle with rad. AM)

= FM : MG,
so that c (AM), FM = c (BM ), M G

= the segment ABE'.

We have therefore, from the equation () above,
(the sphere) : (segmt. A f BB') = GE : ED,
whence (segmt. ABB') : (segmt. ABB') = CD : DE.

Diodes' solution.

Diocles starts, like Archimedes, from the property, proved in
Prop. 2, that, if the plane of section cut a diameter AA of the
sphere at right angles in M, and if H, H' be taken on OA, OA'
produced respectively so that

OA' + AM : AM = HM : MA,
OA+ AM : AM=H'M : MA,

then the cones HBB', H'EE' are respectively equal to the
segments ABB', A'EE'.

ON THE SPHEKE AND CYLINDER II.

Then, drawing the inference that

HA :AM=OA':A'M,
H'A' :A'M=OA : AM,

he proceeds to stato the problem in the following form, slightly
generalising it by the substitution of any given straight line for
OA or OA':

Given a straight line AA', its extremities A, A', a ratio C : D,
and another straight line as AK, to divide AA' at M and to find
two points H, H' on A' A and AA' produced respectively so that
tJie following relations may hold simultaneously,

HA :AM=AK:A'M
H'A' : A'M = AK : AM

(7).

Analysis.

Suppose the problem solved and the points M, H, H' all
found.

Place AK at right angles to AA', and draw A'K' parallel
and equal to AK. Join KM, K'M, and produce them to meet
K'A'> KA respectively in E, F. Join KK' y draw EG through
E parallel to A' A meeting KF in G, and through M draw QMN
parallel to AK meeting EG in Q and KK' in N.

Now HA : A M = A'K 1 : A'M, by (),

= FA : A M, by similar triangles,
whence HA = FA.

Similarly H'A' = A'E.

Next,

FA + AM : A'K' + A'M = AM : A'M

= AK+ AM : EA' + A'M, by similar triangles.

ARCHIMEDES

Therefore

(FA + AM) . (EA f + A'M) = (KA + AM) . (K'A' + AM).
Take AR along AH and A'R' along AH' such that

Then, since FA + A M = HM, EA' + A'M = MH', we have
HM.MH' = RM.MR' .................. (8).

(Thus, if .R falls between A and IT, R f falls on the side of H'
remote from A' y and vice versa.)

: ME', by hypothesis,

Now

= RM. MR' : MH'\ by (8).

Measure -ZlfF along MN so that MV=A'M. Join 4'F and
produce it both ways. Draw RP, R'P' perpendicular to RR'
meeting A'V produced in P, P' respectively. Then, the angle
MA'V being half a right angle, PP' is given in position, and,
since R, R' are given, so are P, P 7 .

And, by parallels,

P'V:PV=R'M :MR.

ON THE SPHERE AND CYLINDER II. 77

Therefore PV . P'F: PF a = RM . MR' : RM\

But PIT = 2RM*.

Therefore PF. P'F = *RM . MR'.

And it was shown that

RM.MR' :MH'* = C:D.

Hence PF. P'F : MH'* = 2C:D.
But MH' = A'M + A'E = VM + MQ = QF

Therefore QF 8 : PF. P'F = D : 2(7, a given ratio.
Thus, if we take a line p such that

and if we describe an ellipse with PP' as a diameter and p as
the corresponding parameter [= DD'*/PP' in the ordinary
notation of geometrical conies], and such that the ordinates to
PP' are inclined to it at an angle equal to half a right angle,
i.e. are parallel to QF or AK, then the ellipse will pass
through Q.

Hence Q lies on an ellipse given in position.
Again, since EK is a diagonal of the parallelogram GK',
GQ.QN=AA'.A'K'.

If therefore a rectangular hyperbola be described with KG,
KK' as asymptotes and passing through A', it will also pass
through Q.

Hence Q lies on a given rectangular hyperbola.

Thus Q is determined as the intersection of a given ellipse

* There is a mistake in the Greek text here which seems to have escaped the
notice of all the editors up to the present. The words are lav apa iroujcrv/ttei', w*
r^v A vpbs rty 8nr\a<rlat> TT)S F, ofJrws ryv TT irp6j oXX^v TWO, wj rty 4>, i.e. (with
the lettering above) " If we take a length p such that D : 2C = PP' : jp." This
cannot he right, because we should then have

Q,V*'.PV.P'V=PP'\p,

whereas the two latter terms should be reversed, the correct property of the
ellipse being

QV* : PV.P'V=p : PP'. [Apollonius I. 21]

The mistake would appear to have originated as far back as Eutocius, but I
think that Eutocius is more likely to have made the slip than Diooles himself,
because any intelligent mathematician would be more likely to make such a slip
in writing out another man's work than to overlook it if made by another.

78 ARCHIMEDES

and a given hyperbola, and is therefore given. Thus M is
given, and H, H''ca,n at once be found.

Synthesis.

Place AA' y AK at right angles, draw A'K' parallel and
equal to AK, and join KK'.

Make AR (measured along A'A produced) and A'R'
(measured along AA produced) each equal to AK, and
through R y R' draw perpendiculars to RR'.

Then through A' draw PP' making an angle (AA 'P) with
A A 1 equal to half a right angle and meeting the perpendiculars
just drawn in P, P' respectively.

Take a length p such that

D:2C = p :PP'*,

and with PP 7 as diameter and p as the corresponding parameter
describe an ellipse such that the ordinates to PP' are inclined
to it at an angle equal to A A'P, i.e. are parallel to AK.

With asymptotes KA t KK' draw a rectangular hyperbola
passing through A 1 .

Let the hyperbola and ellipse meet in Q, and from Q draw
QM VN perpendicular to AA' meeting AA' in M, PP' in V
and KK' in N. Also draw GQE parallel to AA' meeting AK,
A'K' respectively in G, E.

Produce KA, K'M to meet in F.
Then, from the property of the hyperbola,
GQ.QN=AA'.A'K' 9
and, since these rectangles are equal, KME is a straight line.

Measure AH along AR equal to AF, and A'H' along A'R'
equal to A'E.

From the property of the ellipse,

QV*:PV.P'V=p:PP'
= D:2C.

* Here too the Greek text repeats the same error as that noted on p. 77.

ON THE SPHERE AND CYLINDER II. 79

And, by parallels,

PV:P'V=RM:R'M,
or PV.P'V:P'V* = RM.MR':R'M*,

while P'V* = 2R'M*, since the angle RA'P is half a right
angle.

Therefore PV.P'V=1RM. MR',

whence QF" : 1RM. MR' = D:2C.

But QV=EA' + A'M=MH'.

Therefore RM.MR' : MH" = G:D.
Again, by similar triangles,

FA + A M : K'A' + A'M=AM: A'M

= KA+AM:EA' + A'M.
Therefore

(FA + AM).(EA' + A'M) = (KA + AM). (K'A' + A'M)
or HM.MH' = RM.MR'.

It follows that

HM.MH':MH'* = C:D,

or HM : MH' = C: D (a).

Also HA : AM = FA: AM,

= A'K':A'M, by similar

triangles... (/3),
and H'A':A'M=EA':A'M

= AK:AM ( 7 ).

Hence the points M, H, H' satisfy the three given
relations.]

Proposition 5. (Problem.)

To construct a segment of a sphere similar to one segment
and equal in volume to another.

Let ABB' be one segment whose vertex is A and whose
base is the circle on BB' as diameter; and let DEF be another
segment whose vertex is D and whose base is the circle on EF

ARCHIMEDES

as diameter. Let AA', DD' be diameters of the great circles
passing through BB' y EF respectively, and let 0, C be the
respective centres of the spheres.

Suppose it required to draw a segment similar to DE F and
equal in volume to ABB'.

Analysis. Suppose the problem solved, and let def be the
required segment, d being the .vertex and ef the diameter of
the base. Let dd' be the diameter of the sphere which bisects
e/at right angles, c the centre of the sphere.

Let M, G, g be the points where BB', EF, ef are bisected
at right angles by A A, DD' t dd' respectively, and produce 0-4,
<7jD, cd respectively to H, K, k, so that

OA' + A'M:A'M=HM:MA\

cd' + d'g:d'g = kg:gd

and suppose cones formed with vertices H, K, k and with the
same bases as the respective segments. The cones will then be
equal to the segments respectively [Prop. 2],

Therefore, by hypothesis,

the cone HBB' = the cone kef.

ON THE SPHERE AND CYLINDER II. 81

Hence

(circle on diameter BB') : (circle on diameter ef) = kg : HM,
so that BB' s :ef* = kg:HM .................. (1).

But, since the segments DBF, 'def are similar, so are the
cones KEF, kef.

Therefore KG : EF =* kg : ef.

And the ratio KG : EF is given. Therefore the ratio kg : ef
is given.

Suppose a length R taken such that

kg:ef=HM:R ..................... (2).

Thus R is given.

Again, since kg : HM= BB' 2 : ef 3 = ef:R 9 by (1) and (2),
suppose a length 8 taken such that

or B'*:ef*

Thus BB' : ef=ef: 8 = 8 : R,

and ef, 8 are two mean proportionals in continued proportion
between BB', R.

Synthesis. Let ABB', DEF be great circles, AA', DD'
the diameters bisecting BB', EF at right angles in M, G
respectively, and 0, C the centres.

Take H, K in the same way as before, and construct the
cones HBB', KEF, which are therefore equal to the respective
segments ABB', DEF.

Let R be a straight line such that

KG:EF=HM: R,
and between BB', R take two mean proportionals ef, 8.

On ef as base describe a segment of a circle with vertex d
and similar to the segment of a circle DEF. Complete the
circle, and let dd' be the diameter through d, and c the centre.
Conceive a sphere constructed of which def is a great circle,
and through ef draw a plane at right angles to dd'.
H. A. 6

82 ARCHIMEDES

Then shall def\>e the required segment of a sphere.

For the segments DBF, def of the spheres are similar, like
the circular segments DEF, def.

Produce cd to k so that

cd' + d'g : d'g = kg : gd.
The cones KEF, kef&cQ then similar.

Therefore kg : ef= KG : EF= HM : R,
whence kg : HM = ef:R.

But, since BB', ef, S, R are in continued proportion,
BB": ef* = BB':S

= ef:R
= kg:HM.

Thus the bases of the cones HBB', kef are reciprocally
proportional to their heights. The cones are therefore equal,
and def is the segment required, being equal in volume to the
cone kef. [Prop. 2]

Proposition 6. (Problem.)

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.

Let ABB' be the segment to whose surface the surface of
the required segment is to be equal, ABA'B' the great circle
whose plane cuts the plane of the base of the segment A BB' at
right angles in BB'. Let A A' be the diameter which bisects
BB' at right angles.

Let DEF be the segment to which the required segment
is to be similar, DED'F the great circle cutting the base of the
segment at right angles in EF. Let DD' be the diameter
bisecting EF at right angles in 0.

Suppose the problem solved, def being a segment similar
to DEF and having its surface equal to that of ABB'\ and

ON THE SPHERE AND CYLINDER II.

complete the figure for def as for DEF, corresponding points
being denoted by small and capital letters respectively.

Join AB, DF, df.

Now, since the surfaces of the segments def, ABB' are equal,
so are the circles on df, AB as diameters ; [I. 42, 43]

that is, df=AB.

From the similarity of the segments DEF, def we obtain

d'd : dg = D'D : DO,

and dff:df=DG:DF;

whence d'd : df= D'D : DF,

or d'd:AB = D'D:DF.

But AB, D'D, DF are all given ;

therefore d'd is given.
Accordingly the synthesis is as follows.
Take d'd such that

d'd:AB = D'D:DF. (1).

Describe a circle on d'd as diameter, and conceive a sphere
constructed of which this circle is a great circle.

84 ARCHIMEDES

Divide d'd at g so that

and draw through g a plane perpendicular to d'd cutting off
the segment defof the sphere and intersecting the plane of the
great circle in ef. The segments def, DEF are thus similar,
and dg\df=DG:DF.

But from above, componendo,

Therefore, ex aequali, d'd : df= D'D : DF,
whence, by (1), df= AB.

Therefore the segment def has its surface equal to the
surface of the segment ABB' [I. 42, 43], while it is also similar
to the segment DEF.

Proposition 7. (Problem.)

From a given sphere to cut off a segment by a plane so that
the segment may have a given ratio to the cone which has the same
base as the segment and equal height

Let AA' be the diameter of a great circle of the sphere.
It is required to draw a plane at right angles to A A' cutting
off a segment, as ABB', such that the segment ABB' has to
the cone ABB' a given ratio.

Analysis.

Suppose the problem solved, and let the plane of section
cut the plane of the great circle in BB', and the diameter
AA' in M. Let be the centre of the sphere.

Produce OA to H so that

OA' + A'M : A'M = HM : MA.

.(1).

ON THE SPHERE AND CYLINDER II. 85

Thus the cone HBB' is equal to the segment ABB'. [Prop. 2]

Therefore the given ratio must be equal to the ratio of the
cone HBB' to the cone ABB', i.e. to the ratio HM : MA.

Hence the ratio OA 1 -f A'M : A'M is given ; and therefore
A'M is given.

Now OA':A'M>OA' :A'A,

so that OA' + A'M : A'M > OA' + A' A : A' A

>3:2.

Thus, in order that a solution may be possible, it is a
necessary condition that the given ratio must be greater than
3:2.

The synthesis proceeds thus.

Let A A' be a diameter of a great circle of the sphere, the
centre.

Take a line DE, and a point F on it, such that DE : EF is
equal to the given ratio, being greater than 3 : 2.

Now, since OA' + A' A : A' A = 3:2,

DE : EF > OA' + A'A : A'A 9
so that DF:FE> OA' : A'A.

Hence a point M can be found on AA' such that

DF : FE = OA' : A'M. ................... (2).

Through M draw a plane at right angles to A A' intersecting
the plane of the great circle in BB', and cutting off from the
sphere the segment ABB'.

As before, take H on OA produced such that

OA' + A'M : A'M^HM : MA.
Therefore HM :MA=DE : EF, by means of (2).

It follows that the cone HBB', or the segment ABB', is to
the cone ABB' in the given ratio DE : EF.

ARCHIMEDES

Proposition 8.

If a sphere be cut by a plane not passing through the. centre
into two segments ABB', ABB', of which ABB' is the greater,
then the ratio

(segmt. ABB') : (segmt. ABB')

< (surface of ABB J : (surface of ABB J
but > (surface of ABB J : (surface of ABB')**.

Let the plane of section cut a great circle A BAB' at right
angles in BE', and let AA be the diameter bisecting BB' at
right angles in M.

Let be the centre of the sphere.
Join AB, AB.

As usual, take H on OA produced, and H' on OA' produced,
so that

OA + AM:AM = HM:MA (1),

OA+AM\AM=H'M:MA (2),

and conceive cones drawn each with the same base as the two
segments and with apices H, H' respectively. The cones are
then respectively equal to the segments [Prop. 2], and they
are in the ratio of their heights HM, H'M.

Also

(surface of ABB f ) : (surface of ABB') = Aff : AB* [I. 42, 43]

= A'M:AM.

* This is expressed in Archimedes 1 phrase by saying that the greater seg-
ment has to the lesser a ratio "less than the duplicate (dur\d<rtoi>) of that which
the surface of the greater segment has to the surface of the lesser, but greater
than the sesquialterate (i)fu6\iov) [of that ratio]."

ON THE SPHERE AND CYLINDER II. 87

We have therefore to prove
(a) that H'M : MH < A'M* : MA*,

(6) that H'M : MH > A'M* : MA*.

(a) From (2) above,

A'M : AM = H'M -.OA+AM

= H'A' : OA', since OA = OA'.

Since A'M > AM, H'A' > OA' ; therefore, if we take K on
H'A' so that OA' = A'K, K will fall between H' and A'.

And, by (1), A'M : AM = KM : MH.

Thus KM : MH = H'A' : A'K, since A'K - OA,

> H'M : MK.

Therefore H'M . MH < KM*.

It follows that

H'M. MH : MH* < KM 1 : MH*,
or H'M : MH < KM 1 : MH*

< A'M* : AM 3 , by (1).

(b) Since OA' = OA,

A'M.MA<A'O.OA,
or A'M : OA' < OA : AM

< H'A' : A'M, by means of (2).
Therefore A'M 1 < H'A' . OA'

< H'A'. A'K.
Take a point N on A' A such that

A'N 1 = H'A'. A'K.

Thus H' A' : A'K = A'N*: A'K* (3).

Also H'A' : A'N = A'N : A'K,

and, componendo,

H'N : A'N= NK : A'K,
whence A'N* : A'K 1 = H'lf* : NK*.

88 ARCHIMEDES

Therefore, by (3),

Now H'M : MK>H'N: NK.

Therefore H'M* : MK* > H'A' : A'K

> H'A' : OA'

> A'M: MA, by (2), as above,
>OA' + A'M:MH,\>y(l),
>KM:MH.

Hence H'M* : MH* = (H'M * : MK*) . (KM 9 : MH*)

>(KM:MH).(KM*-.MH*}.
It follows that

[The text of Archimedes adds an alternative proof of this
proposition, which is here omitted because it is in fact neither
clearer nor shorter than the above.]

Proposition 9.

Of all segments of spheres which have equal surfaces the
hemisphere is the greatest in volume.

Let ABA'B' be a great circle of a sphere, AA' being
a diameter, and the centre. Let the sphere be cut by
a plane, not passing through 0, perpendicular to AA' (at Jl/),
and intersecting the plane of the great circle in BB'. The
segment ABB 1 may then be either less than a hemisphere as
in Fig. 1, or greater than a hemisphere as in Fig. 2.

Let DED'E' be a great circle of another sphere, DD'
being a diameter and C the centre. Let the sphere be cut by
a plane through C perpendicular to DD' and intersecting the
plane of the great circle in the diameter EE'.

ON THE SPHERE AND CYLINDER II.

Suppose the surfaces of the segment ABB' and of the
hemisphere DEE' to be equal.

Since the surfaces are equal, AB = DE.
Now, in Fig. 1, AB 2 > 2AM 2 and < 2AO\
and, in Fig. 2, AB 2 < 2AM 2 and > 2A 0\

Hence, if R be taken on AA' such that

[I. 42, 43]

R will fall between and M.

Also, since AB 2 = DE 2 , AR = CD.

Produce OA' to K so that OA' = 4'
# so that

, and produce A 9 A to

or, componendo, A'K+A'M:A'M=HM:MA ............ (1).

Thus the cone HBB' is equal to the segment ABB f .

[Prop. 2]

Again, produce CD to F so that CD = DJP, and the cone
FEE 1 will be equal to the hemisphere DEE'. [Prop. 2]

Now A R . RA' > AM. MA f t

and

90 ARCHIMEDES

Hence

AR . RA' + RA* > AM. MA + AM. A'K,
or AA'.AR>AM.MK

>HM.A'M, by (1).
Therefore AA : AM > HM : AR,
or AB* : BM 2 > HM : AR,

i.e. AR 9 : BM* > HM : 2AR, since AB 2

>HM:CF.

Thus, since A R = CD, or CE,

(circle on diam. EE') : (circle on diam. BB') > HM : CF.
It follows that

(the cone FEE') > (the cone HBB'),

and therefore the hemisphere DEE' is greater in volume than
the segment ABB'.

MEASUBEMENT OF A CIRCLE.

Proposition 1.

The area of any circle is equal to a right-angled triangle in
which one of the sides about the right angle is equal to the radius,
and the other to the circumference, of the circle.

Let ABCD be the given circle, K the triangle described.

Then, if the circle is not equal to K, it must be either
greater or less.

I. If possible, let the circle be greater than K.

Inscribe a square ABCD, bisect the arcs AB, BC, CD, DA,
then bisect (if necessary) the halves, and so on, until the sides
of the inscribed polygon whose angular points are the points of
division subtend segments whose sum is less than the excess of
the area of the circle over K.

92 ARCHIMEDES

Thus the area of the polygon is greater than K.

Let AE be any side of it, and ON the perpendicular on A E
from the centre 0.

Then ON is less than the radius of the circle and therefore
less than one of the sides about the right angle in K. Also the
perimeter of the polygon is less than the circumference of the
circle, i.e. less than the other side about the right angle in K.

Therefore the area of the polygon is less than K ; which is
inconsistent with the hypothesis.

Thus the area of the circle is not greater than K.

II. If possible, let the circle be less than K.

Circumscribe a square, and let two adjacent sides, touching
the circle in E, H, meet in T. Bisect the arcs between adjacent
points of contact and draw the tangents at the points of
bisection. Let A be the middle point of the arc EH, and FAG
the tangent at A.

Then the angle TAG is a right angle.
Therefore TG > GA

>GH.

It follows that the triangle FTG is greater than half the area
TEAH.

Similarly, if the arc AH be bisected and the tangent at the
point of bisection be drawn, it will cut off from the area GAH
more than one-half.

Thus, by continuing the process, we shall ultimately arrive
at a circumscribed polygon such that the spaces intercepted
between it and the circle are together less than the excess of
K over the area of the circle.

Thus the area of the polygon will be less than K.

Now, since the perpendicular from on any side of the
polygon is equal to the radius of the circle, while the perimeter
of the polygon is greater than the circumference of the circle,
it follows that the area of the polygon is greater than the
triangle K\ which is impossible.

MEASUREMENT OF A CIRCLE. 93

Therefore the area of the circle is not less than K.
Since then the area of the circle is neither greater nor less
than K 9 it is equal to it.

Proposition 2.

The area of a circle is to the square on its diameter as 11
to 14.

[The text of this proposition is not satisfactory, and Archi-
medes cannot have placed it before Proposition 3, as the
approximation depends upon the result of that proposition,]

Proposition 3.

The ratio of the circumference of any circle to its diameter
is less than 3f but greater than

[In view of the interesting questions arising out of the
arithmetical content of this proposition of Archimedes, it is
necessary, in reproducing it, to distinguish carefully the actual
steps set out in the text as we have it from the intermediate
steps (mostly supplied by Eutocius) which it is convenient to
put in for the purpose of making the proof easier to follow.
Accordingly all the steps not actually appearing in the text
have been enclosed in square brackets, in order that it may be
clearly seen how far Archimedes omits actual calculations and
only gives results. It will be observed that he gives two
fractional approximations to \/3 (one being less and the other
greater than the real value) without any explanation as to how
he arrived at them ; and in like manner approximations to the
square roots of several large numbers which are not complete
squares are merely stated. These various approximations and
the machinery of Greek arithmetic in general will be found
discussed in the Introduction, Chapter IV.]

I. Let AB be the diameter of any circle, its centre, AC
the tangent at A ; and let the angle A OC be one-third of a
right angle.

94 ARCHIMEDES

Then OA : AC\=f& : 1]>265 : 153 (1),

and OC : CA [=2 : 1] = 306 : 153 (2).

First, draw OD bisecting the angle AOG and meeting AC
inD.

Now CO : OA = CD: DA, [Eucl. VI. 3]

so that [CO + OA:OA = CA: DA, or]

CO + OA :CA = OA : AD.
Therefore [by (1) and (2)]

OA :AD>671 : 153 (3).

Hence OD* : A D* [= (OA* + AD*) : AD*

> (571 s + 153') :153 2 ]

> 349450 : 23409,

so that OD :DA>591% : 153 (4).

Secondly, let OE bisect the angle AOD, meeting AD in E.
[Then DO : OA = DE : EA,

so that DO + OA : DA = OA : AE.]

Therefore OA : AE [> (591 + 571) : 153, by (3) and (4)]

(5).

MEASUREMENT OF A CIRCLE. 95

[It follows that

OE* : EA* > {(1162) 2 + 153 2 } : 153 s

> (1350534|f + 23409) : 23409

> 1373943ff : 23409.]

Thus OE:EA> 1172 : 153 ..................... (6).

Thirdly, let OF bisect the angle AOE and meet AE in F.

We thus obtain the result [corresponding to (3) and (5)

above] that

OA : AF[> (1162 + 1172) : 153]

>2334: 153 ..................... (7).

[Therefore OF* : FA 2 > {(2334J) 2 + 153 2 } : 153 2

> 5472132^ : 23409.]
Thus OF:FA>23M:153 ..................... (8).

Fourthly, let OG bisect the angle AOF, meeting AF in (?.
We have then

OA : AG [> (2334J + 2339J) : 153, by means of (7) and (8)]
> 4673^:153.

Now the angle AOC, which is one-third of a right angle,
has been bisected four times, and it follows that

Z A 00 ? ^ (a right angle).

Make the angle AOH on the other side of OA equal to the
angle A OG, and let OA produced meet OH in H.

Then Z GOH=^ (a right angle).

Thus GH is one side of a regular polygon of 96 sides cir-
cumscribed to the given circle.

And, since OA : AG > 4673 : 153,
while AE = 20 A, GH = 2AG,

it follows that

A B : (perimeter of polygon of 96 sides) [> 4673J : 153 x 96]

^ 4673 J: 14688.

AKCHIMEDES

14688 _ 667*
~

<3f

Therefore the circumference of the circle (being less than
the perimeter of the polygon) is a fortiori less than 3f times
the diameter AB.

II. Next let AB be the diameter of a circle, and let AC,
meeting the circle in C y make the angle CAB equal to one-third
of a right angle. Join BG.

Then AC : CB [= V3 : 1] < 1351 : 780.

First, let AD bisect the angle BAG and meet BC in d and
the circle in D. Join BD.

Then

and the angles at -D, C are both right angles.

It follows that the triangles ADB, [ACd], BDd are similar.

Therefore

AD:DB = BD: Dd

[=AC:Cd]

= AB:Bd

[Eucl. VI. 3]

=AB+AC:BC
BA+AC:BC=AD:DB.

MEASUREMENT OF A CIRCLE. 97

[But AC : CB< 1351 : 780, from above,

while BA :5tf=2:l

- 1560 : 780.]

Therefore 4D :ZXB< 2911 : 780 (1).

[Hence A B* : BD* < (291 1* + 780*) : 780*

< 9082321 : 608400.]
Thus AB :BD< 3013f : 780 (2).

Secondly, let AE bisect the angle BAD, meeting the circle
in E\ and let BE be joined.

Then we prove, in the same way as before, that
AE: EB[=BA + AD-.BD

< (3013| + 2911) : 780, by (1) and (2)]

< 5924f : 780

< 5924f x js : 780 x &

<1823 : 240 (3).

[Hence A B* : BE 9 < (1823 2 + 240 2 ) : 240*
< 3380929 : 57600.]

Therefore AB : BE < 1838 T \ : 240 (4).

Thirdly, let AF bisect the angle BAE t meeting the circle
in F.

Thus AF : FB [= BA + AE : BE

< 3661.fr : 240, by (3) and (4)]

< 3661 T 9 T x ii : 240 x U
<1007 :66 (5).

[It follows that

AB 9 : BF* < (1007 s + 66') : 66*

< 1018405 : 4356.]
Therefore AB : BF< 1009 : 66 (6).

Fourthly, let the angle BAF be bisected by AG meeting the
circle in G.

Then AG : GB [= BA + AF : BF]

< 2016J : 66, by (5) and (6).

H. A. 7

98 ARCHIMEDES

[And Aff : BQ* < {(2016J)' + 66 2 } : 66'

< 4069284^ : 4356.]
Therefore AB : BQ < 2017* : 66,
whence BG : AB > 66 : 2017J (7).

[Now the angle SAG which is the result of the fourth bisection
of the angle BAG, or of one- third of a right angle, is equal to
one-fortyeighth of a right angle.

Thus the angle subtended by BG at the centre is
Jj (a right angle).]

Therefore BG is a side of a regular inscribed polygon of 9(>
sides.

It follows from (7) that

(perimeter of polygon) : AB [> 96 x 66 : 20l7]

> 6336: 2017 J.

A A 6336 010

And 2bl7i >8 W-

Much more then is the circumference of the circle greater than
times the diameter.

Thus the ratio of the circumference to the diameter
< 3 but >

ON CONOIDS AND SPHEKOIDS.

Introduction *.

" ARCHIMEDES to Dositheus greeting.

In this book I have set forth and send you the proofs of the
remaining theorems not included in what I sent you before, and
also of some others discovered later which, though I had often
tried to investigate them previously, I had failed to arrive at
because I found their discovery attended with some difficulty.
And this is why even the propositions themselves were not
published with the rest. But afterwards, when I had studied
them with greater care, I discovered what I had failed in
before.

Now the remainder of the earlier theorems were propositions
concerning the right-angled conoid [paraboloid of revolution] ;
but the discoveries which I have now added relate to an obtuse-
angled conoid [hyperboloid of revolution] and to spheroidal
figures, some of which I call oblong (Trapapatcea) and others fiat

I. Concerning the right-angled conoid it was laid down
that, if a section of a right-angled cone [a parabola] be made to
revolve about the diameter [axis] which remains fixed and

* The whole of this introductory matter, including the definitions, is trans-
lated literally from the Greek text in order that the terminology of Archimedes
may be faithfully represented. When this has once been set out, nothing will
be lost by returning to modern phraseology and notation. These will accordingly
be employed, as usual, when we come to the actual propositions of the treatise.

100 ARCHIMEDES

return to the position from which it started, the figure compre-
hended by the section of the right-angled cone is called a right-
angled conoid, and the diameter which has remained fixed
is called its axis, while its vertex is the point in which the
axis meets (airrerai) the surface of the conoid. And if a plane
touch the right-angled conoid, and another plane drawn parallel
to the tangent plane cut off a segment of the conoid, the base
of the segment cut off is defined as the portion intercepted by
the section of the conoid on the cutting plane, the vertex
[of the segment] as the point in which the first plane touches
the conoid, and the axis [of the segment] as the portion cut
off within the segment from the line drawn through the vertex
of the segment parallel to the axis of the conoid.

The questions propounded for consideration were

(1) why, if a segment of the right-angled conoid be cut off
by a plane at right angles to the axis, will the segment so cut
off be half as large again as the cone which has the same base
as the segment and the same axis, and

(2) why, if two segments be cut off from the right-angled
conoid by planes drawn in any manner, will the segments so cut
off have to one another the duplicate ratio of their axes.

II. Respecting the obtuse-angled conoid we lay down the
following premisses. If there be in a plane a section of an
obtuse-angled cone [a hyperbola], its diameter [axis], and the
nearest lines to the section of the obtuse-angled cone [i.e. the
asymptotes of the hyperbola], and if, the diameter [axis]
remaining fixed, the plane containing the aforesaid lines be
made to revolve about it and return to the position from which
it started, the nearest lines to the section of the obtuse-angled
cone [the asymptotes] will clearly comprehend an isosceles cone
whose vertex will be the point of concourse of the nearest lines
and whose axis will be the diameter [axis] which has remained
fixed. The figure comprehended by the section of the obtuse-
angled cone is called an obtuse-angled conoid [hyperboloid of
revolution], its axis is the diameter which has remained fixed,
and its vertex the point in which the axis meets the surface

ON CONOIDS AND SPHEROIDS. 101

of the conoid. The cone comprehended by the nearest lines to
the section of the obtuse-angled cone is called [the cone]
enveloping the conoid (irepiexcov TO fcwvoeiSes), and the
straight line between the vertex of the conoid and the vertex
of the cone enveloping the conoid is called [the line] adjacent
to the axis (iroTeovtra rc5 agovi). And if a plane touch the
obtuse-angled conoid, and another plane drawn parallel to the
tangent plane cut off a segment of the conoid, the base of
the segment so cut off is defined as the portion intercepted by
the section of the conoid on the cutting plane, the vertex [of
the segment] as the point of contact of the plane which touches
the conoid, the axis [of the segment] as the portion cut off
within the segment from the line drawn through the vertex of
the segment and the vertex of the cone enveloping the conoid ;
and the straight line between the said vertices is called
adjacent to the axis.

Right-angled conoids are all similar; but of obtuse-angled
conoids let those be called similar in which the cones enveloping
the conoids are similar.

The following questions are propounded for consideration,

(1) why, if a segment be cut off from the obtuse-angled
conoid by a plane at right angles to the axis, the segment so
cut off has to the cone which has the same base as the segment
and the same axis the ratio which the line equal to the sum
of the axis of the segment and three times the line adjacent
to the axis bears to the line equal to the sum of the axis of
the segment and twice the line adjacent to the axis, and

(2) why, if a segment of the obtuse-angled conoid be cut
off by a plane not at right angles to the axis, the segment so
cut off will bear to the figure which has the same base as
the segment and the same axis, being a segment of a cone*
(atrorp.afjLa tcwvov), the ratio which the line equal to the sum
of the axis of the segment and three times the line adjacent
to the axis bears to the line equal to the sum of the axis of the
segment and twice the line adjacent to the axis.

* A segment of a cone is defined later (p. 104).

102 ARCHIMEDES

III. Concerning spheroidal figures we lay down the follow-
ing premisses. If a section of an acute-angled cone [ellipse] be
made to revolve about the greater diameter [major axis] which
remains fixed and return to the position from which it started,
the figure comprehended by the section of the acute-angled
cone is called an oblong spheroid (trapap.aK^ <r<<upoei8e?).
But if the section of the acute-angled cone revolve about the
lesser diameter [minor axis] which remains fixed and return
to the position from which it started, the figure comprehended
by the section of the acute-angled cone is called a flat spheroid
(eTTiTrXarv ffQaipoeiSes). In either of the spheroids the axis
is defined as the diameter [axis] which has remained fixed, the
vertex as the point in which the axis meets the surface of the
spheroid, the centre as the middle point of the axis, and the
diameter as the line drawn through the centre at right angles
to the axis. And, if parallel planes touch, without cutting,
either of the spheroidal figures, and if another plane be drawn
parallel to the tangent planes and cutting the spheroid, the
base of the resulting segments is defined as the portion inter-
cepted by the section of the spheroid on the cutting plane, their
vertices as the points in which the parallel planes touch the
spheroid, and their axes as the portions cut off within the
segments from the straight line joining their vertices. And
that the planes touching the spheroid meet its surface at one
point only, and that the straight line joining the points of
contact passes through the centre of the spheroid, we shall
prove. Those spheroidal figures are called similar in which
the axes have the same ratio to the 'diameters/ And let
segments of spheroidal figures and conoids be called similar if
they are cut off from similar figures and have their bases
similar, while their axes, being either at right angles to the
planes of the bases or making equal angles with the corre-
sponding diameters [axes] of the bases, have the same ratio
to one another as the corresponding diameters [axes] of the

The following questions about spheroids are propounded for
consideration,

(1) why, if one of the spheroidal figures be cut by a plane

ON CONOIDS AND SPHEROIDS. 103

through the centre at right angles to the axis, each of the
resulting segments will be double of the cone having the same
base as the segment and the same axis ; while, if the plane of
section be at right angles to the axis without passing through
the centre, (a) the greater of the resulting segments will bear
to the cone which has the same base as the segment and the
same axis the ratio which the line equal to the sum of half the
straight line which is the axis of the spheroid and the axis of
the lesser segment bears to the axis of the lesser segment, and
(6) the lesser segment bears to the cone which has the same
base as the segment and the same axis the ratio which the line
equal to the sum of half the straight line which is the axis
of the spheroid and the axis of the greater segment bears to the
axis of the greater segment ;

(2) why, if one of the spheroids be cut by a plane passing
through the centre but not at right angles to the axis, each of
the resulting segments will be double of the figure having the
same base as the segment and the same axis and consisting of a
segment of a cone*.

(3) But, if the plane cutting the spheroid be neither
through the centre nor at right angles to the axis, (a) the
greater of the resulting segments will have to the figure
which has the same base as the segment and the same axis
the ratio which the line equal to the sum of half the line
joining the vertices of the segments and the axis of the lesser
segment bears to the axis of the lesser segment, and (6) the
lesser segment will have to the figure with the same base
as the segment and the same axis the ratio which the line
equal to the sum of half the line joining the vertices of the
segments and the axis of the greater segment bears to the axis
of the greater segment. And the figure referred to is in these
cases also a segment of a cone*.

When the aforesaid theorems are proved, there are dis-
covered by means of them many theorems and problems.
Such, for example, are the theorems
(1) that similar spheroids and similar segments both of

* See the definition of a segment of a cone (MrnafM K&*OV) on p. 104.

104 ARCHIMEDES

spheroidal figures and conoids have to one another the triplicate
ratio of their axes, and

(2) that in equal spheroidal figures the squares on the
' diameters ' are reciprocally proportional to the axes, and, if in
spheroidal figures the squares on the ' diameters 1 are reciprocally
proportional to the axes, the spheroids are equal.

Such also is the problem, From a given spheroidal figure
or conoid to cut off a segment by a plane drawn parallel to a
given plane so that the segment cut off is equal to a given cone
or cylinder or to a given sphere.

After prefixing therefore the theorems and directions (eVi-
Tayiuvra) which are necessary for the proof of them, I will
then proceed to expound the propositions themselves to you.
Farewell.

DEFINITIONS.

If a cone be cut by a plane meeting all the sides [generators]
of the cone, the section will be either a circle or a section of an
acute-angled cone [an ellipse]. If then the section be a circle,
it is clear that the segment cut off from the cone towards the
same parts as the vertex of the cone will be a cone. But, if
the section be a section of an acute-angled cone [an ellipse], let
the figure cut off from the cone towards the same parts as the
vertex of the cone be called a segment of a cone. Let the
base of the segment be defined as the plane comprehended by
the section of the acute-angled cone, its vertex as the point
which is also the vertex of the cone, and its aads as the straight
line joining the vertex of the cone to the centre of the section
of the acute-angled cone.

And if a cylinder be cut by two parallel planes meeting all
the sides [generators] of the cylinder, the sections will be either
circles or sections of acute-angled cones [ellipses] equal and
similar to one another. If then the sections be circles, it is
clear that the figure cut off from the cylinder between the
parallel planes will be a cylinder. But, if the sections be
sections of acute-angled cones [ellipses], let the figure cut off
from the cylinder between the parallel planes be called a
frustum (TO/UO?) of a cylinder. And let the bases of the

ON CONOIDS AND SPHEROIDS. 105

frustum be defined as the planes comprehended by the sections
of the acute-angled cones [ellipses], and the axis as the straight
line joining the centres of the sections of the acute-angled
cones, so that the axis will be in the same straight line with
the axis of the cylinder."

Lemma.

If in an ascending arithmetical progression consisting of the
magnitudes A Jt A^, ... A n the common difference be equal to the
least term A lt then

and

[The proof of this is given incidentally in the treatise On
Spirals, Prop. 11. By placing lines side by side to represent
the terms of the progression and then producing each so as to
make it equal to the greatest term, Archimedes gives the
equivalent of the following proof.

If Sn^Ai+Ai+.^+A^ + An,

we have also S n = A n + A n ^ + A n ^ + ...4- -4,.

And A l + -4 n _j = A 2 + A n ^ 2 = ... = A n .

Therefore 2S n = (n + 1) A nt

whence n . A n < 2S nt

and n.A n >2S n - l .

Thus, if the progression is a, 2a, . . . na,

and /i*a<2S n ,

but >2S IM .]

Proposition 1.

If A lf B lt C l9 .../fj and A^, jB 2 , (7 3 , ...K^ be two series of
magnitudes such that

A , D __ A . JJ \

/Ij . JDj = -/Ij x^ji I

J?, : C^ss-Bj : (7,, a?id so on j ^ '

106 ARCHIMEDES

and if A 3 , B 3 , d> ...K z and A 4 , B 4 , d, ...K 4 be two other series
such that

A\ I A$ = ^1 2 : -"-4*

B l :B S =B Z :B 4 , and soon
then (

1 /g\

) .............. ^ P ''

The proof is as follows.

Since A 3 : A l = A 4 : A.,,

and A l :B l ^A 9 :B. 9

while Bi-.Bi^Bt-.Bt,

we have, ex aequali, A 3 : B 3 = A 4 : B 4 . \

Similarly B 3 :C 3 =B^:C 4t and so on j

Again, it follows from equations (a) that

Therefore

A, : A,^(A, + B t + d + ... + /TO : (A 9 +B 9 -h ... +
or (A + B l + d + -.. + #0 : ^i = (A, + B 2 + C..+ ... +
and A l :A 3 = A: A^

while from equations (7) it follows in like manner that

By the last three equations, ex aequali,

COR. If any terms in the third and fourth series corre-
sponding to terms in the first and second be left out, the
result is the same. For example, if the last terms K 3 , K 4 are
absent,

where I immediately precedes K in each series.

OK CONOIDS AND SPHEROIDS.

107

Lemma to Proposition 2.

[On Spirals, Prop. 10.]

// A 19 A*> A 3t ...A n be n lines forming an ascending
arithmetical progression in which the common difference is equal
to the least term A lt then

A l (A l

Ai A 2

n- 3 A-2A n _,

AH Aft_i A..,j AB Aj AI

Let the lines A n , A n - lt -4 n _ 2 , ...A l be placed in a row
from left to right. Produce A n - lt A n _, ...Ai until they are
each equal to A ni so that the parts produced are respectively
equal to A l9 A. 2 , ...4 n _,.

Taking each line successively, we have

(A, + A n _tf = A? +

\A Z 4* -"-n a) == A^ 4* <

"t" ^^2' "n a

108 ARCHIMEDES

And, by addition,

+ 24, . An-i + 24 2 . 4 n _ 2 4- . . . + 24^ . 4, .

Therefore, in order to obtain the required result, we have to
prove that

+ ... + A n * ............ (a).

Now 24 3 . 4 n _ 2 as A l . 44 n _ 2 , because 4 2 = 24i ,
24 8 . A n -t = A l . 6 4 n _ s , because A s = 34j ,

It follows that

And this last expression can be proved to be equal to

A* + A*+... + A n *.

For An^A^n.An)

because (n 1) A n = -4 n _, 4-

4- -"-n 2 4-
f

Similarly 4 Vi = ^! {^n^ -f 2(4 7l _ 2 -f 4 n _ 8 4-

whence, by addition,

A n t

1 + 5A n - t + ... +(2n-

ON CONOIDS AND SPHEROIDS. 109

Thus the equation marked (a) above is true ; and it follows
that

COR. 1. From this it is evident that

n.A n ><3(A' + A*+...+A n *) ............. (1).

Also An^A^An + ^A^ + An-i+.-.+At)}, as above,
so that An > A^An + A n ^ + ... -Mi),
and therefore

It follows from the proposition that

_ 1 ) .................. (2).

COR. 2. All these results will hold if we substitute similar
figures for squares on all the lines ; for similar figures are in the
duplicate ratio of their sides.

[In the above proposition the symbols A lt A^, ...A n have
been used instead of a, 2a, 3a, ...na in order to exhibit the
geometrical character of the proof ; but, if we now substitute
the latter terms in the various results, we have (1)

(n + 1) nV + a(a + 2a + ... 4- na)

= 3 {a 2 4- (2a) 2 + (3a)* + . . . + (a) f j.

Therefore a 9 + (2a) a + (3a) 2 + . . . + (na) 2

n ( n + !)}

a }

a * (/ , -i \

8l ( " +1) "

_. _

Also(2) i'
and (3) n'

110 ARCHIMEDES

Proposition 3.

IfA lt A 2 ... A n be any number of areas such that*
A l = 00 + s ,

A n a. nx + (nx)* y

(* )~iff\
2 + T j '

(ft 7i^?*\
S + -Q-
L O /

For, by the Lemma immediately preceding Prop. 1,

n . anx < 2 (ax + a . 20 + . . . + a . nx) y
and > 2 (ax + a . 20 + . . . + a . /T^T #).

Also, by the Lemma preceding this proposition,
n . (nx)* < 3 {x* + (2#)' + (30)" + . . . + (/w?)"}
and > 3 {0 2 + (20) 1 + . . . + (n~^~T a;) 2 }.

Hence

ClfflQC VL ( fUE I

-y + -y- ; - < [(00 4- 0>) + {a . 20 -f (20/J + . . . + {a . n0 ^(710)")
and
> [(00 + s ) + {a . 2# -t- (2#) 2 j + ... + {a . ti^l x -f (?i-"l 0) 1 }],

an*x

and

It follows that

or n.A n :(A l

also n.^ n :(^ 1

* The phraseology of Archimedes here is that associated with the traditional
method of application of areas: rf K a...irap Ard^ra* atrav wapairfov n xupiov
vvtppd\\ov cfdci TTpay6vt? t if to each of the lines there be applied a space
[rectangle] exceeding by a square figure. " Thus A l is a rectangle of height x ap-
plied to a line a but overlapping it so that the base extends a distance x beyond a.

ON CONOIDS AND SPHEROIDS.

Ill

Proposition 3.

(1) If TP, TP' be two tangents to any conic meeting in T,
and if Qq, Q'q' be any two chords parallel respectively to TP,
TP' and meeting in 0, then

QO.Oq:Q'O.Oq'=TP*: TP".
" And this is proved in the elements of conies*."

(2) If QQ' be a chord of a parabola bisected in V by the
diameter PV, and if PV be of constant length, then the areas of
the triangle PQQ' and of the segment PQQ' are both constant
whatever be the direction of QQ'.

Let ABB' be the particular segment of the parabola whose
vertex is A, so that BE' is bisected perpendicularly by the axis
at the point H, where AH = PV.

Draw QD perpendicular to PV.

Let p a be the parameter of the principal ordinates, and let
p be another line