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[Illustration: _Aristippus Philosophus Socraticus, nausragio cum ejectus
ad Rhodiensium litus animadvertisses Geometrica Schemata descripta,
exclamavisse ad comites ita dicitur_, Bene Speremus, Hominum enim
vestigia video.
_Vitruv. Architect lib. 6. Prief_.
delin MBurghers sculptUniv. Oxon.]
PIONEERS OF PROGRESS
MEN OF SCIENCE
EDITED BY S. CHAPMAN, M.A., D.SC., F.R.S.
ARCHIMEDES
BY
SIR THOMAS HEATH
K.C.B., K.C.V.O., F.R.S.; SC.D., CAMB.
HON. D.SC., OXFORD
[Greek: Dos moi pou sto, kai kino ten gen]
LONDON:
SOCIETY FOR PROMOTING
CHRISTIAN KNOWLEDGE
NEW YORK: THE MACMILLAN CO.
1920
CONTENTS.
CHAP. PAGE
I. ARCHIMEDES 1
II. GREEK GEOMETRY TO ARCHIMEDES 7
III. THE WORKS OF ARCHIMEDES 24
IV. GEOMETRY IN ARCHIMEDES 29
V. THE SANDRECKONER 45
VI. MECHANICS 50
VII. HYDROSTATICS 53
BIBLIOGRAPHY 57
CHRONOLOGY 58
CHAPTER I.
ARCHIMEDES.
If the ordinary person were asked to say off-hand what he knew of
Archimedes, he would probably, at the most, be able to quote one or
other of the well-known stories about him: how, after discovering the
solution of some problem in the bath, he was so overjoyed that he ran
naked to his house, shouting [Greek: eureka, eureka] (or, as we might
say, "I've got it, I've got it"); or how he said "Give me a place to
stand on and I will move the earth"; or again how he was killed, at the
capture of Syracuse in the Second Punic War, by a Roman soldier who
resented being told to get away from a diagram drawn on the ground which
he was studying.
And it is to be feared that few who are not experts in the history of
mathematics have any acquaintance with the details of the original
discoveries in mathematics of the greatest mathematician of antiquity,
perhaps the greatest mathematical genius that the world has ever seen.
History and tradition know Archimedes almost exclusively as the inventor
of a number of ingenious mechanical appliances, things which naturally
appeal more to the popular imagination than the subtleties of pure
mathematics.
Almost all that is told of Archimedes reaches us through the accounts by
Polybius and Plutarch of the siege of Syracuse by Marcellus. He perished
in the sack of that city in 212 B.C., and, as he was then an old man
(perhaps 75 years old), he must have been born about 287 B.C. He was
the son of Phidias, an astronomer, and was a friend and kinsman of King
Hieron of Syracuse and his son Gelon. He spent some time at Alexandria
studying with the successors of Euclid (Euclid who flourished about 300
B.C. was then no longer living). It was doubtless at Alexandria that he
made the acquaintance of Conon of Samos, whom he admired as a
mathematician and cherished as a friend, as well as of Eratosthenes; to
the former, and to the latter during his early period he was in the
habit of communicating his discoveries before their publication. It was
also probably in Egypt that he invented the water-screw known by his
name, the immediate purpose being the drawing of water for irrigating
fields.
After his return to Syracuse he lived a life entirely devoted to
mathematical research. Incidentally he became famous through his clever
mechanical inventions. These things were, however, in his case the
"diversions of geometry at play," and he attached no importance to them.
In the words of Plutarch, "he possessed so lofty a spirit, so profound a
soul, and such a wealth of scientific knowledge that, although these
inventions had won for him the renown of more than human sagacity, yet
he would not consent 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 practical utility, he placed
his whole ambition in those speculations in the beauty and subtlety of
which there is no admixture of the common needs of life".
During the siege of Syracuse Archimedes contrived all sorts of engines
against the Roman besiegers. There were catapults so ingeniously
constructed as to be equally serviceable at long or short range, and
machines for discharging showers of missiles through holes made in the
walls. Other machines consisted of long movable poles projecting beyond
the walls; some of these dropped heavy weights upon the enemy's ships
and on the constructions which they called _sambuca_, from their
resemblance to a musical instrument of that name, and which consisted of
a protected ladder with one end resting on two quinqueremes lashed
together side by side as base, and capable of being raised by a
windlass; others were fitted with an iron hand or a beak like that of a
crane, which grappled the prows of ships, 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 with this geometrical Briareus who uses our ships like cups to
ladle water from the sea, drives our _sambuca_ off ignominiously with
cudgel-blows, and, by the multitude of missiles that he hurls at us all
at once, outdoes the hundred-handed giants of mythology?" 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 'there it is,' 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 fighting and assault, putting
all his hope in a long siege".
Archimedes died, as he had lived, absorbed in mathematical
contemplation. The accounts of the circumstances of his death differ in
some details. 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, his mind and his eyes being alike
fixed 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 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, threatening to kill him; and, when
Archimedes saw him, he begged him earnestly to wait a little while 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 he carried gold in the vessel, killed him."
The most picturesque version of the story is 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.
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 (3/2) which the
cylinder bears to the sphere; from which we may infer that he himself
regarded the discovery of this ratio as his greatest achievement.
Cicero, when quaestor in Sicily, found the tomb in a neglected state and
restored it. In modern times not the slightest trace of it has been
found.
Beyond the above particulars of the life of Archimedes, we have nothing
but a number of stories which, if perhaps not literally accurate, yet
help us to a conception of the personality of the man 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 anointing
himself, in the oil on his body. Of the same kind is the story mentioned
above, that, having discovered while in a bath the solution of the
question referred to him by Hieron as to whether a certain crown
supposed to have been made of gold did not in fact contain a certain
proportion of silver, he ran naked through the street to his home
shouting [Greek: eureka, eureka].
It was in connexion with his discovery of the solution of the problem
_To move a given weight by a given force_ that Archimedes uttered the
famous saying, "Give me a place to stand on, and I can move the earth"
([Greek: dos moi pou sto kai kino ten gen], or in his broad Doric, as
one version has it, [Greek: pa bo kai kino tan gan]). Plutarch
represents him as declaring to Hieron that any given weight could be
moved by a given force, and boasting, in reliance on 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 show him some great
weight moved by a small force, he fixed on a ship of burden with three
masts from the king's arsenal which had only been drawn up by the great
labour of many men; and loading her with many passengers and a full
freight, sitting himself the while afar off, with no great effort but
quietly setting in motion with his hand a compound pulley, he drew the
ship towards him smoothly and safely as if she were moving through the
sea." Hieron, we are told elsewhere, was so much astonished that he
declared that, from that day forth, Archimedes's word was to be accepted
on every subject! Another version of the story describes the machine
used as a _helix_; this term must be supposed to refer to a screw in the
shape of a cylindrical helix turned by a handle and acting on a
cog-wheel with oblique teeth fitting on the screw.
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 he gives a description of it,
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. It may have been moved
by water, for 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 "unicus
spectator caeli siderumque". Hipparchus says, "From these observations
it is clear that the differences in the years are altogether small, but,
as to the solstices, I almost think that both I and Archimedes have
erred to the extent of a quarter of a day both in observation and in the
deduction therefrom." It appears, therefore, that Archimedes had
considered the question of the length of the year. Macrobius says that
he discovered the distances of the planets. Archimedes himself describes
in the _Sandreckoner_ the apparatus by which he measured the apparent
diameter of the sun, i.e. the angle subtended by it at the eye.
The story that he set the Roman ships on fire by an arrangement of
burning-glasses or concave mirrors is not found in any authority earlier
than Lucian (second century A.D.); but there is no improbability in the
idea that he discovered some form of burning-mirror, e.g. a paraboloid
of revolution, which would reflect to one point all rays falling on its
concave surface in a direction parallel to its axis.