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Archimedes of Syracuse

Archimedes of Syracuse

Born: 287 BC in Syracuse, Sicily

Died: 212 BC in Syracuse, Sicily

Archimedes' father was Phidias, an astronomer. We know
nothing else about Phidias other than this one fact and we only know this since
Archimedes gives us this information in one of his works, The Sandreckoner. A
friend of Archimedes called Heracleides wrote a biography of him but sadly this
work is lost. How our knowledge of Archimedes would be transformed if this lost
work were ever found, or even extracts found in the writing of others.

Archimedes was a native of Syracuse, Sicily. It is
reported by some authors that he visited Egypt and there invented a device now
known as Archimedes' screw. This is a pump, still used in many parts of the
world. It is highly likely that, when he was a young man, Archimedes studied
with the successors of Euclid in Alexandria. Certainly he was completely
familiar with the mathematics developed there, but what makes this conjecture
much more certain, he knew personally the mathematicians working there and he
sent his results to Alexandria with personal messages. He regarded Conon of
Samos, one of the mathematicians at Alexandria, both very highly for his
abilities as a mathematician and he also regarded him as a close friend.

In the preface to On spirals Archimedes relates an
amusing story regarding his friends in Alexandria. He tells us that he was in
the habit of sending them statements of his latest theorems, but without giving
proofs. Apparently some of the mathematicians there had claimed the results as
their own so Archimedes says that on the last occasion when he sent them
theorems he included two which were false:-

... so that those who claim to discover everything,
but produce no proofs of the same, may be confuted as having pretended to
discover the impossible.

Other than in the prefaces to his works, information
about Archimedes comes to us from a number of sources such as in stories from
Plutarch, Livy, and others. Plutarch tells us that Archimedes was related to
King Hieron II of Syracuse (see for example):-

Archimedes ... in writing to King Hiero, whose friend
and near relation he was....

Again evidence of at least his friendship with the
family of King Hieron II comes from the fact that The Sandreckoner was
dedicated to Gelon, the son of King Hieron.

There are, in fact, quite a number of references to
Archimedes in the writings of the time for he had gained a reputation in his
own time which few other mathematicians of this period achieved. The reason for
this was not a widespread interest in new mathematical ideas but rather that
Archimedes had invented many machines which were used as engines of war. These
were particularly effective in the defence of Syracuse when it was attacked by
the Romans under the command of Marcellus.

Plutarch writes in his work on Marcellus, the Roman
commander, about how Archimedes' engines of war were used against the Romans in
the siege of 212 BC:-

... when Archimedes began to ply his engines, he at
once shot against the land forces all sorts of missile weapons, and immense
masses of stone that came down with incredible noise and violence; against
which no man could stand; for they knocked down those upon whom they fell in
heaps, breaking all their ranks and files. In the meantime huge poles thrust
out from the walls over the ships and sunk some by great weights which they let
down from on high upon them; others they lifted up into the air by an iron hand
or beak like a crane's beak and, when they had drawn them up by the prow, and
set them on end upon the poop, they plunged them to the bottom of the sea; or
else the ships, drawn by engines within, and whirled about, were dashed against
steep rocks that stood jutting out under the walls, with great destruction of
the soldiers that were aboard them. A ship was frequently lifted up to a great
height in the air (a dreadful thing to behold), and was rolled to and fro, and
kept swinging, until the mariners were all thrown out, when at length it was
dashed against the rocks, or let fall.

Archimedes had been persuaded by his friend and
relation King Hieron to build such machines:-

These machines [Archimedes] had designed and
contrived, not as matters of any importance, but as mere amusements in
geometry; in compliance with King Hiero's desire and request, some little time
before, that he should reduce to practice some part of his admirable
speculation in science, and by accommodating the theoretic truth to sensation
and ordinary use, bring it more within the appreciation of the people in
general.

Perhaps it is sad that engines of war were appreciated
by the people of this time in a way that theoretical mathematics was not, but
one would have to remark that the world is not a very different place at the
end of the second millenium AD. Other inventions of Archimedes such as the
compound pulley also brought him great fame among his contemporaries. Again we
quote Plutarch:-

[Archimedes] had stated [in a letter to King Hieron]
that given the force, any given weight might be moved, and even boasted, we are
told, relying on the strength of demonstration, that if there were another
earth, by going into it he could remove this. Hiero being struck with amazement
at this, and entreating him to make good this problem by actual experiment, and
show some great weight moved by a small engine, he fixed accordingly upon a
ship of burden out of the king's arsenal, which could not be drawn out of the
dock without 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 head of the pulley in his hand and drawing the cords by
degrees, he drew the ship in a straight line, as smoothly and evenly as if she
had been in the sea.

Yet Archimedes, although he achieved fame by his
mechanical inventions, believed that pure mathematics was the only worthy
pursuit. Again Plutarch describes beautifully Archimedes attitude, yet we shall
see later that Archimedes did in fact use some very practical methods to
discover results from pure geometry:-

Archimedes possessed so high a spirit, so profound a
soul, and such treasures of scientific knowledge, that though these inventions
had now obtained him the renown of more than human sagacity, he yet would not
deign to leave behind him any commentary or writing on such subjects; but,
repudiating as sordid and ignoble the whole trade of engineering, and every
sort of art that lends itself to mere use and profit, he placed his whole
affection and ambition in those purer speculations where there can be no
reference to the vulgar needs of life; studies, the superiority of which to all
others is unquestioned, and in which the only doubt can be whether the beauty
and grandeur of the subjects examined, of the precision and cogency of the
methods and means of proof, most deserve our admiration.

His fascination with geometry is beautifully described
by Plutarch:-

Oftimes Archimedes' servants got him against his will
to the baths, to wash and anoint him, and yet being there, he would ever be
drawing out of the geometrical figures, even in the very embers of the chimney.
And while they were anointing of him with oils and sweet savours, with his
fingers he drew lines upon his naked body, so far was he taken from himself,
and brought into ecstasy or trance, with the delight he had in the study of
geometry.

The achievements of Archimedes are quite outstanding.
He is considered by most historians of mathematics as one of the greatest
mathematicians of all time. He perfected a methods of integration which allowed
him to find areas, volumes and surface areas of many bodies. Chasles said that
Archimedes' work on integration (see):-

... gave birth to the calculus of the infinite
conceived and brought to perfection by Kepler, Cavalieri, Fermat, Leibniz and
Newton.

Archimedes was able to apply the method of exhaustion,
which is the early form of integration, to obtain a whole range of important
results and we mention some of these in the descriptions of his works below.
Archimedes also gave an accurate approximation to p and showed that he could
approximate square roots accurately. He invented a system for expressing large
numbers. In mechanics Archimedes discovered fundamental theorems concerning the
centre of gravity of plane figures and solids. His most famous theorem gives
the weight of a body immersed in a liquid, called Archimedes' principle.

The works of Archimedes which have survived are as
follows. On plane equilibriums (two books), Quadrature of the parabola, On the
sphere and cylinder (two books), On spirals, On conoids and spheroids, On
floating bodies (two books), Measurement of a circle, and The Sandreckoner. In
the summer of 1906, J L Heiberg, professor of classical philology at the
University of Copenhagen, discovered a 10th century manuscript which included
Archimedes' work The method. This provides a remarkable insight into how
Archimedes discovered many of his results and we will discuss this below once
we have given further details of what is in the surviving books.

The order in which Archimedes wrote his works is not
known for certain. We have used the chronological order suggested by Heath in
in listing these works above, except for The Method which Heath has placed
immediately before On the sphere and cylinder. The paper [47] looks at
arguments for a different chronological order of Archimedes' works.

The treatise On plane equilibriums sets out the
fundamental principles of mechanics, using the methods of geometry. Archimedes
discovered fundamental theorems concerning the centre of gravity of plane
figures and these are given in this work. In particular he finds, in book 1,
the centre of gravity of a parallelogram, a triangle, and a trapezium. Book two
is devoted entirely to finding the centre of gravity of a segment of a
parabola. In the Quadrature of the parabola Archimedes finds the area of a
segment of a parabola cut off by any chord.

In the first book of On the sphere and cylinder
Archimedes shows that the surface of a sphere is four times that of a great
circle, he finds the area of any segment of a sphere, he shows that the volume
of a sphere is two-thirds the volume of a circumscribed cylinder, and that the
surface of a sphere is two-thirds the surface of a circumscribed cylinder
including its bases. A good discussion of how Archimedes may have been led to
some of these results using infinitesimals is given in . In the second book of
this work Archimedes' most important result is to show how to cut a given
sphere by a plane so that the ratio of the volumes of the two segments has a
prescribed ratio.

In On spirals Archimedes defines a spiral, he gives
fundamental properties connecting the length of the radius vector with the
angles through which it has revolved. He gives results on tangents to the
spiral as well as finding the area of portions of the spiral. In the work On
conoids and spheroids Archimedes examines paraboloids of revolution,
hyperboloids of revolution, and spheroids obtained by rotating an  ellipse either about its major axis or about
its minor axis. The main purpose of the work is to investigate the volume of
segments of these three-dimensional figures. Some claim there is a lack of
rigour in certain of the results of this work but the interesting discussion
in  attributes this to a modern day
reconstruction.

On floating bodies is a work in which Archimedes lays
down the basic principles of hydrostatics. His most famous theorem which gives
the weight of a body immersed in a liquid, called Archimedes' principle, is
contained in this work. He also studied the stability of various floating
bodies of different shapes and different specific gravities. In Measurement of
the Circle Archimedes shows that the exact value of p lies between the values 310/71
and 31/7. This he obtained by circumscribing and
inscribing a circle with regular polygons having 96 sides.

The Sandreckoner is a remarkable work in which
Archimedes proposes a number system capable of expressing numbers up to 8x1016
in modern notation. He argues in this work that this number is large enough to
count the number of grains of sand which could be fitted into the universe.
There are also important historical remarks in this work, for Archimedes has to
give the dimensions of the universe to be able to count the number of grains of
sand which it could contain. He states that 
Aristarchus has proposed a system with the sun at the centre and the
planets, including the Earth, revolving round it. In quoting results on the
dimensions he states results due to 
Eudoxus, Phidias (his father), and to 
Aristarchus. There are other sources which mention Archimedes' work on
distances to the heavenly bodies. For example in  Osborne reconstructs and discusses:-

...a theory of the distances of the heavenly bodies
ascribed to Archimedes, but the corrupt state of the numerals in the sole
surviving manuscript [due to Hippolytus of Rome, about 220 AD] means that the
material is difficult to handle.

In the Method, Archimedes described the way in which
he discovered many of his geometrical results (see):-

... certain things first became clear to me by a
mechanical method, although they had to be proved by geometry afterwards
because their investigation by the said method did not furnish an actual proof.
But it is of course easier, when we have previously acquired, by the method,
some knowledge of the questions, to supply the proof than it is to find it
without any previous knowledge.

Perhaps the brilliance of Archimedes' geometrical results
is best summed up by Plutarch, who writes:-

It is not possible to find in all geometry more
difficult and intricate questions, or more simple and lucid explanations. Some
ascribe this to his natural genius; while others think that incredible effort and
toil produced these, to all appearances, easy and unlaboured results. No amount
of investigation of yours would succeed in attaining the proof, and yet, once
seen, you immediately believe you would have discovered it; by so smooth and so
rapid a path he leads you to the conclusion required.

 Heath adds his
opinion of the quality of Archimedes' work:-

The treatises are, without exception, monuments of
mathematical exposition; the gradual revelation of the plan of attack, the
masterly ordering of the propositions, the stern elimination of everything not
immediately relevant to the purpose, the finish of the whole, are so impressive
in their perfection as to create a feeling akin to awe in the mind of the
reader.

There are references to other works of Archimedes
which are now lost.  Pappus refers to a
work by Archimedes on semi-regular polyhedra, Archimedes himself refers to a
work on the number system which he proposed in the Sandreckoner,  Pappus mentions a treatise On balances and
levers, and Theon mentions a treatise by Archimedes about mirrors. Evidence for
further lost works are discussed in  but
the evidence is not totally convincing.

Archimedes was killed in 212 BC during the capture of
Syracuse by the Romans in the Second Punic War after all his efforts to keep
the Romans at bay with his machines of war had failed. Plutarch recounts three
versions of the story of his killing which had come down to him. The first
version:-

Archimedes ... was ..., as fate would have it, intent
upon working out some problem by a diagram, and having fixed his mind alike and
his eyes upon the subject of his speculation, he never noticed the incursion of
the Romans, nor that the city was taken. In this transport of study and
contemplation, a soldier, unexpectedly coming up to him, commanded him to
follow to Marcellus; which he declining to do before he had worked out his
problem to a demonstration, the soldier, enraged, drew his sword and ran him
through.

The second version:-

... a Roman soldier, running upon him with a drawn
sword, offered to kill him; and that Archimedes, looking back, earnestly
besought him to hold his hand a little while, that he might not leave what he
was then at work upon inconclusive and imperfect; but the soldier, nothing
moved by his entreaty, instantly killed him.

Finally, the third version that Plutarch had heard:-

... as Archimedes was carrying to Marcellus
mathematical instruments, dials, spheres, and angles, by which the magnitude of
the sun might be measured to the sight, some soldiers seeing him, and thinking
that he carried gold in a vessel, slew him.

Archimedes considered his most significant
accomplishments were those concerning a cylinder circumscribing a sphere, and
he asked for a representation of this together with his result on the ratio of
the two, to be inscribed on his tomb. 
Cicero was in Sicily in 75 BC and he writes how he searched for
Archimedes tomb (see for example):-

... and found it enclosed all around and covered with
brambles and thickets; for I remembered certain doggerel lines inscribed, as I
had heard, upon his tomb, which stated that a sphere along with a cylinder had
been put on top of his grave. Accordingly, after taking a good look all around
..., I noticed a small column arising a little above the bushes, on which there
was a figure of a sphere and a cylinder... . Slaves were sent in with sickles
... and when a passage to the place was opened we approached the pedestal in
front of us; the epigram was traceable with about half of the lines legible, as
the latter portion was worn away.

It is perhaps surprising that the mathematical works
of Archimedes were relatively little known immediately after his death. As
Clagett writes in:-

Unlike the Elements of  Euclid, the works of Archimedes were not widely known in
antiquity. ... It is true that ... individual works of Archimedes were
obviously studied at Alexandria, since Archimedes was often quoted by three
eminent mathematicians of Alexandria: 
Heron,  Pappus and  Theon.

Only after 
Eutocius brought out editions of some of Archimedes works, with
commentaries, in the sixth century AD were the remarkable treatises to become
more widely known. Finally, it is worth remarking that the test used today to
determine how close to the original text the various versions of his treatises
of Archimedes are, is to determine whether they have retained Archimedes'
Dorian dialect.

J J O'Connor and E F Robertson
Список
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