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Archimedes of Syracuse
287 - 212 BC
The work done by
Archimedes, a Greek mathematician, was wide ranging, some of it
leading to what has become integral calculus. He is considered
one of the greatest mathematicians of all time.
Archimedes probably was born in the seaport city of Syracuse, a
Greek colony on the island of Sicily. He was the son of an
astronomer, Phidias, and may have been related to Hieron, King
of Syracuse, and his son Gelon. Archimedes studied in Alexandria
at the school established by Euclid and then settled in his
native city.
To the Greeks of this time, mathematics was considered one of
the fine arts - something without practical application but
pleasing to the intellect and to be enjoyed by those with the
requisite talent and leisure. Archimedes did not record the many
mechanical inventions he made at the request of King Hieron or
simply for his own amusement, presumably because he considered
them of little importance compared with his purely mathematical
work. These inventions did, however, make him famous during his
life.
Fact and Fancy
The many stories that are told of Archimedes are the prototype
of the absentminded-professor stories. A famous one tells how
Archimedes uncovered a fraud attempted on Hieron. The King
ordered a golden crown and gave the goldsmith the exact amount
of gold needed. The goldsmith delivered a crown of the required
weight, but Hieron suspected that some silver had been used
instead of gold. He asked Archimedes to consider the matter.
Once Archimedes was pondering it while he was getting into a
bathtub full of water. He noticed that the amount of water
overflowing the tub was proportional to the amount of his body
that was being immersed. This gave him an idea for solving the
problem of the crown, and he was so elated he ran naked through
the streets repeatedly shouting "Heureľka, heureľka!" (I have
discovered it!)
There are several ways Archimedes may have determined the
proportion of silver in the crown. One likely method relies on a
proposition which Archimedes later wrote in a treatise, On
Floating Bodies, and which is equivalent to what is now called
Archimedes' principle: a body immersed in a fluid is buoyed up
by a force equal to the weight of fluid displaced by the body.
Using this method, he would have first taken two equal weights
of gold and silver and compared their weights when immersed in
water. Next he would have compared the weight of the crown and
an equal weight of pure silver in water in the same way. The
difference between these two comparisons would indicate that the
crown was not pure gold.
On another occasion Archimedes told Hieron that with a given
force he could move any given weight. Archimedes had
investigated properties of the lever and pulley, and it is on
the basis of these that he is said to have asserted, "Give me a
place to stand and I can move the earth." Hieron, amazed at
this, asked for some physical demonstration. In the harbor was a
new ship which the combined strength of all the Syracusans could
not launch. Archimedes used a mechanical device that enabled
him, standing some distance away, to move the ship. The device
may have been a simple compound pulley or a machine in which a
cogwheel with oblique teeth moves on a cylindrical helix turned
by a handle.
Hieron saw that Archimedes had a most inventive mind in such
practical matters as constructing mechanical aids. At this time
one use for such inventions was in the military field. Hieron
persuaded Archimedes to construct machines for possible use in
warfare, both defensive and offensive.
A Time of War
Plutarch in his biography of the Roman general Marcellus
describes the following incident. After the death of Hieron,
Marcellus attacked Syracuse by land and sea. Now the instruments
of warfare made at Hieron's request were put to use. "The
Syracusans were struck dumb with fear, thinking that nothing
would avail against such violence and power. But Archimedes
began to work his engines and hurled against the land forces all
sorts of missiles and huge masses of stones, which came down
with incredible noise and speed; nothing at all could ward off
their weight, but they knocked down in heaps those who stood in
the way and threw the ranks into disorder. Furthermore, beams
were suddenly thrown over the ships from the walls, and some of
the ships were sent to the bottom by means of weights fixed to
the beams and plunging down from above; others were drawn up by
iron claws, or crane-like beaks, attached to the prow and were
plunged down on their sterns, or were twisted round and turned
about by means of ropes within the city, and dashed against the
cliffs. … Often there was the fearful sight of a ship lifted out
of the sea into mid-air and whirled about as it hung there,
until the men had been thrown out and shot in all directions,
when it would fall empty upon the walls or slip from the grip
that had held it."
Later writers tell how Archimedes set the Roman ships on fire by
focusing an arrangement of concave mirrors on them he basic idea
is that the mirror reflects to one point all the sun's light
entering parallel to the mirror axis.
Marcellus, according to Plutarch, gave up trying to take the
city by force and relied on a siege. The city surrendered after
8 months. Marcellus gave orders that the Syracusan citizens were
not to be killed, taken as slaves, or mistreated. But some Roman
soldier did kill Archimedes. There are different accounts of his
death. One version is that Archimedes, now 75 years old, was
alone and so absorbed in examining a diagram that he was unaware
of the capture of the city. A soldier ordered him to go to
Marcellus, but Archimedes would not leave until he had worked
out his problem to the end. The soldier was so enraged, he
killed Archimedes. Another version is that Archimedes was
bringing Marcellus a box of his mathematical instruments, such
as sundials, spheres, and angles adjusted to the apparent size
of the sun, when he was killed by soldiers who thought he was
carrying valuables in the box. "What is, however, agreed,"
Plutarch says, "is that Marcellus was distressed, and turned
away from the slayer as from a polluted person, and sought out
the relatives of Archimedes to do them honor."
Archimedes had requested his relatives to place upon his tomb a
drawing of a sphere inscribed within a cylinder with a notation
giving the ratio of the volume of the cylinder to that of the
sphere - an indication of what Archimedes considered to be his
greatest achievement. The Roman statesman and writer Cicero
tells of finding this tomb much later in a state of neglect.
Other Inventions
Perhaps while in Egypt, Archimedes invented the water screw, a
machine for raising water to irrigate fields. Another invention
was a miniature planetarium, a sphere whose motion imitated that
of the earth, sun, moon, and the five other planets then known
(Saturn, Jupiter, Mars, Venus, and Mercury); the model may have
been kept in motion by a flow of water. Cicero tells of seeing
it over a century later and claimed that it actually represented
the periods of the moon and the apparent motion of the sun with
such accuracy that it would, over a short period, show the
eclipses of the sun and moon. Since astronomy was a branch of
mathematics in Archimedes' time, he undoubtedly considered this
and his other astronomical inventions much more important than
those which could be put to practical use.
Archimedes is said to have made observations of the solstices to
determine the length of the year and to have discovered the
distances of the planets. In The sand Reckoner he describes a
simple device for measuring the angle subtended by the sun at an
observer's eye.
Contributions to Mathematics
Euclid's Elements had catalogued practically all the results of
Greek geometry up to Archimedes' time. Archimedes adopted
Euclid's uniform and rigorously logical form: axioms followed by
theorems and their proofs. But the problems Archimedes set
himself and his solutions were on another level from any that
preceded him.
In geometry Archimedes continued the work in Book XII of
Euclid's Elements. In Book XII the method of exhaustion,
discovered by Eudoxus, is used to prove theorems on areas of
circles and volumes of spheres, pyramids, and cones. Two of the
theorems are mentioned by Archimedes in the preface to On the
Sphere and Cylinder. After stating the result concerning the
ratio of the volumes of a cylinder and an inscribed sphere, he
says that this result can be put side by side with his previous
investigations and with those theorems of Eudoxus on solids,
namely: the volume of a pyramid is one-third the volume of a
prism with the same base and height; and the volume of a cone is
one-third the volume of a cylinder with the same base and
height.
There was no direct computation of areas and volumes enclosed by
various curved lines and surfaces, but rather a comparison of
these with each other or with the areas and volumes enclosed by
rectilinear figures such as rectangles and prisms. The reason
for this is that the area, for a simple example, of a circle
with radius of length one cannot be expressed exactly by any
fraction or integer. It is possible, however, to say as is done
in Proposition 2 of Book XII of the Elements that the ratio of
the area of one circle to another is exactly equal to the ratio
of the squares of their diameters, or, in a more concise form
closer to the Greek, circles are to one another as the squares
of the diameters. The proof of this theorem relies on
(theoretically) being able to "exhaust" the circle by inscribing
in it successively polygons whose sides increase in number and
hence which fit closer to the circle. Thus the curved line, the
circle, can be closely approximated by a rectilinear figure, a
polygon.
Recognizing this, it would be easy to conclude that the circle
itself is a polygon with "infinitely" many "infinitesimal"
sides. Even by Euclid's time this concept had a long history of
philosophic controversy beginning with the well-known Zeno's
paradoxes discussed by Aristotle. Archimedes, aware of the
logical problems involved in making such a facile statement,
avoids it and proceeds in his proofs in an invulnerable manner.
However, a student with a knowledge of integral calculus today
would find Archimedes' method very cumbersome. It should
nevertheless be remembered that the theorems which make the work
almost trivial to any modern mathematician were obtained only in
the 17th, 18th, and 19th centuries, about 2000 years after
Archimedes.
In modern terminology, the area of a circle with radius of
length one is the irrational number denoted by p, and although
Archimedes knew it could not be calculated exactly, he knew how
to approximate it as closely as desired. In his treatise
Measurement of a Circle, using the method of exhaustion,
Archimedes proves that p is between 3 1/7 and 3 10/71 (it is
actually 3.14159).
Large numbers seem to have some fascination of their own. A
common Greek proverb was to the effect that the quantity of sand
eludes number, that is, is infinite. To the Greeks this might
seem especially true since their numeral system did not include
a zero. Numbers were represented by letters of the alphabet, and
for large numbers this notation becomes clumsy. In The Sand
Reckoner Archimedes refutes the idea expressed by the proverb by
inventing a notation which enables him to calculate in a
reasonably concise way the number of grains of sand required to
fill the "universe." He takes the universe to be the size of a
sphere centered at the earth and having as radius the distance
from the earth to the sun. After saying this he also points out
an alternative view of the universe that had been expressed by a
contemporary astronomer, Aristarchus of Samos, namely, that the
sun is fixed, the earth revolves about the sun, and the stars
are fixed a long distance beyond the earth. Astronomical data,
together with the assumption that there are no more than 10,000
grains of sand in a volume the size of a poppyseed, are the
basis of calculations leading up to the conclusion that the
number of grains of sand which could be contained in a sphere
the size of the universe is less than 10 51, in modern notation.
Other known works by Archimedes that are purely geometrical are
On Conoids and Spheroids, On Spirals, and Quadrature of the
Parabola. The first is concerned with volumes of segments of
such figures as the hyperboloid of revolution. The second
describes what is now known as Archimedes' spiral and contains
area computations. The third is on finding areas of segments of
the parabola.
Another of Archimedes' works in mechanics, besides On Floating
Bodies mentioned previously, is On the Equilibrium of Planes.
From such simple postulates as "Equal weights at equal distances
balance," positions of centers of gravity are determined for
parabolic segments.
As is true of all other mathematicians of antiquity, Archimedes
usually wrote in a way which left no indication of how he
arrived at the theorems; all the reader sees is a theorem
followed by a proof. But in 1906 a hitherto-lost treatise by
Archimedes, The Method, was found. In it Archimedes explains a
certain method by which it is possible to get a start in
investigating some of the problems in mathematics by means of
mechanics. "For," Archimedes writes, "certain things first
became clear to me by a mechanical method, although they had to
be demonstrated by geometry afterwards because their
investigation by the said method did not furnish an actual
demonstration." Thus Archimedes is careful to distinguish
between a heuristic approach to verifying a theorem and the
proof of the theorem. The Method utilizes theorems from his
mechanical treatise On the Equilibrium of Planes and provides an
excellent example of the interplay between pure and applied
mathematics.
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This web page was last updated on:
08 December, 2008
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