* Biography *
Archimedes |
The work done by Archimedes (ca.
287-212 B.C.), 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 *
Archimedes' Principle |
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 "Heureμka, heureμka!"
(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 π, 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 π 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.
Archimedes."
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