“Mathematics reveals its secrets only to those who
approach it with pure love, for its own beauty.”
Archimedes,
who probably was related to the royal family at Syracuse, was born there in 287
B.C. and died in 212 B.C. He went to the University of Alexandria and attended
the lectures of Conon, but, as soon as he had finished his studies, returned to
Sicily where he passed the remainder of his life. He took no part in public affairs,
but his mechanical ingenuity was astonishing, and, on any difficulties which
could be overcome by material means arising, his advice was generally asked by
the government.
Archimedes was a lonely sort
of eagle. As a young man he had studied for a short time at Alexandria, Egypt,
where he made two life-long friends, Conon, a gifted mathematician for whom
Archimedes had a high regard both personal and intellectual, and Eratosthenes,
also a good mathematician but quite a fop. These two, particularly Conon, seem
to have been the only men of his contemporaries with whom Archimedes felt he
could share his thoughts and be assured of understanding. Some of his finest
work was communicated by letters to Conon. Later, when Conon died, Archimedes
corresponded with Dositheus, a pupil of Conon.
Most
mathematicians are aware that the Archimedean screw was another of his
inventions. It consists of a tube, open at both ends, and bent into the form of
a spiral like a corkscrew. If one end be immersed in water, and the axis of the
instrument (i.e. the axis of the cylinder on the surface of which the tube lies)
be inclined to the vertical at a sufficiently big angle, and the instrument
turned round it, the water will flow along the tube and out at the other end.
In order that it may work, the inclination of the axis of the instrument to the
vertical must be greater than the pitch of the screw. It was used in Egypt to
drain the fields after an inundation of the Nile, and was also frequently
applied to take water out of the hold of a ship.
The
work for which he is perhaps now best known is his treatment of the mechanics
of solids and fluids; but he and his contemporaries esteemed his geometrical
discoveries of the quadrature of a parabolic area and of a spherical surface,
and his rule for finding the volume of a sphere as more remarkable; while at a
somewhat later time his numerous mechanical inventions excited most attention:
(i) On plane geometry the extant
works of Archimedes are three in number, namely:
(a) The Measure of the Circle.
(b) The Quadrature of the
Parabola.
(c) The work on Spirals.
(ii) On geometry of three
dimensions the extant works of Archimedes are two in number, namely:
(a) The Sphere and Cylinder.
(b) Conoids and Spheroids.
He invented general methods
for finding the areas of curvilinear plane figures and volumes bounded by curved
surfaces, and applied these methods to many special instances, including the
circle, sphere, any segment of a parabola, the area enclosed between two radii
and two successive whorls of a spiral, segments of spheres, and segments of
surfaces generated by the revolution of rectangles (cylinders), triangles
(cones), parabolas (paraboloids), hyperbolas (hyperboloids), and ellipses (spheroids)
about their principal axes. He gave a method for calculating π (the ratio of
the circumference of a circle to its diameter), and fixed π as lying between 3
1/7 and 3 10/71; he also gave methods for approximating to square roots which
show that he anticipated the invention by the Hindus of what amount to periodic
continued fractions. In arithmetic, far surpassing the incapacity of the
unscientific Greek method of symbolizing numbers to write, or even to describe,
large numbers, he invented a system of numeration capable of handling numbers
as large as desired. In mechanics he laid down some of the fundamental
postulates, discovered the laws of levers, and applied his mechanical
principles (of levers) to calculate the areas and centers of gravity of several
flat surfaces and solids of various shapes. He created the whole science of
hydrostatics and applied it to find the positions of rest and of equilibrium of
floating bodies of several kinds.
He
was also the first to give a method of calculating 
to any desired degree of
accuracy. It is based on the fact that the perimeter of a regular polygon of n
sides inscribed in a circle is smaller than the circumference of the
circle, whereas the perimeter of a similar polygon circumscribed about the
circle is greater than its circumference (see figure below).
By
making n sufficiently large, the two perimeters will approach the
circumference arbitrarily closely, one from above, the other from below.
Archimedes started with a hexagon, and progressively doubling the number of
sides, he arrived at a polygon of 96 sides, which yielded
That Archimedes did this without trigonometry, and without decimal (or other positional) notation is an illustration of his tenacity. However, we shall use both of these to go through the calculation.
A regular polygon of 40 sides. No internal
circle has been drawn.
It seems that Archimedes,
despising applied mathematics himself, had nevertheless yielded in peace time
to the importunities of Hieron, and had demonstrated to the tyrant’s
satisfaction that mathematics can, on occasion, become devastatingly practical.
To convince his friend that mathematics is capable of more than abstract
deductions. Archimedes had applied his laws of levers and pulleys to the
manipulation of a fully loaded ship, which he himself launched singlehanded.
Remembering this feat when the war clouds began to gather ominously near,
Hieron begged Archimedes to prepare a suitable welcome for Marcellus. Once more
desisting from his researches to oblige his friend, Archimedes constituted
himself a reception committee of one to trip the precipitate Romans.
When they arrived his ingenious deviltries stood grimly waiting to greet them.The harp-shaped turtle affair on the eight quinqueremes (five banks) lasted no longer than the fame of the conceited Marcellus. A succession of stone shots, each weighing over a quarter of a ton, hurled from the super catapults of Archimedes, demolished the unwieldy contraption.Crane-like beaks and iron claws reached over the walls for the approaching ships, seized them, spun them round, and sank or shattered them against the jutting cliffs.
The land forces, mowed down by the Archimedean artillery, fared no better. Camouflaging his rout in the official bulletins as a withdrawal to a previously prepared position in the rear, Marcellus backed off to confer with his staff. Unable to rally his mutinous troops for an assault on the terrible walls, the famous Roman leader retired.
Archimedes was killed during the sack of the city which followed its capture, in spite of the orders, given by the consul Marcellus who was in command of the Romans, that his house and life should be spared.
It is said that a soldier entered his study while he was regarding a geometrical diagram drawn in sand on the floor, which was the usual way of drawing figures in classical times. Archimedes told him to get o_ the diagram, and not spoil it. The soldier, feeling insulted at having orders given to him and ignorant of who the old man was, killed him. According to another and more probable account, the cupidity of the troops was excited by seeing his instruments, constructed of polished brass which they supposed to be made of gold.
The
Romans erected a splendid tomb to Archimedes, on which was engraved (in
accordance with a wish he had expressed) the figure of a sphere inscribed in a
cylinder, in commemoration of the proof he had given that the volume of a
sphere was equal to two-thirds that of the circumscribing right cylinder, and
its surface to four times the area of a great circle. Cicero1 gives a charming
account of his efforts (which were successful) to rediscover the tomb in 75 B.C.
References
1- Ball, W. W. R. (1960). A
Short Account of the History of Mathematics. New York: Dover Publications,
Inc.
2- Bell, E. T. (1986). Men of Mathematics.
New York: Simon and Schuster.
3- Beckman, P. (1971). The
history of PI. New York: St. Martin’s Press