## 9.2. Archimedes (287? -212 B.C.)

*great*geometer" and his works and inventions brought him fame that lasts to this very day. He was one of the last great Greek mathematicians.

Born in 287 B.C., in Syracuse, a Greek seaport colony in Sicily, Archimedes was the son of Phidias, an astronomer. Except for his studies at Euclid's school in Alexandria, he spent his entire life in his birthplace. Archimedes proved to be a master at mathematics and spent most of his time contemplating new problems to solve, becoming at times so involved in his work that he forgot to eat. Lacking the blackboards and paper of modern times, he used any available surface, from the dust on the ground to ashes from an extinguished fire, to draw his geometric figures. Never giving up an opportunity to ponder his work, after bathing and anointing himself with olive oil, he would trace figures in the oil on his own skin.

Much of Archimedes fame comes from his relationship with Hiero, the king of Syracuse, and Gelon,
Hiero's son. The great geometer had a close friendship with and may have been related to the monarch.
In any case, he seemed to make a hobby out of solving the king's most complicated problems to the utter
amazement of the sovereign. At one time, the king ordered a gold crown and gave the goldsmith the
exact amount of metal to make it. When Hiero received it, the crown had the correct weight but the
monarch suspected that some silver had been used instead of the gold. Since he could not prove it, he
brought the problem to Archimedes. One day while considering the question, "the wise one" entered his
bathtub and recognized that the amount of water that overflowed the tub was proportional the amount of
his body that was submerged. This observation is now known as Archimedes' Principle and gave him the
means to solve the problem. He was so excited that he ran naked through the streets of Syracuse shouting
"*Eureka! eureka!*" (I have found it!). The fraudulent goldsmith was brought to justice. Another
time, Archimedes stated "Give me a place to stand on and I will move the earth." King Hiero, who was
absolutely astonished by the statement, asked him to prove it. In the harbor was a ship that had proved
impossible to launch even by the combined efforts of all the men of Syracuse. Archimedes, who had been
examining the properties of levers and pulleys, built a machine that allowed him the single-handedly
move the ship from a distance away. He also had many other inventions including the Archimedes'
watering screw and a miniature planetarium.

Though he had many great inventions, Archimedes considered his purely theoretical work to be his true
calling. His accomplishments are numerous. His approximation of
between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate
square roots. Unhappy with the unwieldy Greek number system, he devised his own that could
accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery
of the Archimedes' Principle. However, his greatest invention was *integral calculus.* To
determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes
broke the sections into an infinite number of rectangles and added the areas together. This is known as
*integration.* He also anticipated the invention of* differential calculus* as he devised ways
to approximate the slope of the tangent lines to his figures. In addition, he also made many other
discoveries in geometry, mechanics and other fields.

The end of Archimedes life was anything but uneventful. King Hiero had been so impressed with his friend's inventions that he persuaded him to develop weapons to defend the city. These inventions would prove quite useful. In 212 B.C., Marcellus, a Roman general, decided to conquer Syracuse with a full frontal assault on both land and sea. The Roman legions were routed. Huge catapults hurled 500 pound boulders at the soldiers; large cranes with claws on the end lowered down on the enemy ships, lifted them in the air, and then threw them against the rocks; and systems of mirrors focused the sun rays to light enemy ships on fire. The Roman soldiers refused to continue the attack and fled at the mere sight of anything projecting from the walls of the city. Marcellus was forced to lay siege to the city, which fell after eight months. Archimedes was killed by a Roman soldier when the city was taken. The traditional story is that the mathematician was unaware of the taking of the city. While he was drawing figures in the dust, a Roman soldier stepped on them and demanded he come with him. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

For related information on Archimedes, see: Euclid , Archimedien property.

**Sources**

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- Bell, E.T.
*Men of Mathematics.*New York: Simon and Schuster, Inc., 1937. pp. 28-34. - Lewis, Albert C. "Archimedes."
*Encyclopedia of World Biography.*New York: McGraw- Hill, Inc., 1973. vol. 1, pp.219-223. - Muir, Jane.
*Of Men and Numbers.*New York: Dodd, Mead & Company, 1962. - Turnbull, Herbert Westren.
*The Great Mathematicians*. New York: New York University Press, 1961.

**Paul Golba**