Archimedes of Syracuse Biography

World of Mathematics on Archimedes of Syracuse

Archimedes of Syracuseis considered one of the greatest thinks of the ancient world. He established the principles of plane and solid geometry, discovered the concept of specific gravity, conducted experiments on buoyancy, demonstrated the power of mechanical advantage, and invented the Archimedes Screw, an auger-like device for raising water.

Archimedes was born in the Greek city of Syracuse, on the island of Sicily, in 287 B.C. He was the son of Phidias, the astronomer and mathematician. What we know of Archimedes' life comes from his extant writings, and from the histories authored by Plutarch, Cicero, and other historians several centuries after his death. Due to the length of time between Archimedes' death and his biographers' accounts, as well as inconsistencies among their writings, details of his life must remain subject to question.

Plutarch records that Archimedes was a relative of King Hieron, but Cicero claims he was of low birth. It is believed he obtained his early schooling in Syracuse, then traveled to Alexandria to study with Conon, the Egyptian mathematician and astronomer. Archimedes was close friends with both Conon and the custodian of the Alexandrian library, Eratosthenes. He corresponded with them about his mathematical and scientific discoveries long after he had completed his formal studies and returned to Syracuse. Although much of Archimedes' work was applied to practical ends, he himself was more interested in pure thought, and supposedly believed things connected with daily needs were ignoble and vulgar. Reports of his personal habits reflect his lack of concern with the mundane. He was known for becoming so engrossed in his thoughts that he would forget to eat and bathe. Many of the equations he developed were scratched out first in ashes or traced with after-bath oil on his skin.

Archimedes' contributions to mathematical knowledge were diverse. On the subject of plane geometry three of the treatises he wrote have survived, Measurement of a Circle, Quadrature of the Parabola, and On Spirals. In Measurement of a Circle, he described his method for calculating , the ratio between the circumference of a circle and its diameter. By a method that involved measuring the perimeter of inscribed and circumscribed polygons, Archimedes correctly determined that the value of was somewhere between 3.1408 and 3.1428. In the same treatise he set forth the formula r² for determining a circle's area. In Quadrature of the Parabolaand On Spirals, Archimedes advanced his technique for determining the area under curves, a sophisticated version of the method of exhaustion, originally developed by the Egyptians. His use of this technique, elaborated upon in another volume, The Method, anticipated the development of integral calculus by two thousand years.

Archimedes dealt with the topic of solid geometry in his writings On the Sphere and Cylinderand On Conoids and Spheroids. On the Sphere and Cylinder contains several famous proofs, including his demonstration that the volume of a sphere is equal to 4/3 r³. Archimedes also showed in this work that the volume of a sphere is two-thirds the volume of a cylinder surrounding it, as long as the cylinder's height and width are equal to the sphere's diameter. So proud was he of this latter discovery that he requested its illustration be engraved on his tombstone, a wish that was eventually fulfilled.

On the subject of arithmetic Archimedes wrote several essays, of which only The Sand Reckonerremains. Addressed to the son of King Hieron, it proposed the problem of determining the number of grains of sand in the universe, and contained a special notation for estimating and expressing very large numbers. Archimedes was famous for another complicated arithmetic puzzle called the Cattle Problem, in which one had to determine the number of bulls and cows of various colors, given that each cattle color was represented in a particular ratio to the others. There are an infinite number of possible solutions to the problem, but deriving them was especially challenging to the ancient Greeks, who had no knowledge of algebra.

According to the ancient historians, Archimedes was frequently called upon by King Hieron to solve practical problems. Perhaps the most famous of these was the task of determining whether King Hieron's crown was made of pure gold. Hieron believed his jeweler had stolen a portion of the gold intended for the crown, substituting an equal weight of another metal. Legend has it that Archimedes stepped into a bath and, noticing the displacement of water, conceived of a vertical buoyancyforce. Realizing that this concept would allow him to measure the density of the king's crown in comparison to pure gold, he supposedly jumped from the tub and ran naked through the streets yelling "eureka, eureka!" Archimedes not only proved the crown was tainted, but went on to describe the idea of specific gravity and develop a generalized concept of buoyancy known today as Archimedes' Principle.

The Archimedes Screwwas reportedly invented in order to empty water from the hold of one of King Hieron's ships. This device consisted of a screw, encased in a cylinder, that was turned by a hand-crank. As the screw spiraled upward it carried water. A similar device is still used today to lift water in the Nile Delta of Egypt. Archimedes recognized the mechanical advantage that could be gained by using levers, and it is said he boasted that with a long enough lever he could move the Earth. He determined the inverse mathematical relationship between the effort required to raise a load with a lever and the distance of the load from the lever's pivot point or fulcrum. The story is told that King Hieron, skeptical of the power of mechanical advantage, challenged Archimedes to move a three-mast ship, laden with passengers and freight, that lay aground near Syracuse Harbor. To meet the challenge Archimedes is said to have designed a system of compound pulleys. With a relatively effortless pull of a rope he was able to guide the vessel into the water. Archimedes was also interested in astronomy and made several accomplishments in this field. He built a device to estimate the size of the sun and constructed a model planetarium to demonstrate the motion of the planets.

During Archimedes' lifetime the first two of the three Punic Wars between the Romans and the Carthaginians were fought. Syracuse allied itself with Carthage, and when the Roman general Marcellus began a siege on the city in 214 B.C., Archimedes was called upon by King Hieron to aid in its defense. The historical accounts of Archimedes' war-faring inventions are vivid and possibly exaggerated. It is claimed that he devised catapult launchers that threw heavy beams at the Roman ships, grappling cranes that hoisted ships out of water, and burning-glasses that reflected the sun's rays and set ships on fire. Marcellus had given orders that when Syracuse was finally conquered, Archimedes, whose reputation was widely known, should be taken alive. When the Romans finally sacked the city in 212 B.C., a soldier found Archimedes quietly etching equations in the sand, absorbed in a mathematical problem. Reportedly, Archimedes ordered the soldier not to disturb the figures in the sand. Enraged, the soldier drew his sword and impaled him.