Archimedes was born in Syracuse, a Greek city in Sicily, in 287 B.C., and lived there for most of his life. He was so far ahead of his time that some of his principles did not become established until the fifteenth century. He invented the Archimedean screw (a water screw) which is used in Egypt even today, and he explained the theory of the lever. He declared that, if he had a lever long enough and a prop strong enough, he could, single-handedly, move the world (this is true, although in practice highly impractical). His most celebrated work is that on the sphere and the cylinder, which he requested should be inscribed on his tombstone. He found that the value of π was somewhere between 3 10/71 and 3 1/7 (or between 3.1408 to 3.1428) by finding the perimeter of inscribed and circumscribed many-sided polygons. If the two numbers are averaged together, we get 3.1418, which is π correct to three decimal places.
In his book The Sand Reckoner, which he addressed to King Gelon of Syracuse, he described his own system of counting immense numbers. Ancient Greek mathematicians described numbers between 1 and 99,999,999 as "the first numbers". The number 100,000,000 was described as "one in the second numbers". The "second numbers" ran from 100,000,000 to 1016 − 1. Archimedes extended this system up to the "ten thousand times ten thousandth numbers", which is able to express any number up to (but not including) 10800,000,000. He then used this notation to estimate the number of grains of sand required to fill the entire universe.
Archimedes discovered the law of specific gravity, which states that any body weighs just as much less when held under water as the weight of the water which it crowds out of place. The story goes that Hieron II, King of Syracuse, suspected a goldsmith of putting some other metal than gold in his crown, asked Archimedes to ascertain whether this was so. Archimedes, while thinking over the matter one day, got into his bath (which was full to the brim) and realized that the volume of water that would run over the edge of the tub was the same as the volume of his body. He then saw that if he put the crown into a vessel, and weighed the water which overflowed, and then put a piece of gold of the same mass of the crown, the water overflowed by the pure gold ought to equal in weight that of the crown if it were also of pure gold (As it turned out, it didn't). He was so overjoyed at this discovery that he ran home naked, crying "Eureka!" ("I have found it!") through the streets. (Of course, this probably wasn't too astonishing for the citizens of Syracuse, since the Greeks habitually exercised in the nude, and the sight of a naked male would have meant little to them).
He helped Hieron defended Syracuse against the Romans by inventing machines of war (although some of the machines described in the historical record seem unlikely). Supposedly, he invented machines that lifted ships out of the water and dropped them with so much force that they sank, and burned ships by concentrating on them the rays of the sun with mirrors.
When Syracuse was taken in 212 B.C., the Roman general Marcellus ordered his soldiers not to hurt Archimedes, and offered a reward to whoever should bring him safe to him. A Roman soldier found him in his studio, so busy working on a mathematics problem that he did not even know that the enemy had entered the gates. The soldier ordered Archimedes to come with him; when Archimedes asked the soldier to wait while he finished the math problem he was working on, the soldier killed him, to the grief of Marcellus. Marcellus ordered an honourable burial for Archimedes and built a monument over his grave inscribed as he had desired.
About Archimedes from Vitruvius' On Architecture.
Sources used (see bibliography page for titles corresponding to numbers): 14.