As a young man, Archimedes traveled to Egypt for it is said that while there, he invented what is today known as the Archimedes' screw.

It is believed that he studied with the students of Euclid in Alexandria because he mastered the ideas of Euclid and he references one Alexandrian mathematician, Conon of Samos, as a close friend. In his work On Spirals, he mentions a practice in Alexandria where mathematicians would give results but would not include proofs for those results. Archimedes disapproved of this practice and on one occasion sent false results to see how the

Alexandrian mathematicians responded.

Throughout his life, Archimedes made numerous engineering inventions. Despite his rising fame, Archimedes was most proud of his work in pure mathematics. Plutarch writes:

*Archimedes is responsible for contributing the term "eureka" into the English language. Eureka is Greek for "I found it." Archimedes was working on a problem for King Heiron when he figured out the solution while taking a bath. He was so excited, that he ran around Syracuse naked shouting "Eureka! Eureka!" which means "I have found it! I have found it!".*

Archimedes possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, he yet would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life; studies, the superiority of which to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, of the precision and cogency of the methods and means of proof, most deserve our admiration.

In his day, Archimedes was very well known. There are many references to him from contemporary writers. Interestingly, he was not famous for his mathematics but for his many mechanical devices that could be used in warfare. Plutarch writes about the use of these war machines when Marcellus led the Roman army against Syracuse in 212 BC:

Ultimately, the Romans succeeded in taking Syracuse and Archimedes was killed during the invasion. Plutarch writes about three versions of the death of Archimedes. The most famous version is that he didn't notice the Roman soldier because he was intense on a math problem. In another version, he is confronted by the same soldier who tells him that he has come to kill him. Archimedes pleads for his life by saying that he has many mathematical works that have not yet been finished. The soldier is unmoved by this plea and kills him. In the third version, Archimedes carries his equipment to show Marcellus what can be done. The soldiers stop him and slay him because they believe that Archimedes has gold hidden inside the equipment.

... when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over the ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air(a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.

Cicero writes that when Marcellus returned to Rome, he brought with him two inventions of Archimedes: a star globe, that mapped the sky onto a sphere, and the other was an orrery, an object that predicts the motions of the sun, moon, and planets.

Today, historians agree that Archimedes, Newton, and Gauss are the greatest mathematicians who have ever lived.

He made great advances in the method of integration that enabled him to find areas, volumes, and surface areas of geometric forms. He gave one of the first accurate approximations of π and found a method for estimating square roots. He invented a method for representing very large numbers. He invented the compound pulley, the law of the lever, he found a method for estimating the weight of an object immersed in a liquid, he came up with the concept of center of gravity. His favorite theorem was that the volume of a sphere is two-thirds the volume of a circumscribed cylinder.

Heath writes about Archimedes:

*References*

The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.

- Archimedes, Wikipedia
- Archimedes, MacTutor
- T. L. Heath,
*A history of Greek mathematics***II**(Oxford, 1931). - Plutarch, Marcellus, MIT's on-line version