Archimedes was a native of Syracuse, Sicily. Some authors have said that he visited Egypt and invented a device there now known as Archimedes’ screw. This screw is a pump, still used in many parts of the world. When Archimedes was a young man, he studied with the descendants of Euclid in Alexandria. He was familiar with the mathematics used there, and he knew personally the mathematicians working there and he sent his results to Alexandria with personal messages. He regarded Conon of Samos, one of the mathematicians at Alexandria, very highly for his abilities as a mathematician and he also regarded him as a close friend.

In the preface to “On Spirals” Archimedes told a story about his friends in Alexandria. He said that he was in the habit of sending them statements of his latest theorems, but without giving proofs. Some of the mathematicians there claimed the results as their own so Archimedes said that on the last occasion when he sent them theorems he included two which were false. Other than in the prefaces to his works, information about Archimedes comes to us from a number of sources such as in stories from Plutarch, Livy, and others.

Plutarch tells us that Archimedes was related to King Hieron II of Syracuse. Evidence of his friendship with the family of King Hieron II comes from the fact that “The Sandreckoner” was dedicated to Gelon, the son of King Hieron. There are many of references to Archimedes in the writings of the time. He was probably the only mathematician of his time with such a high reputation. This is because he was not interest in new mathematical ideas but had invented war machines. They were particularly effective in the defense of Syracuse when it was attacked by the Romans and Marcellus.

Plutarch writes in his work on Marcellus, the Roman commander, about how Archimedes’ war machines were used against the Romans in the siege of 212 BC; “… 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, 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. ” Archimedes had been persuaded by his friend and King Hieron to build these machines; These machines Archimedes had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero’s desire and request, some little time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of the people in general. ”

Although Archimedes became famous becasue of his mechanical inventions, he believed that mathematics was the only worthwhile objective. Plutarch described Archimedes with far-fetched attitude, but Archimedes did use some very practical methods to discover results from pure geometry. In “On Spirals” Archimedes defines a spiral. He gives properties connecting the length of the radius vector with the angles through which it has revolved. He also gives results on tangents to the spiral and finding the area of portions of the spiral.

In the work “On Conoids and Spheroids” Archimedes examines paraboloids of revolution, hyperboloids of revolution, and spheroids performed by rotating an ellipse either about its major axis or about its minor axis. The purpose of the work was to study the volume of segments of 3-D figures. Archimedes considered his most significant accomplishments were about a cylinder circumscribing a sphere. He asked for a representation of this together with his result on the ratio of the two, to be inscribed on his tomb.

Cicero was in Sicily in 75 BC and he writes how he searched for Archimedes tomb; “… nd found it enclosed all around and covered with brambles and thickets; for I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been put on top of his grave. Accordingly, after taking a good look all around … , I noticed a small column arising a little above the bushes, on which there was a figure of a sphere and a cylinder… . Slaves were sent in with sickles … and when a passage to the place was opened we approached the pedestal in front of us; the epigram was traceable with about half of the lines legible, as the latter portion was worn away. “