ARCHIMEDESThe Ancient World’s Genius

BIOGRAPHY Archimedes was born around

287 B.C. in the Greek city-state of Syracuse on the island of Sicily.

He died around 212 B.C. at the age of 75.

He is a Greek mathematician, engineer, inventor, physicist, and astronomer

He was the son of Phidias, an astronomer.

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• According to Plutarch, Archimedes came from the same royal family as the city’s ruler, King Hieron II.

• Archimedes stayed in Syracuse his whole life, except for the time when he went to Alexandria.

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• It was said that while he was in Alexandria, he studied with the pupils of Euclid and became friends with Conon of Samos and with Eratosthenes.

• Upon his return to Syracuse from Egypt, he devoted his life to the study of mathematics.

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• It seemed that it was the devotion to Hieron that induced Archimedes to divert his mathematical studies to his engineering skills.

• In fact, many of his inventions were created at Hieron’s request.

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• He also invented various war machines in defending his city against the Romans.

• Because of these machines, Roman soldiers were in abject terror and refused to advance.

…continuation• When the defenders, had feasted

and drank their fill a religious festival, pro-Roman sympathizers inside the city directed the enemy to a weak point in the walls. Marcellus gave explicit orders to his officers that the life and household of Archimedes should be spared; but before they could locate the great scientist, he had been slain by a common soldier.

…continuation• Accounts of his death has

been told in various forms:– Traditional Story• He was absorbed in a

geometrical problem whose diagram was drawn in the sand. As the shadow of an approaching Roman soldier fell over his diagrams, the agitated mathematician called out, “Don’t disturb my circles!” The soldier, insulted at having orders thus given to him, retaliated by drawing his sword

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–Another legend• Archimedes was slain by looters who supposed

that his astronomical instruments, constructed of polished brass, were actually made of gold.

DISCOVERIES and INVENTIONS

MATHEMATICAL CONTRIBUTIONOF ARCHIMEDES

MEASUREMENT OF A CIRCLE

• Archimedes was the first one to precisely calculate the value of pi. He accomplished this by finding the areas of 2 polygons: the polygon that was inscribed inside the circle, and the polygon in which a circle was circumscribed.

MEASUREMENT OF A CIRCLE

• Archimedes didn’t calculate the exact value of pi, but rather came up with a very close approximation – he used 96-sided polygons to come up with a value that fell between 3.1408 and 3.14285

ON SPHERE AND CYLINDERWith cylinder circumscribing a sphere, he showed that the surface area of a sphere is four times that of a great circle. He also finds the area of any segment of a sphere and shows that the volume of a sphere is 2/3 the volume of a circumscribed cylinder.

THE SAND RECKONER

• Greek mathematical notation was not positional; it utilized many symbols and was cumbersome to work with.

THE SAND RECKONER

• The "M" is a myriad, and represents 10,000. The Greek work is murious  (uncountable, pl. murioi). The Romans converted to this to myriad.

• The Sand Reckoner is a remarkable work in which Archimedes proposes a number system that uses powers of a myriad (base 100,000,000) and is capable of expressing numbers up to 8 x 1063 in modern notation.

• He argues in this work that this number is large enough to count the number of grains of sandwhich could be fitted into the universe.

ARCHIMEDEAN SPIRAL

• This spiral was studied by Archimedes in about 225 BC in a work On Spirals. It had already been considered by his friend Conon.

• Archimedes was able to work out the lengths of various tangents to the spiral. It can be used to trisect an angle and square the circle.

OTHER ACCOMPLISHMENTS

BOUYANCY

King Hieron II had given the goldsmith a particular amount of gold to melt down and make into a crown. When the crown was made and returned to the king, the king was suspicious that the goldsmith had stolen some of the gold and replaced it with an equal weight of silver.

The king turned to Archimedes for help…

BOUYANCYArchimedes happened to go to the bath, and on getting a tub observed that the more water ran out over the tub. As this pointed out the way to explain the case in question, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for he, as he ran, shouted repeatedly in Greek, ‘EUREKA, EUREKA,’ meaning “I have found it.”

THE LAW OF LEVER

• Archimedes did not invent the lever, however he gave an explanation about the principle

• Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle.

• According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth.“

• Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.

ARCHIMEDES SCREW• A machine for raising water, allegedly invented

by Archimedes for removing water from the hold of a large ship. One form consists of a circular pipe enclosing a helix and inclined at an angle of about 45 degrees to the horizontal with its lower end dipped in the water; rotation of the device causes the water to rise in the pipe. Other forms consist of a helix revolving in a fixed cylinder or a helical tube wound around a shaft.

ARCHIMEDES SCREW

• Modern screw pumps, consisting of helices rotating in open inclined troughs, are effective for pumping sewage in wastewater treatment plants. The open troughs and the design of the screws permit the passage of debris without clogging.

DEATH RAY

GIANT CLAW

CATAPULTS

SOURCES

• http://www.techhive.com/article/191389/a-brief-history-of-pi.html

• http://www.redstoneprojects.com/trebuchetstore/archimedes_1.html

• http://physics.weber.edu/carroll/archimedes/sand.htm

• http://mathworld.wolfram.com/ArchimedesSpiral.html

• http://www.britannica.com/EBchecked/topic/32831/Archimedes-screw