3RD NATIONAL

CONFERENCE ON

MATHEMATICS

CONDUCTED BY : N.A.M.T

THE MAN WHO DARED

TO MOVE THE EARTH

PAPER PRESENTER:SANJEEV TUMMALA

SRI PRAKASH SYNERGY SCHOOL

PEDDAPURAM, A.P

E-MAIL: sanjeevt2010ind@gmail.com

“give me a place to stand

and I will move the earth”

ARCHIMEDESTHE MAN WHO DARED TO MOVE THE EARTH

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.

BIOGRAPHY

• 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.

BIOGRAPHY

• 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.

BIOGRAPHY

• 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.

BIOGRAPHY

• 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.

BIOGRAPHY• When the defenders, had

feasted and drank their fill of 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.

BIOGRAPHY• 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

Archimedes being killed by a roman soldier

DISCOVERIES and INVENTIONS

MATHEMATICAL

CONTRIBUTION

OF 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 CYLINDER

With 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

• The Sand Reckoner (Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the contemporary model, and invent a way to talk about extremely large numbers. The work, also known in Latin as Archimedis Syracusani Arenarius & Dimensio Circuli, which is about 8 pages long in translation, is addressed to the Syracusan king Gelon II (son of Hiero II), and is probably the most accessible work of Archimedes; in some sense, it is the first research-expository paper.

• There are some people , king Gelon for instance

who think that the number of grains of sand is

infinite in multitude and Archimedes of course is

an exception, here they mean not only the sand

grains in Syracuse but in the entire universe.

• Even though he set out to count the grains of

sand in the universe, the Greek system of

counting was not easy to play with and hence he

had to devise a new system of counting.

THE SAND RECKONER

THE SAND RECKONER

• Greek mathematical notation was not

positional; it utilized many symbols and

was cumbersome to work with.

• First, Archimedes had to invent a system of naming large numbers. The number system in use at that time could express numbers up to a myriad (10,000), and by utilizing the word "myriad" itself, one can immediately extend this to naming all numbers up to a myriad myriads (108).

• Archimedes called the numbers up to 108 "first numbers" and called 108 itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad-myriad times, 108·108=1016.

• This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad-myriad times the unit of the 108-th numbers, i.e., .

• After having done this, Archimedes called the numbers he had defined the "numbers of the first period", and called the last one, , the "unit of the second period".

THE SAND RECKONER

THE SAND RECKONER

• The "M" is a myriad, and represents 10,000. The Greek word is murious (uncountable, pl. murioi). The Romans converted 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 sand which 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.

• Modern Archimedes' screws which have replaced some of the

windmills used to drain the polders at Kinderdijk in the

Netherlands

DEATH RAY

•The death ray involves a

simple principle of setting

up fire by focusing of light

onto the desired object.

•Archimedes allegedly set

up an entire fleet of Roman

ships by using parabolic

reflectors

•Each mirror is capable of

generating heat of up to

6000K (surface temp of

sun)

GIANT CLAW

The Claw of Archimedes is a

weapon that he is said to have

designed in order to defend

the city of Syracuse. Also

known as "the ship shaker,"

the claw consisted of a crane-

like arm from which a large

metal grappling hook was

suspended. When the claw

was dropped onto an attacking

ship the arm would swing

upwards, lifting the ship out of

the water and possibly sinking

it.

CATAPULTS

• A catapult is a ballistic device used to launch a projectile a great distance without the aid of explosive devices—particularly various types of ancient and medieval siege engines. Although the catapult has been used since ancient times, it has proven to be one of the most effective mechanisms during warfare. The word 'catapult' comes from the Latin 'catapulta', which in turn comes from the Greek itself from , "downwards“ "to toss, to hurl". Catapults were invented by the ancient Greeks.

LEGACY

•Galileo praised Archimedes many times, and referred to him as

a "superhuman". Leibniz said "He who understands Archimedes

and Apollonius will admire less the achievements of the

foremost men of later times."

•There is a crater on the Moon named Archimedes (29.7° N,

4.0° W) in his honour, as well as a lunar mountain range,

the Montes Archimedes (25.3° N, 4.6° W).[

•The Fields Medal for outstanding achievement in mathematics

carries a portrait of Archimedes, along with a carving illustrating

his proof on the sphere and the cylinder. The inscription around

the head of Archimedes is a quote attributed to him which reads

in Latin: "Transire suum pectus mundoque potiri" (Rise above

oneself and grasp the world).

•Archimedes has appeared on postage stamps issued by East

Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971),

San Marino (1982), and Spain (1963).

Fields medal carrying a portrait of Archimedes