Archimedes

ARCHIMEDES
The 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.

…continuation
• 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.

…continuation
• 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.

…continuation
• 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.

…continuation
• 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

…continuation
–Another legend
• Archimedes was slain by looters who supposed
that his astronomical instruments, constructed
of polished brass, were actually made of gold.

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
• 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 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.

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…

BOUYANCY
Archimedes 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.

SOURCES
• http://www.techhive.com/article/191389/a-brief-
history-of-pi.html
• http://www.redstoneprojects.com/trebuchetstor
e/archimedes_1.html
• http://physics.weber.edu/carroll/archimedes/san
d.htm
• http://mathworld.wolfram.com/ArchimedesSpiral
.html
• http://www.britannica.com/EBchecked/topic/328
31/Archimedes-screw

Archimedes

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