Eureka
Archimedes was born and mainly lived in Syracuse on the
eastern coast of Sicily. He is believed to have been close to
Hieron II, King of Syracuse, who tried to encourage Archimedes
to use his extraordinary talents for practical purposes. The
King had commissioned a gold wreath that he wished to
consecrate to the gods. When the wreath was delivered, it
weighed the correct amount for which he had been charged.
However, he was concerned that some of the gold had been
replaced by an equal weight of a metal of lesser value, such
as silver or lead. Because of its consecrated nature,
dissection or analysis was impossible.
Archimedes was presumably pondering this quandary while
visiting the public baths in Syracuse. As he sank into the
bath, water overflowed: the further he sank, the more water
poured out. Upon realising that the amount of water displaced
was a direct measure of his volume, legend has it that he
leapt from the bath and ran, naked and dripping, to the King,
repeatedly shouting "Eureka" (I have found it). He knew that
if he immersed the crown in water and measured the overflow,
he could find its volume. Whatever its shape, if the crown
were pure gold it should have an equal volume to an equal
weight of pure gold, whatever shape that had. When Archimedes
measured the volume of the crown it was greater than the
volume of a kilo of gold, and Hieron saw that he had been
cheated.

... he wrote
down nothing other than mathematics. 

He then gave Archimedes the challenge of launching the
Syracusia, an enormous ship weighing 4 200 tons that had been
built as a gift for Ptolemy, King of Egypt and which was so
heavy that all previous launch attempts had failed. According
to Plutarch, Archimedes used a polypaston, or block and
tackle, with a large number of sheaves. This effectively
created many ropes, with the weight of the ship divided
between them. Consequently each rope, including the final rope
that was being pulled, had only to support a fraction of the
weight: a force of only a fraction of the weight was
sufficient to lift the weight. Archimedes realised that with a
perfect lever there was no theoretical limit to how large a
load could be shifted with any given weight.
Apart from an account of his elegant planetariums, he wrote
down nothing other than mathematics. This work included an
early approximation of the value for pi, the calculation of
relative volumes of spheres and cylinders, and an original
system of notation (with a basevalue of 100 million) to
express enormous numbers. He was also an outstanding
astronomer, and invented 'Archimedes' Screw', a device for
raising water.
When the Romans invaded Syracuse in 214 BC, Archimedes
invented 'engines of war' to defend the city, including cranes
to drop rocks, claws to lift ships from the water, and
machines to fire missiles. Most famous were the burning
mirrors, with which Archimedes is supposed to have set ships
on fire. This was theoretically straightforward: a parabolic
mirror could be used to focus the rays of the sun onto one
point, which would then reach temperatures sufficient to set
alight anything at its focal point. The construction was more
complicated, and Archimedes possibly approximated a parabolic
mirror with a large number of small mirrors: the more mirrors,
the closer the approximation.
The siege of Syracuse was, however, to cost him his life.
He was reportedly absorbed in his mathematics when captured.
Instructions that he be kept alive were ignored when he
ordered the Roman soldier to stay away from his work, and he
was killed immediately.