Archimedes of Syracuse
Born: 287 BC in Syracuse, Sicily
Died: 212 BC in Syracuse, Sicily
Archimedes, the greatest mathematician of antiquity, made his greatest contributions in geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz.
He was the son of the astronomer Phidias and was close to King Hieron and his son Gelon, for whom he served for many years.
He was an accomplished engineer but loved pure mathematics.
Stories from Plutarch, Livy,
and others describe machines invented by Archimedes for the defense of Syracuse. These include the catapult, the compound pulley and a burning-mirror.
Among Archimedes most famous works is Measurement of the Circle, in which he determined the exact value of 
to be between the values 
and 
. This he obtained by circumscribing and inscribing a circle with regular polygons having 96 sides.
However, he required the proof of two fundamental relations about the perimeters and areas
of inscribed and
circumscribed regular polygons.
The computation. With respect to a circle of radius r, let
Further, let 
denote the regular inscribed
polygons,
similarly, 
for the circumscribed polygons.
The following formulae give the relations between the perimeters and areas of
these 
polygons.
Using n-gons up to 96 sides he derives
Archimedes -- Books Extant
On the Sphere and Cylinder
Measurement of the Circle
On Conoids and Spheroids
On Spirals
On Plane Equilibria, Two Books
The Sand-Reckoner
Quadrature of a Parabola
On Floating Bodies: two Books
Stomachion, fragments only
The Method
Archimedes proved, among many other geometrical results, that the volume of a sphere is two-thirds the volume of a circumscribed cylinder.
This he considered his most significant accomplishments, requesting that a representation of a cylinder circumscribing a sphere be inscribed on his tomb.
His fascination with geometry is beautifully described by Plutarch.
Often times Archimedes' servants got him against his will to the baths, to wash and anoint him, and yet being there, he would ever be drawing out of the geometrical figures, even in the very embers of the chimney. And while they were anointing of him with oils and sweet savors, with his figures he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.
Archimedes literally invented the whole study of hydrostatics. In one particular result he was able to compute the maximum angle that a (paraboloid) ship could list before it capsized -- and he did it without calculus!
The case of the fraudulent gold crown. King Hiero II commissioned the manufacture of a gold crown. Suspecting the goldsmith may have substituted silver for gold, he asked Archimedes to determine its authenticity. He was not allowed to disturb the crown in any way.
What follows is a quote from Vitruvius (first cent BC):
The solution which occurred when he stepped into his bath and caused it to overflow was to put a weight of gold equal to the crown and know to be pure, into a bowl which was filled with water to the brim. Then the gold would be removed and the king's crown put in, in its place. An alloy of lighter silver would increase the bulk of the crown and cause the bowl to overflow.
There are some technical exceptions to this method. A better solution applies Archimedes' Law of Buoyancy and his Law of the Lever:
Suspend the wreath from one end of a scale and balance it with an equal mass of gold suspended from the other end. Then immerse the suspended wreath and gold into a container of water. If the scale remains in balance then the wreath and the gold have the same volume, and so the wreath has the same density as pure gold. But if the scale tilts in the direction of the gold, then the wreath has a greater volume than the gold.
He explores very large numbers in the Sand Reckoner by determining the number of grains of sand required to fill the universe of Aristarchus. To do this he needs new numbers and notations for magnitude. He constructed numbers up to 
.
Archimedes discovered fundamental theorems concerning the center of gravity of plane figures and solids. His most famous theorem gives the weight of a body immersed in a liquid, called Archimedes' principal.
In The Method Archimedes reveals how he discovered some of his theorems. The method is basically a ``geometric method of the lever." He balances lines as one might balance weights. This work was found relatively recently, having being rediscovered only in 1906.
Archimedes' mechanical skill together with his theoretical knowledge enabled him to construct many ingenious machines.
Archimedes spent some time in Egypt, where he invented a device now known as Archimedes' screw. This is a pump, still used in many parts of the world.
Quadrature of a Parabola. Archimedes proved, with the Method of Exhaustion, that
and gave two proofs.
The Spiral. 
Archimedes squared the circle using the
spiral.
He does this by proving that, in length, 
.
Note, PQ is tangent to the spiral at P and
is a right angle.
He also determined the area of one revolution
( 
) of 
to by
That is, the area enclosed by the spiral arc of one revolution is one third of the area of the circle with center at the origin and of radius at the terminus of the spiral arc.
He also showed how to trisect the angles using the spiral.
Simply construct circles of radii with constant successive differences. These circles will cut the spiral at equal angles. To trisect a particular angle, simply trisect a radial line segment corresponding to the values of the radii at the intersection of the spiral and lines that make the angle (with vertex at the origin). Construct the circles at the trisection points, with centers at the origin.
Archimedes was killed during the capture of Syracuse by the Romans in the Second Punic War. Plutarch recounts this story of his killing: As fate would have it, Archimedes was intent on working out some problem by a diagram, and having fixed both his mind and eyes upon the subject of his speculation, he did not notice the entry of the Romans nor that the city was taken. In this transport of study a soldier unexpectedly came up to him and commanded that he accompany him. When he declined to do this before he had finished his problem, the enraged soldier drew his sword and ran him through.