The burning mirror. Image by Stanzino delle Matematiche

Archimedes was a famous mathematician whose theorems and philosophies became world known. He was born 287 BC in Syracuse, Sicily and gained a reputation in his own time which few other mathematicians of this period achieved. He is considered by most historians of mathematics as one of the greatest mathematicians of all time. Most of the facts about his life come from a biography about the Roman soldier Marcellus written by the Roman biographer Plutarch.

Archimedes probably spent some time in Egypt early in his career, but he resided for most of his life in Syracuse, the principal city-state in Sicily, where he was on intimate terms with its king, Hieron II. Archimedes published his works in the form of correspondence with the principal mathematicians of his time, including the Alexandrian scholars Conon of Samos and Eratosthenes of Cyrene.

He played an important role in the defense of Syracuse against the siege laid by the Romans in 213 BC by constructing war machines so effective that they long delayed the capture of the city. But the Roman general Marcus Claudius Marcellus eventually captured Syracuse in the autumn of 212 or spring of 211 BC, and Archimedes was killed in the sack of the city.

According to Plutarch, Archimedes had so low an opinion of the kind of practical invention at which he excelled and to which he owed his contemporary fame that he left no written work on such subjects. While it is true that -apart from a dubious reference to a treatise, “On Sphere-Making"- all of his known works were of a theoretical character, nevertheless his interest in mechanics deeply influenced his mathematical thinking. Not only did he write works on theoretical mechanics and hydrostatics, but his treatise Method Concerning Mechanical Theorems shows that he used mechanical reasoning as a heuristic device for the discovery of new mathematical theorems.

Legend has it that Archimedes discovered his famous theory of buoyancy - Archimedes Principle - while taking a bath. He was so excited that he ran naked through the streets of Syracuse shouting “Eureka, eureka (I have found it)!”

Another legend describes how Archimedes uncovered a fraud against King Hieron II of Syracuse using his principle of buoyancy. The king suspected that a solid gold crown he ordered was partly made of silver. Archimedes first took two equal weights of gold and silver and compared their weights when immersed in water. Next he compared the weights of the crown and a pure silver crown of identical dimensions when each was immersed in water. The difference between these two comparisons revealed that the crown was not solid gold.

The works of Archimedes which have survived are as follows.
On plane equilibriums (two books),
Quadrature of the Parabola,
On the Sphere and Cylinder (two books),
On Spirals,
On Conoids and Spheroids,
On Floating Bodies (two books), Measurement of a Circle,
and The Sandreckoner.

Archimedes screw
This machine for raising water, allegedly invented by the ancient Greek scientist 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.

The burning mirror
Archimedes invented many machines which were used as engines of war. These were particularly effective in the defence of Syracuse when the Romans under the command of Marcellus attacked it. During the Roman siege of Syracuse, he is said to have single-handedly defended the city by constructing lenses to focus the Sun’s light on Roman ships and huge cranes to turn them upside down.

The compound pulley
Archimedes had stated in a letter to King Hieron that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labour and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.

The lever
Only his phrase here I think is enough: “Give me a place to stand and rest my lever on, and I can move the Earth.”

The spheres
Archimedes is supposed to have made two “spheres” that Marcellus took back to Rome - one a star globe and the other a device (the details of which are uncertain) for mechanically representing the motions of the Sun, Moon, and planets. One was a solid sphere on which were engraved or painted the stars and constellations, which Marcellus placed in the Temple of Virtue. Such celestial globes predate Archimedes by several hundred years and Cicero credits the famed geometers Thales and Eudoxos with first constructing them.

The second sphere, which Marcellus kept for himself, was much more ingenious and original. It was a planetarium: a mechanical model, which shows the motions of the sun, moon, and planets as, viewed from the earth. Cicero writes that Archimedes must have been “endowed with greater genius that one would imagine it possible for a human being to possess” to be able to build such an unprecedented device.

Derek De Solla Price of Yale University, who concluded that it was an ancient planetarium in which the positions of the heavenly bodies were indicated by dials on the face of the device, analyzed the device, now called the Antikythera mechanism.

The gearworks are about as complicated as those in a modern mechanical clock and represent the earliest physical evidence of an advanced metallic mechanism. Price gives evidence that this mechanism was in the Archimedean tradition and strongly suggests that Archimedes’ planetarium was its forerunner.

Geometry - Archimedes discovered pi (?)
He performed numerous geometric proofs using the rigid geometric formalism outlined by Euclid, excelling especially at computing areas and volumes using the method of exhaustion.
Archimedes, although he achieved fame by his mechanical inventions, believed that pure mathematics was the only worthy pursuit. He was a brilliant mathematician who helped develop the science of geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz.

Although many solid figures having all kinds of surfaces can be conceived, those which appear to be regularly formed are most deserving of attention. Those include not only the five figures found in the godlike Plato, that is, the tetrahedron and the cube, the octahedron and the dodecahedron, and fifthly the icosahedron, but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons.

The death of Archimedes
Archimedes was killed in 212 B.C., at the capture of Syracuse by the Roman general, Marcellus, during the Second Punic War. All of his marvelous machinery of war -invented by Archimedes to keep the Romans at bay- had finally failed. His death is recounted by Plutarch:
Archimedes...was..., as fate would have it, intent upon working out some problem
by a diagram, and having fixed his mind alike and his eyes upon the subject of his
speculations, he never noticed the incursion of the Romans, nor that the city was
taken. In this transport of study and contemplation, a soldier, unexpectedly coming
up to him, commanded him to follow to Marcellus; which he declined to do before
he had worked out his problem to a demonstration. The soldier, enraged, drew his
sword and ran him through.

His legendary last words: “Don’t disturb my circles” ( mi mou tous kiklous taratte ).

Archimedes was buried Syracuse, where he was born, were he grew up, where he worked, and where he died. On his grave their is an inscription of pi, his most famous discovery. They also placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.