MATHEMATICIAN

Life
Archimedes, the greatest mathematician of the ancient world and the greatest mathematical genius in Europe until Newton, was born, lived and died in the Greek city-state of Syracuse, in Sicily. He was the son of an astronomer called Phidias, and was closely associated with (and possibly related to) the city's ruler, Hieron II. He studied at Alexandria, in Egypt, where he met Euclid's successors Eratosthenes and Dositheus; he was also a friend and associate of Conon of Samos. His output was prodigious, both in quantity and in quality; and his enquiring mind explored many different fields: geometry, mirrors and lenses, hydraulics, mechanics, architecture, siege craft. His name is inextricably associated with the genesis of engineering in ancient Greece, and with the resolution of many famous mathematical problems. He is perhaps most celebrated, however, for his part in the defence of Syracuse against a besieging Roman fleet. Legend has it that when an act of treachery finally delivered the city to the enemy, a Roman soldier came upon Archimedes working out the answer to a problem in mathematics and killed him on the spot. His burial site was discovered by Cicero in 75 BC.

A commentary on Archimedes is found in the work of 6th century writer Eutocius of Ascalon, who also mentions a biography (now lost) of the celebrated mathematician written by Heraclides.


Work
The principal extant works are:

"On the Sphere and Cylinder": books I and II. Book I: Quadrature of the parabola, 6 definitions, 5 propositions (the 5th is the axiom of continuity), 44 theorems.

"Measurement of the Circle": 3 theorems are preserved.

"On Conoids and Spheroids": The book contains 32 theorems and 1 corollary following the 6th theorem. Some of the theorems use trigonometric functions and integral calculus.

"On Spirals": The book contains 28 theorems and 6 corollaries.

"On the Equilibrium of Planes or Centres of Gravity of Planes or Mechanics": Books I and II.

Book I: Theory of levers. Contains 7 postulates, 15 theorems and 2 corollaries following the 5th theorem.

Book II: Centre of gravity of paraboloid plane sections. 10 theorems.

"The Sand-Reckoner"

"Quadrature of the Parabola"

"On Floating Bodies" (Hydrostatics)

"On the Method of Mechanical Theorems"

"Book of Lemmas"

"The Cattle Problem"

"On the Heptagon in a Circle"

"The Water Clock"

"On Tangential Circles"

"Principles of Geometry"

Lost works:

"On Triangles"

"On Quadrangles"

"On 13 semi-regular solids"

"Arithmetic"

"On Balances or Levers"

"Centres of Gravity"

"On Sundials"

"Catoptrica" (Optics)

"Isoperimetrics"

"Elements of Mechanics"

"Equilibria"

"On Sphere-Making"

"Elements on Supports"

"On Parallel Lines"

"On Gravity and Buoyancy" (pycnometers, areometers)

"On Burning Glasses"

"Perspective"

"On the Calendar"

"Winches, Hydroscopes, Pneumatics"

"Burning by mirror"

"On Architecture"

"On Odometers".

Many of Archimedes' inventions are celebrated in the history of mechanics and technology, but since only fragments of his vast work have been preserved, a number of these cannot be properly described. Some of his most important inventions include: the areometer/pycnometer, an astronomical device for calculating the size and distance of stars, the winch, various instruments operated by wind or steam power, reflecting lenses (which he used to set Roman ships ablaze), the endless screw (a system for drawing water up from lower levels), an odometer, (an instrument for measuring distances), a planisphere (a sphere representing the cosmos, within which the movements of the sun, the moon, the planets and the constellations could be reproduced), block-and-tackle and multiple pulley systems (machines for raising weights using ropes), siphons, water clocks and a type of cannon. This latter invention was reconstructed by engineer Ioannis Sakkas in 1990, as were Archimedes' water clock and burning mirrors. Archimedes has had a tremendous influence on European thinking, as he had on the Arab scientists who translated his works into Arabic, thus ensuring the survival of many of his works despite the loss of the originals.

ARCHIMEDES: CRANES, CATAPULTS, MIRRORS

Archimedes played a major part in defending his natal city of Syracuse against a protracted Roman siege, as the designer of a host of weapons and machines to repulse the attackers. These fall into three main categories: a) cranes (or 'claws') that lifted enemy ships out of the water and dashed them against the rocks, b) catapults of every size and description that hurled bolts and stones varying distances, and c) the mirrors that focused sunlight on the ships and set them alight. This latter invention has become legendary, and much has been written about whether such a thing could in fact have been possible in the time of Archimedes. Most experts, and particularly foreign experts, were persuaded that the construction of such a system was a myth, despite the weight of literary evidence supporting the story, until engineer Ioannis Sakkas succeeded in demonstrating that it was indeed possible. Sakkas used 70 copper-plated glass lenses, with diameters ranging from 1.70 to 0.70 metres, and his experiment was carried out at the Palaska Training Centre on the island of Salamina on November 6, 1973. Sakkas placed his 70 lenses in a circle, and succeeded in focusing the sun's rays on a small boat, built in the same way as Roman craft and equipped with the same sort of materials, lying 55 metres away. In less than three minutes the boat was ablaze. Sakkas' experiment was reported around the world, and caused quite a stir. Three previous tests had also produced satisfactory results, and together they confirmed that Archimedes did indeed set fire to Roman ships. While we do not know the full effect of this conflagration, the psychological impact on the enemy must have been terrible. That, of course, is why his feat acquired the status of a legend and is still talked about to this day.

ARCHIMEDES: THE ENDLESS SCREW

Another of Archimedes' major inventions was the endless screw, a simple but important device that has helped solve many ordinary everyday problems. Although from time to time the attribution of the invention of this device to Archimedes has been challenged, in the end it has been generally accepted. The philological evidence, indeed, is overwhelming: Archimedes was indisputably the inventor and constructor of the screw that bears his name. Moschion declares that it was invented by Archimedes, and that with this device he towed the huge luxury vessel "Syracusa". The discovery is attributed to Archimedes by Agatharchides of Cnidus, as attested by Diodorus Siculus, and by Posidonius of Syria, as attested by Strabo, who indeed adds further information about it. The screw, he tells us, was the result of a visit Archimedes paid to Egypt, at the invitation of Ptolemy II Philadelphus. While there, he devised and built the screw to help the peasants draw water from the Nile to irrigate their fields.

Very soon the screw was in use all around the Mediterranean basin and the Middle East, and it continued to be used in the same form for centuries: in fact, in some parts of North Africa, including Egypt, it is still employed. The spread of this device was facilitated by the Pax Romana, when information, knowledge and experience could circulate freely and unimpeded throughout the Empire. The same was true after the Arab conquest of North Africa and part of Europe, which explains the presence and use of the screw in many parts of Europe right up to the Middle Ages.

ARCHIMEDES: THE WATER CLOCK

Time-keeping was a major problem for the people of the ancient world. The Greeks divided the day into 12 parts, and measured the length of the daytime hour. But how to measure time at night, especially when the relative lengths of day and right changed with the seasons? This problem was solved by Archimedes' invention of the water clock, which permitted accurate calculation of the time both by day and by night. (The margin of error of his clock was just 2 minutes either way!). This complex mechanism was four metres tall, could sound the hours, was ornately decorated and was driven by water-power.

Archimedes' treatise on clocks survived only in the Arabic translation made in about the 9th century, manuscripts of which still exist in libraries in London, Paris and Oxford. From these a German translation was made, which was published in the Acts of the Academy of Natural Sciences of Al-Saal in 1918. This text was translated into Greek by Evanghelos Stamatis, and thus this lost work by Archimedes once again exists in its original language. The translator stresses the fact that Archimedes' clock was the first water-powered clock in history, and that it was the fore-runner both of Hero's automata and of the mechanical figures ornamenting the throne of Byzantium.

Archimedes' water clock has recently been reconstructed, to a smaller scale, by Ioannis Sakkas on the basis of the description given by Stamatis. Finally, it should be noted that contemporary researchers into the history and development of clocks, like Derek de Sola Price, believe that the inventor of the specific clock may not have been Archimedes, because the Arabic manuscript of the treatise also mentions Philo and Hero. He does not suggest any other engineer of the period as capable of constructing such a device.