Archimedes
One of the Greatest Greek Mathematicians of Antiquity

287 BC - 212 BC

Give me a place to stand and rest my lever on, and I can move the Earth.
Archimedes

Archimedes, who combined a genius for mathematics with a physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics.
Alfred North Whitehead

Archimedes was the finest scientist and mathematician of the ancient world but little firmly known of his life, although legends exist. He is known to have used experiments to test his theories, which he then expressed mathematically.
The Cambridge Dictionary of Scientists (Second Edition) 2002

It is no exaggeration to describe Archimedes as the creator of the science of mechanics. Naturally before his time many isolated facts had been discovered, but it was only with him that mechanics became a unified body of theory capable of yielding new and unexpected practical applications.
A Dictionary of Scientists, Oxford University Press, 1999.

The word “Eureka” is a Greek word (heureka) for “I have found”. Today it means “to find”, “to discover” and used as an exclamation inserted into an utterance without grammatical connection to it. As a noun it means an important discovery. The word has found place in English dictionary because of exclamation supposedly uttered by Archimedes when he discovered a way to determine the purity of gold by applying the principle of relative density or specific gravity. Archimedes is regarded as one of the greatest “working scientists” and mathematicians of the antiquity. His approximation of (?) between 3×1/2 and 3×10/71 was the most accurate of his time and he devised a new way to approximate square root. He had anticipated the invention of differential calculus as he devised ways to approximate the slope of tangent lines of his figures. Archimedes revolutionized mechanics, founded the scientific discipline called hydrostatics and established the precise study of more complex solids. He invented an early form of calculus and developed an advanced understanding of numerology. Archimedes was as much an applied mathematician as a pure mathematician.

In his own time he used to be known as “the wise one”, “the master” and “the great geometer.” The fame of Archimedes in his own time was mainly due to his proximity to King Hieron II, the then ruler of Syracuse and his son Gelon. It is believed that Archimedes was related to the monarch. He was also the tutor of Gelon. It seems Archimedes made a hobby out of solving the king’s most complicated problems to the utter amazement of the sovereign. Today Archimedes is best known for the following:

i. For his discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder.

ii. For his formulation of a hydrostatic principle known as Archimedes’ principle.

iii. For his invention of the Archimedes’ screw—a device for raising water by means of a rotating broad-threaded screw or spiral bent tube within an inclined hollow cylinder.

Archimedes designed all sorts of pumps and the Archmedean water screw is still in use in some parts of the world. The story of Archimedes jumping from the bath naked is usually linked with his discovery of the principle of hydrostatics. One really does not know whether this incident was actually responsible for formulating his hydrostatic principle. The story has several sources and we do not know which is the correct one. The description of the incident by the Roman architect Vitruvius is considered as the most reliable one by many. But then we should remember that Vitruvius wrote two centuries after the event took place. According to the version of the story given by Vitruvius, King Hieron decided to get a gold wreath prepared for dedicating it to the gods. (Some version says it was a crown). This way he had decided to celebrate his continuing good fortune. The king gave a lump of gold to a local artist for the purpose. However, when the artist returned with the completed gold wreath the king felt that the artist did not use all the gold given to him. The weight of the gold wreath was same as that of the gold given to him by the king. The king thought that the artist had mixed less expensive silver with gold. The king asked Archimedes to look into the problem. Archimedes also did not have a ready-made answer for the king but he promised to think about it. The legend says that one morning, several days later, Archimedes was still thinking about the problem when lowered himself in a bath. While doing so Archimedes noticed that some of the water was displaced by his body and flowed over the edge of the tub. In a flash he understood how to solve the problem posed by the king. Archimedes solved the problem by grasping the concept of relative density. He was so excited at his sudden discovery that he leapt from the bath and started running naked through the streets while shouting “ureka! Eureka!” (I have found it). He requested the king to make him available a lump of gold weighing the same as the wereath. After receiving the piece of gold, he immersed it in a tub of water filled to the brim and measured the water displaced by it. In the same way he immersed the wreath and measured the overflow. The water displaced by the wreath was more than the gold. In this way Archimedes found that the wreath was mixed with other metal of lower density.

Archimedes discovered the laws of levers and used pulleys. After the discovery of the laws of the levers, he boasted “Give me a place to stand and rest my lever on, and I can move the Earth.” As such it was not possible to challenge the statement directly. So he was asked to move a ship which had required a large group of labourers to put into position. Archimedes moved the ship easily by using a compound pulley system.

Archimedes was born in 287 B.C. in a noble family of Syracuse, Cicily. Syracuse was the most powerful city-state in Cicily. Besides being an aristocrat, Archimedes’ father was an astronomer and a mathematician. This information comes from one of Archimedes’ works, The Sandreckoner. Most of the information about Archimedes’ life comes from the writing of Plutarch (in Greek Plutarchos c.46-c.120 AD), who lived three centuries later. Plutarch’s best known work Parallel Lives compares eminent Greeks with their Roman counterparts. Plutrach’s Lives concentrate on the moral character of each subject rather than on the political events of the time. As a result a minor incident or anecdote will acquire a greater importance in the narrative than it would in a standard history or biography. Plutarch in his Life of Alexander wrote: “I am writing biography, not history, and the truth is that the most brilliant exploits often tell us nothing of the virtues or vices of the men who reformed them, while on the other hand a chance remark or a joke may reveal far more of a man’s character than the mere feat or winning battles in which thousands fall, or of marshalling great armies or laying siege to the cities.” In Plutarch’s writings Archimedes finds mention as a mere insertion in the biography of the Roman General, Marcus Claudius Marcellus (c. 268-c.208 B.C.), who was known as the Sword of Rome. Of course, there were other sources including Archimedes’ own stray comments here and there in his prefaces to the treatises he wrote.

Archimedes studied at Alexandria in Egypt. The city of Alexandria was founded in B.C. by the Alexander the Great. It is at Alexandria that Alexander the Great was buried in a resplendent gold coffin in BC. Its location is not known today. By the early 2nd century BC Alexandria was emerging as the greatest centre of learning in the Mediterranean world. In this regard it surpassed even Athens. The famous library of Alexandria attracted scholars from all over Hellenistic world. Its collection of manuscripts included Aristotle’s extensive collection—the greatest private collection of the Greek era. Euclid worked at Alexandria. However, Euclid probably had died before Archimedes arrived at Alexandria. But then Archimedes would certainly have read Euclid’s geometry textbook Elements. This famous book laid the foundation of geometry. It was also likely that Archimedes had studied with one Euclid’s pupils. At Alexandria,

Archimedes befriended two fellow students with whom he was to remain in correspondence throughout his life. These two friends—Conon of Samos and Eratosthenes of Cyrene were fine mathematicians.

According to one legend Archimedes visited Spain after leaving Alexandria. A story, mentioned by Leonardo da Vinci in his notebooks, narrates that Archimedes acted as a military engineer for King Ecliderides of Cliodastri. Diodorus, a historian of Cicily and who lived in the first century BC, speaks of Archimedes’ Screw being used for pumping water from the silver mines of Rio Tinto in southern Spain. According to Diodorus, Archimedes invented his screw just for this purpose. Some other legends speak of Archimedes returning to Egypt for a second time. During his second trip to Egypt he was said to have employed on the largescale irrigation works as a measure to control the flooding of the Nile Delta. And as per these reports Archimedes
screw was invented during this time.

Among the treatises of Archimedes which have survived are: 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. Another of his work titled The Method has been found in a tenth century manuscript discovered by J L Heiberg, a Professor of Philology at the University of Copenhagen. There are references to Archimedes’ other works which have been lost. The surviving works of Archimedes give a unique insight the workings of him. Most of these works are easy to follow even for nonmathematicians. All of his known works were of a theoretical character. He left no written work on his practical inventions. As he had a very low opinion about these inventions he did not consider it worth writing about them. Thus Plutarch wrote: “Archimedes possessed so high a spirit, so profound a soul, and such treasure of scientific knowledge, that though these inventions had now obtained him the renown of more than human sagacity, yet he did not leave behind him any commentary or writing on such subjects…”. However, it is certain that his interest in mechanics deeply influenced his mathematical thinking. He not only wrote works on theoretical mechanics and hydrostatics but he also used mechanical reasoning as a heuristic device for the discovery of new mathematical theorems as evident in his Method Concerning Mechanical Theorems. He published his works in the form of correspondence with important mathematicians of his time. The only manuscript that Archimedes wrote on practical matters was On Spheremaking. The manuscript, which is now lost, is referred to by the Greek mathematician Pappus of Alexandria, who lived in the 4th century AD.

Archimedes used to remain engrossed in some problem or the other all the time. He was most interested in geometry. Even while taking bath (which used to be a rare occurrence for Archimedes), he used to draw geometrical figures even on his naked body. Thus Plutarch wrote: “oftimes 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 savours, with his fingers 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 invented many machines, which were used as engines of war. Among his war machines were enormous mirrors to focus the Sun’s rays and set fire to the Roman ships, and a variety of catapults. His huge catapults hurled 500 pound boulders at the enemy soldiers. He played an important role in the defense of Syracuse against the siege laid by the Romans in 213 BC by effectively deploying his war machines. His single handed effort long delayed the capture of the city. This is how Plutarch described the impact of Archimedes’ war machines. “…when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence; against which no man could stand; for they knocked down those upon whom they fell in heaps, breaking all their ranks and files. In the meantime huge poles thrust out from the walls over ships and sunk some by great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane’s beak and when, they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by the engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashes against the rock, or let fall.” Syracuse was eventually captured by the Roman General Marcellus in the autumn of 212 or spring of 211 BC.

It is believed that Archimedes created two spheres, which were brought to Rome by Marcellus. Among these two spheres, one was a solid one on which were engraved or painted the stars and constellations. It should be mentioned that Archimedes was not the first to construct such a celestial globe. Perhaps the Greek geometers Thales and Eudoxos first constructed such globes. Marcellus placed this sphere in the Temple of Virtue.

The second sphere was an original and ingenious work. It was a miniature planetarium—a mechanical model showing the motions of the Sun, Moon, and planets as viewed from the Earth. Archimedes’ planetarium was an intricate device. While constructing the planetarium, Archimedes accepted the Earth-centred view of the universe—the universe, with the Earth at its centre. Archimedes’ device was capable of tracing the motions of the Sun, Moon and planets about the Earth with reference to the spheres of fixed stars during the course of the day. With its help the successive phases of the Moon and the lunar eclipses could also be illustrated. Cicero (106-43 BC), the Roman statesman, philosopher, and a great orator, was very much impressed by this ingenious device by Archimedes. Cicero thought that Archimedes was “endowed with greater genius that one would imagine it possible for a human being to possess” to able to construct such a device. Archimedes’ planetarium has been quoted by many ancient writers in prose as well as in verse. Many considered it as one of the first Christian proofs of existence of God or a divine creator. The logic was very simple for such an argument—just as Archimedes’s planetarium required a creator, there must be a creator of greater intelligence to be capable of producing the cosmos—the object which the human intelligence attempted to imitate.

Archimedes was killed by a Roman soldier when the City of Syracuse was taken by the Romans. The year was 212 B.C. It is said that Marceless, the Roman General in charge had issued orders to his soldiers not to harm Archimedes and to treat him with respect. The legend goes to state that Archimedes was found while engaged in drawing a geometrical diagram in the sand, the city burning around him. Archimedes was unaware of the taking of the city by the Romans. There are many versions of the story of his killing. Plutarch recounts three versions which had come down to him.

The first version says: “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 speculation, 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 declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through.”

The second version: “…a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him,” The third version: “…as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him.” The Romans 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 Archimedes considered his greatest achievement. Cicero, while describing how he searched for Archimedes tomb wrote: “…and found it enclosed all around and covered with brables and thickets; for I remembered certain doggerel lines inscribed, as I had heard, upon his tomb, which stated that a sphere along with a cylinder had been put on top of his grave. Accordingly, after taking a good look around…, I noticed a small column arising a little above the bushes, on which there was a figure of a sphere and a cylinder…Slaves were sent in with sickles…and when a passage to the place was opened we approached the pedestal in front of us; the epigram was traceable with about half of lines legible, as the latter portion was worn away.”

For Further Reading

1. Archimedes & The Fulcrum by Paul Strathern. London: Arrow Books, 1998.
2. The History of Science: From the Ancient Greeks to the Scientific Revolutions by Ray Spangenburg and Diane K. Moser. Hyderabad: Universities Press (India) Limited, 1999.
3. The Cambridge Dictionary of Scientists (Second Edition) by David, Ian, John & Margaret Millar, Cambridge: Cambridge University Press, 2002.
4. A Dictionary of Scientists Oxford: Oxford University Press, 1999.
5. The Macmillan Encyclopaedia. London: Macmillan London Limmited, 1981.
6. Chambers Biographical Dictionary (Centenary Edition). Edinburgh:Chambers Harrap Publishers Ltd.
7. http://scidiv.bcc.ctc.edu/Math/Archimedes.html
8. http://www.crystallinks.com/archimedes.html