Archimedes

If the ordinary person were asked to say off-hand what he knew of Archimedes, he would probably, at the most, be able to quote one or other of the well-known stories about him: how, after discovering the solution of some problem in the bath, he was so overjoyed that he ran naked to his house, shouting ε?ρηκα, ε?ρηκα (or, as we might say, "I've got it, I've got it"); or how he said "Give me a place to stand on and I will move the earth"; or again how he was killed, at the capture of Syracuse in the Second Punic War, by a Roman soldier who resented being told to get away from a diagram drawn on

the ground which he was studying.

And it is to be feared that few who are not experts in the history of mathematics have any acquaintance with the details of the original discoveries in mathematics of the greatest mathematician of antiquity, perhaps the greatest mathematical genius that the world has ever seen.

History and tradition know Archimedes almost exclusively as the inventor of a number of ingenious mechanical appliances, things which naturally appeal more to the popular imagination than the subtleties of pure mathematics.

Almost all that is told of Archimedes reaches us through the accounts by Polybius and Plutarch of the siege of Syracuse by Marcellus. He perished in the sack of that city in 212 B.C., and, as he was then an old man (perhaps 75 years old), he must have been born about 287 B.C. He was the son of Phidias, an astronomer, and was a friend and kinsman of King Hieron of Syracuse and his son Gelon. He spent some time at Alexandria studying with the successors of Euclid (Euclid who flourished about 300 B.C. was then no longer living). It was doubtless at Alexandria that he made the acquaintance of Conon of Samos, whom he admired as a mathematician and cherished as a friend, as well as of Eratosthenes; to the former, and to the latter during his early period he was in the habit of communicating his discoveries before their publication. It was also probably in Egypt that he invented the water-screw known by his name, the immediate purpose being the drawing of water for irrigating fields.

After his return to Syracuse he lived a life entirely devoted to mathematical research. Incidentally he became famous through his clever mechanical inventions. These things were, however, in his case the "diversions of geometry at play," and he attached no importance to them. In the words of Plutarch, "he possessed so lofty a spirit, so profound a soul, and such a wealth of scientific knowledge that, although these inventions had won for him the renown of more than human sagacity, yet he would not consent to leave behind him any written work on such subjects, but, regarding as ignoble and sordid the business of mechanics and every sort of art which is directed to practical utility, he placed his whole ambition in those speculations in the beauty and subtlety of which there is no admixture of the common needs of life".

During the siege of Syracuse Archimedes contrived all sorts of engines against the Roman besiegers. There were catapults so ingeniously constructed as to be equally serviceable at long or short range, and machines for discharging showers of missiles through holes made in the walls. Other machines consisted of long movable poles projecting beyond the walls; some of these dropped heavy weights upon the enemy's ships and on the constructions which they called sambuca, from their resemblance to a musical instrument of that name, and which consisted of a protected ladder with one end resting on two quinqueremes lashed together side by side as base, and capable of being raised by a windlass; others were fitted with an iron hand or a beak like that of a crane, which grappled the prows of ships, then lifted them into the air and let them fall again. Marcellus is said to have derided his own engineers and artificers with the words, "Shall we not make an end of fighting with this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives our sambuca off ignominiously with cudgel-blows, and, by the multitude of missiles that he hurls at us all at once, outdoes the hundred-handed giants of mythology?" But the exhortation had no effect, the Romans being in such abject terror that, "if they did but see a piece of rope or wood projecting above the wall they would cry 'there it is,' declaring that Archimedes was setting some engine in motion against them, and would turn their backs and run away, insomuch that Marcellus desisted from all fighting and assault, putting all his hope in a long siege".

Archimedes died, as he had lived, absorbed in mathematical contemplation. The accounts of the circumstances of his death differ in some details. Plutarch gives more than one version in the following passage: "Marcellus was most of all afflicted at the death of Archimedes, for, as fate would have it, he was intent on working out some problem with a diagram, and, his mind and his eyes being alike fixed on his investigation, he never noticed the incursion of the Romans nor the capture of the city. And when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; whereat the soldier was so enraged that he drew his sword and slew him. Others say that the Roman ran up to him with a drawn sword, threatening to kill him; and, when Archimedes saw him, he begged him earnestly to wait a little while in order that he might not leave his problem incomplete and unsolved, but the other took no notice and killed him. Again, there is a third account to the effect that, as he was carrying to Marcellus some of his mathematical instruments, sundials, spheres, and angles adjusted to the apparent size of the sun to the sight, some soldiers met him and, being under the impression that he carried gold in the vessel, killed him." The most picturesque version of the story is that which represents him as saying to a Roman soldier who came too close, "Stand away, fellow, from my diagram," whereat the man was so enraged that he killed him.

Archimedes is said to have requested his friends and relatives to place upon his tomb a representation of a cylinder circumscribing a sphere within it, together with an inscription giving the ratio (3/2) which the cylinder bears to the sphere; from which we may infer that he himself regarded the discovery of this ratio as his greatest achievement. Cicero, when quaestor in Sicily, found the tomb in a neglected state and restored it. In modern times not the slightest trace of it has been found.

Beyond the above particulars of the life of Archimedes, we have nothing but a number of stories which, if perhaps not literally accurate, yet help us to a conception of the personality of the man which we would not willingly have altered. Thus, in illustration of his entire preoccupation by his abstract studies, we are told that he would forget all about his food and such necessities of life, and would be drawing geometrical figures in the ashes of the fire, or, when anointing himself, in the oil on his body. Of the same kind is the story mentioned above, that, having discovered while in a bath the solution of the question referred to him by Hieron as to whether a certain crown supposed to have been made of gold did not in fact contain a certain proportion of silver, he ran naked through the street to his home shouting ε?ρηκα, ε?ρηκα.

It was in connexion with his discovery of the solution of the problem To move a given weight by a given force that Archimedes uttered the famous saying, "Give me a place to stand on, and I can move the earth" (δ?? μοι πο? στ? κα? κιν? τ?ν γ?ν, or in his broad Doric, as one version has it, π? β? κα? κιν? τ?ν γ?ν). Plutarch represents him as declaring to Hieron that any given weight could be moved by a given force, and boasting, in reliance on the cogency of his demonstration, that, if he were given another earth, he would cross over to it and move this one. "And when Hieron was struck with amazement and asked him to reduce the problem to practice and to show him some great weight moved by a small force, he fixed on a ship of burden with three masts from the king's arsenal which had only been drawn up by the great labour of many men; and loading her with many passengers and a full freight, sitting himself the while afar off, with no great effort but quietly setting in motion with his hand a compound pulley, he drew the ship towards him smoothly and safely as if she were moving through the sea." Hieron, we are told elsewhere, was so much astonished that he declared that, from that day forth, Archimedes's word was to be accepted on every subject! Another version of the story describes the machine used as a helix; this term must be supposed to refer to a screw in the shape of a cylindrical helix turned by a handle and acting on a cog-wheel with oblique teeth fitting on the screw.

Another invention was that of a sphere constructed so as to imitate the motions of the sun, the moon, and the five planets in the heavens. Cicero actually saw this contrivance, and he gives a description of it, stating that it represented the periods of the moon and the apparent motion of the sun with such accuracy that it would even (over a short period) show the eclipses of the sun and moon. It may have been moved by water, for Pappus speaks in one place of "those who understand the making of spheres and produce a model of the heavens by means of the regular circular motion of water". In any case it is certain that Archimedes was much occupied with astronomy. Livy calls him "unicus spectator caeli siderumque". Hipparchus says, "From these observations it is clear that the differences in the years are altogether small, but, as to the solstices, I almost think that both I and Archimedes have erred to the extent of a quarter of a day both in observation and in the deduction therefrom." It appears, therefore, that Archimedes had considered the question of the length of the year. Macrobius says that he discovered the distances of the planets. Archimedes himself describes in the Sandreckoner the apparatus by which he measured the apparent diameter of the sun, i.e. the angle subtended by it at the eye.

The story that he set the Roman ships on fire by an arrangement of burning-glasses or concave mirrors is not found in any authority earlier than Lucian (second century A.D.); but there is no improbability in the idea that he discovered some form of burning-mirror, e.g. a paraboloid of revolution, which would reflect to one point all rays falling on its concave surface in a direction parallel to its axis.

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