Analysis: Calculating Areas and Volumes

Abstract

In 216 B.C.E., the Sicilian city of Syracuse made the mistake of allying itself with Carthage during the second Punic war, and thus was attacked by Rome, portending what would ultimately happen to the entire classical Greek world. During a long siege, soldiers of the Roman general Marcellus were terrified by ingenious war machines defending the city, invented by the Syracusan Archimedes, greatest mathematician of the ancient world, born in 287 B.C.E. These included catapults to hurl great stones, as well as ropes, pulleys, and hooks to raise and smash Marcellus’s ships, and perhaps even burning mirrors setting fire to their sails. Finally, though, probably through betrayal, Roman soldiers entered the city in 212 B.C.E., with orders from Marcellus to capture Archimedes alive. Plutarch relates that “as fate would have it, he was intent on working out some problem with a diagram and, having fixed his mind and his eyes alike 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” [93, p. 97]. Despite the great success of Archimedes’ military engineering inventions, Plutarch says that “He would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life” [93, p. 100]. Perhaps the best indication of what Archimedes truly loved most is his request that his tombstone include a cylinder circumscribing a sphere, accompanied by the inscription of his amazing theorem that the sphere is exactly two-thirds of the circumscribing cylinder in both surface area and volume!