II. ARCHIMEDES

The greatest of ancient scientists was born at Syracuse about 287 B.C., son of the astronomer Pheidias, and apparently cousin to Hieron II, the most enlightened ruler of his time. Like many other Hellenistic Greeks who were interested in science and could afford the expense, Archimedes went to Alexandria; there he studied under the successors of Euclid, and derived an inspiration for mathematics that gave him two boons—an absorbed life and a sudden death. Returning to Syracuse, he devoted himself monastically to every branch of mathematical science. Often, like Newton, he neglected food and drink, or the care of his body, in order to pursue the consequences of a new theorem, or to draw figures in the oil on his body, the ashes on the hearth, or the sand with which Greek geometers were wont to strew their floors.² He was not without humor: in what he considered his best book, The Sphere and the Cylinder, he deliberately inserted false propositions (so we are assured), partly to play a joke upon the friends to whom he sent the manuscript, partly to ensnare poachers who liked to appropriate other men’s thoughts.³ Sometimes he amused himself with puzzles that brought him to the verge of inventing algebra, like the famous Cattle Problem that so beguiled Lessing;⁴ sometimes he made strange mechanisms to study the principles on which they operated. But his perennial interest and delight lay in pure science conceived as a key to the understanding of the universe rather than as a tool of practical construction or expanding wealth. He wrote not for pupils but for professional scholars, communicating to them in pithy monographs the abstruse conclusions of his research. All later antiquity was fascinated by the originality, depth, and clarity of these treatises. “It is not possible,” said Plutarch, three centuries later, “to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius; others think that these easy and unlabored pages were the result of incredible effort and toil.’⁵

Ten of Archimedes’ works survive, after many adventures in Europe and Arabia. (1) The Method explains to Eratosthenes, with whom he had formed a friendship in Alexandria, how mechanical experiments can extend geometrical knowledge. This essay ended the ruler-and-compass reign of Plato and opened the door to experimental methods; even so it reveals the different mood of ancient and modern science: the one tolerated practice for the sake of theoretical understanding, the other tolerates theory for the sake of possible practical results. (2) A Collection of Lemmasdiscusses fifteen “choices,” or alternative hypotheses, in plane geometry. (3) The Measurement of a Circle arrives at a value between  and  for π—the ratio of the circumference to the diameter of a circle—and “squares the circle” by showing, through the method of exhaustion, that the area of a circle equals that of a right-angled triangle whose perpendicular equals the radius, and whose base equals the circumference, of the circle. (4) The Quadrature of the Parabola studies, by a form of integral calculus, the area cut off from a parabola by a chord, and the problem of finding the area of an ellipse. (5) On Spirals defines a spiral as the figure made by a point moving from a fixed point at a uniform rate along a straight line which is revolving in a plane at a uniform rate about the same fixed point; and finds the area enclosed by a spiral curve and two radii vectores by methods approximating differential calculus. (6) The Sphere and the Cylinder seeks formulas for the volume and surface area of a pyramid, a cone, a cylinder, and a sphere. (7) On Conoids and Spheroids studies the solids generated by the revolution of conic sections about their axes. (8) The Sand-Reckoner passes from geometry to arithmetic, almost to logarithms, by suggesting that large numbers may be represented by multiples, or “orders,” of 10,000; by this method Archimedes expresses the number of grains of sand which would be needed to fill the universe—assuming, he genially adds, that the universe has a reasonable size. His conclusion, which anyone may verify for himself, is that the world contains not more than sixty-three “ten-million units of the eighth order of numbers”—or, as we should put it, 10⁶⁸. References to lost works of Archimedes indicate that he had also discovered a way of finding the square root of nonsquare numbers. (9) On Plane Equilibriums applies geometry to mechanics, studies the center of gravity of various bodily configurations, and achieves the oldest extant formulation of scientific statics. (10) On Floating Bodies founds hydrostatics by arriving at mathematical formulas for the position of equilibrium of a floating body. The work begins with the then startling thesis that the surface of any liquid body at rest and in equilibrium is spherical, and that the sphere has the same center as the earth.

Perhaps Archimedes was led to the study of hydrostatics by an incident almost as famous as Newton’s apple. King Hieron had given to a Syracusan Cellini some gold to be formed into a crown. When the crown was delivered it weighed as much as the gold; but some doubt arose whether the artist had made up part of the weight by using silver, keeping the saved gold for himself. Hieron turned over to Archimedes his suspicion and the crown, presumably stipulating that the one should be resolved without injuring the other. For weeks Archimedes puzzled over the problem. One day, as he stepped into a tub at the public baths, he noticed that the water overflowed according to the depth of his immersion, and that his body appeared to weigh—or press downward—less, the more it was submerged. His curious mind, exploring and utilizing every experience, suddenly formulated the “principle of Archimedes”—that a floating body loses in weight an amount equal to the weight of the water which it displaces. Surmising that asubmerged body would displace water according to its volume, and perceiving that this principle offered a test for the crown, Archimedes (if we may believe the staid Vitruvius) dashed out naked into the street and rushed to his dwelling, crying out “Eureka! eureka!”—I have found it! I have found it! Home, he soon discovered that a given weight of silver, since it had more volume per weight than gold, displaced more water, when immersed, than an equal weight of gold. He observed also that the submerged crown displaced more water than a quantity of gold equaling the crown in weight. He concluded that the crown had been alloyed with some metal less dense than gold. By replacing gold with silver in the gold weight which he was using for comparison, until the compound displaced as much water as the crown, Archimedes was able to say just how much silver had been used in the crown, and how much gold had been stolen.

That he had satisfied the curiosity of the King did not mean so much to him as that he had discovered the law of floating bodies, and a method for measuring specific gravity. He made a planetarium representing the sun, the earth, the moon, and the five planets then known (Saturn, Jupiter, Mars, Venus, and Mercury), and so arranging them that by turning a crank one could set all these bodies in motions differing in direction and speed;⁶ but he probably agreed with Plato that the laws that govern the movements of the heavens are more beautiful than the stars.* In a lost treatise partly preserved in summaries, Archimedes so accurately formulated the laws of the lever and the balance that no advance was made upon his work until A.D. 1586. “Commensurable magnitudes,” said Proposition VI, “will balance at distances inversely proportional to their gravities”⁸—a useful truth whose brilliant simplification of complex relationships moves the soul of a scientist as the Hermes of Praxiteles moves the artist. Almost intoxicated with the vision of power which he saw in the lever and the pulley, Archimedes announced that if he had a fixed fulcrum to work with he could move anything: “Pa bo, kai tan gan kino” he is reported to have said, in the Doric dialect of Syracuse: “Give me a place to stand on, and I will move the earth.”⁹ Hieron challenged him to do as well as say, and pointed to the difficulty which his men were experiencing in beaching a large ship in the royal fleet. Archimedes arranged a series of cogs and pulleys in such wise that he alone, sitting at one end of the mechanism, was able to draw the fully loaded vessel out of the water onto the land.¹⁰

Delighted with this demonstration, the King asked Archimedes to design some engines of war. It was characteristic of the two men that Archimedes, having designed them, forgot them, and that Hieron, loving peace, never used them. Archimedes, says Plutarch,

possessed so high a spirit, so profound a soul, and such treasures of scientific knowledge, that though these inventions had now obtained for him the renown of more than human sagacity, he yet would not deign to leave behind him any writing on such subjects; but, repudiating as sordid and ignoble . . . 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—studies whose superiority to all others is unquestioned, and in which the only doubt can be whether the beauty and grandeur of the subjects examined, or the precision and cogency of the methods and means of proof, most deserve our admiration.¹¹

But when Hieron was dead Syracuse became embroiled with Rome, and the doughty Marcellus assailed it by land and sea. Though Archimedes was now (212) a man of seventy-five, he superintended the defense on both fronts. Behind the walls that protected the harbor he set up catapults able to hurl heavy stones to a considerable distance; their rain of projectiles was so devastating that Marcellus retreated until he could advance by night. But when the ships were seen near the shore the sailors were harassed by bowmen who shot at them through the holes that Archimedes’ aides had pierced in the wall. Moreover, the inventor had arranged within the walls great cranes which, when the Roman vessels came within reach, were turned by cranks and pulleys so as to drop upon the ships heavy weights of stone or lead that sank many of them. Other cranes, armed with gigantic hooks, grasped vessels, lifted them into the air, dashed them against the rocks, or plunged them end-foremost into the sea.*¹² Marcellus withdrew his fleet, and put his hopes in an attack by land. But Archimedes bombarded the troops with large stones thrown by catapults to such effect that the Romans fled, saying that they were being opposed by gods; and they refused to advance again.¹⁴ “Such a great and marvelous thing,” comments Polybius, “does the genius of one man show itself to be when properly applied. The Romans, strong both by sea and by land, had every hope of capturing the town at once if one old man of Syracuse were removed; as long as he was present they did not venture to attack.”¹⁵

Abandoning the idea of taking Syracuse by storm, Marcellus resigned himself to a slow blockade. After a siege of eight months the starving city surrendered. In the slaughter and pillage that followed Marcellus gave orders that Archimedes should not be injured. During the sack a Roman soldier came upon an aged Syracusan absorbed in studying figures that he had traced in the sand. The Roman commanded him to present himself at once to Marcellus. Archimedes refused to go until he had worked out his problem; he “earnestly besought the soldier,” says Plutarch, “to wait a little while, that he might not leave what he was at work upon inconclusive and imperfect, but the soldier, nothing moved by this entreaty, instantly killed him.”¹⁶ When Marcellus heard of it he mourned, and did everything in his power to console the relatives of the dead man.¹⁷ The Roman general erected to his memory a handsome tomb, on which was engraved, in accordance with the mathematician’s expressed wish, a sphere within a cylinder; to have found formulas for the area and volume of these figures was, in Archimedes’ view, the supreme achievement of his life. He was not far wrong; for to add one significant proposition to geometry is of greater value to humanity than to besiege or defend a city. We must rank Archimedes with Newton, and credit him with “a sum of mathematical achievement unsurpassed by any one man in the world’s history.”¹⁸

But for the abundance and cheapness of slaves Archimedes might have been the head of a veritable Industrial Revolution. A treatise on Mechanical Problems wrongly attributed to Aristotle, and a Treatise on Weights wrongly ascribed to Euclid, had laid down certain elementary principles of statics and dynamics a century before Archimedes. Strato of Lampsacus, who succeeded Theophrastus as head of the Lyceum, turned his deterministic materialism to physics, and (about 280) formulated the doctrine that “nature abhors a vacuum.”¹⁹ When he added that “a vacuum can be created by artificial means,” he opened the way to a thousand inventions. Ctesibius of Alexandria (ca. 200) studied the physics of siphons (which had been used in Egypt as far back as 1500 B.C.), and developed the force pump, the hydraulic organ, and the hydraulic clock. Archimedes probably improved—and unwittingly gave his name to—the ancient Egyptian water screw, which literally made water flow uphill.²⁰ Philon of Byzantium, about 150, invented pneumatic machines, and various engines of war.²¹ The steam engine of Heron of Alexandria, which came after the Roman conquest of Greece, brought this period of mechanical development to a climax and close. The philosophical tradition was too strong; Greek thought went back to theory, and Greek industry contented itself with slaves. The Greeks were acquainted with the magnet, and the electrical properties of amber, but they saw no industrial possibilities in these curious phenomena. Antiquity unconsciously decided that it was not worth while to be modern.