The Life of Greece

  The gods were not the chief interest of the sculptors who brightened the evening of Greek art. These men looked upon Olympus as a quarry of subjects, and no more. When that quarry had been worn down by repetition they turned to the earth and took delight in representing the wisdom and loveliness, the strangeness and absurdities of human life. They carved or cast impressive heads of Homer, Euripides, and Socrates. They made a number of smooth and delicate Hermaphrodites, whose equivocal beauty arrests the eye in the Archeological Museum at Constantinople, or the Borghese Gallery in Rome, or the Louvre. Children offered refreshingly natural poses, like the boy who removes a thorn from his foot, and another who struggles with a goose,* and—finest of this class—the trustful Praying Youth attributed to Lysippus’ pupil Boëthus.† Or the sculptors went to the woods and depicted sylvan sprites like the Barberini Faun of Munich, or hilarious satyrs like the Drunken Silenus of the Naples Museum. And here and there, with jolly frequency, they inserted among their figures the rosy cheeks and impish pranks of the god of love.

  IV. COMMENTARY

  This sudden irruption of humor into the once formal sanctuaries of Greek sculpture is a distinctive mark of Hellenistic art. Every museum has preserved from the ruins of the age some laughing faun, some singing Pan, some rioting Bacchus, some urchin serving as a fountain with alarming indecency. Perhaps the return of Greek art to Asia restored to it the variety, feeling, and warmth which it had almost lost in its classic subordination to religion and the state. Nature, which had been adored, began now to be enjoyed. Not that classic moderation disappeared: the Youth of Subiaco in the Museo delle Terme, the Sleeping Ariadne of the Vatican, the Sitting Maiden of the Palace of the Conservatori continue the delicate tradition of Praxiteles; and in Athens, throughout this period, many sculptors fought the “modernistic” tendencies of their time by deliberately going back to fourth-and fifth-century styles, even, now and then, to the archaic dignity of the sixth. But the spirit of the age was for experiment, individualism, naturalism, and realism, with a strong countercurrent toward imagination, idealism, sentiment, and dramatic effect. Artists carefully followed the progress of anatomy, and worked more from models in studios; sculptors carved their figures to be seen not only from in front, but from all sides. They used novel materials—crystal, chalcedony, topaz, glass, dark basalt, black marble, porphyry—to imitate the pigment of Negroes or the ruddy faces of satyrs illumined with wine.

  Their fertility of invention equaled their mastery of technique. They were tired of repeating types; they anticipated Ruskin’s criticism,* and were resolved to show the reality and individuality of the persons and objects they portrayed. They no longer confined themselves to the perfect and the beautiful, to athletes, heroes, and gods; they made genre pictures or terra cottas of workingmen, fishermen, musicians, market men, jockeys, eunuchs; they sought unhackneyed subjects in children and peasants, in characterful features like those of Socrates, in bitter old men like Demosthenes, in powerful, almost brutal faces like that of Euthydemus the Greco-Bactrian king, in desolate derelicts like the Old Market Woman of the Metropolitan Museum in New York; they recognized and relished the variety and complexity of life. They did not hesitate to be sensual; they were not parents anxious about the chastity of their daughters, nor philosophers disturbed by the social consequences of an epicurean individualism; they saw the charms of the flesh, and carved them into a beauty that might for a while laugh at wrinkles and time. Freed from the conventions of the classic age, they indulged themselves in tender sentiment, and pictured, possibly with sincere feeling, shepherds dying of undisillusioned love, pretty heads lost in romantic reverie, mothers fondly contemplating their children: these, too, seemed to them a part of the reality they would record. And finally they faced the facts of pain and grief, of tragic catastrophes and untimely death; and they resolved to find a place for them in their representation of human life.

  No student with a mind of his own will join in any sweeping judgment about Hellenistic decay; a general conclusion to this effect serves too easily as an excuse for ending the story of Greece before the task is done. We feel in this period a slackening of creative impulse, but we are compensated by the lavish abundance of an art now completely master of its tools. Youth cannot last forever, nor are its charms supreme; the life of Greece, like every life, had to have a natural subsidence, and accept a ripe old age. Decadence had set in, it had bitten into religion, morals, and letters, and had left its stigmata upon individual works here and there; but the impetus of the Greek genius kept Greek art, like Greek science and philosophy, near their zenith to the end. And never in its isolated youth had the Greek passion for beauty, or the Greek power and patience to embody it, spread so triumphantly, or with such rich stimulation and result, into the sleeping cities of the East. There Rome would find it, and pass it on.

  CHAPTER XXVIII

  The Climax of Greek Science

  I. EUCLID AND APOLLONIUS

  THE fifth century saw the zenith of Greek literature, the fourth the flowering of philosophy, the third the culmination of science. The kings proved more tolerant and helpful to research than the democracies. Alexander sent to the Greek cities of the Asiatic coast camel loads of Babylonian astronomical tablets, most of which were soon translated into Greek; the Ptolemies built the Museum for advanced studies, and gathered the science as well as the literature of the Mediterranean cultures into the great Library; Apollonius dedicated his Conies to Attalus I, and under the protection of Hieron II Archimedes drew his circles and reckoned the sand. The fading of frontiers and the establishment of a common language, the fluid interchange of books and ideas, the exhaustion of metaphysics and the weakening of the old theology, the rise of a secularly minded commercial class in Alexandria, Rhodes, Antioch, Pergamum, and Syracuse, the multiplication of schools, universities, observatories, and libraries, combined with wealth, industry, and royal patronage to free science from philosophy, and to encourage it in its work of enlightening, enriching, and endangering the world.

  About the opening of the third century—perhaps long before it—the tools of the Greek mathematician were sharpened by the development of a simpler notation. The first nine letters of the alphabet were used for the digits, the next letter for 10, the next nine for 20, 30, etc., the next for 100, the next for 200, 300, and so forth. Fractions and ordinals were expressed by an acute accent after the letter; so, according to the context, l’ stood for one tenth or tenth; and a small l under a letter indicated the corresponding thousand. This arithmetical shorthand provided a convenient system of computation; some extant Greek papyri crowd complicated calculations, ranging from fractions to millions, into less space than similar reckonings would require in our own numerical notation.*

  Nevertheless the greatest victories of Hellenistic science were in geometry. To this period belongs Euclid, whose name would for two thousand years provide geometry with a synonym. All that we know of his life is that he opened a school at Alexandria, and that his students excelled all others in their field; that he cared nothing about money, and when a pupil asked, “What shall I profit from learning geometry?” bade a slave give him an obol, “since he must make a gain out of what he learns”;1 that he was a man of great modesty and kindliness; and that when, about 300, he wrote his famous Elements, it never occurred to him to credit the various propositions to their discoverers, because he made no pretense at doing more than to bring together in logical order the geometrical knowledge of the Greeks.* He began, without preface or apology, with simple definitions, then postulates or necessary assumptions, then “common notions” or axioms. Following Plato’s injunctions, he confined himself to such figures and proofs as needed no other instruments than ruler and compasses. He adopted and perfected a method of progressive exposition and demonstration already familiar to his predecessors: proposition, diagrammatic illustration, proof, and conclusion. Despite minor flaws the total result was a mathematical architecture that rivaled the Parthenon as a symbol of the Greek mind.
Actually it outlived the Parthenon as an integral form; for until our own century the Elements of Euclid constituted the accepted textbook of geometry in nearly every European university. One must go to the Bible to find a rival for it in enduring influence.

  A lost work of Euclid, the Conies, summarized the studies of Menaechmus, Aristaeus, and others on the geometry of the cone. Apollonius of Perga, after years of study in Euclid’s school, took this treatise as the starting point of his own Conies, and explored in eight “books” and 387 propositions the properties of the curves generated by the intersection of a cone by a plane. To three of these curves (the fourth being the circle) he gave their lasting names—parabola, ellipse, and hyperbola. His discoveries made possible the theory of projectiles, and substantially advanced mechanics, navigation, and astronomy. His exposition was laborious and verbose, but his method was completely scientific; his work was as definitive as Euclid’s, and its seven extant books are to this day the most original classic in the literature of geometry.

  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 Lemmas discusses 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 a submerged 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 Pro
position 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