I. THE SCIENTISTS
COMPARED with the bold advance of the fifth century, and the revolutionary achievements of the third, science in the fourth marked time, and contented itself, in great part, with recording its accumulations. Xenocrates wrote a history of geometry, Theophrastus a history of natural philosophy, Menon a history of medicine, Eudemus histories of arithmetic, geometry, and astronomy.1 The problems of religion, morals, and politics appearing to be more vital and pressing than the problems of nature, men turned with Socrates from the objective study of the material world to a consideration of the soul and the state.
Plato loved mathematics, dipped his philosophy into it deeply, dedicated the Academy to it, almost, in Syracuse, gave a kingdom for it. But arithmetic was for him a half-mystical theory of number; geometry was not a measuring of the earth, it was a discipline of pure reason, a portal to the mind of God. Plutarch tells of Plato’s “indignation” at Eudoxus and Archytas for carrying on experiments in mechanics, “as the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help . . . from matter.” In this way, Plutarch continues, “Mechanics came to be separated from geometry, and, repudiated or neglected by philosophers, took its place as a military art.”2 Nevertheless, in his own abstract way, Plato served mathematics well. He redefined the point as the beginning of a line,3 formulated a rule for finding square numbers that are the sum of two squares,4 and invented or developed mathematical analysis5—i.e., the proof or disproof of a proposition by considering the results that follow from assuming it; the reductio ad absurdum is one form of this method. The emphasis on mathematics, in the curriculum of the Academy, helped the science if only by training such creative pupils as Eudoxus of Cnidus and Heracleides of Pontus.
Plato’s friend Archytas, besides being seven times chosen strategos of Taras, and writing several tracts of Pythagorean philosophy, developed the mathematics of music, doubled the cube, and wrote the first known treatise on mechanics. Antiquity credited him with three epochal inventions—the pulley, the screw, and the rattle; the first two laid the foundations of machine industry, the third, says the grave Aristotle, “gave children something to occupy them, and so prevented them from breaking things about the house.”6 In this same age Dinostratus “squared the circle” by using the quadratrix curve. His brother Menaechmus, a pupil of Plato, founded the geometry of conic sections,* doubled the cube, formulated the theoretical construction of the five regular solids,†advanced the theory of irrational numbers, and gave the world a famous phrase. “O King,” he said to Alexander, “for traveling over the country there are both royal roads and roads for common citizens; but in geometry there is one road for all.”‡8
The great name in fourth-century science is Eudoxus, who helped Praxiteles to give Cnidus a niche in history. Born there about 408, he set out at the age of twenty-three to study medicine with Philistion at Locri, geometry with Archytas at Taras, and philosophy with Plato at Athens. He was poor, and lived cheaply at the Piraeus, whence he walked to the Academy every scholastic day. After a stay in Cnidus he went to Egypt and spent sixteen months studying astronomy with the priests of Heliopolis. We find him next in Propontine Cyzicus, lecturing on mathematics. At the age of forty he moved with his pupils to Athens, opened there a school of science and philosophy, and for a time rivaled Plato. Finally he returned to Cnidus, set up an observatory, and was entrusted with the task of giving the city a new code of laws.9
His contributions to geometry were fundamental. He invented the theory of proportion,§ and most of the propositions, transmitted to us in the fifth book of Euclid; and he devised the method of exhaustion which made it possible to calculate the area of the circle and the volume of the sphere, the pyramid, and the cone; without this preliminary work Archimedes would have been impossible. But the absorbing interest of Eudoxus was in astronomy. We catch the spirit of the scientist in his remark that he would gladly be consumed like Phaethon if he might thereby discover the nature, size, and form of the sun.10 The word astrology was then used to include what we call astronomy, but Eudoxus advised his pupils to ignore the Chaldean theory that a person’s fortune could be told by noting the position of the stars at the time of his birth. He longed to reduce all celestial motions to fixed laws; and in his Phainomena—which antiquity considered its greatest book on astronomy—he laid the foundation for the scientific predicton of the weather.
His most famous theory was a brilliant failure. He suggested that the universe was composed of twenty-seven transparent and therefore invisible spheres, revolving in diverse directions and at various speeds about the center of the earth; and that the heavenly bodies were fixed upon the periphery or shell of these concentric spheres. The system now seems fantastic, but it was one of the first attempts to give a scientific explanation of celestial behavior. In accord with it Eudoxus calculated with considerable accuracy (if we may rashly take our present “knowledge” of these matters as a norm) the synodic and zodiacal periods of the planets.* The theory did more than any other in antiquity to stimulate astronomic research.
Ecphantus of Syracuse wrote, about 390: “The earth moves about its own center in an eastward direction.”12 Heracleides of Pontus, one of the great polymaths of antiquity—the author of famous works on grammar, music, poetry, rhetoric, history, geometry, logic, and ethics—took up the suggestion, or advanced it independently, arguing that instead of the whole universe revolving about the earth, the relevant phenomena can be explained by supposing that the earth itself rotates daily upon its axis.13 Venus and Mercury, said Heracleides, revolve around the sun. For one brilliant moment, perhaps, Heracleides anticipated Aristarchus and Copernicus, for we read in the fragments of Geminus (ca. 70 B.C.): “Heracleides of Pontus said that, even on the assumption that the earth moves in a certain way, while the sun is in a certain way at rest, the apparent irregularity with reference to the sun can be saved.”14 We shall probably never know just what Heracleides meant.
Meanwhile a modest progress was being made in the sciences. In geography Dicaearchus of Messana, the biographer of Greece, measured the height of mountains, established the circumference of the earth at some thirty thousand miles, and noted the influence of the sun upon the tides. In 325 Nearchus, one of Alexander’s generals, sailed from the mouth of the Indus along the southern coast of Asia to the Euphrates; his log, partly preserved in Arrian’s Indica,15 was one of the classics of ancient geography. Geodesy—the measurement of land surfaces, elevations, depressions, positions, and volumes—had already been christened (geodaisia) as distinct from geometry.16 Philistion of Italian Locri, at the beginning of the century, practiced animal dissection, and called the heart the main regulator of life, the seat of the pneuma, or soul. Diocles of Euboean Carystus, about 370, dissected the womb of animals, described human embryos of twenty-seven to forty days, advanced anatomy, embryology, gynecology, and obstetrics, and scotched a favorite Greek error by announcing that both sexes contribute “seed” to form the embryo.17 A second Aspasia became one of the famous physicians of fourth-century Athens, renowned for her work in women’s diseases, surgery, and other branches of medicine.18 And lest medical science should lower the death rate too fast for the means of subsistence, Aeneas Tacticus, the Arcadian, published about 360, in time for Philip and Alexander, the first Greek classic on the art of war.