THE cultural activity of Periclean Greece takes chiefly three forms-art, drama, and philosophy. In the first, religion is the inspiration; in the second it is the battleground; in the third it is the victim. Since the organization of a religious group presumes a common and stable creed, every religion sooner or later comes into opposition with that fluent and changeful current of secular thought that we confidently call the progress of knowledge. In Athens the conflict was not always visible on the surface, and did not directly affect the masses of the people; the scientists and the philosophers carried on their work without explicitly attacking the popular faith, and often mitigated the strife by using the old religious terms as symbols or allegories for their new beliefs; only now and then, as in the indictments of Anaxagoras, Aspasia, Diagoras of Melos, Euripides, and Socrates, did the struggle come out into the open, and become a matter of life and death. But it was there. It ran through the Periclean age like a major theme, played in many keys and elaborated in many variations and forms; it was heard most distinctly in the skeptical discourses of the Sophists and in the materialism of Democritus; it sounded obscurely in the piety of Aeschylus, in the heresies of Euripides, even in the irreverent banter of the conservative Aristophanes; and it was violently recapitulated in the trial and death of Socrates. Around this theme the Athens of Pericles lived its mental life.
I. THE MATHEMATICIANS
Pure science, in fifth-century Greece, was still the handmaiden of philosophy, and was studied and developed by men who were philosophers rather than scientists. To the Greeks higher mathematics was an instrument not of practice but of logic, directed less to the conquest of the physical environment than to the intellectual construction of an abstract world.
Popular arithmetic, before the Periclean period, was almost primitively clumsy.* One upright stroke indicated 1, two strokes 2, three 3, and four 4; 5, 10, 100, 1000, and 10,000 were expressed by the initial letter of the Greek word for the number—pente, deka, hekaton, chilioi, myrioi.Greek mathematics never achieved a symbol for zero. Like our own it betrayed its Oriental origin by taking from the Egyptians the decimal system of counting by tens, and from the Babylonians, in astronomy and geography, the duodecimal or sexagesimal system of counting by twelves or sixties, as still on our clocks, globes, and charts. Probably an abacus helped the people with the simpler calculations. Fractions were painful for them: to work with a complex fraction they reduced it to an accumulation of fractions having I as their common numerator; so 
was broken down into 
.1
Of Greek algebra we have no record before the Christian era. Geometry, however, was a favorite study of the philosophers, again less for its practical value than for its theoretical interest, the fascination of its deductive logic, its union of subtlety and clarity, its imposing architecture of thought. Three problems particularly attracted these mathematical metaphysicians: the squaring of the circle, the trisection of the angle, and the doubling of the cube. How popular the first puzzle became appears in Aristophanes’ Birds, in which a character representing the astronomer Meton enters upon the stage armed with ruler and compasses, and undertakes to show “how your circle may be made a square”—i.e., how to find a square whose area will equal that of a given circle. Perhaps it was such problems as these that led the later Pythagoreans to formulate a doctrine of irrational numbers and incommensurable quantities.* It was the Pythagoreans, too, whose studies of the parabola, the hyperbola, and the ellipse prepared for the epochal work of Apollonius of Perga on conic sections.2 About 440 Hippocrates of Chios (not the physician) published the first known book on geometry, and solved the problem of squaring the lune.† About 420 Hippias of Elia accomplished the trisection of an angle through the quadratrix curve.3 About 410 Democritus of Abdera announced that “in constructing lines according to given conditions no one has ever surpassed me, not even the Egyptians;”4 he almost made the boast forgivable by writing four books on geometry, and finding formulas for the areas of cones and pyramids.5 All in all, the Greeks were as excellent in geometry as they were poor in arithmetic. Even into their art geometry entered actively, making many forms of ceramic and architectural ornament, and determining the proportions and curvatures of the Parthenon.
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