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.