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A. Greek Mathematics The Greeks adopted elements of mathematics from both the Babylonians and the Egyptians. The new element in Greek mathematics, however, was the invention of an abstract mathematics founded on a logical structure of definitions, axioms, and proofs. According to later Greek accounts, this development began in the 6th century scaps bc with Thales of Miletus and Pythagoras of S¨¢mos, the latter a religious leader who taught the importance of studying numbers in order to understand the world. Some of his disciples made important discoveries about the theory of numbers and geometry, all of which were attributed to Pythagoras. In the 5th century scaps bc, some of the great geometers were the atomist philosopher Democritus of Abdera, who discovered the correct formula for the volume of a pyramid, and Hippocrates of Chios, who discovered that the areas of crescent-shaped figures bounded by arcs of circles are equal to areas of certain triangles. This discovery is related to the famous problem of squaring the circle¡ªthat is, constructing a square equal in area to a given circle. Two other famous mathematical problems that originated during the century were those of trisecting an angle and doubling a cube¡ªthat is, constructing a cube the volume of which is double that of a given cube. All of these problems were solved, and in a variety of ways, all involving the use of instruments more complicated than a straightedge and a geometrical compass. Not until the 19th century, however, was it shown that the three problems mentioned above could never have been solved using those instruments alone. In the latter part of the 5th century scaps bc, an unknown mathematician discovered that no unit of length would measure both the side and diagonal of a square. That is, the two lengths are incommensurable. This means that no counting numbers n and m exist whose ratio expresses the relationship of the side to the diagonal. Since the Greeks considered only the counting numbers (1, 2, 3, and so on) as numbers, they had no numerical way to express this ratio of diagonal to side. (This ratio, , would today be called irrational.) As a consequence the Pythagorean theory of ratio, based on numbers, had to be abandoned and a new, nonnumerical theory introduced. This was done by the 4th-century scaps bc mathematician Eudoxus of Cnidus, whose solution may be found in the Elements of Euclid. Eudoxus also discovered a method for rigorously proving statements about areas and volumes by successive approximations. Euclid was a mathematician and teacher who worked at the famed Museum of Alexandria and who also wrote on optics, astronomy, and music. The 13 books that make up his Elements contain much of the basic mathematical knowledge discovered up to the end of the 4th century scaps bc on the geometry of polygons and the circle, the theory of numbers, the theory of incommensurables, solid geometry, and the elementary theory of areas and volumes. The century that followed Euclid was marked by mathematical brilliance, as displayed in the works of Archimedes of Syracuse and a younger contemporary, Apollonius of Perga. Archimedes used a method of discovery, based on theoretically weighing infinitely thin slices of figures, to find the areas and volumes of figures arising from the conic sections. These conic sections had been discovered by a pupil of Eudoxus named Menaechmus, and they were the subject of a treatise by Euclid, but Archimedes' writings on them are the earliest to survive. Archimedes also investigated centers of gravity and the stability of various solids floating in water. Much of his work is part of the tradition that led, in the 17th century, to the discovery of the calculus. Archimedes was killed by a Roman soldier during the sack of Syracuse. His younger contemporary, Apollonius, produced an eight-book treatise on the conic sections that established the names of the sections: ellipse, parabola, and hyperbola. It also provided the basic treatment of their geometry until the time of the French philosopher and scientist Ren¨¦ Descartes in the 17th century. After Euclid, Archimedes, and Apollonius, Greece produced no geometers of comparable stature. The writings of Hero of Alexandria in the 1st century scaps ad show how elements of both the Babylonian and Egyptian mensurational, arithmetic traditions survived alongside the logical edifices of the great geometers. Very much in the same tradition, but concerned with much more difficult problems, are the books of Diophantus of Alexandria in the 3rd century scaps ad. They deal with finding rational solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis). |