al solutions to kinds of problems that lead immediately to equations in several unknowns. Such equations are now called Diophantine equations (see Diophantine Analysis). Applied Mathematics in Greece Paralleling the studies described in pure mathematics were studies made in optics, mechanics, and astronomy. Many of the greatest mathematical writers, such as Euclid and Archimedes, also wrote on astronomical topics. Shortly after the time of Apollonius, Greek astronomers adopted the Babylonian system for recording fractions and, at about the same time, composed tables of chords in a circle. For a circle of some fixed radius, such tables give the length of the chords subtending a sequence of arcs increasing by some fixed amount. They are equivalent to a modern sine table, and their composition marks the beginnings of trigonometry. In the earliest such tables-those of Hipparchus in about 150 BC-the arcs increased by steps of 71, from 0 to 180. By the time of the astronomer Ptolemy in the 2nd century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his Almagest a table of chords in a circle for steps of 3, which, although expressed sexagesimally, is accurate to about five decimal places. In the meantime, methods were developed for solving problems involving plane triangles, and a theorem-named after the astronomer Menelaus of Alexandria-was established for finding the lengths of certain arcs on a sphere when other arcs are known. These advances gave Greek astronomers what they needed to solve the problems of spherical astronomy and to develop an astronomical system that held sway until the time of the German astronomer Johannes Kepler. Medieval and Renaissance Mathematics Following the time of Ptolemy, a tradition of study of the mathematical masterpieces of the preceding centuries was established in various centers of Greek learning. The preservation of such works as have su...
