B. 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 7°„, from 0°„ to 180°„. By the time of the astronomer Ptolemy in the 2nd century scaps 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 1°„, 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.