Trigonometry, the study of the properties and proportions of triangles, has been an important branch of mathematics for thousands of years. The origins of trigonometry can be traced to Greece in the third century b.c. Among ancient civilizations, accurate navigation required precise knowledge about the positions and paths of celestial bodies, so the need for greater accuracy in astronomical calculation became the most important early motivation for developing trigonometry. Trigonometry later became useful as an aid in geography, map-making, surveying and a host of other fields. About 100 b.c., during the period of the Alexandrian Greeks, the Greek astronomer Hipparchus developed what is today called spherical trigonometry. The Greek astronomers used spherical trigonometry to determine the time of day, direction of motion, and the positions of ships or reference points. Greek trigonometry flourished in the following centuries, reaching its highest point with the astronomer Menelaus (first century...