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 a.d.) and his successor, Ptolemy. Following the decline of Greek civilization about 640 a.d., little progress occurred until the advancement of science and mathematics recommenced during the Renaissance, beginning approximately in 1400.
About 1450, at a time when surveying work took on new importance, a type of trigonometry known as plane trigonometry began to be used. Plane trigonometry deals with calculations in two dimensions, considering triangles in one plane such as on a map or a tract of land, and is of great use in measuring lengths and distances that cannot be measured directly. Distances across lakes or land, and even the heights of mountains can be determined by employing plane trigonometry.
John Napier's 1594 invention of logarithms was inspired by his desire to simplify the spherical trigonometry calculations used in various astronomical problems faced at that time. His work with trigonometric functions, particularly the logarithms of sines, is of particular significance in the study of trigonometry. In the seventeenth and eighteenth century, the imprecise trigonometric, logarithmic and nautical tables of past centuries were revised and expanded by mathematicians such as George Rhaeticus (1514-1576), Nicholas Copernicus, Francois Viete and Bartholomaus Pitiscus (1561-1613). In fact, it was in Pitiscus' 1613 book, Thesaurus , that the term "trigonometry" was first used. It is derived from two Greek words, trigonos, meaning "triangle," and metria, meaning "to measure." This period marked the metamorphosis of trigonometry from a practical technique applied to astronomy and surveying into a branch of mathematics in its own right. By the late seventeenth century, Abraham de Moivre had begun applying analytical methods to trigonometry, a lead that was followed by Leonard Euler, who advanced spherical trigonometry by demonstrating the importance of trigonometric functions. Trigonometry continued to find new uses in the following centuries, and remains an important mathematical field today, with applications in surveying, map-making, military artillery calculation, and astronomy.
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