Halley, Cotes, and the nautical meridian (Q1842105)

In the middle of the 17th century P. Nuñes was the first to make a thorough mathematical investigation of the loxodromic curves, i.e. curves with a constant bearing, topical question in nautical cartography. The concept of a loxodromic chart, in which an intended sailing course would appear as a straight line, was centuries old. The publication in 1569 by Gerhard Kremer (called Mercator) of a world chart on a conformal modified cylindrical projection was a very substantial advance. The major problem in this type of construction is the correct spacing of the parallels of latitude, it became known as the problem of ``the true division of the nautical meridian''. Many mathematicians tried to solve this problem, among the others Stevin, Wallis, Gregory, Leibniz. In this paper the author illustrates the solutions of Edmund Halley (1696) and Roger Cotes (1714). Halley gave a solution by relating the geometry of the loxodrome on the sphere to the properties of the Mercator chart, through the intermediacy of the properties of the logarithmic spiral. Cotes showed how his own logometric methods yielded Halley's rule more economically and further that the use of the logarithmic spiral was not necessary.