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# An Application of Geography to Mathematics: History of the Integral of the Secant

by V. Frederick Rickey (Bowling Green State University) and Philip M. Tuchinsky (Ford Motor Company)

Mathematics Magazine
May, 1980

Subject classification(s): Integration | Single Variable Calculus | Calculus | Trigonometry | Spheres | Solid Geometry | Geometry and Topology
Applicable Course(s): 4.8 History of Math | 3.2 Mainstream Calculus II

This article is part of the Mathematics of Planet Earth 2013 Collection. Sailors wanted a map where curves of fixed compass direction were drawn as straight lines. Gerardus Mercator accomplished this by designing his famous world map in 1569. However, he did not describe how he did this. In 1599 Edward Wright showed that a line segment at latitude $$\theta$$ was stretched by a factor of $$\sec \theta$$ in the creation of Mercator’s map. Then the length of a vertical line segment along a longitudinal meridian is represented by an integral of $$\sec \theta$$. In 1645 Henry Bond conjectured that an antiderivative of $$\sec \theta$$ is $$\ln|\tan(\theta)/2+ \pi/4)|$$, a trigonometric variant of the result used in most calculus books today $$\ln|\sec \theta + \tan \theta|$$. This was first proved by James Gregory in 1668. The first proof intelligible to modern calculus students was given by Isaac Barrow (1630-1677).

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