100 Years to Solve an Integral (2020)

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The integral of sec(x) was a major mathematical problem first introduced by Geradus Mercator in 1569 for map-making purposes.
The exact solution to the integral of sec(x) was found accidentally in 1645, 86 years after Mercator's introduction, without the use of calculus.
A formal proof for the integral of sec(x) was provided in 1668, 99 years after the problem was first proposed.
The integral of sec(x) is crucial for the Mercator map projection, which is used in modern online maps like Google Maps and Apple Maps.
Mercator's projection was designed to make rhumb lines (lines of constant bearing) appear straight, aiding navigation.
The integral was discovered by Henry Bond in 1645 through numerical comparison with logarithmic tables, before calculus was formalized.
The integral of sec(x) has multiple equivalent forms, including ln|sec(x) + tan(x)| + c and ln|tan(x/2 + 45°)| + c.
The Mercator projection distorts areas, making regions near the poles appear larger than they are, which has been criticized for its colonial and racist implications.
Alternative map projections like the Winkel Triple and Robinson Projection offer compromises between form and scale.
The history of the integral of sec(x) highlights how mathematical concepts often have complex and non-linear developments.

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