100 Years to Solve an Integral (2020)

Summary

The integral of sec(x) is well known to any beginners calculus student. Yet this integral was once a major outstanding maths problem. It was first introduced by Geradus Mercator who needed it to make his famous map in 1569. He couldn’t find it and used an approximation instead. The exact solution was found accidentally 86 years later without calculus in 1645. It then took another two decades until a formal proof was given in 1668, 99 years after Mercator first proposed the problem. Update 13 March 2021: added a note on how Napier calculated logarithm trigonometry tables. This was prompted by a correction raised in a discussion of this post on HackerNews. Update 10 October 2021: the great circle and rhumb line images are now made with a script that uses Cartopy. Previously they were made with a Matlab application. You can see the new script at my Github repository and enter your co-ordinates to generate your own lines. As this comic by SMBC rightly teases, the history of mathematics is often not so straightforward. Theorems, formulas and notation that are routinely discussed in class, were once insights or accidents themselves. This is the story of one such formula, the integral of the secant. I first read about it almost a decade ago when I got interested in cartography: the science and art of map making. This integral is of vital importance to the Mercator map and therefore many online maps that use it like Apple Maps and Google Maps. This story has been told several times before: see 1, 2 or 3. But these are all journal articles, consigned mostly to academics. I want to present it here in a less formal and more colourful setting to make it more accessible. This is an article about mathematics so familiarity with the following is helpful: algebra, trigonometry, radians and basic calculus. These are usually covered in advanced high school maths classes or first year maths courses. First year maths In first year maths at university after a month of differentiation we...