100 years to solve an integral: The history of the Mercator map

&gt; In 1614, John Napier introduced logarithms.<p>&gt; Napier’s main motivation was to find an easier way to do multiplication and division.<p>&gt; Next, mathematicians decided to combine these tables. If you wanted to multiply trigonometric functions, you could find the values in a trigonometric table and then convert them to logarithms.<p>Actually, Napier&#x27;s 1614 <i>Mirifici Logarithmorum Canonis Descriptio</i> contains tables of −10⁷ ln(sin x&#x2F;10⁷) [0]. Non-trigonometric log tables appeared later.<p>[0] <a href="https:&#x2F;&#x2F;jscholarship.library.jhu.edu&#x2F;bitstream&#x2F;handle&#x2F;1774.2&#x2F;34187&#x2F;31151005337641.pdf#page=25" rel="nofollow">https:&#x2F;&#x2F;jscholarship.library.jhu.edu&#x2F;bitstream&#x2F;handle&#x2F;1774.2...</a>

I did a course on cartography a couple of years ago, and one of the better assignments was to find a suitable projection and make a proper map describing a comparison in distances between two points. The projection I used was of course Two-Point Equidistant Projection (the only possible one).<p>My resulting map is on the last page here: <a href="https:&#x2F;&#x2F;adelie.antarkt.is&#x2F;aron&#x2F;l0020b.pdf" rel="nofollow">https:&#x2F;&#x2F;adelie.antarkt.is&#x2F;aron&#x2F;l0020b.pdf</a><p>EDIT: Apparently it was one of my GIS courses, not the cartographic one, which were more color theory and labels.<p>One bad thing about the projection is that great circles are curved, so my straight lines should have been curved as well.

BTW, is there still no online map that would use some better projection (i.e. anything but [Web-]Mercator)? I mean, there is Google Earth, of course, but it has too much visual effects added, to use it as a go-to tool, as I use OSM or Google Maps. But mercator makes large scale distance and area comparison absolutely unintelligible, and I would rather much prefer to be able to use Kavrayskiy VII&#x2F;Natural Earth or something like that with OSM. No way it could be <i>too</i> computationally expensive in 2020, right? (I mean, once again, we <i>do</i> have Google Earth.)

<i>If you are looking for directions in a city, what matters most to you is that the roads look correct. This is why the Mercator map is used.</i><p>Whoa, wait a second...is this saying that roads which look straight on a map are actually not straight, in the sense that they&#x27;re not actually following the shortest path between two points?<p>It&#x27;s slightly appalling to me to think that people all over the world are building crooked roads so that they appear straight on a Mercator map, and then using Mercator maps so that existing roads continue to look straight.<p>I guess it probably doesn&#x27;t make much difference on a local level, where the distances are relatively short, but there&#x27;s still something a little bit horrifying about it, if I&#x27;m understanding all of this correctly.

The author tells the story of typical undergraduate instruction for the integrals of <i>tan(x)</i> and <i>sec(x)</i>. I would have thought that such a setting would have included that these were improper integrals, because of the infinities in the functions. i.e. if you evaluate the definite integral for any particular interval, it will give you the right answer except if you&#x27;ve gone through the part where the graph goes up to plus infinity and back through minus infinity.<p>Can someone more mathematically literate than me shed any light on whether it matters? I guess it&#x27;s still useful even if the integral is undefined at certain points off the edges of the map.

Buckminster Fuller invented a map projection called the Dymaxion [1], imagine peeling an orange peel as one intact piece and placing the North pole at the very center.<p>1 <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dymaxion_map" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Dymaxion_map</a>

In modern shipping is this used or do ships just follow the shortest distance between points? Perhaps GPS has made that simple?