
100 years to solve an integral: The history of the Mercator map - the_origami_fox
https://liorsinai.github.io/mathematics/2020/08/27/secant-mercator.html
======
mci
> In 1614, John Napier introduced logarithms.

> Napier’s main motivation was to find an easier way to do multiplication and
> division.

> 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.

Actually, Napier's 1614 _Mirifici Logarithmorum Canonis Descriptio_ contains
tables of −10⁷ ln(sin x/10⁷) [0]. Non-trigonometric log tables appeared later.

[0]
[https://jscholarship.library.jhu.edu/bitstream/handle/1774.2...](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf#page=25)

~~~
the_origami_fox
Yes you're right. It has to do with how he derived the approximation formula
for the natural logarithm. He needed a function y=sin(x) for his log(y)
calculations. But I am not sure when the log(f(x))) tables for the other
trigonometric functions came about. As far I understood, initially a single
log(y) table sufficed.

~~~
mci
Well, I suppose Napier presaged the concept of a single source of truth.
Aren't other log-trig tables a waste of paper if you can look up log(cos x)
under log(sin(90°−x)) and quickly calculate log(tan x) as log(sin x)−log(cos
x)?

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widforss
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).

My resulting map is on the last page here:
[https://adelie.antarkt.is/aron/l0020b.pdf](https://adelie.antarkt.is/aron/l0020b.pdf)

EDIT: Apparently it was one of my GIS courses, not the cartographic one, which
were more color theory and labels.

One bad thing about the projection is that great circles are curved, so my
straight lines should have been curved as well.

~~~
082349872349872
LLA: the optimal hub — at least in summer. (and modulo transpacific routes?)

~~~
widforss
The LLA freight proposal was given by the teacher in the assignment, but I
guess there is a real benefit. Isn't Anchorage preferred over Honolulu for
most trans-pacific flights?

~~~
082349872349872
Hej. Sorry, I have no idea.

[https://www.flightradar24.com/](https://www.flightradar24.com/) makes it look
like most flights are direct (too far south for ANC, too far north for HNL),
but I couldn't quickly find any way to filter out cargo from pax.

PS. Having just seen
[https://www.youtube.com/watch?v=5hdUKZLQIuQ](https://www.youtube.com/watch?v=5hdUKZLQIuQ)
I have to ask if you all really chew snus? What savages — tobacco is for
schnupfen.
[https://www.youtube.com/watch?v=oOa1cz7O3kU](https://www.youtube.com/watch?v=oOa1cz7O3kU)

~~~
widforss
Haha, if you drive EPA you most definitely use snuff, in the mouth.

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krick
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/Natural Earth or something like that with OSM. No way it could
be _too_ computationally expensive in 2020, right? (I mean, once again, we
_do_ have Google Earth.)

~~~
bdon
I don't think there is an area-preserving projection that can work for all the
scales online map services are designed for.

If you are creating your own global-extent map at a limited range of scales,
most of the OSM tooling for example PostGIS lets you specify a custom
projection like
[http://shadedrelief.com/ee_proj/](http://shadedrelief.com/ee_proj/)

~~~
krick
If I'm creating my own map I can kinda draw whatever the fuck I want. I need a
web-map (preferably OSM-based) that I can use as a web-map, but with a
projection less fucked up than Mercator, of which I can name at least 10 in
addition to 2 I already named. Also, web-mercator generally is not a problem
on small scales, so there isn't really a problem of alternating between
projections depending on scale. (Which is not really necessary too, since
other projections are generally more or less ok even on small scale. You can
totally use globe view on google maps on city-scale and barely notice any
difference at all.)

~~~
elliekelly
Perhaps it was unintentional but the tone of your comment comes across as over
the top hostile in response to someone who was just trying to help you.

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NickM
_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._

Whoa, wait a second...is this saying that roads which look straight on a map
are actually not straight, in the sense that they're not actually following
the shortest path between two points?

It'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.

I guess it probably doesn't make much difference on a local level, where the
distances are relatively short, but there's still something a little bit
horrifying about it, if I'm understanding all of this correctly.

~~~
jcranmer
Any physical infrastructure, such as roads, is going to be distorted by local
topography, which furthermore may not be entirely consistent from year to year
(land can creep forward or hillsides erode somewhat). This deviation will tend
to enormously outweigh the errors caused by map projection.

Another note is that legal boundaries tend to be defined on the basis of
actual surveyed boundary markers, not on the logical definition of the line.
So the US-Canada border, while originally defined as exactly 49°N, the actual
border is actually generally about a few hundred feet south of that line.
These surveys were generally conducted by people running literal chains in a
constant bearing for some distance (maybe several hundred yards at most) and
then taking another reading of their position to correct the line. Of note is
the Mason-Dixon Line, which was precise enough to note that the errors in
their surveying had a systematic error to them--which was realized to be the
physical mass of the Appalachian Mountains ever-so-slightly deviating plumb
lines from vertical.

Also note that historical boundaries were more often specified as "reference
point and bearing" (or along line meridians/parallels, which amounts to the
same thing) then "from point A to point B". In this regard, the straight line
on a Mercator map is more accurate than the geodesic shortest-distance-
between-two-points.

~~~
dhosek
Then there's the history of the Tennessee-Kentucky border (and that weird jog
in the otherwise straight line). [https://www.tngenweb.org/campbell/hist-
bogan/surveyor.html](https://www.tngenweb.org/campbell/hist-
bogan/surveyor.html)

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rm445
The author tells the story of typical undergraduate instruction for the
integrals of _tan(x)_ and _sec(x)_. 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've
gone through the part where the graph goes up to plus infinity and back
through minus infinity.

Can someone more mathematically literate than me shed any light on whether it
matters? I guess it's still useful even if the integral is undefined at
certain points off the edges of the map.

~~~
the_origami_fox
It matters here in the sense that to show the poles, you need to stretch the
Mercator map to infinity, because the function sec(x) is undefined at 90
degrees. This is clearly absurd, so map makers get around the problem by
cutting the top of the map off, usually around the 85 degrees parallel. I
chose not to include this detail in the article.

To slightly misquote the great Richard Feynman, "It turns out that it's
possible to sweep the infinities under the rug by a certain crude skill." :)

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sova
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.

1
[https://en.wikipedia.org/wiki/Dymaxion_map](https://en.wikipedia.org/wiki/Dymaxion_map)

~~~
shard
The irregular shape of Dymaxion always bothered me. I much prefer the
Cahill–Keyes projection:
[https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projectio...](https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projection)

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

~~~
tgb
Even back "in the day" the routes weren't actually following those lines.
Here's a neat animation of the routes many ships took (from their logbooks)
starting in the 1750's:

[http://sappingattention.blogspot.com/2012/04/visualizing-
oce...](http://sappingattention.blogspot.com/2012/04/visualizing-ocean-
shipping.html)

