It might seem straightforward to pull out a map and measure the distance between two distant points, however, the larger the distance the bigger the distortion caused by traveling on the curved surface of the Earth as opposed to flat 2D space. So while the distance you measure to your neighbouring town won't be too bad, if you're measuring between London and Rio the curvature of the Earth will make a big difference to the distance that you'll travel. To help figure out the correct distance there's the haversine formula.

The haversine formula allows you to calculate the shortest distance between two points on a sphere using their latitudes and longitudes — this will be the arc between them on the great circle that includes both points. A great circle is a circle on a sphere with the same centre as the sphere, like the Equator. The haversine formula isn't perfect in practice, as Earth isn't a perfect sphere — see the 3 tallest mountains.

It was invented around two hundred years ago, together with tables to speed any calculations, to help sailors navigate.

The haversine formula

All maps involve decisions. Whenever you compress real life onto something smaller and less detailed you have to choose what to keep in and what to leave out. And in the case of printing a map of the Earth, you have to figure out how to get something that is curved onto something that is flat, and this involves trade-offs and decisions as well. 3 dimensional space just doesn’t flatten to 2 dimensional paper without choices.

In 1569 Gerardus Mercator created a map that brilliantly solved a pressing problem — that of being able to follow a straight line while sailing and it correspond to a straight line on the map. His map also did a good job of preserving the shape of countries. But to do this you have to stretch the areas at the top and bottom of the Earth. Because most of the land on Earth is in the Northern hemisphere, and because that land is generally further north than the land in the southern hemisphere is south, it has the effect of enlarging Northern countries such as Europe, the US and Canada, Scandinavia and Russia as compared to countries closer to the equator and in the Southern hemisphere.

Most world maps you see are still similar to Mercator’s projection. In fact Google Maps desktop used it until just recently. It could well be that this distorts our worldview by emphasizing Northern countries in size and therefore importance. When I see a more size accurate Gall-Peters projection I have to say it’s hard to argue.

The Mercator projection

Where the Northern Hemisphere simply allows you to find the North Star, around which all the stars will rotate, finding your way in the Southern Hemisphere is a little more involved. Unlike in the North, there’s no obvious star due south in the Southern Hemisphere. Instead, you have to use two handy indicators: the Southern Cross — easily spotted, fortunately — and two stars known as The Pointers. Find where these intersect, as in the sketch, and you’ve found due south.

Or, if you prefer, you can follow the direction the Southern Cross is pointing for a distance equal to 4.5x the length of the Southern Cross to arrive at the spot.

Incidentally, I wondered about reliable methods of finding south in the Northern Hemisphere and north in the Southern Hemisphere. And I think the answer is to find north or south and turn around.

Navigation sketches