A message "You have proven that the earth is round" up when you are the first to travel around the whole world.
Is this just for fun, or does it give any benefits?
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This message shows up once you have discovered (at least) one tile in each vertical column of the map. They don't have to connect horizontally. This way you 'connect' the two sides of the world and 'prove' the earth is round.
Once you have received this message, you will be able to scroll around the world. Also, your minimap might change a bit to correctly display your new world orientation. It doesn't change anything else though.
Someone wrote this in a comment that is now deleted.
It does not give any in-game effects per se, other than that you know how wide the world is. But it does give your opponents a small hint of your progress. On some maps it can be a very good sign that you have discovered astronomy. In some situations it can even be a 100% proof of this.