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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.

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    Other players also receive a message that someone has done it, whether or not that means anything to you. It can let you make presumptions about their tech at least – n_plum Jan 7 at 23:33
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    You need to discover (at least) one tile in each vertical column of the map. They don't have to connect horizontally - you can have a unit in the north and another in the south, or any other combination, and it still works. – Cadence Jan 8 at 0:09
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    @Cadence You could write your own answer to include that fact. – Broman Jan 8 at 6:36
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    In a situation like this it is better to comment or add a new post rather than edit. I rarely edit others posts other than formatting or minor errors. One important reason is that you may be wrong, and if you are wrong you should definitely not put those words in someone else's mouth. – Broman Jan 8 at 8:09
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    @Rogem In Civ IV there was a +1 movement to naval for circumnavigating. There is no bonus in Civ 5 however. – Bilkokuya Jan 8 at 10:50
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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.

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