When i need to build a settlers? There is a choice between stopping city growth and early expansion.
My suggestion is if time to build is less than time to grow. But it's just a kind of random guess.
In original Civilization i started to build at city size no less than 4.
Best Answer
Summary: Early game, build a settler basically as soon as you can. Continue to grow as quickly as you can defend your settlers, or until you hit your desired size for your long term strategy.
In terms of the early game, it is pretty easy to apply some science to this. Let's say that I'm in a typical starting situation. I build my first city on turn 1, and I start working a space that gives me two extra food. That means it will take 8 turns for my city to grow to population 2.
On turn 8, I start working another square that gives me 2 food. At this point I can start building a settler. It will take me 18 turns to complete that settler. On turn 30, I have completed the settler, and moved him to a new location to build a city. So far I have accumulated (8*4)+(22*5)=142 science, and I am now working 5 tiles (3 population, plus 1 for each city) and gaining 6 science per turn. Both of my cities are 8 turns from growing.
After those 8 more turns, I have accumulated 134 +(8*7) = 190 science. I am now working 7 tiles and gaining 7 science per turn.
Let's say that instead I decided to just grow my capital. My capital grows to size 3 on turn 20. When turn 30 rolls around, I'm still at size 3. I have accumulated (8*4)+(12*5)+(10*6)=152 science, 10 more than scenario 1. I am now gaining 6 science per turn, but only working 4 tiles. My capital is 6 turns from growing.
Moving forward 8 more turns, I have accumulated 144 + (6*6) + (2*7) = 194 science, 4 more than scenario 1. But now I'm only working 5 tiles, and gaining 7 science per turn. As soon as my scenario 1 cities grow again the science advantage is gone, my capital is only 1 population larger, I'm working fewer tiles and gaining less science and money, not to mention having fewer resources in my borders.
There are a few other benefits to having a large capital, but I think they are far outweighed by the benefits of having multiple cities growing simultaneously. Even if you're going for a cultural victory, getting those first couple cities down quickly will put you way ahead later on.