Does someone describe to me how science is working in the game? I know that it gets more expensive for each level. But does the cost for research increase when your empire gets bigger? If yes how much?
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Yes, the cost for research increases depending on how many provinces you have. I am not sure of the actual calculation, but it is a significant amount. I find that if you are playing a small one or two province country it is often a good idea to wait to expand until you have a few technologies since they are so cheap when you are small.
I think research costs are directly proportional to your number of provinces, irrespective or province population and income.
What goes up must come down? E=MC2? 
Tech costs increase as your empire increases. And, it appears the new patch will increase the costs if you get too far ahead in research (to keep you from getting too far ahead of an historicially-realistic timeline.)
Tech costs increase as your empire increases. And, it appears the new patch will increase the costs if you get too far ahead in research (to keep you from getting too far ahead of an historicially-realistic timeline.)