What extent did the idea of the enlightenment influence the politics of eighteenthcetury erope

I don't think it is possible to overstate how significant those ideas were.

That was the time in which the absolutist governments of Europe began to fall, being replaced with Constitutional Monarchies or other representative governments. This led to upheaval in much of Europe as the "old guard" of nations faced revolutions or wars with nations (like France) that wanted to change their own political orders as well.