Explain which one of the atlantic revolutions lives up to the ideals of the enlightenment

The Atlantic revolutions (such as the American and French Revolutions) were influenced most by the Enlightenment idea of reason and rationality.

Enlightenment thinkers felt that science and reason should be applied to the analysis of all aspects of life.  They felt that nothing should be accepted simply because it was traditional to do so.  They believed that pursuing reason and natural law would lead to progress in human society.
These ideas clearly showed in the ideas of the Atlantic revolutions.  These revolutions did things like attacking the idea of monarchy in favor of democracy.  This fit nicely with Enlightenment thought because monarchy was a system that was not rational and was only adhered to because of tradition.  The French Revolution was strongly anti-clerical.  This, too, was consistent with Enlightenment thought since giving power to the Church and clergy was a traditional idea with no basis in science and reason.

Thus, Enlightenment ideas of rationality and reason and science strongly influenced the Atlantic revolutions.