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.
The Age of Enlightenment was about challenging the traditional lines of authority that existed at that time. Focus was directed to individualism and reason with issues thought out logically. According to Immanuel Kant the idea behind the Enlightenment can be expressed as the freedom for one to use one's own intelligence. Intelligence meant challenging existing political and social systems with regard to authority, justice and equality.
The Atlantic revolutions in the Americas and Europe were geared towards establishing Enlightenment ideals within the participating countries. These ideals included equality of all men and the rejection of imperialism, nobility and titles as seen in the American Declaration of Independence and Constitution. Most countries that participated in the revolution (such as Haiti) came up with their own versions of Enlightenment ideals in their push for independence.