The Enlightenment idea of Natural Law influenced both the American and French Revolutions. In this theory, law derived from nature and was innately moral. Parliament had no right to impose immoral ("unnatural") laws on the American colonies, such as the Stamp Act, which John Adams declared "of no binding force" because it violated Natural Law (not to mention English Common Law). Natural Laws were "self evident," as stated in the Declaration of Independence and included the concepts of liberty and the dignity of mankind. While Americans tended to ground ideas of Natural Law in the word of a Biblical God, the French revolutionaries wanted to eradicate Christianity in their pursuit of Natural Law, which they located in notions of liberty, equality and the brotherhood of all people. 

Republicanism was another Enlightenment notion embraced by both the French and the Americans. The French revolutionaries abolished the monarchy, beheading the king and queen, while the Americans refused to establish a hereditary monarchy, opting instead for an elected President. This completely turned upside down medieval and Renaissance notions of a great chain of being, in which power was top-down, passed by God through his anointed monarch, and from thence to the aristocracy and the people. It is difficult for those of us in the United States especially, where a republican form of government seems "natural" and unthreatening, to understand how frightening this concept was to the established elites in Europe at the time. However, both the American and French revolutionaries wanted to get rid of the "tyranny" of monarchy, which they found oppressive and in violation of their natural rights.

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.

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.

How did Enlightenment ideas contribute to the Atlantic revolutions?

Ideas in Enlightenment played a role in the Atlantic revolutions as shown by the objectives of these revolutions among them being republicanism. Republicanism is a type of government where the leader is a representative of the people unlike in monarchies and aristocracies where the people are subjects to the leader. This ideology comes from the idea of Enlightenment which states that all men are created equal and nature accords them equal opportunities. Atlantic revolutions were thus motivated by the need to change governance of territories from monarchies and aristocracies to republicanism.

An example of Enlightenment ideals can be found in statements made in the American Declaration of Independence “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable rights that among these are Life, Liberty and the pursuit of Happiness.” Similar statements were documented in other countries that were involved in the Atlantic revolution and in other revolutions throughout the world.

The ideas of the Enlightenment contributed to the Atlantic revolutions in at least two important ways.

First, these ideas gave the revolutionaries a logic for challenging the regimes that ruled them.  They were not simply trying to overthrow one king for another.  Instead, they were questioning an entire system based on the idea that the system was not based on reason and natural law.

Second, these ideas gave the revolutionaries an alternative system to strive for.  The rebels did not simply have to be against the current system.  Instead, they had ideas of things like government by the consent of the governed and separation of powers.  These ideas had come to them through such Enlightenment thinkers as Locke and Montesquieu.

In these ways, Enlightenment ideas contributed greatly to the Atlantic revolutions.