Q:

What philosophical problem did Newton solve?

In a philosophy 101 lecture I listened to recently, the lecturer said that if a philosopher is successful and is able to solve a philosophical question for good, the topic of that question branches out of philosophy and becomes a separate field of study. 
He gave as an example mechanics: Newton was a philosopher who was successful, and in the process created the science of mechanics. 
If that is the case, what was the purely philosophical problem that Newton was trying to solve (as opposed to the physical/mathematical problem of how to quantify the laws of motion)?   

A:

The problems of motion and gravity were philosophical problems before the scientific revolution. People followed Aristotle's teachings.
https://en.wikipedia.org/wiki/Aristotelian_physics
Look at how Aristotle described motion and gravity. There's no mathematics involved... no empiricism really. It's all argued from first principles, starting with all sorts of assumptions. More philosophical than scientific.
Bacon, Descartes, Galileo, Newton... they all had a hand in overthrowing Aristotelian natural philosophy. So Newton along with others turned physics (and science) into a mathematical and empirical matter. 
With Newton, we get his laws of motion, and his law of gravity. Mathematical and empirical.
Look at the various aspects of the scientific method described here:
https://en.wikipedia.org/wiki/Scientific_revolution#Scientific_method
It all goes against Aristotle.

A:

A science is a well-defined systematic way of organizing and extending a body of knowledge.  What makes it a science is its reliability, not its subject.
Philosophy explores subjects that are outside the realm of science, which is to say, that do not have a well-established, reliable, systematic way of approaching them. Your lecturer is stating that when a philosopher successfully creates a reliable, systematic way of creating a body of knowledge around a particular discipline, that discipline passes out of the realm of philosophy and into the realm of science.
While the consideration of the laws of motion seems, from our perspective, to be the very paradigm of a science, it only looks that way post-Newton (and his contemporaries).  It wasn't a mathematical problem previously until Newton (and Leibniz) discovered the mathematical relationship.  He approached the subject as a philosopher, and left it as a scientist. 
(PLEASE NOTE: This answer is substantively the same as Ameet's, just with a different emphasis)

