Newton (1642 – 1727)

Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles Of Natural Philosophy") (1687)

Newton takes Descartes' philosophy (the Cartesian Method), formalizes and perfects it, and "proves it" by combining it with Empiricism (observation, Locke etc).

Newton:

a) Formally applies mathematical knowledge to explain physical phenomena.  He is able to focus on purely factual laws describing physical phenomena: showing how strictly mathematical principles (Descartes' method) can explain natural phenomena/universe without recourse to faith or authority.

Of course people had used complex mathematical formulas to describe elements of nature -- the movements of the planets and stars especially -- for thousands of years, but Newton is able to explain the forces guiding those phenomena.

b) Uses experiment (empiricism: observation) as basis of knowledge. This is the final break between both philosophy and theology from this new thing called "science"; scientific  knowledge will not be based on hypothesis/logic/reason etc alone but on observation verified through symbolic logic or math; in other words, he applies Locke et al's empirical philosophy with Descartes et al's analytical/logical method.

c) Invents calculus: in many ways this perfects the analytical method Descartes' said we should use to simplify human thinking, as calculus allowed us to greatly extend our understanding of natural phenomena (note: calculus is simultaneously created by Leibnitz)

In so doing he develops what is often called the first “mind without metaphysics”; or we could say he takes the "meta" out of "metaphysics": that is to say, Newton is the first to explain the workings of the universe without defaulting to faith or God or "the Gods" etc. 

When pushed to explain why the laws of physics are what they are (a metaphysical question), he famously replies "I don't know." In other words, he adopts Descartes' view that we should not call assumptions knowledge: we only "know" those things that we are able to prove and accept as certain; he sticks to self evident truths.