On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*

Question

When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the necessary calculus is being described in geometric language. Fortunately for us lesser mortals, the mathematician-astrophysicist Chandrasekhar has written a book that rephrases Newton's arguments in the language of modern-day calculus. However, if we stick to Newton's way of doing things, how can we improve the rigor of his synthetic-geometric arguments?