In formulating his physical theories, Newton developed and used mathematical methods now included in the field of calculus. But the language of calculus as we know it was largely absent from the Principia; Newton gave many of his proofs in a geometric form of infinitesimal calculus, based on limits of ratios of vanishing small geometric quantities.
I see what you did there.
But seriously, the role of calculus is kinda interesting because he did it all geometrically, apparently: http://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica