The conflict between the Newtonians and the Leibnizians is often portrayed as a dispute over priority for the invention of the calculus. It was that, but it was much more, since the participants differed on matters of substance concerning theology, natural philosophy, and mathematics. The first and second editions of Newton’s Principia were published during this protracted conflict. By examining a few propositions about orbits and some continental reactions to them we can infer some of the differences between the two schools concerning mathematical methods and values.