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In contrast with the Cartesian mechanical philosophy, in Newton’s physics, material objects were not required to be in direct contact in order to influence each other's motion. Instead,forces could act at a distance. To explain both falling bodies and the motions of the moon and planets, Newton posited a '''gravitational force''' that acted as the inverse square of the distance between objects. He claimed to have derived this relationship from Kepler's observational laws of planetary motion. But, he was unable to supply a mechanical explanation for how gravity worked. Newton wrote that "I have not yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses". [[CiteRef::Smith (2009)|p. 7]] The mathematical language used in The the ''Principia'' was geometry, which was also the basis for prior major models in celestial mechanics, including the works of Ptolemy, Copernicus and Kepler. None of these earlier works however, offered any rigorously mathematical explanation of the motions they described. [[CiteRef::Smith (2009)]]

Even though Newton presented his arguments in the ''Principia'' using the language of geometry, in the course of his work on forces and motion he invented '''integral and differential calculus'''. Although Newton circulated manuscripts, he did not actually publish his work on calculus until the first decade of the eighteenth century. [[CiteRef::Cohen and Smith (Eds.) (2002)| p. 20]] He is co-credited independently for the calculus alongside his contemporary and rival natural philosopher, Leibniz.[[CiteRef::Cohen and Smith (Eds.) (2002)|pp. 13-14]] As a mathematical technique, calculus was the first method capable of articulating the quantity of accelerationdealing with constantly changing quantities, unlocking a new world of calculations which geometry alone had been incapable of solving.[[CiteRef::Friedman (2002)]] Eventually, in the 18th century , mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using Newton's own technique of calculus, but using and the symbolic expressions that Leibniz had developed.[[CiteRef::Smith (2009)|p. 29]] In following years, calculus became indispensable tool for scientists in the Newtonian mosaic to solve problems in physics, and to predict the behaviour of material objects with an unprecedented degree of accuracy.[[CiteRef::Smith (2009)]] Although geometry is still taught in schools today, calculus is the primary mathematical technique learned and used in physics and engineering classrooms.

Out of these four rules a new, engaged method for conducting science emerged that stood in stark contrast to the previous passive and theoretical Cartesian and Aristotelian-scholastic methods. Propositions formulated based on observations of the natural world and placed back into the natural world to be tested empirically.[[CiteRef::Smith (2002)]] The calculus became deeply incorporated in into the experimental method, as it was used to mathematically calculate empirical predictions from natural laws, and then evaluate how exactly the prediction matched the observed reality. Newton claimed to have derived his law of universal gravitation using this method as applied to Kepler's laws of planetary motion. In the Cartesian natural philosophy, the centripetal force had already been defined as the agent that pulled the moon towards the Earth, keeping its orbit circular rather than linear. Newton appealed to rules 1) and 2) to claim that the centripetal force, and the force that compelled objects to move downwards towards the Earth, were merely two different expressions of the same thing. Newton then went on to apply the third rule, and argue that this force, which he called gravity, must be a universal property of all material objects. From here, he went on to argue for the unification of superlunary and sublunary phenomena.[[CiteRef::Harper (2002)|pp. 183-184]]

|Criticism=Although many natural philosophers in the 17th century were convinced by Newton’s views on the the proper method To proponents of conducting science, many were not willing to abandon the Cartesian mechanical philosophy, it was important that all motion in the universe be given a cause involving direct physical contact.  Contemporary philosopher Leibniz in particular was concerned that the theory of gravity as a regression in natural philosophy, as Newton could not account for the source of gravity. To the Cartesians, it was more important that all motion in the universe could be given a direct cause, which was only possible under the mechanical philosophy, even if this amounted to a larger gap between theory and experimental evidence.[[CiteRef::Janiak (2016)]]Although many natural philosophers in the 17th century were convinced by Newton’s views on the the proper method of conducting science, many were not willing to abandon the Cartesian mechanical philosophy.