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=== Newton on Mathematics and Natural Philosophy ===
Newton's two most important works of natural philosophy were the ''Principia'', published in 1687, which dealt with his theories of motion and universal gravitation, and ''Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions, and Colours of Light'', or simply ''Opticks'', which was published in 1704 and dealt with his theories of light and color. [[CiteRef::Westfall(1999)]] More than Descartes, Newton made mathematics central to the conduct of natural philosophy, by producing a general mathematical theory of the motion of bodies. [[CiteRef::Janiak (2016)]] He posited three mathematical laws of motion, together with a law of universal gravitation. Changes in the state of motion of objects were caused by forces acting on them. Quantities of force and amounts of matter were measurable. The laws specified the mathematical relationship between the acceleration experienced by an object, the quantity of matter composing it, and the magnitude of the forces acting on it. [[CiteRef::Smith (2009)]]
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 ''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 published presented his key work arguments in the ''Principia'' using the language of geometry, as a mathematician in the course of his work on forces and motion he is primarily role in inventing 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 had been was the first method that was capable of articulating the quantity of acceleration, unlocking a new world of calculations which geometry as a technique alone had been incapable of solving.[[CiteRef::Friedman (2002)]] Eventually, in the 18th century that mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using Newton's own technique of calculus, but in using the symbolic expression 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.
=== Newton on Methodology ===
