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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 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 The works of Ptolemy, he was unable to supply a mechanical explanation for how gravity worked. The Copernicus, and Kepler used the mathematical language used in of geometry. In the ''Principia'' was geometryNewton likewise presented his arguments geometrically. Newton's predecessors however, which was also the basis for prior major models offered only a descriptive account of celestial motions. Newton sought to explain these motions in celestial mechanicsterms of gravitational forces acting between bodies. In order to do this, including the works Newton needed to invent a new branch of Ptolemymathematics, Copernicus ''integral and Keplerdifferential calculus''. None of these earlier works howeverCalculus deals with mathematical quantities that are continuously changing, offered any rigorously mathematical explanation such as the magnitude and direction of the motions they describedgravitational forces acting on an orbiting body. [[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 dealing 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 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)]]
