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Like DescartesIn contrast with the Cartesian mechanical philosophy, Newton believed that mathematics was an imperative part of conducting natural philosophy.[[CiteRef::Janiak (2016)]] In in Newton’s physics, material objects were not required to be in direct contact with in order to influence each other in order for 's motion to occur. Instead, objects react to each other via forces could act at a forcedistance. To explain both falling bodies and the motions of the moon and planets, Newton posited a new concept which Newton envisioned gravitational force that acted as a quantifiable property contained in all material objects, the amount inverse square of which is directly proportional to the quantity of matter contained in the objectdistance between objects. Quantities of force and matter were thus introduced He claimed to the mosaic as ontological entities that were measurable. By applying Newton’s three have derived this relationship from Kepler's observational laws of planetary motion. But, material objects in the universe were compelled he was unable to accelerate through action at supply a distancemechanical explanation for how gravity worked. Additionally, the laws outlined the mathematical relations between this acceleration and Newton wrote that "I have not yet been able to deduce from phenomena the quantities of force and matter could be explained and predicted mathematically, thereby giving mathematics a new central role in the study of natural philosophy. In the ''Principia'', Newton made extensive use of mathematics in his argument reason for the unified theory 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 many of the prior major models for in celestial mechanics that were studied at the time, 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)]]
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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.
Even though Newton published his key work in the language of geometry, as a mathematician he is primarily role in inventing integral and differential calculus. 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 the first method that was capable of articulating the quantity of acceleration, unlocking a new world of calculations which geometry as a technique had been incapable of solving.[[CiteRef::Friedman (2002)]] Eventually, 18th century that mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using Newton's own technique of calculus, but in the symbolic expression 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.
