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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'' which was published in 1704 and dealt with his theories of light and color. [[CiteRef::Westfall(1999)]] More than Descartes, Newton made mathematics much more central to the conduct of natural philosophythan Descartes, 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. 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. The works of Ptolemy, Copernicus, and Kepler used the mathematical language of geometry in their descriptive accounts of celestial motions. In the ''Principia'' Newton likewise presented his arguments geometrically. Newton sought not simply to describe celestial motions, but to explain these motions in terms of gravitational forces acting between bodies. In order to do this, Newton invented a new branch of mathematics, '''integral and differential calculus'''. Calculus deals with mathematical quantities that are continuously changing, such as the magnitude and direction of gravitational forces acting on an orbiting body. [[CiteRef::Friedman (2002)]][[CiteRef::Smith (2009)]] He developed the basic concept of calculus during 1665-6, while Cambridge University was closed due to a plague. [[CiteRef::Cohen and Smith (Eds.) (2002)|p. 10]]
# In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypothesis, until yet other phenomena make such propositions either more exact or liable to exceptions.[[CiteRef::Newton (1999)|pp. 794-796]]</blockquote>
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 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, 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, which Aristotle had deemed to be distinct realms.[[CiteRef::Harper (2002)|pp. 183-184]]
Historical research indicates that the scientific community did not use Newton's own criteria in evaluating his work. Newton's theories did not become accepted outside of England until after its prediction of the oblate spheroid shape of the Earth was confirmed by expeditions to Lapland and Peru. Thus, Newton's theories became accepted via a hypothetico-deductive method based on confirmed novel predictions that distinguished it from the rival Cartesian vortices, rather than via Newton's own inductive methodology. [[CiteRef::Barseghyan (2015)|p. 48-49]][[CiteRef::Terrall (1992)]][[CiteRef::McMullin (2001)]] According to McMullin, Newton's methodology ran contrary to the consensus that had been emerging among natural philosophers of his time, in favor of hypothesis. [[CiteRef::McMullin (2001)]] Christiaan Huygens and John Locke are known to have taken the experimental philosophy, if not necessarily the full content of Newton’s theories, to heart.[[CiteRef::Janiak (2016)]]
