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While both Newton’s physics and philosophy were heavily influenced by Descartes’ ideas, they were also a challenge to what had, by then, become the new Cartesian orthodoxy. Descartes' '''mechanical natural philosophy''' was derived from ancient Greek atomism. He was the most prominent member of a community of '''corpuscularist''' thinkers, who maintained that visible objects were made of unobservably tiny particles, whose relations and arrangement were responsible for the properties of visible bodies. Particles influenced one another only by direct physical contact, which was the cause of all motion, and ultimately all change.[[CiteRef::Disalle (2004)]] Aristotle had explained the properties of visible bodies in terms of their form, rather than in terms of the arrangement of their constituent parts. He maintained that heavy objects, composed of the element earth, tended towards their natural place; the center of the universe. The concept of a sphere of earth at rest in the center of the universe was central to Aristotle's entire cosmology. Motion in the terrestrial and celestial realms were seen as fundamentally different.[[CiteRef::Bodnar (2016)]] Descartes' theories explained gravity , qualitatively, as due to a swirling vortex of particles around the Earth, which pushed things towards its center. Celestial motions were not different in kind. In accord with Copernican heliocentrism, Descartes posited that a larger vortex surrounded the sun, with the smaller planetary vorticies caught in a larger solar vortex.[[CiteRef::Garber (1992)]][[CiteRef::Disalle (2004)]] In Newton's time, major champions of the mechanical natural philosophy included Christiaan Huygens (1629-1695) and Gottfried Wilhelm Leibniz (1646-1716), who was to become a major rival of Newton's. By the time Newton published his magnum opus, ''Philosophiae Naturalis Principia Mathematica'' (''Mathematical Principles of Natural Philosophy'')in 1687, Descartes' views had been accepted at Cambridge. The title of Newton's work suggests he intended it to be in dialog with Descartes' ''Principia Philosophiae'' (''Principles of Philosophy'') published in 1644.[[CiteRef::Janiak (2016)]] Newton contested Cartesianism as the orthodoxy he sought to overturn.

Descartes saw the ultimate justification of knowledge claims to lie with human reason and the absence of doubt. He relied on classical methods of theorizing and conjectured hypotheses in order to construct scientific propositions.[[CiteRef::Janiak (2016)]] Such a '''rationalist''' approach to knowledge was also championed by Baruch Spinoza (1632-1677), Nicolas Malebranche (1638-1715), and by Gottfried Wilhelm Leibniz.[[CiteRef::Lennon and Dea (2014)]] But, by the early 17th century, experimental researchers like Galileo Galilei and Robert Boyle (1627-1691) had begun to elaborate and practice a very different approach to knowledge based on experimentation and extensive use of mathematics. Following the '''inductive methodology ''' advocated by [[Francis Bacon]](1561-1626), they maintained that theoretical principles emerged from experimental data by a process of '''inductive generalization'''. However, there were also dissenters like Newton's contemporary Christiaan Huygens, who believed that most experimental work involved formulating hypotheses about unobservable entities, which were tested by their observable consequences. This was an early form of '''hypothetico-deductivism'''. Newton rejected Cartesian rationalism, and argued that the Cartesians did not sufficiently employ mathematics and experimentation in their work. He rejected the method of hypotheses outright. [[CiteRef::McMullin (2001)]][[CiteRef::Janiak (2016)]] He supported '''inductivism''', and held epistemological views similar to those of his contemporary and friend [[John Locke]](1632-1704), who maintained that all knowledge came from experience.[[CiteRef::Rogers (1982)]]

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 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 acceleration, 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 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.