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|Historical Context=When Isaac Newton began his studies at Cambridge University's prestigious Trinity College in 1661, more than a century had passed since Nicolaus Copernicus (1473-1543) had proposed a '''heliocentric cosmology''' in his 1543 ''De Revolutionibus Orbium Coelestium'' (''On the Revolutions of Heavenly Spheres''). It had been fifty years since Galileo Galilei (1564-1642) had published his observations with the telescope in 1610, which uncovered dramatic evidence favoring the Copernican system. Around the same time, Johannes Kepler (1571-1630) had published his laws of planetary motion, indicating that the planets revolved around the sun on elliptical paths, replacing the circular motion and complex epicycles of Copernicus and Ptolemy.[[CiteRef::Westfall (1980)|pp. 1-7]] According to Westfall, "by 1661 the debate on the heliocentric universe had been settled; those who mattered had surrendered to the irresistible elegance of Kepler's unencumbered ellipses, supported by the striking testimony of the telescope, whatever the ambiguities might be. For Newton, the heliocentric universe was never a matter in question".[[CiteRef::Westfall (1980)|p. 6]] A planetary Earth that rotated on its axis and revolved around the sun was incompatible with the accepted physics of [[Aristotle]] (384-322 BCE). The community of the time was engaged with the question of how it could be that the Earth itself was in motion through space, and with the question of how one could hope to gain reliable knowledge in the face of the failure of Aristotelian scholastic knowledge accepted for centuries.
Newton’s education at Cambridge was classical, focusing on Aristotelian rhetoric, logic, ethics, and physics. Bound to '''Aristotelian scholasticism''' by statutory rules,[[CiteRef::Christianson (1984)|p. 33]] the curriculum had changed little in decades.[[CiteRef::Westfall (1980)|pp. 81-90]][[CiteRef::Smith (2009)]] Like many of the more ambitious students, Newton distanced himself from classical metaphysics and instead studied the works of the French natural philosopher [[René Descartes]](1596-1650) on his own. By 1664, Newton is known to have read the 1656 Latin edition of Descartes' ''Opera Philosophica'', a one volume compilation of Descartes' major works.[[CiteRef::Smith (2009)]] Newton is known to have been profoundly influenced by Descartes views of space, matter, and God, and by commentaries on Descartes by Henry More (1614-1687). [[CiteRef::Janiak (2014)]] Descartes had died just over a decade earlier, and his works had first been published within the preceding thirty years. They were gaining in popularity and by about 1680 would become the [[Theory Acceptance|accepted]] centerpiece of the Cambridge curriculum, as they also would in Paris by 1700.[[CiteRef::Barseghyan (2015)|p. 190]] When Newton published his magnum opus, the ''Philosophiae Naturalis Principia Mathematica'' (''Mathematical Principles of Natural Philosophy'')in 1687, he was challenging a Cartesian orthodoxy. The full 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)]]
Descartes 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. In this '''mechanical natural philosophy''', particles influenced one another only by direct physical contact, which was the cause of all motion, and ultimately all change.[[CiteRef::Disalle (2004)]] One of the attractions of these ideas is that, unlike Aristotle's, they allowed for a movable planetary Earth, and celestial motions weren't different in kind from terrestrial motions. They explained gravity, in qualitative terms, as due to a swirling vortex of particles around the Earth, which pushed things towards its center. 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.
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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[[CiteRef::Newton (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'' [[CiteRef::Newton (1704)]] which was published in 1704 and dealt with his theories of light and color. [[CiteRef::Westfall(1999)]] Newton made mathematics much more central to the conduct of natural philosophy than 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]]
