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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, Galileo Galilei and Robert Boyle (1627-1691) had begun to elaborate and practice an experimental approach to knowledge. Much of Newton's natural philosophy was adapted from Descartes' views, but he was skeptical of Descartes' rationalism and rejected his method of hypotheses outright.[[CiteRef::Janiak (2016)]] Instead, his epistemological views drew from Galileo and Boyle and were similar to those of his contemporary and friend John Locke (1632-1704), who maintained that all knowledge came from experience.[[CiteRef::Rogers (1982)]]

Whereas Descartes did not rely on mathematical reasoning for his deductions of scientific propositions, Newton believed that mathematics was an imperative part of conducting natural philosophy.[[CiteRef::Janiak (2016)]] In Newton’s physics, material objects were not required to be in direct contact with each other in order for motion to occur. Instead, objects react to each other via a force, a new concept which Newton envisioned as a quantifiable property contained in all material objects, the amount of which is directly proportional to the quantity of matter contained in the object. Quantities of force and matter were thus introduced to the mosaic as ontological entities that were measurable. By applying Newton’s three laws of motion, material objects in the universe were compelled to accelerate through action at a distance. Additionally, the laws outlined the mathematical relations between this acceleration and 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 for the unified theory of gravity.[[CiteRef::Smith (2009)]] The mathematical language used in The ''Principia'' was geometry, which was also the basis for many of the major models for celestial mechanics that were studied at the time, including the works of Ptolemy, Copernicus and Kepler.[[CiteRef::Smith (2009)]]

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.

Prior to the publication of The ''Principia'', the philosophy of motion and change in the universe was largely a theoretical and non-mathematical enterprise. The dominating methodological approach to natural philosophy both in the Aristotelian-scholastic and Cartesian mosaic, was one in which truths about the natural world were proposed as conjectural hypotheses. Newton explicitly rejected the method of hypotheses, and instead demanded that all propositions be deduced from the phenomena and then converted into general principles via induction. In the second edition of The ''Principia'', Newton states:

Using these principles, Newton was able to derive the law of universal gravity in the context of his method. In the Cartesian mosaic, the centripetal force had already been defined as the agent that pulled the moon towards the Earth, keeping its orbit circular rather than linear. Newton applied rules 1) and 2) to determine 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.[[CiteRef::Harper (2002)|pp. 183-184]]

Although not all of the ontological changes to the mosaic described in The ''Principia'' were immediately accepted, the new experimental philosophy that he described influenced contemporary scientists within the same century of it’s publication. [Newtons philosophy] Both prominent 17th century natural philosophers 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)]] By 1700 the acceptance of “experimental philosophy” methodological structure had overtaken that of Cartesianism in England.[[CiteRef::Janiak (2016)]]