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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, 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 that outlined the mathematical relations between force and matter, the motion of all material objects 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.[Newtons principia] 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]] The calculus is a mathematical technique that is capable of solving problems in physics involving acceleration, which is a quantity that lay at the heart of Newton’s theory of motion.[[CiteRef::Friedman (2002)]] It was only in the 18th century that mathematicians Jacob Hermann and Leonhard Euler expressed Newton’s laws of motion using calculus.[[CiteRef::Smith (2009)|p. 29]] In preceding 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.
'''Newton on method'''
