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1) “No more cause of natural things should be admitted than are both true and sufficient to explain their phenomena”
3) Those qualities of bodies that cannot be intended and remitted (i.e. qualities that cannot be increased and diminished) and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally”
4) “In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true nonwithstanding any contrary hypothesis, until yet other phenomena make such propositions either more exact or liable to exceptions”[[CiteRef::Newton (1999)]]
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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, the calculus I is the primary mathematical technique learned and used in physics and engineering classrooms.
'''Newton on method'''
“I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this experimental philosophy, propositions are deduced from the phenomena and are made general by induction. The impenetrability, mobility, and impetus of bodies and the laws of motion and law of gravity have been found by this method. And it is enough that gravity should really exist and should act according to the laws that we have set forth and should suffice for all the motions of the heavenly bodies and of our sea.”[[CiteRef::Newton (1999)]]
Newton called his method the experimental philosophy, because theories about the behavior of empirical objects can only be refuted via experimental procedures.[[CiteRef::Smith (20042002)]] He expressed the core beliefs from which he derived his method in a set of four “rules for the study of natural philosophy,” which he stated in book III of The Principia as follows: "1) No more cause of natural things should be admitted than are both true and sufficient to explain their phenomena
2) Therefore, the causes assigned to natural effects of the same kind must be, so far as possible, the same
3) Those qualities of bodies that cannot be intended and remitted (i.e. qualities that cannot be increased and diminished) and that belong to all bodies on which experiments can be made should be taken as qualities of all bodies universally 4) In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true nonwithstanding any contrary hypothesis, until yet other phenomena make such propositions either more exact or liable to exceptions.”[[CiteRef::Newton (1999)]] Out of these four rules a new, active method for conducting science emerged that stood in stark contrast the previous passive and theoretical Cartesian and Aristotelian-scholastic methods. Propositions are born from natural sources and placed back into the natural world to be tested empirically.[[CiteRef::Smith (20042002)]] As the four rules were absorbed into the ensuing mosaic, the calculus became deeply incorporated in the experimental method, as it was used to mathematically calculate from natural laws an empirical prediction, and then evaluate how exactly the prediction matched the observed reality.
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 centripedal 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. [Cambridge, chpt. 5 pp. 183-184]
