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I came across this today while looking for works on the method of Newton's reasoning. It is a fascinating subject and the article is illuminating.

Newton seems to have inferred, and tested by many experiments, that certain constrained motions consisted of two components: an initial velocity; and a centripetal motion.

He seems to have been (or was) the first person to derive a mathematical statement of the centripetal motion. He found this was also true of the celestial motions observed by Kepler and Halley, and inferred it was true of any orbital motion.

I was astonished to find in Principia that Newton explicitly does not define a cause, seen in Definition 8. He says that "force" is only a mathematical statement of effect, and not an explanation of cause. Gravity is not a cause, but an effect, the cause of which is completely unknown.

The Axioms or Laws seem to play the role of a theory, which Newton demonstrates to be true of all known observations of motion. The real axioms come in Book III Rules of Philosophizing, because there is no reason to accept that celestial motion has the same "cause" unless you accept Rule 2: "Accordingly, to natural effects of the same kind, the same causes should be assigned, as far as possible". Otherwise the celestial motion could be an interesting coincidence.

John Heinmiller

Newton calls his principles the Three Axioms and not the Three Laws for a very simple reason. He calls them axioms because they are fundamental to his thinking. They are necessarily first principles, much like Euclid's five axioms are, they can never be proved to perfection though they can be disproved by a single observation, again like Euclid's five axioms and all of his calculations are dependent upon these, again like Euclid's five axioms.

If we get rid of even one of his three first principles, we have to redo almost all of Newton's calculations, for they will suddenly not make any sense. We have to assume that bodies remain in a specific state of motion unless a force is applied to it, we have to assume that the force acts in a specific way and we have to assume that there is an equal and opposite reaction, to do otherwise is to render each and every one of the calculations in Newton's great Principia becomes mere exercises of mathematics, without any meaning in real life. This gives the three principles a fundamental importance beyond anything in the rest of the book.

I know that a lot of people prefer to call his three principles laws instead of axioms. But Newton was entirely correct because, just as Euclid's five axioms were so fundamental to his entire geometry, Newton's three axioms are absolutely and totally fundamental to his physics. They are the first principles, the root foundation, without which all of his calculations and analysis are absolutely meaningless.

Eric Schliesser

Newton calls his three Axioms also Laws: Axiomata sive leges Motus.

Brian Ellis

Newton's concept of motive force is not the modern concept of force. I wrote a paper on this in the Journal of the History of Ideas Vol xxiii, Np.2. It is in fact the integral form f = ma. What Newton actually said in his Principia was
The change of motion of a body (= mass x change of velocity) is proportional to the motive force impressed; and made in the direction of the right line in which that force is impressed.
In explanation of Axiom 2, (in the very next sentence), Newton writes:
If any force generates a motion, a double force will generate double the motion, a triple force, triple the motion, whether that force be impressed all together and at once, or gradually and successively. (my italics).
The is just one of infinitely many integral forms, because there should be a constant of integration, if he were deriving it from f = ma. But we know that this constant is zero, because Newton includes this information in his Axiom 1, which is Descartes' principle of natural motion. The integral form is also more intuitive. Newton's concept of motive force is that of 'a cause of change of motion'. It is a whole temporally extended event, not today's concept of motive force. Of course, Newton uses the modern concept to, e.g. in his law of gravitation.
Brian Ellis, Emeritus Professor of Philosophy, La Trobe University

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