Axiomata Sive Leges Motus--Newton, Principa, p. 12.

In the *Principia*, after Newton introduces his definitions, and his scholium to the definitions, he turns the page and offers three "laws of motion." And while Newton's own formulation was displaced during the development of 'Newtonian' or 'classical' mechanics (Newton works with proportions not equalities), they have become familiar to us as "laws." But it is notable that Newton also introduces them as "Axioms."

*Why?*

I have never read a satisfactory account of this oddity, and here I want to start developing an answer.

Newton's use of *'Sive'* is, I propose, like the Spinozistic 'sive' (as in *Deus sive natura*), where it really means, 'that is,' rather than a disjunctive 'or.' So, Newton is suggesting that one can treat the same entities as axioms and as laws of motion. Before I explain why he would suggest that, I first offer an interpretation of each on its own terms.

In the early modern period, an 'axiom' *can *be understood as an established principle in a science (see, for example, *Chambers' Cyclopedia, 182*). A 'principle," in turn, is a foundational, established ground or (general) cause. Newton is suggesting, thereby, that he is taking for granted a body of work (in 17th century mechanics). In fact, the three "laws" are, as Newton himself implies, implicit in and used by Huygens in his 1672 *Horologium*. To be sure, Newton's own way of presenting them, and interpreting their reach, was not standard; it was unprecedented to apply them to celestial affairs; nor was his way of understanding the third law of motion common (as became clear over its application to systems of bodies that are not in contact, but far apart).

That Newton treats them as a *laws of motion* is a firm nod to Descartes (as is widely recognized in the scholarly literature due to influence of I.B. Cohen and Howard Stein.) Newton's *Principia* is meant to displace Descartes' *Principia -- *with Newton proudly announcing that his are truly mathematical principles (with the implication being that Descartes only paid lip-serve to the role of mathematics in natural philosophy)* -- *and Newton's laws are meant to displace Descartes' *purported* laws.

But "laws" talk has a further implication: they are, in terminology of the age, *second* causes. That is, God is the first cause, but God's agency works by way of second causes (that is, laws of nature or laws of motion). Now, in Descartes scholarship it is an open debate what the status of laws are. I happen to agree with Hattab that for Descartes, laws really are genuine secondary causes. Similarly, I have argued that for Newton laws are really genuine causes -- akin (but not identical) to formal causes in the traditional, Scholastic conception, while forces are efficient causes for Newton. [To be sure there are alternative interpretations, but it is worth alerting the general reader that theses days few Newton scholars defend the once popular idea that Newton was an instrumentalist about either the laws or forces.]

So, by using 'Axioms, that is, laws of motion' Newton is signaling both (a) that he is developing his account from shared premises (this is also clear in his treatment of what counts as "phenomena" in Book 3 of the *Principia*), that's why he uses 'axioms,' as well as that these (b) may also be understood as a description of God's creation rather than a *mere* hypothetical model (as Descartes and Huygens tended to suggest); (b) fits Newton's language (he considers his own system as "true") as codified in his fourth rule of reasoning (but that was added only in third edition. As Andreas Hüttemann suggested to me in correspondence, (a) connects to the fact that Newton is offering a mathematical treatment (as 'axioms' are familiar from that use), while (b) reminds us that it is not a mere mathematical model, but identifying genuine causes.

As an aside, it is sometimes suggested (e.g., by Andrew Janiak and Steffen Ducheyne in their books on Newton) that Book 1 of the P*rincipia* is a mathematical treatment and Book 3 offers a physical (or causal) treatment; there is a way of reading Newton this way. But that has always struck me as misleading (not just because Book 2 ends up having an odd status). For, at the start of section 11 (of Book I), Newton acknowledges that the preceding results are all based on an un-physical assumption: They concern ‘‘bodies attracted toward an immovable center, such as, however, hardly exists in the natural world.’’ In light of the third law, Newton then introduces more realistic assumptions into his model that (in propositions 58–61) apply to Keplerian motion (of Earth–Moon–Sun systems). Subsequent sections relax even more of the simplifying assumptions, and the models become increasingly less idealized. This suggests that Newton recognized that his mathematical models had differing relationships to reality. This is especially evident in the scholia that Newton included in Book I— these often deal with issues of traditional natural philosophy; for example, the last two scholia and the surrounding propositions in the final section (14) of Book I treat of Snell’s law, the nature of lenses, and the speed of light. (For more on this, see my handbook paper with Chris Smeenk, or this review.)

I do not claim that the above is the final word on these matters. The true test for my interpretation lies in the nitty gritty detail of the wordings and applications of the 'laws.' For example, I have been silent here about one peculiarity of the laws of motion: they also hint at an account of* body* (although that is less complete than the treatment of motion). But that's for another time.

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.

Posted by: Anthony | 03/10/2017 at 10:02 PM

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.

Posted by: John Heinmiller | 05/12/2018 at 06:33 PM

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

Posted by: Eric Schliesser | 05/16/2018 at 12:28 PM

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

Posted by: Brian Ellis | 02/01/2020 at 12:59 AM