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Are there any traces of the third law of motion before Newton's Principia? wiki says

Newton arrived at his set of three laws incrementally. In a 1684 manuscript written to Huygens, he listed four laws: the principle of inertia, the change of motion by force, a statement about relative motion that would today be called Galilean invariance, and the rule that interactions between bodies do not change the motion of their center of mass. In a later manuscript, Newton added a law of action and reaction, while saying that this law and the law regarding the center of mass implied one another. Newton probably settled on the presentation in the Principia, with three primary laws and then other statements reduced to corollaries, during 1685.

Are there really no hints about this principle (not really conservation of momentum) before 1684?

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    $\begingroup$ Before 1684 and before Newton are two different things. Newton formulated an equivalent of the third law as early as 1665, but e.g. Dugas flatly states:"This principle belongs properly to Newton: none of his predecessors had formulated it." Indeed, nothing of the sort is to be found even in Wallis, Wren and Huygens, the trio that Newton himself credited for it in Principia (to facilitate its acceptance, no doubt), see Home, Third Law in Newton's Mechanics. The prevailing view was essentially that the equality of forces is obviously false. $\endgroup$
    – Conifold
    Sep 5, 2022 at 0:37
  • $\begingroup$ @Conifold : Have you got a specific reason to suggest that Newton's crediting of Wallis, Wren & Huygens was less than straightforward? Among them they account for substantial work and results on collisions, which is widely thought to be a significant element in Newton's development of the 3rd law. Why is that not sufficient to justify an acknowledgement? How is it appropriate to suggest insincerity on Newton's part in the absence of evidence? $\endgroup$
    – terry-s
    Sep 5, 2022 at 1:01
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    $\begingroup$ @terry-s "He apparently felt little need to justify his first two laws... yet he went to some lengths to justify the third law... Newton knew that his third law was by no means as universally received as he had claimed... Most notably, the law was not an obvious consequence of the principle of conservation of momentum. Because Newton's predecessors confused "forces of percussion" with the "force" a body had by virtue of its motion, they were unable to move from the principle to the law". Newton's credits were not exactly straightforward with the inverse square law either. $\endgroup$
    – Conifold
    Sep 5, 2022 at 2:23
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    $\begingroup$ @Conifold : I suggest you're confusing the (idiosyncratic) views of modern interpreters (or misinterpreters) with actual evidence that bears on Newton's intent. Home's opinion is not generally accepted, the quotation also does not show the relevance (to Newton's intent here) of conservation of momentum. Newton acknowledged people who had worked on collision theory, the 'Waste book' content shows this was also background to N's development of the 3rd law, the quotation shows no sign of impropriety about that. The imputation about the inv-sq law is a separate matter needing address elsewhere. $\endgroup$
    – terry-s
    Sep 5, 2022 at 9:30
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    $\begingroup$ @terry-s Newton opens the Scholium with, in his own words:"Hitherto I have laid down such principles as have been received by mathematicians", and later writes "By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion..." If his intention was to credit them for collisions only and then bridge to his third law, "as have been received" and "together with did determine" are a strange way of putting it. And "greatest geometers of our times" is smooth too. $\endgroup$
    – Conifold
    Sep 6, 2022 at 7:58

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Much early thinking about what became the laws of motion in the Principia, including the third law, can be found in Newton's "Waste Book". It is a notebook started in 1665, and it is discussed by a number of authors, including D Fraser (2005), "The third law in Newton's Waste book" (Studies in History and Philosophy of Science Part A 36(1):43-60). There is also some related material in another notebook of the period ('Questiones quaedam philosophicae', 'Certain philosophical questions').

The 'Waste Book' seems to be the only Newton manuscript source offering advance hints of the third law before the 1684 manuscript 'De motu' (an initial precursor of 'Principia', written on the stimulus of questions put to Newton by Edmund Halley, and sent to Halley and to the Royal Society). The 'Waste Book' casts some light on the origins and justification of the third law, as discussed in the Fraser paper cited above (p.50): this mentions that while Newton's words suggest he thought it was obvious that action and reaction were opposite in direction, e.g. when two bodies press against each other, it was not so obvious at first that the action and reaction were also equal in size.

Much of the ingredient thinking on dynamics that went into the 'Waste Book' appears to have related to the problem of formulating laws that would account acceptably for the phenomena of bodies in collision. As also argued by Fraser (2005), some of the relevant arguments relate to the equal actions on bodies at both ends of a compressed spring.

All of these sources may give insight into Newton's thinking, but there remain always dangers of anachronism and misinterpretation in reading such documents; and on some points, the language of the sources can permit different authors to draw different conclusions about their content and implications. For example, R S Westfall (1971), 'Force in Newton's Physics' (at p.348) considering the Waste book, wrote that

"Except for the special case in which [two interacting bodies] p and r have equal motions, the conclusion that they press each other equally had seemed obviously false to the great majority of those who before 1665 had attempted to analyze the force of percussion. The Waste book suggests that Newton came to deny the intuitively obvious by recognising that for every impact, there is a frame of reference, that of the common centre of gravity, in which the two bodies do have equal forces. That is, his treatment of impact depended on his accepting a relativity of motion in terms of which the idea of an absolute force of a body's motion is meaningless."

On the other hand, Fraser (2005:53) thought that

"the evidence suggests that the third law originated in the empirical or theoretical investigation of elasticity and that, for Newton, its justification lay in his application of an elastic model of collisions."

And then Westfall (1971:472) had already pointed out that

"in demonstrating the validity of the third law in the case of two bodies attracting each other, Newton {for the sake of argument} imagined B to attract A more strongly than A attracts B. If that were the case, there would be an unbalanced force in the system of two bodies, and the system would move in the line AB ‘in infinitum with a motion perpetually accelerated’ -- a situation contradicting the first law." (with quote from the Scholium to the laws of motion (1729:p37)).

In this way numerous lines of argument converge on the third law. Whichever of them accounts better for the history of its development, the answer remains a 'yes', there are several antecedent hints or traces of the third law found in Newton's writing before the 'Principia' and before the 1684 'De motu', with clear signs that he gave attention to the subject matter in the 1660s.

As a postscript, it's also relevant that Newton made no claim that the laws of motion he stated in the 'Principia' were his own. In the 'Scholium' to the laws of motion Newton credited several others (Galileo, Wren, Wallis, Huygens, Mariotte). He also called the laws 'axioms', indicating that he thought they made generally acceptable starting-points not in need of proof: he wrote (1729 English translation of Principia, p.31) that they were "such principles as have been receiv'd by Mathematicians, and are confirm'd by abundance of experiments", i.e. largely experiments of others. (The naming of the laws as Newton's was made by others long after his lifetime: probably in recognition that as a set and a compilation they were not in use before Newton: he selected and combined them from among competing prior-proposed principles, and also amended their expression and scope. Concepts of action and reaction do predate Newton's work, but the third law gave precision to the concept of this paired interaction, e.g. R S Westfall (1971), 'Force in Newton's Physics': especially discussion of Newton's various drafts of laws of motion, pp.448-456.)

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