Newton's Corollary #1 to the Laws of Motion (Principia)

Question

I'm currently working through selected portions of Newton's Principia, but I'm already stuck in trying to understand his explanation for the first corollary (i.e., Corollary I) to the laws of motion. Here's a link for quick reference: https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/Axioms,_or_Laws_of_Motion

While I believe I understand the corollary as a whole, I'm primarily baffled by his last sentence in his explanation, namely, "...Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I."

As many of you might recognize, Law I states the law of inertia: "Every body preserves in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon." In other words, objects experiencing no external forces either remain at rest or continue their motion in a straight line.

With Corollary I considering motion for a body under two simultaneous forces while Law I considers motion in absence of forces, I don't see how Law I is even relevant for drawing that conclusion in that last sentence. Maybe one of you might know?

On a further note, in the Scholium following the corollaries, Newton credits Galileo for the first two Laws and the first two Corollaries; however, I'm unable to find anything direct from Galileo regarding the first corollary.

Answer 1

The question you raise is a good one, and it has to do with two of the laws of motion, both the first law (about inertia and the lack of need for any external influence to maintain an existing straight-line motion), and the second law (about the effect of impressed force to change motion).

There has been some published controversy (and likely misunderstanding as to Newton's intent) about the second law -- and a question discussed has been whether the second law is only about force impressed in an instant, as an impulse, or whether it is also intended to cover the effects of continuous impressed forces. I suggest that a good place from which to consider the variety of published views about this would be a 2006 paper in Archives for the History of Exact Sciences (vol. 60 pp.157-207), by Bruce Pourciau (mathematical professor at Lawrence University, who has written extensively on Newton and mathematical physics), entitled "Newton's Interpretation of Newton's Second Law". Pourciau argues strongly and (as I believe) effectively, that Newton's intent was that according to the occasion or context the second law was intended to apply directly both to impulsive and to continuous forces. (A similar conclusion was reached on this particular point by Michael Nauenberg (Am. J. Phys. 80, 931-3 (2012) -- though it was in the context of disagreement with Pourciau on other points.)

In the context that you raise -- i.e. the first corollary to the laws of motion -- I believe you are quite correct in identifying this as an illustration of impulsive force. For an example in support of that, the corollary begins by mentioning 'a force M impressed apart in the place A' (1729 translation). 'In the place A' is wording that clearly (for this example) localises to a point the particular force that is spoken of here. So your own consequential reasoning looks quite right, that after the localised force that is mentioned, the motion that it generated is thereafter governed by the law of inertia (and the same is additively true for the other force mentioned).

So I agree with your clarification for the context in which it is made, adding only, for the generality of cases, a suggestion to read the Pourciau paper and some of the others he cites before making any general conclusions about whether in other contexts the second law has to concern impulses only -- which for the reasons given in the cited paper would be very arguably a mistake.

Answer 2

Seems silly to answer my own question, but maybe this will help someone later on.

After some digging around, I realized my confusion was due to a misunderstanding of Newton's use of "impressed force." I was able to find commentary by James C. Maxwell in his book "Matter and Motion" in which he clarifies that "impressed force" is actually understood to be "impulse." With that interpretation, Corollary 1 becomes much clearer since the motion illustrated is understood to be the result of an impulse that is acted on the body at point A.

Answer 3

There are draft versions of the manuscript in which Newton works with a set of six laws of motion, with corrolaries. In the end Newton opted to designate three of them as 'laws of motion', with corrolaries.

The presentation style in the Principia follows the common practice of the time of using the type of structure of Euclid's Elements, as that work was regarded as the pinnacle of scholarly achievement. As we know: Euclid presents axioms, and develops theorems.

However: over the centuries it has become evident that even in mathematics axiomatization is problematic, in the sense that arriving at an exhaustive set of axioms is more challenging than earlier assumed. There is the example of Hilbert's axioms Hilbert had set out to arrive at an exhaustive set of axioms for Euclidean geometry. Hilbert's axioms are a set of 20.

From a modern perspective:
The laws of motion as presented in the Principia should not be expected to provide an exhaustive set of axioms.

In my opinion: the designation 'Axioms' is an overreach. In effect a set of 'laws of motion' serves as a focus, it is a way of communicating: these concepts are important.

From a modern perspective we can recognize the parallelogram rule as a statement asserting that all vector quantities add according to vector addition in Euclidean geometry.

For vector quantities to not add according to vector-addition-in-Euclidean-geometry space itself would have to be non-euclidean. Well, none of the contemporaries of Newton would raise the possibility that space may be non-euclidean

It seems likely that Newton reassigned the parallellogram rule from law to corrolary because he and everyone of his contemporaries totally expected the parallellogram rule to hold good.

Recommended reading:
The answers to the question here on hsm stackexchange:
What were Newton's six laws of motion?