Distinguishing Newton's First and Second Laws

Isaac Newton's three laws of motion are: (1) if F = 0 then a = 0; (2) F = ma; (3) actions have reactions.

The first law is a trivial consequence of the second. Why did Newton bother to state it as a separate law?

A new paper answers the question:

On the independence problem of Newton's first law Ido Yavetz, Ehud Aharoni Newton's laws of motion pose an apparent problem, sometimes referred to as "the independence problem": the first law seems to be a simple consequence of the second law, raising the question of why it was included as a separate law. Numerous answers to this question have been proposed in the literature. The main contribution of this paper is a novel answer which we call "the formal explanation." Unlike previous accounts it relies on mathematical formalism and argues that the definitions of Euclidean geometry necessitate the inclusion of the first law. We provide evidence in support of this claim. A second contribution is a comprehensive review of previously suggested explanations, which so far have often been treated in a fragmented manner, and a discussion of the plausibility of the various answers.

It turns out that many people have addressed this, and this paper has a new explanation.

Roughly, mathematicians did not always treat zero as a number. There are many examples in Euclid's Elements where an argument gets restated for the zero and nonzero cases, when a modern textbook would unify them. This paper shows that Newton's usage is consistent with Euclid's.

Newton did not have vectors either, so he had to state the directional components of F = ma separately. One version of his second law was:

”A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.”

This is similar to Euclid, as he only wrote about nonzero quantities being proportional.