As we usually learn it, Newton’s second law of motion is:
F=ma
If we try to interpret Newton’s first law of motion in terms of algebraic equations, it’s very natural to take it as simply describing the case where the acceleration, and thus the resultant force, is zero. The first law then becomes a special case of the second law; and you will find that many physics textbooks state this.
This is quite right and reasonable if we mean by Newton’s first and second laws what most physics textbooks mean. But it’s worth noting that if we take the laws as actually stated in the Principia, this conclusion is impossible: the first law can’t be a special case of the second law, if we take them in Newton’s own formulation. Newton’s own second law, of course, is not the equation F=ma.
Continue reading this post here. Correction of any details (either here or there) is very welcome. The sort of reflection in this post has grown as a side issue out of another project on early modern Newtonianism; one of the major problems of studying the impact of Newtonianism in its early context is the constant danger of anachronism. Working out the difference between Newtonian physics as Newton knew it and Newtonian physics as it came to be is important to such a project, and this is an attempt to contribute to that.
Whewell's Ghost 