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.
It is easy enough to prove, however, that if we use the right combination of units, the first law, the second law, and the definition of quantity of motion (definition II), then F=ma for cases where mass is constant. To do it you use the method of construction The first law, as Newton states it, is:
Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.
The second law is:
The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
(1) Posit an alteration of motion.
(2) Use Law I to conclude that there is a force impressed upon the object whose motion is altered.
(3) Since Law II tells you how the change of motion is related to this impressed force, you can use Law II along with the definition of quantity of motion (Definition II) to infer F=ma. So F=ma follows from the second law plus some basic assumptions.
Definition II tells us that quantity of motion is the measure of motion from velocity and quantity of matter (mass). Therefore suppose we start at time ti with a body of mass mi whose motion is measured at a velocity vi. And suppose it changes, so that at tf we have mf at vf. Then (mfvf - mivi)/(tf-ti) is our change of motion for those times. According to Law I, there is a force F for this; according to Law II, the change of motion is proportional to F. Let us assume that mass doesn't arbitrarily (or even non-arbitrarily) change: mf=mi, so both can just be called m, and m can be factored out, since it is unaffected by change of time, to get the claim that m((vf-vi)/(tf-ti)) is proportional to F. That's mass times change of velocity over change of time; change of velocity over change of time is acceleration, which gives ma, which is proportional to F. That should look familiar.
Newton's first law can't be a special case of his second because they aren't in the same category. What Law I in effect does is tell us that an alteration of motion requires a particular kind of cause, namely, an impressed force, or a combination of impressed forces; and (depending on how it is read) it tells us that we do not need to look for such a cause unless the motion is altered. Law II extends this by telling us how, precisely, the alteration of motion is related to the impressed force causing it, assuming that there is both an alteration of motion and an impressed force. One of Newton's major projects in the Principia is to develop a method for properly accounting for the difference between true and apparent motions and the causes underlying that difference; he needs Law I to do this properly. But if we interpret them in this light, Law I is not a special case of Law II; it simply tells us when a certain sort of cause exists, while Law II tells us how the effect is related to that cause when it does exist. These are two completely distinct things, and you have to use both of them to get the usual equations. If we took Law I simply to say that when force is zero, acceleration is zero, it would be a special case of F=ma; but Law I as Newton formulates it doesn't say that. It tells us when we must appeal to forces and when we don't need to do so. We need to know this before equations about forces are even possible. This is something I think even physicists sometimes forget: equations are never fundamental. We don't start with equations; we start with rules of inference and means of measurement, then use those to get equations.
It's interesting in this light to look at William Whewell's interpretation of Newton's laws of motion, because Whewell is very sensitive to the fact that we start with rules of inference. Whewell argues in his Philosophy of the Inductive Sciences and in his 1834 essay, "On the Nature and Truth of the Laws of Motion," that the laws of motions are actually general causal principles to which we add certain experimental facts to obtain more restricted causal principles suitable for discussing physical motion. Thus Newton's First Law is, on his view, nothing other than the claim Every change is produced by a cause, given that we have experimentally ruled out certain things as causes for motion (time is the major one that needs to be ruled out; but location, which Whewell considers but takes to be ruled out by the fact that we are considering no external forces, and the object's own mass, which Whewell does not, also need to be ruled out). That is, Law I is just "Velocity does not change without a cause" plus "The time for which a body has already been in motion is not a cause of change of velocity." With the specific causal inferences provided by Newton's three laws, we can then produce the whole of Newtonian dynamics (add rules governing how we reason about equilibrium and you have statics as well).
In any case, the case of Newton's laws of motion shows one way in which science changes by drifting: because of the usefulness of F=ma, over time it has come to be treated as indistinguishable from Law II, even though they are logically distinct.
