F=ma
Trying to interpret Newton's first law of motion in 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 many physics textbooks stating this.
This is quite right and reasonable given the way the two laws are usually understood. 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 on the reasoning shown, if we take them in Newton's own formulation. One minor but important point is that Newton's own second law is not the equation F=ma. I say 'minor' because it is, in fact, easy enough to prove that if we use the right combination of units, F=ma for cases where mass is constant and time is not, given the first law, the second law, and the definition of quantity of motion (definition II). To do it you use the method of construction: you posit an alteration of motion, use the first law to conclude that there is a force, and use the second law and the definition of quantity of motion to infer that the change of velocity divided by the change of time is equal to some constant times the impressed force, divided by the mass. The constant can be set to one with the right combination of units, and we therefore have an equation equivalent to the standard one. So F=ma follows from the second law; but this is not the same as to say that it is the second law -- in part because you need to assume things besides the second law to make the proof work.
More importantly, however, Newton's first law can't be a special case of his second because they don't discuss the same thing. The first law 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.
What the first law in effect does is tell us that an alteration of motion requires a cause, namely, an impressed force, or a combination of impressed forces. The second law extends this by telling us how, precisely, the alteration of motion is related to the impressed force causing it. (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.) But if we interpret them in this light, the first law is not a special case of the second law; it simply tells us when a certain sort of cause exists, while the second law tells us how the effect is related to that cause when it does exist. These are two completely distinct things, and when you try to do proofs with them, you have to use both of them to get the usual equations. If we took the first law simply to say that when force is zero, acceleration is zero, it would be a special case of F=ma; but the first law doesn't simply say that, but gives a causal role to forces.
This is a very small sort of change, since it doesn't change anything structurally speaking -- you can get the same equations, in somewhat different ways, on both interpretations. But it is a striking example of how the role Newton himself assigns to a feature of his physics might be somewhat different from that assigned by later physicists with somewhat different interests.
