Usually I see an inertial reference frame being defined as a reference frame in which Newton's first and second laws holds. That means that if a particle is at rest, it stays at rest unless some external force acts upon it and that if it is in uniform motion on a straight line it stays that way unless some external force changes that.
Most books I've seem until today after defining one inertial frame that way says that Newton's laws are to valid on inertial frames. That is, they are laws, they are supposed to hold. Is not something like if they hold, it is an affirmation: "they do hold when we work on inertial reference frames".
Now, when reading about Special Relativity, some books says that prior to Einstein there was one "principle of relativity" that could be stated as follows:
The laws of Mechanics are invariant in every inertial reference frame
and that this is the result of Galileo's discussion about Salviatti's ship. I'm having a hard time with this because of the following line of thought:
If Newton's laws do hold on inertial reference frames by the definition of inertial reference frames and by the statement of the laws themselves, why this principle is hanging arround anyway? I mean, it seems to me like something automatic from Newton's laws already.
My question is then: "where this principle of relativity from Galileo enters Classical Mechanics? Is it something that already follows trivially from the laws like I'm supposing or it is something that must be added as another axiom of the theory?"