Further to Timaeus's Answer, the second postulate follows from the first postulate if we know about light. Otherwise, the second cannot follow from the first in a strict sense.
However, even if you don't know about light, there is still a way whereby the second postulate can be strongly motivated by the first, as follows.
The first postulate is essentially Galileo's notion of relativity as explained by his Allegory of Salviati's Ship.
If you assume:
- The first relativity postulate; and
- A concept of absolute time, i.e. that the time delay between two events will be measure to be the same for all inertial observers; and
- Homogeneity of space and time so that linear transformation laws between inertial frames are implied (see footnote)
Then these three assumptions alone uniquely define Galilean Relativity.
However, if you ask yourself "what happens to Galileo's relativity if we relax the assumption of absolute time" but we keep 1. and 3. above, then instead we find that a whole family of Lorentz transformations, each parametrised by a parameter $c$, are possible. Galilean relativity is the limiting member of this family as $c\to\infty$. The study of this question was essentially Einstein's contribution to special relativity. You can think of it as Galileo's relativity with the added possibility of an observer-dependent time. I say more about this approach to special relativity in my answer to the Physics SE Question "What's so special about the speed of light?".
It follows from this analysis that if our Universe has a finite value of $c$, then something moving at this speed will be measured to have this speed by all inertial observers. However, there is nothing in the above argument to suggest that there actually is something that moves at this speed, although we could still measure $c$ if we can have two inertial frames moving relative to each other at an appreciable fraction of $c$. It becomes a purely experimental question as to whether there is anything whose speed transforms in this striking way.
Of course, the Michelson Morley experiment did find something with this striking transformation law.
Footnote: The homogeneity of space postulate implies the transformations act linearly on spacetime co-ordinates, as discussed by Joshphysic's answer to the Physics SE question "Homogeneity of space implies linearity of Lorentz transformations". Another beautiful write-up of the fact of linearity's following from homogeneity assumptions is Mark H's answer to the Physics SE question "Why do we write the lengths in the following way? Question about Lorentz transformation".
My question is (1) how Maxwell's equations contradicted Galilean
principle of relativity.
Maxwell's equations have wave solutions that propagate with speed $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$.
Since velocity is relative (speed c with respect to what?), it was initially thought that the what is an luminiferous aether in which electromagnetic waves propagated and which singled out a family of coordinate systems at rest with respect to the aether.
If so, then light should obey the Galilean velocity addition law. That is, a lab with a non-zero speed relative to the luminiferous aether should find a directionally dependent speed of light.
However, the Michelson–Morley experiment (original and follow-ups) failed to detect such a directional dependence. Some implications are
(1) there is no aether and electromagnetic waves propagate at an invariant speed. This conflicts with Galilean relativity for which two observers in relative uniform motion will measure different speeds for the same electromagnetic wave. This path leads to special relativity theory.
(2) there is an aether but it is undetectable. This path leads to Lorentz aether theory.
Best Answer
The first does not imply the second, but you are nevertheless on the track of something which I will try to spell out.
The second postulate can be made in a more minimal way. Instead of mentioning all inertial frames, one need only mention one:
One can then bring in the first postulate and argue that if there is just one inertial frame with this property, then it must also be true in all inertial frames. One can also make a further argument that the speed observed for light in vacuum must be the same in all frames (as long as the term 'vacuum' is not being used to refer to some medium which itself could have detectable motion). So your suspicion was almost right: the first postulate does have something to say about the universality of the speed of light. But nevertheless the principle of relativity (i.e. postulate 1) on its own cannot be used to derive anything about the speed of light nor the maximum speed for signals. For, don't forget, Newtonian physics respects the principle of relativity (your postulate number 1), and in Newtonian (or Galilean) physics there is no finite maximum speed. In the Galilean version of spacetime, signals can in principle go infinitely fast, and everyone agrees about which events are simultaneous and which are not. Therefore the second postulate certainly does not entirely follow from the first.
Finally, I would like to note that you do not need to mention light when stating the second postulate, and arguably it is useful not to bring it in. It suffices to say that there is a finite maximum speed at which causation can happen, and leave it at that. By "causation can happen" I mean there is a finite maximum speed at which an influence from any given event can carry effects to other events. Sometimes we use the word "signals" for such influences; then the statement would be that there is a finite maximum speed for signals. Or, to hone it down a little more: