Other than his numerous experiments on acceleration and the motion of objects, Galileo was also very observant, and he probably liked to ponder deeply about the things he see.
For example, he wondered whether it is possible for sailors in a boat to tell whether the ship is actually moving or not, without looking at the surrounding seascape and shoreline. To investigate this, he devised a simple experiment: the sailor will drop a ball when the boat is docking in bay, and when the boat is moving with uniform velocity in the open sea.
Of course, in practice there will be complications: depending on the weather the sea can be quite rough, and the light boats in Galileo's time will be heaved up and down by the choppy waters, and the result of the experiment will be quite different from that performed in bay, where the waters are calmer.
So Galileo imagined an ideal windless day where the waters in the open sea is calm and smooth, as would the waters in the bay. In such an ideal situation, Galileo argued that there will no difference in the outcomes of the experiments, be it done in the bay or out in the open sea. Now, if the sailor is locked in the cabin with no portholes to look out from, what would he conclude from the experiment?
What else?
"The boat is not moving, when I did my two experiments."
said the sailor, even though the boat is indeed moving with uniform velocity in the second case.
Actually, this was not what he said exactly. However, in his book Dialogue Concerning the Two Chief World Systems Galileo wrote a parable on a dialogue between two of the three chief characters, SALVATIUS and SAGREDUS in which SALVATIUS painted a hypothetical situation in which one lock himself or herself in the main cabin of a large ship together with a zoo, literally, and another person do the same in an identical ship. One of this ship is moving with uniform velocity on a smooth sea and the other one at rest.
In the parable, SALVATIUS's conclusion is that if both persons make careful observations on the two zoos on the moving ship and the stationary ship, then none of their observations about the motion of butterflies and bees flying and people throwing balls at each other, will differ from each other.
This leads to the Galilean Principle of Relativity, which was first stated as
The mechanical laws of physics are the same for every observer moving uniformly with constant speed in a straight line.
Ah... but this seems to imply that there must an observer who is absolutely at rest, as postulated by Aristotle, otherwise how do we know that any other observer is moving with a uniform velocity?
But what the Galilean Principle of Relativity did is precisely to remove the need for such an absolutely resting observer. We may argue for this by referring to Galileo's ship parable, in which the uniformly moving observer sees the same physics as the stationary observer. Therefore, if there is any absolutely resting observer, he will share the same physics will other uniformly moving observers in the sense all physical experiments performed in their closed main cabin will obtain the same results, relative to coordinates and times measured in this closed cabin.
However, if all physical measurements coincide, what serves to distinguish the absolutely resting observer's cabin from the uniformly moving observer's cabin? To tell the truth, nothing! If there is something that allows us to tell them apart, then clearly the two cabins are not equivalent physically, implying there must be at least one physical experiment whose results are different for the absolutely resting observer and the uniformly moving observer. But this would mean that Galileo's conclusion about the equivalence of physics is wrong!
Therefore, we have two choices:
Galilean Principle of Relativity TRUE = No absolutely resting observer.
Galilean Principle of Relativity FALSE = Absolutely resting observer possible.
Since then many empirical tests on the validity of this Principle has been carried out, and it is now clear to us that, at low velocities (much less than the speed of light) and not near strongly gravitating bodies, Galilean's Principle of Relativity has to be accepted as valid, and it is on this principle that Newton's Laws of Motions are based upon.