In Jim Doyle's page on Galileo's Relativity he says that it is commonly understood that the central problem of relativity deals with how people who view an event from different locations can agree on when and where the event happened.

The Galilean relativity hypothesis is typically stated thusly:

"Any two observers moving at constant speed and direction with respect to one another will obtain the same results for all mechanical experiments."

In his Notes on Special Relativity Michael Fowler says that physicists reformulated Galileo's observation in a slightly more technical, but equivalent, way: they said the laws of physics are the same in a uniformly moving room as they are in a room at rest.

Much of our deliberations about relativity is tied up in ideas about inertial frames and with what one observer can know about what goes on in other inertial frames. There are, however, considerations which bring this particular emphasis, at least as it pertains to Galileo, into question.

The purpose behind the tests to be done in Galileo's Parable of the Ship was to examine evidence which he claimed supported the idea that even though the Earth rotates (roughly 1000 miles per hour at the equator), we do not sensibly perceive the motion. The issues of (1)why we don't get flung off the Earth by centrifugal acceleration or (2)why we don't get blown away by howling winds, had been discussed earlier, in the dialog, but were not thoroughly resolved. )

[Note that there is a small problem in the above web quote. Salvatius (Galileo's mouthpiece) actually says to hang up a bottle that empties drop-by-drop into a narrow-mouthed vessel, rather than into a wide vessel. ]

The following few paragraphs are intended to add a bit of light on the unresolved issues(1) (2) mentioned above. [Thanks to Bert Schreiber of Bellair, Texas for recommending the additions.]

(1) Apparently in 1632 natural philosophers had not yet devised a way of calculating so-called centrifugal accelerations which could be compared to gravitational accelerations. (Also, gravitational acceleration science was in its crawling phase.) We now know that the centrifugal acceleration associated with the Earth's equatorial rotation speed is about one third of one percent of the gravitational acceleration.

(Side Note: In 1687, or a little earlier, a seafaring navigator named Richer(*) reported to Isaac Newton that his pendulum clock, which ran accurately at latitudes near that of London, consistently lost two minutes a day near the equator. Newton is said to have ascribed the problem to an increase in Earth's radius at the equator.)
(*) The link: http://www.mines.edu/fs_home/tboyd/GP311/MODULES/GRAV/NOTES/latitude.html to an article about Richer no longer works. [22 August 2005.]

In the 1960's chronometers [navigational time keeping devices] on sea going vessels were spring driven (no pendulums) and their clock works were constrained to move in the horizontal plane. External gimbals and weights kept the devices parallel to the earth's surface. This way of doing time keeping would tend to minimize latitude related variations in clock speeds. [Added 22 August 2005.]

A thorough study of pendulum behavior could have provided Galileo with evidence that his environment, the Earth's surface, was rotating. It would take a really fine instrument or a very fast ship to differentiate ship's motion from that of the ocean/world.

(2) The old idea that Earth's rotation would lead to winds on the order of 1000 miles per hour winds at the equator must go back to ancient ideas akin to that of the firmament as described in Genesis 1:6 - 8 and 1: 14 - 18. The firmament seems to have been the Earth's atmosphere which had stars (lights in the firmament) embedded in its upper reaches.

It is not generally appreciated, but Galileo made it plain that the reason for the correspondences of observations (in a stationary versus moving shipboard cabin) hinged on the fact that everything in the ship's cabin, especially the air, was moving with the observers.

Regarding these tests Galileo does not mention unaccelerated motion in a straight line. On the First Day discussions he makes it plain that he cannot abide the idea of motion in a straight line. (Straight line travel, according to Salvatius, would soon put the traveler into outer space with no place to go.) It may be said, tongue-in-cheek, that Galileo preferred circular arguments. [Elliptical orbits and oblate spheroids hadn't been invented yet.]

Regarding the importance of the ambient air co-moving with the observers, consider what would happen if Galileo's moving laboratory had a means to allow the free movement of outside air through the compartment. Assume that provisions are still in place to prevent optical and acoustic information from the external world from getting in.

Under this condition (external air moving throughout the cabin), if a ship were to be moving forward with speed v (Salvatius says you can have the moving ship to proceed at "any speed you like."), the stationary air would appear (to the onboard observers) to be flowing through their compartment with speed -v. As a result, every one, or most, of the tests that Galileo proposed for his moving shipboard observer would yield a different result than for the observers in the same cabin back at the pier.

[Salvatius, in fact, addresses the non-co-moving air issue. He talks about observations being done in the open air (on deck) and says that when exposed to the elements, you'd expect to see some noticeable differences from what happens inside the sheltered cabin. That consideration is omitted in the web quote above. It is also generally ignored when modern day relativity theorists discuss so-called Galilean relativity.]

Galileo's observers were prevented (by design) from seeing or measuring anything outside of their shipboard cabin. Their discernment of relative positions and velocities related to physical processes going on in other cabins (also traveling in constant straight line motion) was not the issue.

Finally, if one subscribes to the idea that the air which cloaks the Earth's surface bears a certain functional resemblance to the air that was trapped in Galileo's ship cabin, i.e., co-moving with the observers, and (for a given pressure and density) that light travels at more-or-less a constant speed with respect to the dispersive medium through which it passes, then it can be suggested that the null-result of the Michelson-Morley experiment is in accord with Galileo's shipboard tests.

We may need to give Galileo some slack regarding the central problem of relativity.

Heretical Postscript

References

(2) Fox, J.G., Evidence Against Emission Theories, Am. J. of Phys, 33, 1, (1965).