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Galileo's writings describe two experiments involving ships. These are summarized in the Wikipedia article Galileo's ship. (A lot of the text in the article is mine, and if there are things I'm misunderstanding, then they probably appear in the article as well.) In one of these experiments, a cannonball is dropped from the mast of a moving ship, and is observed to land at the base of the mast rather than behind it. In his letter to Ingoli, he insists strenuously both on the fact that he really carried out the experiment and on the outcome.

This experiment is basically identical with one shown in the classic PSSC film Frames of Reference. For 20 years, I've been showing students this film, and accompanying it with a description of Galileo's version, and an anecdote I heard somewhere about a bet. The story is that Galileo got in a bet with two noblemen about the outcome of the experiment, but after the noblemen agreed to the bet, Galileo was unable to collect his money because they didn't agree with him that an experiment was the proper way to settle who was right. As I've learned more about the history in recent years, I've failed to come up with any information to support the anecdote about the bet. Is it totally bogus?

As circumstantial evidence against it, secondary sources such as Drake devote considerable energy to discussing the evidence that the experiment was real rather than a thought experiment, and, e.g., Drake never mentions a bet. It seems that if there were historical sources for the story of the bet, then a specialist like Drake would have considered them important evidence on this point.

Many other people from that era appear to have discussed the experiment or claimed to have done it. Maybe the story about the bet is really about one of them, not Galileo?

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You have correctly pointed at Wiki's quotation from Galileo's Dialogo sopra i due massimi sistemi del mondo (The Dialogue Concerning the Two Chief World Systems, 1632) where the thought experiments regarding the "composition of motions" in what we today call "inretial frames" (e.g. the fall of a ball from the mast of a sailing ship) are discussed at lenght (for about 50 pages in the Second Day).

The Dialogue is "acted" by three fictional characters :

  • Salviati : the Copernican presenting Galileo's views

  • Sagredo : the "intelligent layman" who is initially neutral

  • Simplicio : the follower of Ptolemy and Aristotle, presenting the traditional views regarding the natural world.

It seems to me that the story of "bet with two noblemen about the outcome of the experiment" is only a way to simplify for a child audience the literary dialogic form adopted by Galileo (following his "theacher" Plato).


Addendum

From Galileo Galilei, Dialogues on two world systems, (1632, Engl.tr. Thomas Salusbury, 1661), page 125-126 :

SALV. Now tell me; if the stone let fall from the Round­ top, when the ship is in a swift course, should fall exactly in the same place of the ship, in which it falleth when the ship is at anchor, what service would these experiments do you, in order to the ascertaining whether the vessel doth stand still or move? [...] Have you ever tryed the experiment of the Ship?

SIMPL. I have not; but yet I believe that those Authors which alledg the same, have accurately observed it; besides that the cause of the disparity is so manifestly known, that it admits of no question.

SALV. That it is possible that those Authors instance in it, without having made tryal of it, you your self are a good testi­mony, that without having examined it, alledg it as certain, and in a credulous way remit it to their authority; as it is now not onely possible, but very probable that they likewise did; I mean, did remit the same to their Predecessors, without ever arriving at one that had made the experiment: for whoever shall examine the same, shall find the event succeed quite contrary to what hath been written of it: that is, he shall see the stone fall at all times in the same place of the Ship, whether it stand still, or move with any whatsoever velocity. [...]

SIMPL. How is this? You have not made an hundred, no nor one proof thereof, and do you so confidently affirm it for true? I for my part will return to my incredulity, and to the confidence I had that the Experiment hath been tried by the principal Au­thors who made use thereof, and that the event succeeded as they affirm.

SALV. I am assured that the effect will ensue as I tell you; for so it is necessary that it should.

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    $\begingroup$ This seems plausible. It hadn't occurred to me before that the number of noblemen would match up in this way with the number of characters in the dialog. $\endgroup$ – Ben Crowell Feb 2 '15 at 16:59

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