a. Let us first consider Galileo’s own presentation of his thought experiment.

Let us first make this assumption, I argue as follows in proving that bodies of the same material but of unequal volume move [in natural motion] with the same speed. Suppose there are two bodies of the same material, the larger a and the smaller b, and suppose, if it is possible, as
asserted by our opponent, that a moves [in natural motion] more swiftly than b. We have, then, two bodies of which one moves more swiftly. Therefore according to our assumption, the combination of the two bodies will move more slowly than that part which by itself moved
more swiftly than the other. If, then, a and b are combined, the combination will move more slowly than a alone. But the combination of a and b is larger than a is alone. Therefore, contrary to the assertion of our opponents, the larger body will move more slowly than the
smaller. But this would be self-contradictory.

The argument inevitably leads to its conclusion: bodies of the same material have the same speeds in free fall. Following Gendler’s neat reconstruction we can summarize the argumentative structure as
follows: (1) natural speed is mediative (the natural speed of a combined body will fall between the natural speeds of the component bodies); (2) weight is additive (the weight of a combined body will be the sum of the weights of the component bodies); hence (3) natural speed is not directly proportional to weight; and, moreover the only way to hold on to (1) – (3) simultaneously is by asserting that (4) natural speed is independent of weight.

The crux of the argument seems to lie in premise (1). One could wonder how Galileo can claim to know that this is a valid assumption. A first possible answer is provided by the following note which he wrote in a margin in the original manuscript: “Aristotle makes this same assumption in the solution of the 24th Mechanical Problem.” Now, this is a little bit of a stretch on Galileo’s part. The 24th Mechanical Problem deals with the famous paradox of Aristotle’s wheel, not at all with the
natural speeds of falling bodies. The importation of that assumption, in the context of the thought experiment would require a much more substantial argument. It is not at all obvious that rolling wheels and falling bodies partake in the same principles. Moreover, if this assumption were accepted only on Aristotle’s authority, then it might well function in a reduction of the Aristotelian theory, but
not in an argument which seeks to establish an alternative theory. For the conclusion (4) to hold generally, independent grounds for accepting premise (1) must be present. Such grounds are provided by Galileo, however:

Thus, if we consider two bodies, e.g., a piece of wax and an inflated bladder, both moving upward from deep water, but the wax more slowly than the bladder, our assumption is that if both are combined, the combination will rise more slowly than the bladder alone, and more
swiftly than the wax alone. Indeed it is quite obvious. For who can doubt that the slowness of the was will be diminished by the speed of the bladder, and, on the other hand, that the speed of the bladder will be retarded by the slowness of the wax, and that some motion will result intermediate between the slowness of the wax and the speed of the bladder?

The same argument is then repeated for a piece of wood and an inflated bladder falling downward in air. These are of course very revealing examples. The first thing to notice is that they involve bodies of
different material. Now, since Galileo wants to conclude that for bodies of the same material the speed of fall is equal, it would have been clearly self-defeating if he could have adduced empirical examples
of this kind to illustrate his assumption. But this also points toward the fact that Galileo considered his assumption to be an empirical fact of the matter, possibly following a theoretical principle, but surely recognizable without such a principle at hand. Secondly, the provenance of this empirical fact of the matter is easily recognizable. Take two bodies of different material and compare their behaviour with the behaviour of a mixture of these materials…

Once again we find Galileo translating the situation of La bilancetta by having natural speeds mirror the positions of the counterweight on the hydrostatic balance. These positions on the balance arm had indeed undeniably shown that “specific weight” is mediative. But this implies that the proportionality of speed with “specific weight” is a hidden assumption of his thought experiment. The thought experiment thus accomplishes the transformation from absolute to “specific” weights by
presupposing the latter.

b. The dynamical conundrum

Once that the conclusion of the thought experiment is reached, it becomes impossible to hold on to a proportionality between speed and absolute weight. However, this leaves Galileo without any intelligible dynamics, as the balance is his paradigm case of a situation in which the force of weight can be immediately understood. In La bilancetta, he had been able to take these forces, as measured by absolute weights, as the starting point for analysing specific weights, by exploiting the fact that any body is always opposed by an equal volume of water in a hydrostatic balance. At this point he thus did also not consider specific weights as giving rise to forces directly. That he still holds on to this indirect relation in De motu is clear if we remember that at several places (after already having presented the thought experiment), Galileo does set speeds proportional to forces which are measured by differences in absolute weights – differences which then can be transformed into differences of “specific” weights by pretending (on the basis of the thought experiment) that the results hold independently of the volumes. But if we are not mistaken in imputing to Galileo a dynamics which still refers back to experiences with absolute weights, then the conclusion of the thought experiment must have presented
a potential conundrum for him.

The absence of an explicit concept of specific weight undoubtedly helped to mask the dynamical problem. By not explicitly thematizing the dimensional differences within the undifferentiated concept of “grave”, the conundrum might have seemed less pressing (and indeed seems to have been largely ignored by most Galileo scholars). There was of course also the attempt at explaining the equality of speeds by considering the alleviation effect of a medium, which might have
eased Galileo’s mind at this point – provided he did not realize himself that he was being incoherent. But it must anyway have been clear to him that this was insufficient. This can be judged from the fact
that after that he has established the possibility of motion in a void, he proclaims that the thought experiment must also be valid in this situation.49 Given that the argument is supposed to remain
precisely the same, it is clear that the effect of the medium can not be operative in reaching the desired conclusion. This helps us to pinpoint more precisely the gap that remains in Galileo’s dynamical
conceptualization of motion. As the transformation procedure which he used to such great effect in La bilancetta completely breaks down in the void, he is left without any way to connect his mathematical
scheme with the shared experiences that had to secure its intelligibility. What he offers instead is his thought experiment, which supposedly can provide for an equally incontestable experience that could
possibly anchor his explanatory scheme (albeit it does this, as we saw, by actually presupposing
further experiences which go back to phenomena involving dense media). That it is indeed supposed
to render the dynamics of free fall immediately intelligible is further proved by the following passage,
which follows almost directly after the presentation of the thought experiment:
And who, I ask, will not recognize the truth at once, if he looks at the matter simply and
naturally? For if we suppose that bodies a and b are equal and are very close to each other, all
will agree that they will move with equal speed. And if we imagine that they are joined
together while moving, why, I ask, will they double the speed of their motion, as Aristotle
held, or increase their speed at all?50

The question is to the point, and it will be the starting point for a successful solution of the conundrum
in the postils to Rocco, but at this point it must remain a rhetorical question. If a balance does indeed
measure a body’s tendency for downward motion, as repeatedly implied by Galileo in De motu, then
the only natural response to the question would be: why not? This is not to deny that Galileo was
convinced that they do not: he clearly believed that specific gravity provided a much better measure
for the speed of fall. But it is the argumentative structure of De motu itself that leaves a gap at exactly
this point.

One might wonder whether it is really justified to call this gap a “conundrum”, as there are no
clear signs that Galileo was puzzled by it in any significant respect.51 As far as De motu goes, this
might be true, but as will become clear in the second part of the present paper, at a later time Galileo
indeed began to wonder about how to connect the behaviour of the bodies in his thought experiment
with their behaviour on a balance. At this point he has clearly become aware of the gap that exists
between his explanatory scheme and the basic experiences that were first thought to render it
intelligible. If we would not be allowed to think of this gap as a conundrum, we might hence loose the
means to understand the dynamics behind Galileo’s thinking, as it seems that it really did trigger
Galileo’s rethinking of the thought experiment in a fundamental new way. As was already noticed,
once the gap is perceived as a conundrum, the crucial question becomes why bodies of the same
material would have to move with the same speed in the void. Indeed, in this situation the empirical
examples which were adduced by Galileo to justify the first premise of his thought experiment loose
their intuitive plausibility, which was based on the experience with the behaviour of mixtures in dense
media. This shows that, although he does not need to change the argument itself, he would need some
other kind of justification for the mediative character of natural speeds. In the later presentations of the
thought experiment (to be discussed in Sections 6 and 7) in the postils to Rocco and in the Discorsi,
exactly such a justification will be provided, which will be clearly dynamical in character. As we will
see, once that he has provided the dynamical justification for the first premise, Galileo will also be in a
better position to solve the conundrum raised by the conclusion.

Recapitulating our long analysis of De motu, Galileo’s thought experiment plays a crucial role
therein in at least two respects. It enables him to make the transition from absolute to “specific” weight
as the relevant factor for the natural motion of bodies, without having to define the latter explicitly. At
the same time, it covers up the fact that Galileo by his own standards misses a fully intelligible
dynamics for free fall. It is indeed clear that this transition from absolute to “specific” weight cannot
be based on the effect of a medium on the weight of bodies, while Galileo nowhere gives a hint of how
to understand “specific” weight as a primordial and immediately intelligible dynamical factor: the only
model which he possesses for understanding forces is the balance which measures absolute weights;
and all his dynamical thinking is based on the idea that speeds are caused by such forces.