Stud. Hist. Phil. Sci. 35 (2004) 115–136
www.elsevier.com/locate/shpsa
Galileo and prior philosophy
David Atkinson , Jeanne Peijnenburg
Faculty of Philosophy, University of Groningen, 9718 CW Groningen, The Netherlands
Received 19 August 2002; received in revised form 13 December 2002
Abstract
Galileo claimed inconsistency in the Aristotelian dogma concerning falling bodies and stated that all bodies must fall at the same rate. However, there is an empirical situation where
the speeds of falling bodies are proportional to their weights; and even in vacuo all bodies do
not fall at the same rate under terrestrial conditions. The reason for the deficiency of Galileo’s
reasoning is analyzed, and various physical scenarios are described in which Aristotle’s claim
is closer to the truth than is Galileo’s. The purpose is not to reinstate Aristotelian physics at
the expense of Galileo and Newton, but rather to provide evidence in support of the verdict
that empirical knowledge does not come from prior philosophy.
# 2003 Elsevier Ltd. All rights reserved.
Keywords: Aristotle; Galileo; Thought experiments; Falling bodies
1. Introduction
The thought experiment by which Galileo destroyed the Aristotelian dogma that
heavier bodies fall faster than lighter ones is a classic in the field. It sets the example,
and as such it features prominently in all contemporary studies of scientific thought
experiments (Brown, 1991; Norton, 1991; Sorensen, 1992; Norton, 1996; McAllister,
1996; Gendler, 1998; Brown, 2000). However, as we will show, it does not attain the
impeccable standard that is generally assumed. At first sight, it appears to refute the
Aristotelian paradigm in a decisive and even awe-inspiring manner. But in fact it is
flawed, both in its attempted refutation of the old, as in its attempted demonstration
of the new ideas on falling bodies. One may therefore not cogently claim, as Brown
Corresponding author.
E-mail addresses: [email protected] (D. Atkinson); [email protected] (J. Peijnenburg).
0039-3681/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.shpsa.2003.12.006
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and others have tried to do, that this thought experiment offers us a glimpse into a
Platonic world of verities.
Let us begin with an adaptation of the thought experiment in question. Suppose
we have two pieces of the same material but of different weight; a rock weighing 8
kilograms and another weighing only 4 kg. Suppose we drop them from a tower.
Aristotle, who claims that the rate of fall of a body is proportional to its weight,
must now infer that the heavier rock falls twice as fast as the lighter one, and thus
takes half as much time to reach the ground. Suppose that we next pick up the two
rocks, bind them together with a string, and drop this bound system from the
tower. Then it can be shown that the Aristotelian system leads to a contradiction:
on the one hand, the bound system must fall more slowly than the 8 kg piece, for
the 4 kg rock, which Aristotle says has a natural tendency to fall slowly, will slow
down the rock of 8 kg, which he claims to have a natural tendency to fall more
quickly. Thus the time measured for the bound system to fall to the ground must
be greater than that for the heavier piece alone. On the other hand, the bound system falls faster than the 8 kg rock, for weight is additive: the bound system weighs
12 kg and thus falls one-and-a-half times as fast as the 8 kg piece, and this contradicts the first conclusion. Galileo’s way out of this predicament is to reject the old
idea that the rate of fall is proportional to the weight and replace it by a new
claim, namely that all bodies fall at the same rate, independent of their weight.
It looks as though a pure thought experiment has destroyed an old belief and
replaced it by new knowledge concerning the world, without the need for a real
experiment, that is, without extra empirical input. Such a claim is indeed made by
J. R. Brown. For him, Galileo’s reasoning is the thought experiment par excellence: it gives us ‘a grip on nature just by thinking’ (Brown, 2000, p. 528), it
enables us to ‘go well beyond the old data to acquire a priori knowledge of nature’
(op. cit., p. 529). Thought experiments like Galileo’s are called by Brown ‘Platonic’
(Brown, 1991, p. 77), ‘the truly remarkable ones’ (op. cit., p. 34). The hallmark of
such a thought experiment is that it is simultaneously destructive and constructive;
it destroys an old theory and at the same time establishes a new one:
Galileo showed that all bodies fall at the same speed with a brilliant thought
experiment that started by destroying the then reigning Aristotelian account . . .
That’s the end of Aristotle’s theory: but there is a bonus, since the right account
is now obvious: they all fall at the same speed . . . (Brown, 2000, p. 529)
In Sect. 2, we will take a first look at the textual basis for the Galilean claim,
and at the extant Aristotelian writings on the subject of falling bodies. In the subsequent two sections, we reconstruct Galileo’s reasoning and analyze the deficiencies in it; we show that there is no purely logical objection to the Aristotelian
claim that the rate of fall is proportional to the weight, nor any valid argument for
the Galilean claim that all bodies fall at the same rate. In Sect. 5 we inveigh
especially against Platonists, among whose ranks we must number Brown, and in a
sense Galileo himself.
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We close the paper with some appendices devoted to technical considerations.
Their main purpose is to show that, aside from purely logical matters, Newtonian
physics has definite implications for the correctness or otherwise of Galileo’s conclusions. It is not relevant that Galileo lacked the physics that we possess; the point is
that empirical situations can be envisaged in which Galileo’s claims are correct, and
other situations in which they are not. We can therefore state with confidence that
Galileo’s double claim (namely that Aristotle’s dogma is logically inconsistent, and
that his own dogma is necessarily true) is unfounded. In Appendix A we analyze the
accelerated fall of bodies in a uniform gravitational field, showing that in this situation, Galileo’s conclusion is correct. In Appendix B we consider another situation,
namely that of terminal motion of slowly falling bodies in viscous fluids, and in this
case we show that Aristotle’s conclusion is correct. In Appendix C we return to bodies falling in vacuo, but now taking cognizance of the fact that the earth’s gravitational field is nonuniform. Here it turns out that the details of Galileo’s thought
experiment can be matched, step by step, but that his grand conclusion, that all bodies fall at the same rate (i.e. with the same acceleration) is wrong. Technical details
are given in Appendix D of turbulent fluid motion, these being relevant to a realistic
treatment of musket shot and cannon balls falling from the leaning tower of Pisa, to
cite a possibly apocryphal experiment. Appendix E is devoted to questions of both
Aristotelian and Galilean source material and to commentaries upon them.
2. Galileo contra Aristotle
In his ‘Two new sciences’, Galileo presents his criticism of Aristotle’s dogma
concerning falling bodies with especial clarity:
Salviati: But, even without further experiment, it is possible to prove clearly, by
means of a short and conclusive argument, that a heavier body does not move
more rapidly than a lighter one, provided both bodies are of the same material,
and in short are such as those mentioned by Aristotle . . . If then we take two
bodies whose natural speeds are different, it is clear that on uniting the two, the
more rapid one will be partly retarded by the slower, and the slower will be
somewhat hastened by the swifter. Do you not agree with me in this opinion?
Simplicio: You are unquestionably right.
Salviati: But, if this is true, and if a large stone moves with a speed of, say,
eight, while a smaller stone moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when
tied together make a stone larger than that which before moved with a speed of
eight. Hence the heavier body moves with less speed than the lighter; an effect
which is contrary to your supposition. Thus you see how, from your assumption
that the heavier body moves more rapidly than the light one, I infer that the
heavier body moves more slowly . . . We infer therefore that large and small
bodies move with the same speed, provided they are of the same specific gravity.
(Galileo Galilei, 1638, p. 108)
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To judge Galileo’s critique of Aristotle, let us first study what the Stagirite himself
said. In some places, he writes merely that heavier bodies fall more quickly than
lighter ones:
The mistake common to all those who postulate a single element only is that
they allow for only one natural motion shared by everything . . . But in fact
there are many things which move faster downward the more there is of them.
(Aristotle, De caelo 3.5, 304b 12–19)
We shall call this the weak Aristotelian dogma: it is the qualitative, or comparative
statement that heavier bodies fall faster than lighter ones. But in other places
Aristotle goes further. He writes that times of fall of bodies of differing weights,
from a given point to a lower point, are inversely proportional to those weights:
A given weight moves a given distance in a given time; a weight which is as
great and more moves the same distance in a less time, the times being in inverse
proportion to the weights. For instance, if one weight is twice another, it will take
half as long over a given movement. (Aristotle, De caelo 1.6, 273b30–274a2)
We shall call this the strong Aristotelian dogma: it is the quantitative statement
that the natural motion of a body is proportional to its weight.
Of course, we have to understand here what Aristotle meant by ‘natural motion’,
or perhaps which of the modern descriptive properties of a falling body we should
substitute for the ancient notion of natural motion, before we can reasonably consider the question of inconsistency. Later we will introduce two possible measures
of natural motion, viz. acceleration (Appendices A and C), and terminal velocity
(Sect. 4 and Appendices B and D). For the time being, however, we can make do
with the somewhat vague term ‘natural motion’, which is quantified by the notion
of ‘natural speed’.1
It is not our aim to devalue the great contributions made to physics by Galileo.
However, these contributions have little to do with his claims, via his spokesman
Salviati, that the Aristotelian dogma (whether in its weak or its strong version) is
logically inconsistent. To underscore the fact that Galileo is mistaken, it is sufficient to point to one physical situation in which Aristotle’s dogma, even in its
strong form, is empirically correct. The case which gives Galileo the lie is that of
bodies falling in a fluid (such as air or water) at their terminal velocities in the case
of laminar fluid flow (i.e. when the fluid motion is not turbulent, see Appendices B
and D for the technical details). Consider this passage:
We see that bodies which have a greater impulse either of weight or lightness, if
they are alike in other respects, move faster over an equal space, and in the ratio
1
More particularly, the term ‘natural speeds of falling bodies’ will be taken to mean the speeds
attained, as functions of time, by bodies falling without constraint, except such as is offered by the
medium (if any) in which they find themselves. This general definition allows for accelerated motion, in
which the speeds are nontrivial functions of time, and also for terminal velocities, in which the speeds
depend on the nature of the bodies, and of the medium, but not on time.
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which their magnitudes bear to one another . . . In moving through plena it must
be so; for the greater divides them faster by its force. For a moving thing cleaves
the medium either by its shape, or by the impulse which the body that is carried
along or is projected possesses. (Aristotle, Physica 4.8, 216a14–20)
Under the restriction of laminar flow, the viscous forces on bodies of identical
size and shape are proportional to their velocities, so the terminal rates of fall are
proportional (the times of fall inversely proportional) to the weights. It is not part
of our thesis that Aristotle espoused, or could have espoused this detailed
interpretation, nor that Galileo excluded, or might have excluded the particular
case of terminal fall with laminar flow. The point is simply that, since there is a
situation in which Aristotle’s conclusion is correct, Galileo’s contention that it is
internally inconsistent must be wrong. Somewhere in Galileo’s argument there
must be a flaw. In Section 4 we will see precisely where the flaw lies. But first, in
Section 3, we will offer three reconstructions of Galileo’s argument, with the aim of
finding the weakest set of assumptions that justifies his claim that all bodies fall at
the same rate.
3. Gendler and reconstructing Galileo’s argument
Galileo’s own resolution of the imagined inconsistency in the doctrine that different bodies fall at different rates, as implied by the weak dogma, is that all bodies
must fall at the same rate. Moreover, via the words of Salviati (‘even without further experiment’) he presents this as a truth that is accessible to reason, rendering
experiment unnecessary.
T. S. Gendler analyzes Galileo’s thought experiment with acumen (Gendler,
1998). She first introduces the notion of the mediativity of speeds, which amounts to
the claim that, if two bodies that are moving with different speeds are subsequently
tied together, the bound system will thenceforth move at an intermediate speed. In
order to reconstruct Galileo’s argument, consider the following three claims:
[G1] The natural speeds of falling bodies are mediative
[G2] Weight is additive
[G3] Natural speed is directly proportional to weight
Gendler, arguing on behalf of Galileo, maintains that the only way to maintain
[G1] and [G2], together with the negation of [G3], is to assume that all natural
speeds are the same. For then ‘weight might be additive and natural speed (in a
vacuous sense) mediative, with no contradiction thereby implied’ (Gendler, 1998,
p. 404).
Gendler further maintains that an Aristotelian who wishes to parry the force of
Galileo’s argument, while maintaining that natural speed is nontrivially correlated
with weight, would have to deny [G1] or [G2], or both. He might do this by postulating an essential difference in mechanical behaviour between bodies that are
merely united and bodies that are unified. Gendler even goes so far as to make, for
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the Aristotelian’s intellectual delectation as it were, a mathematical model in which
a continuous function of ‘the degree of connectedness’ of two bodies determines
their physical properties. But Gendler’s conclusion is clear: even a dyed-in-the-wool
Aristotelian, on being confronted with such an exotic way of saving the Master’s
theory, would recant and deny any reality to a distinction between union and unification. Like the Aristotelian, Gendler does not question the validity of [G1] and
[G2].
Although Gendler does not explicitly distinguish between the strong and the
weak Aristotelian dogma, it is clear that her analysis is based on the former rather
than the latter: [G3] represents the strong rather than the weak dogma. We can
nevertheless also reconstruct Galileo’s argument by using merely the weak
Aristotelian dogma. Thus one can show that the following three conditions are
inconsistent with one another unless all natural speeds are the same:
[C1] The natural speeds of falling bodies are mediative
[C2] Weight is additive
[C3] The natural speed of a falling body is a continuous monotonic increasing
function of its weight
Here [C1] and [C2] are identical to Gendler’s [G1] and [G2], whereas [C3] is a formulation of the weak Aristotelian dogma. We have constructed [C3] with a view to
exhibiting a weaker set of conditions that is adequate to the task in hand. [C3]
implies, for instance, that if two falling bodies B(1) and B(2) have weights W(1)
and W(2), with Wð1Þ < Wð2Þ, there exists a continuous, monotonic increasing
function, say U, such that the speeds of the bodies, v(1) and v(2), satisfy
vð1Þ ¼ U½Wð1Þ; and vð2Þ ¼ U½Wð2Þ; with vð1Þ vð2Þ. Let W(12) be the weight
of the composite formed by binding the two bodies together. Then the natural
speed of the composite body is vð12Þ ¼ U½Wð12Þ. However, by [C2],
Wð12Þ ¼ Wð1Þ þ Wð2Þ > Wð2Þ.
The monotonicity of U implies that U½Wð12Þ U½Wð2Þ U½Wð1Þ. Hence
vð12Þ vð2Þ vð1Þ, which is inconsistent with [C1], unless all natural speeds are
equal (this implies that U is a constant, i.e. the trivial monotonic function).
There is however, an even weaker set of conditions which will do the job for
Galileo. It is sufficient to look at bodies that have the same natural speed and the
same weight. Consider the following set:
[Z1] The natural speeds of falling bodies are intensive
[Z2] Weight is extensive
[Z3] The natural speed of a falling body is a continuous function of its weight
In [Z1], by the term intensive we mean that if two bodies with the same natural
speeds are bound together, the natural speed of the composite is the same as that
of each of the two constituent bodies. We use the term here in much the same way
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as when one says that temperature is intensive: two bodies at the same temperature, when brought into thermal contact with one another, constitute a composite
body of the same temperature. [Z1] represents a less powerful constraint than [C1],
in that it only refers to bodies with the same natural speeds. No statement is made
about what happens when bodies with different natural speeds are bound together.
In [Z2], by the term extensive we mean that if two bodies with the same weight are
bound together, the composite has twice the weight of either of the bodies by itself.
Here again no explicit statement is made about what happens when two bodies of
different weights are combined, as it is in [C2], although it is easy to prove that
extensivity implies additivity. In [Z3] the only assumption is that there is a continuous function, U, such that a body of weight W has natural speed v ¼ U½W. Since
no assumption is made now that the continuous function U must be monotonically
increasing, it is clear that [Z3] is weaker than [C3]. We shall now prove that the set
of weak assumptions [Z1]–[Z3] actually implies that U is a constant. That is, U[W]
does not depend on W after all, and therefore all bodies have the same natural
speed, precisely Galileo’s conclusion.
The proof takes the form of a little thought experiment: suppose that we divide a
body of weight W into two pieces, each of equal weight. By [Z2] the pieces each
have weight W/2. By [Z3], since they have equal weight the pieces have equal natural speed, and by [Z1] the natural speed of the original body must be the same as
that of the pieces. Hence:
U½W ¼ U½W=2:
However, we can now repeat the process by dividing one of the pieces into two
equal parts, thereby proving that U[W/4] is equal to U[W/2]. On iterating the
reasoning ad infinitum, we prove that:
U½W ¼ U½W=2 ¼ U½W=4 ¼ U½W=8 ¼ ¼ U½0
where the final step is justified by the fact that U is a continuous function.2 To conclude the proof, apply the same argument to any other body, of weight W0 say,
and show also that U½W0 ¼ U½0. Thus:
U½W ¼ U½W0 for any W and W0 , so U[W] is independent of W, and thus all natural speeds are
equal to one another.
4. Norton and the tacit assumption of Galileo
The foregoing three reconstructions of Galileo’s argument seem unexceptionable.
However, strictly speaking they are all non sequiturs. They are all based on a hid2
It should be admitted that U[0], the natural speed of a falling body in the limit that the weight tends
to zero, is a theoretical construct with no direct physical relevance. It is introduced to allow the proof to
work when W/W0 is irrational.
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den assumption, namely that any other parameters determinative of natural speed
are excluded. We call this ‘Galileo’s tacit assumption’. Without this assumption,
Galileo’s conclusion that all natural speeds are the same does not follow from
[G1]–[G3], nor from [C1]–[C3], nor yet from [Z1]–[Z3]. Even our proof that the
weak set [Z1]–[Z3] implies the constancy of U is invalid. It is correct only if we add
Galileo’s tacit assumption, strengthening [Z3] to:
[Z30 ] The natural speed of a falling body is a continuous function of its weight,
and of nothing else.
The same goes, mutatis mutandis, for [C3] and [G3]. Certainly Galileo knew that
the rate of fall of a gold leaf in air is different from that of a pellet of gold of the
same weight, and therefore that the rate is dependent on the shape as well as the
weight. Moreover, he recognized implicitly that his thought experiment only works
for bodies of the same substance, or at any rate ‘provided they are of the same specific gravity.’ (Galileo, 1638, p. 108) He realized that it would not do to bind two
bodies of different specific weights (or densities) together, for if the natural speed
were to depend on the density, as well as on the weight (as indeed it does for fall in
a viscous medium such as air), his demonstration would fail.3
Brown admits that rate of fall, even in vacuo, could logically depend on other
parameters. He cites chemical composition or colour as possibilities (Brown,
1991), without however taking either of them seriously, nor does he seem to
consider any other possible empirical dependence. In spite of Brown’s dismissal,
the first parameter he mentions (chemical composition) is a serious option, and
indeed experiments of great sensitivity have been performed to measure the accelerations with which different chemical substances fall in vacuo. Does a sphere of
lead fall in vacuo at the same rate as a sphere of aluminium? To phrase the matter more theoretically, is the ratio of gravitational mass and inertial mass the
same for all substances? The gravitational mass of a body may be defined as the
coefficient of proportionality between the body’s weight and the gravitational field
in which it is situated. This is conceptually different from the inertial mass of the
same body, which is the coefficient of proportionality between a force acting on
the body (for example, its weight) and its resulting acceleration. That the two
kinds of mass are numerically equal has been experimentally tested to high accuracy (Eötvös et al., 1922); and the equality was built into the very foundations of
Einstein’s general theory of relativity. Because of the equality, two bodies of different gravitational masses, if placed in the same gravitational field (with no
restraining forces), will suffer the same acceleration, precisely because the ratio of
the two bodies’ inertial masses is the same as the ratio of their gravitational
3
‘Specific weight’ is an intensive property, like density, indeed it is the ratio of the density of the body
in question to that of water. ‘Weight’ is an extensive property, and Galileo did not rule out the possibility that the rate of fall could be influenced by an intensive property like specific weight or density. The
thrust of his argument is that, contra Aristotle, the rate could not be a linear function of weight, which
is an extensive property. This matter is further adumbrated in Appendix E.
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masses. For our purposes, it is important to realize that this equality is an empirical
finding, and not a logical truth. The second parameter that Brown mentioned
(colour) was presumably intended jocularly; but, at the risk of ruining Brown’s
joke, we point out that if two falling bodies of different colours are exposed to
vertically directed light from a laser, tuned to the frequency corresponding to the
colour of one, but not of the other body, then the light pressure experienced by
the two bodies will not be the same, and so the rates of fall of the two bodies will
be unequal.
J. D. Norton has stated explicitly that Galileo’s argument only works if one
adjoins the assumption that the speed of fall of bodies depends only on their
weights. (Norton, 1996, p. 343.) The relevance of Norton’s claim can be illustrated by reconsidering Gendler’s reconstruction of Galileo’s argument. As we
have seen, Gendler does not question the validity of [G1] and [G2]; neither would
a convinced Aristotelian or a Galilean do so. Nevertheless, the matter is more
subtle than Gendler supposes, for denying [G1] is not as bizarre as she seems to
assume. In fact, some falling bodies do not satisfy the mediativity condition [G1].
It is an empirical fact that bodies falling at their terminal velocities in a medium
may, or may not satisfy the mediativity condition [G1]. This can be best explained
on the basis of the set [C1]–[C3] (recall that [G1] is identical to [C1]). For
example, two lead spheres of different weights (and therefore with different
volumes), will have different terminal velocities. If they are tied together sideby-side, the terminal velocity of the united system will lie between the terminal velocities of the constituents, i.e. [C1] applies. In this case, it is [C3] that fails. If, on
the other hand, the spheres are melted and recast as one sphere of weight equal to
the sum of the weights of the two original spheres, then the terminal velocity of
the united system will be greater than those of either of the constituents. The reason is that the retarding viscous force is a function of both the velocity and of the
surface area of the falling body. The smelted sphere falls more quickly than the
united spheres because the surface of the former is smaller than the combined
surfaces of the latter. In this case [C3] applies and [C1] fails (see Appendices B
and D for further details). The situation is analogous with the weaker conditions
[Z1]–[Z3]. For if two identical lead spheres are smelted into one, the terminal
velocity will not be equal to that of the original spheres, but rather will be greater.
Thus [Z1] is incorrect in this situation, while [Z3] is true. Galileo’s Simplicio is
too hasty in agreeing that [C1] is indubitable, and therefore that [C3] must in all
cases be false.
The failure of Galileo’s reasoning can be further illustrated by considering free
fall in a vacuum, and under replacement of the Aristotelian notion of natural speed
by the Galilean notion of acceleration. It is not even true that bodies of different
weights must fall, on earth and in vacuo, with the same acceleration, because these
accelerations may depend on variables other than the weights. To illustrate this
fact, let us return to the tower and the falling pieces of rock, but now in the light
of Newtonian physics. Imagine that I stand at the top of the tower, holding the
8 kg piece in my left hand and the 4 kg piece in my right hand. I stretch out my
arms, so that the two pieces are side by side, at the same height from the ground.
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If I now let them go simultaneously, they will fall with the same acceleration, taking the same time to reach the ground. This is precisely the solution sketched by
Galileo.4 However, if I place one piece just above its companion, but in contact
with it, it will fall with ever so little less acceleration than the other piece, because
the gravitational field at its location is slightly smaller than that at the location of
the lower piece. This means that the two pieces will lose contact: they will separate,
and the lower will reach the ground before the upper. If the two pieces are tied
together (or if they are fused into one another and thus ‘truly one’, it makes no difference), then the gravitational field, averaged over the bound system, lies between
the fields averaged over the lower and over the upper segment separately. Consequently, the time of fall of the bound system will lie between the times for the
upper and the lower pieces. This means that condition [C1] indeed holds nontrivially (and not, to use Gendler’s words, in a vacuous sense). [C1] is relevant
because the position in a nonuniform gravitational field of each point of a body
plays a role in the body’s acceleration.
Of course, the differences described are very small for rocks of a few metres in
diameter. But for mountain sized rocks, or for sizeable asteroids, they would be
considerable. Be that as it may, the size of the effect is not at issue here. In the
presence of a homogeneous gravitational field (and in the absence of air), different
bodies would indeed fall at the same rate. However, this is not an a priori statement about the way bodies fall; indeed, given that the earth’s gravitational field is
inhomogeneous, it is not even an accurate statement. In Appendix C we give further mathematical details, showing precisely how Galileo’s reasoning breaks down
when the gravitational field is nonuniform.
At this juncture, modern apologists for Galileo might remark that the inhomogeneity of the gravitational field could be seen as a disturbing factor, on a par with
air friction. It requires, after all, little effort to postulate a homogeneous gravitational field in order to reinstate Galileo’s thought experiment in all its pristine
splendour. Moreover, aside from the effect of inhomogeneity of the gravitational
field, we can of course cite other instances of disturbing factors. If the rocks contained iron, and there was a magnetic field present, the rate of fall would be influ4
It is generally agreed that Galileo did not in fact drop musket shot and cannon balls from the leaning tower of Pisa. Indeed, Cushing (1998), p. 83, suggests that the effect of air friction, and the technical
difficulties Galileo would have had in recording the times of arrival of balls of different weights, would
probably have rendered such an experiment inconclusive. Galileo could not have verified that the musket shot always arrives after the cannon ball, so there would have been no point in performing the
experiment, he claims. McAllister (1996), pp. 245–248, goes even further. He claims that Galileo was
right to limit his attention to a thought experiment, rather than a real experiment, on the grounds that
irrelevant interfering effects would have nullified the import of a real experiment: ‘I suggest that Galileo
devised thought experiment as a source of evidence about phenomena for use where all feasible concrete
experiments exhibited the shortcoming that distinct performances of them conflicted’ (McAllister, 1996,
p. 245). The claims of Cushing and of McAllister can be contested, for suppose the musket shot to have
had a weight one tenth that of the cannon ball. Then, according to the strong Aristotelian dogma, the
former should have taken more than half a minute to reach the ground. This is a prediction that Galileo
could easily have falsified.
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125
enced, as it would be if the rocks were electrically charged, or if they were electrically conductive, and there was an electrostatic field present. The apologists would
have to suppose the absence of these complicating factors too.
Our answer to these apologists would consist in making a distinction between
specified and unspecified disturbing factors. It would be circular to require all
unspecified disturbing factors to be absent, so that Galileo’s law of falling bodies be
correct. However, for each putative disturbing factor that is specified, one needs a
theory to be able to postulate conditions coherently under which it would be
absent. For instance, one can only identify the inhomogeneity of a gravitational
field as a disturbing factor if one knows enough about it, within the framework of
a theory of gravitation. Galileo lacked such a theory, at least one in which gravitational forces (i.e. weights) drop off as the inverse square of the distance from the
centre of the earth. Such a theory was only invented a generation later by Newton,
who was able to test it quantitatively with the help of his calculus; he was able to
compare the motions of falling apples with that of the moon, as it ‘falls’ endlessly
in its month-long orbit around the earth. Since physical laws are tested by their
empirical implications, specified disturbing factors must likewise be considered,
controlled and rendered manageable within a theory that can be falsified.
Accordingly, thought experiments do not offer us a glimpse into a Platonic world of
verities, as Brown claims. They do not stand on their own, but must be subordinated
to the theories which they inspire (and by which they are inspired). In order to gain
knowledge of the world, Galileo performed, and needed to perform, real experiments
with steel balls and inclined planes.
5. Galileo and prior philosophy
A lively philosophical debate on the nature of thought experiments can be found
in the literature from about 1990. Concerning thought experiments in natural
science, Brown, Norton and Gendler made significant contributions. As we have
seen, Brown adopts a clear stance. For him, the essence of a thought experiment is
that it teaches us new things about the world without the use of new empirical
data. Brown’s world view is denied by Norton, for whom thought experiments are
disguised arguments. On the basis of a careful study of the epistemology of
thought experiments (as opposed to, for instance, their impact on the scientific
community), Norton concludes that thought experiments ‘can do no more than can
ordinary thinking with its standard tools of assumption and argument’ (Norton,
1996, p. 366). In particular, they ‘open no new channels of access to the physical
world’ (op. cit.).
While the distinction between Brown’s view and that of Norton is fairly straightforward (it is the familiar dispute between rationalists and empiricists), the contrast
between Norton and Gendler is not so easy to discern. Gendler opposes both
Norton and Brown. With Norton, and against Brown, she holds that thought experiments are arguments, not reports on perceived vistas in a Platonic world. However,
she departs from Norton in claiming that they are arguments of a special kind,
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namely ones with a particularly strong persuasive power, their ‘justificatory force’.
Apparently taking her inspiration from Mach, 1883, 1887), Gendler ascribes the
justificatory force to the fact that, in a successful thought experiment, instinctive
and hitherto unarticulated empirical knowledge suddenly becomes organised and
manifest. In this vein, Gendler speaks of thought experiments as ‘guided contemplations’, but otherwise underwrites the major points of Norton’s exposé.
Perhaps the best way to pin down the contrast between Norton and Gendler would
be to compare their discussion with the analysis of a joke. The point of a good joke,
like the point of a good thought experiment, is sudden and exhilarating insight;
jokes as well as thought experiments ‘work’ when beliefs that were slumbering in
the background suddenly become manifest. A joke, like a thought experiment, can
however also be explained, so that all the steps are made explicit. Whereas Norton
stresses that thought experiments can without loss of meaning be rewritten as arguments without imaginary particulars, Gendler emphasizes that such an approach
misses the liberating insight that is the hallmark of a good thought experiment
(analogous to ‘getting’ a good joke). Gendler and Norton are merely speaking
about different aspects of a thought experiment—its subject matter, and the
psychological impact on the person who comprehends it, respectively.
Differences between Gendler and Norton aside, our analysis of Galileo’s thought
experiment clearly supports the Norton/Gendler faction rather than the Brown
camp. Being a rationalist, Brown believes the purest thought experiments to be
those that enable us to acquire knowledge of the world without wearying our
physical senses. Indeed, as we have seen, he cites Galileo’s thought experiment as
the paradigmatic example, for in Brown’s view this thought experiment not only
established ‘the end of Aristotle’s theory’, but it also yielded a premium, ‘since the
right account is now obvious: they all fall at the same speed’ (Brown, 2000, p. 529).
We have argued that even this alleged classic case falls short of being a Platonic
thought experiment in Brown’s sense. For the statement that all bodies fall at the
same speed is not always the right account, and of course it is not obvious unless
other conditions are satisfied. These conditions may well not be applicable to particular falling bodies, and whether or not they are applicable is a matter of empirical
research.
This may all sound plausible to empiricists’ ears. However, a towering figure like
Galileo claimed the contrary (‘even without further experiment, it is possible to
prove clearly . . . that a heavier body does not move more rapidly than a lighter
one’); and an astute empiricist like Gendler seems to think that Galileo’s thought
experiment for ever put aside Aristotle’s conclusion as erroneous (‘a simple and
obvious mistake’, Gendler, 1998, p. 402).
We wish to extract from our iconoclastic analysis of one of the most famous
thought experiments of all time this simple, sobering lesson: we can know nothing
of the phenomenal world except by looking at it, theorizing about what we see,
and testing the predictions of our empirical theories by further recourse to Nature
herself. This observation itself cannot be shown to be indubitably correct: it is
rather the painful lesson extracted from millennia of failures of the direct, magical
mode of apperception. The latter day Platonist is hoisted on his own petard: James
D. Atkinson, J. Peijnenburg / Stud. Hist. Phil. Sci. 35 (2004) 115–136
127
Brown’s classic Platonic thought experiment has been shown to be not only logically deficient but also to fail to adequately describe the empirical world.
Acknowledgements
Special thanks are due to Pieter Sjoerd Hasper for an illuminating discussion of
some aspects of Aristotle’s De caelo. We are grateful for the interactive contribution of audiences in Bertinoro, Gent, Groningen, Leusden and Rotterdam, and
in particular for that of James McAllister, who brought Casper’s article to our
attention. We acknowledge also an e-mail from Jim Cushing concerning Galileo’s
reasons for not having attempted the celebrated experiment in Pisa.
Appendix A. Newtonian analysis of accelerated motion
The three reconstructions of Galileo’s argument in Section 3 all use the somewhat vague notion of natural motion, quantified by the scarcely less precise concept of natural speed. In this appendix, we propose ‘acceleration’ as an interpretion
of ‘natural motion’ and as a replacement for ‘natural speed’. Furthermore, we shall
speak here only of falling bodies in vacuo and in a uniform gravitational field. The
aim is to show that, under these conditions, Galileo is right.
Let us investigate, within the formalism of Newtonian physics, not the conditions [C1]–[C3] of Sect. 3, but rather the analogous statements:
[S1] Accelerations of falling bodies are mediative
[S2] Weight is additive
[S3] Accelerations of falling bodies are proportional to weights
Thus ‘natural speeds’ have been replaced by ‘accelerations’, in the spirit of Galileo,
and the strong, rather than the weak form of the Aristotelian dogma has been used
(with the above replacement). Galileo is right in accepting [S1] and [S2] and rejecting [S3]. For bodies falling in a vacuum in a uniform gravitational field, the facts
are that [S1] and [S2] are true and [S3] is false. Such, at least, is the verdict given by
Newton’s laws of motion and gravitation.
The gravitational forces acting on bodies B(1) and B(2), i.e. their weights, are
W(1) and W(2). Let their inertial masses be m(1) and m(2), respectively. According
to Newton’s second law of motion, F ¼ ma, the accelerations of the bodies B(1)
and B(2) are given by að1Þ ¼ Wð1Þ=mð1Þ and að2Þ ¼ Wð2Þ=mð2Þ. It is part of
Newton’s theory that:
[N1a] Forces, and hence in particular gravitational forces, are additive5
[N1b] Inertial masses are additive
5
In general, forces obey the rules of vector addition, but in the present case of forces that are parallel
to one another, vector reduces to scalar addition.
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According to [N1a] and [N1b], the force acting on the composite made by tying the
bodies B(1) and B(2) together (its weight) is Wð1Þ þ Wð2Þ, while the inertial mass
of the composite body is mð1Þ þ mð2Þ. Hence the acceleration of the composite is:
að12Þ ¼ ½W ð1Þ þ W ð2Þ=½mð1Þ þ mð2Þ
¼ ½mð1Það1Þ þ mð2Þ að2Þ=½mð1Þ þ mð2Þ
If we suppose that að1Þ < að2Þ, then the expression on the right is increased if a(1)
is replaced by a(2), whereas it is decreased if a(2) is replaced by a(1). It follows
therefore immediately that að1Þ < að12Þ < að2Þ. In other words, Newton’s claims
[N1a] and [N1b] imply [S1], i.e. accelerations are indeed mediative. This mediativity
is not a mere logical possibility: it is actually realized in a nonuniform gravitational
field, as we show in Appendix C.
As far as [S2] is concerned, the additivity of weights, one might at first think that
it is equivalent to [N1b]. But this is not so, for in Newton’s system weight is a
function of a body’s gravitational mass and the local gravitational field. However,
since weight is a force, [S2] is implied by [N1a]. Conclusion: Newton underwrites
both [S1] and [S2]. Thus, on pain of falling into an Aristotelian contradiction, [S3]
must be wrong. This is all of course exactly in accordance with Galileo’s reasoning.
Appendix B. terminal velocity of fall in a fluid
In the preceding appendix we have been talking about bodies falling in vacuo.
We used the term ‘acceleration’ as an interpretation of ‘natural motion’ and as a
replacement for ‘natural speed’. In the present appendix, however, we shall rather
consider bodies falling in resistive media. Here it is more useful to interpret ‘natural motion’ in terms of ‘speed’ rather than acceleration; and ‘natural speed’ will be
made precise in terms of ‘terminal velocity’. The purpose is to show that, when the
speed is sufficiently slow, Aristotle’s strong dogma is an accurate description of the
empirical state of affairs.
Bodies, falling in a viscous fluid, accelerate at first but approach their terminal
velocities asymptotically. A good case can be made indeed that Aristotle was interested in falling bodies in situations where fluid viscous forces are important:
. . . in the ratio which their magnitudes bear to one another . . . In moving
through plena it must be so. (Aristotle, see above, our italics).
When the speed of the falling body is so small that there is no turbulence in the
flow of air around it, the viscous retarding force is proportional to its velocity, and
Newton’s second law of motion for the acceleration, a, reads:
ma ¼ mg k v
m being the mass (strictly speaking, the inertial mass on the left and the gravitational mass on the right), g the acceleration due to gravity (assumed constant), v
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129
the instantaneous velocity, and k the frictional coefficient due to viscous retardation. This equation has the following solution for the velocity:
v ¼ ½1 exp ðk t =mÞ mg=k
and this yields, for the terminal velocity,
vðterminalÞ ¼ W=k
where W ¼ mg is the weight. Evidently, if we interpret ‘natural speed’ as terminal
velocity in a medium (in plena), then Aristotle is right that the speed of a body is
proportional to its weight. But what has happened then to the Galilean
contradictio? The statement [S1] from Appendix A is replaced by:
[S1’] Terminal velocities of falling bodies are additive (not mediative)
Due to the change of [S1] to [S1’], there is now no inconsistency. Twin sisters suspended from one parachute fall twice as quickly as one sister!
Again, it is not our contention that Aristotle had the above interpretation in
mind, but only that such an interpretation is possible, and it serves, among other
things, to throw further doubt on the worth of Galileo’s thought experiment. The
conclusion applies in special circumstances, namely for two bodies of the same
shape and size, but then Aristotle did write:
We see that bodies which have a greater impulse either of weight or lightness, if
they are alike in other respects, move faster over an equal space, and in the ratio
which their magnitudes bear to one another. (Aristotle, Physica, Book IV/viii/
216a, 13–17)
In this passage, Aristotle is clearly considering the motion of bodies in a medium,
and he is well aware that one needs to compare bodies of the same size and shape.
Appendix C. rate of fall in terrestrial gravity
In Appendix A we looked at bodies falling in vacuo in a uniform gravitational
field. We argued that they all fall with the same acceleration, supporting Galileo’s
conclusion. In Appendix B we considered bodies falling slowly in viscous fluids.
We showed that then it is Aristotle who is correct. In the present appendix we
return to falling bodies in vacuo. However, rather than situating them in a uniform
gravitational field, as we did in Appendix A, we will now place them in the inhomogeneous gravitational field of the earth.
It will be supposed that the bodies have no angular momentum with repect to
the centre of mass of the earth, so that they fall radially. The scenario of Galileo’s
thought experiment, as sketched by Salviati, will be re-enacted, but now with the
two bodies at different distances from the earth. They will in the first instance be
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considered to fall unimpeded, and in the second instance as a bound system, connected by a cord. It will be shown that the accelerations of the systems are mediative, that their weights are additive, but that Galileo’s conclusion that all bodies
must fall at the same rate is incorrect, for the two bodies, as well as the bound system, fall at different rates.
The earth is a spheroid, and the weight of a body depends not only on its gravitational mass, but also on how high it is above the surface of the earth. The acceleration of a falling body at two earth-radii from the centre of the earth is only one
quarter what it is when the body is close to the earth’s surface. The rate at which a
body’s velocity increases is dependent on its position. Consider a body, B(1), of
mass m(1), at distance r from the centre of the earth, and another, B(2), of mass
m(2), at distance r þ d from the centre of the earth, directly above the first one. If
the bodies are not connected, B(1) will fall with acceleration:
gð1Þ ¼ GM=r2
where M is the mass of the earth and G is Newton’s gravitational constant,
whereas B(2) will fall with a smaller acceleration, namely:
gð2Þ ¼ GM=½r þ d2
Suppose now that the two bodies are connected by a light, inextensible cord, of
length d. Since B(1) has a tendency to fall more quickly than B(2), the the cord will
be under tension, say T, which serves to speed up B(2) and to brake B(1). The net
force on the lower body will be mð1Þ gð1Þ T, while that on the upper body will
be mð2Þ gð2Þ þ T. The total force on the composite system is
mð1Þ gð1Þ þ mð2Þ gð2Þ, the tension of the cord having dropped out of this sum.
The total mass of the system is mð1Þ þ mð2Þ, so its acceleration is:
gð12Þ ¼ ½mð1Þ gð1Þ þ mð2Þ gð2Þ=½mð1Þ þ mð2Þ
Since g(1) is larger than g(2), it follows trivially from the above formula that
gð1Þ > gð12Þ > gð2Þ.
Thus the acceleration of the tied, composite system is mediative, lying as it does
between the accelerations that the bodies would have, were they not tied together.
The weight of the composite system, namely ½mð1Þ þ mð2Þ gð12Þ, is indeed additive, being precisely the sum of m(1) g(1) and m(2) g(2). Thus [S1] and [S2] are
applicable in this case, and so [S3] must be untrue. Indeed, the accelerations of the
bodies B(1) and B(2), of weight m(1) g(1) and m(2) g(2), and that of the composite,
of weight ½mð1Þ þ mð2Þ gð12Þ, are respectively g(1), g(2) and g(12), which are not
in the ratio of the weights. In this case the destructive part of Galileo’s reasoning is
valid; but his conclusion, that therefore all bodies fall with the same acceleration, is
patently false. The reason is that the accelerations can, and do depend on
something other than their weights, namely on the distances of their centres of
mass from the centre of mass of the earth.
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131
Appendix D. laminar and turbulent flow
In Appendix B we looked at the terminal velocities of falling bodies in resistive
media, in the domain in which the fluid motion is nonturbulent; and we showed
that the strong dogma of Aristotle is then correct. In the present appendix we will
study falling bodies in resistive media in which turbulence does take place, and we
shall demonstrate that, in this domain, Aristotle’s strong dogma is incorrect, but
his weak dogma still holds.
Consider the legendary experiment that Galileo could have performed from the
leaning tower of Pisa. The parapet at the first clock tower is 55 metres from the
ground. A cannon ball, dropped from the parapet, would take about 3.35 seconds
to reach the ground, while a musket shot takes a few hundredths of a second
longer. Galileo could not have measured such a small difference in times of fall,
indeed, he could not have arranged with reliability to release cannon ball and musket shot within a hundredth of a second of each other. For these, and other reasons, many people have relegated the story of the public experiment to the realm of
myth.
We propose to calculate the velocity of a ball, falling in a viscous fluid, such as
air, as a function of the time. To do this, consider a sphere of radius r moving at
speed v in a fluid of density q and viscosity g. The Reynolds number for this system, a dimensionless quantity, is defined by:
R ¼ 2 q r v=g
ð1Þ
If the velocity is so small that the Reynolds number is not greater than one, we
speak of laminar flow: under these conditions it is found that the fluid moves, near
the sphere, in regular laminae or layers, each having its own velocity. For higher
velocities, at which the Reynolds number is much larger than unity, non-steady or
turbulent flow takes place, with the formation in general of eddies.
When the velocity is so small that R is one or less, the force exerted by the fluid
on the sphere is given by Stokes’ formula:
F ¼ 6pg r v
ð2Þ
but for larger velocities, when the flow is turbulent, the force is given by the semiphenomenological formula of aerodynamics:
F¼
1
Cpq r2 v2
2
ð3Þ
where C is called the drag coefficient—it is the inclusion of this factor that makes
the formula not a purely theoretical one. Experimentally it is found that C is about
1
2 for a Reynolds number between about 1000 and 200,000. Above 200,000, R
drops to about 1/5, but below 1000, C can be much larger than 12 , in fact the
experimental curve (Landau and Lifshitz, 1987) can be roughly fitted up to about
R ¼ 100; 000 by:
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C ¼ 24=R þ
1
2
With this fit to C, the aerodynamic formula (3) reduces to:
F ¼ 6pg r v þ
1
pq r2 v2
4
ð4Þ
which is Stokes’ formula (2), plus an aerodynamic term that corresponds to formula (3) with C ¼ 12 . A sphere of mass m, falling in a fluid, satisfies Newton’s
equation:
1
ma ¼ mg 6pg r v pq r2 v2
4
ð5Þ
in the approximation (4). Here a is the acceleration of the sphere and g is the acceleration due to gravity, which is approximately 9.81 m/s2. We integrate Eq. (5)
exactly, obtaining the following solution for the velocity:
kv ¼ l tanh ðlt þ uÞ j
ð6Þ
with:
j ¼ 3pg r=m
k ¼ 1=4pq r2 =m
l2 ¼ kg þ j2
u¼
and:
1
log ½ðl þ jÞ=ðl jÞ
2
The distance fallen, d, can be obtained by integrating Eq. (6):
kd ¼ log cosh ðlt þ uÞ log cosh u jt
ð7Þ
For very small velocities the aerodynamic term in Eq. (4) can be neglected, and we
recover the situation described in Appendix B for laminar flow, with the identification k ¼ 2mj.
To estimate the relevant orders of magnitude, let us consider dropping two balls
of iron, one of mass 0.5 kg, and the other of mass 50 kg, from the first parapet of
v
the leaning tower of Pisa, 55 metres from the ground. At 20 C, the density and
3
viscosity of air are about q ¼ 1:2 kg=m and g ¼ 0:000 018 kg=ðmsÞ. Given that
the density of iron is 7860 kg/m3, we find the radius of the small ball to be
r ð1Þ ¼ 0:0248 m, and the radius of the large ball to be r ð2Þ ¼ 0:115 m, so:
j ð1Þ ¼ 0:000 008 42 k ð1Þ ¼ 0:001 160
j ð2Þ ¼ 0:000 000 39 k ð2Þ ¼ 0:000 249
in MKS units. Evidently j2 is negligible for both balls, compared to kg, so the
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133
phase u can be neglected, giving l2 ¼ k g to high accuracy, and Eq. (7) becomes:
k d ¼ log cosh lt
From this formula we find that the times of fall of the balls, over 55 m, are:
t ð1Þ ¼ 3:38 s t ð2Þ ¼ 3:35 s
so the heavy ball arrives three hundredths of a second ahead of the light one, a difference that Galileo would indeed not have been able to measure. Eq. (6) becomes,
in good approximation:
v ¼ ðl=kÞ tanh ðltÞ
ð8Þ
and the velocities of arrival of the two balls at the ground are:
vð1Þ ¼ 31:79 m=s
vð2Þ ¼ 32:57 m=s
At these speeds, the Reynolds numbers can be calculated to be:
Rð1Þ ¼ 105 000 Rð2Þ ¼ 500 000
so both balls suffer strongly turbulent air resistance, and the Stokes’ contribution,
the first term in Eq. (4), is negligible as compared to the aerodynamic term, for
which the drag coefficient is close to 12 .
The terminal velocity is the coefficient of the hyberbolic tangent in Eq. (8), and
this is very different for the two balls. However, the heavier ball achieves 95% of its
terminal velocity only after a fall of several kilometres, so one would have to
repeat the experiment from a cliff several km. in height in order to find considerable differences in the velocities of the balls. Note that these terminal velocities are
not in the proportion of their weights. The distance fallen in a given time is a
monotonic decreasing function of k, see Eq. (8). Hence, for given air density, the
velocity is a monotonic increasing function of m/r2. For balls of different radii, r,
but the same density, as in the case of the iron balls, m/r2 is proportional to r, so
the distance fallen in a given time is a monotonic increasing function of r, and so
of mg, the weight (consistent with the weak, but not the strong Aristotelian dogma).
To conclude this appendix, let us estimate how small an iron ‘ball’ would have
to be, in order for its terminal velocity to be ten times that of a wooden ball of the
same size but only one tenth the weight. Stokes’ formula (2) is a good approximation only if the Reynolds number is not in excess of unity. For a sphere of iron
of radius twenty microns (0.02 mm), the terminal velocity is 0.34 m/s, and in this
case R ¼ 0:9. The terminal velocity of the same sized wooden sphere is 0.038 m/s,
just a little greater than one tenth the terminal velocity of the iron ball. For larger,
but still tiny balls, the ratio of the terminal velocities departs further from the ratio
of the weights. For example, with a radius of 1 mm the air flow is turbulent and
the ratio of the terminal velocities is about 1:4 instead of 1:10. It is evident that,
except for balls of microscopic size, and except for the very beginning of the fall,
turbulence is of overriding importance and the strong dogma of Aristotle is not
applicable.
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Appendix E. textual sources
The basic questions to be asked in this appendix are these: did Aristotle actually
claim that the rate of fall of bodies is proportional to their weight (when they move
toward their natural place), given the ambiguities in translating ancient Greek; and
did Galileo in fact state that the rate of fall of all bodies is the same in all circumstances?
Although it seems that the Stagirite did think that a rock that is twice as heavy
as another falls twice as quickly, Barry M. Casper insists that Aristotle must have
intended something else by the term ‘heavier’ than we do:
. . . either Aristotle was a fool or he had something else in mind. (Casper, 1977, p. 329)
In a valiant attempt to defend the Master, Casper refers to the passage:
By lighter or relatively light we mean that one, of two bodies endowed with
weight and equal in bulk, which is exceeded by the other in the speed of its
natural downward movement. (Aristotle, De caelo 4.1, 308a 31–35)
He suggests that we should construe this as an operational definition of relative
lightness and heaviness. As he says, ‘to determine which of two objects is heavier,
one observes their speed of fall; the heavier is the one that falls faster’ (Casper,
1977, p. 328). On this reading, Aristotle’s statement that a heavier body falls faster
than a lighter one (the weak dogma) would be a tautology. This interpretation is
´
supported by Moraux (1965), who, although he translates baqj1 into French as
‘lourd’, renders the above passage thus:
. . . nous parlons de léger relativement à autre chose et de plus léger quand, de
deux corps lourds de volume égal, l’un se porte naturellement vers le bas plus
rapidement que l’autre.6 (Moraux, 1965, p. 137, translator’s italics.)
However, he also gives, as two of the tenets of Aristotle (Moraux, 1965, p. CXLIX):
4. Les poids de plusieurs corps de même nature sont entre eux comme leurs
volumes.
5. Les temps de chute des graves sont inversement proportionnels aux poids.7
In 4, Moraux agrees with Casper that Aristotle defined the relative weight of two
bodies of different material but of the same volume in terms of their relative rates
of fall. However, in 5 he shows that Aristotle also thought that, for two bodies of
the same nature, at the same distance from the earth, one with a volume ten times
that of the other, the weight of the larger is ten times the weight of the smaller
6
. . . we say light compared with something else and lighter when, of two heavy bodies of equal volume,
one travels naturally downwards more rapidly than the other.
7
A free rendering of the French is as follows:
(4) The weights of several bodies of the same nature are proportional to their volumes.
(5) The times of fall of heavy bodies are inversely proportional to their weights.
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135
(correct), and the time of fall of the smaller is ten times longer than that of the larger (incorrect). The matter appears to be more complicated than Casper supposed.
As we have seen, Galileo did not interpret Aristotle in the way that Casper suggests, nor did Philoponus in antiquity (see Wolff, 1971), nor Simon Stevin at the
beginning of the seventeenth century:
. . . Aristotle . . . thinks . . . that when two similar bodies of the same density fall in
air, their rate of fall is in proportion to their relative weights . . . But the experiment against Aristotle is like this: Take two balls of lead (as the eminent man
Jean Grotius . . . and I formerly did in experiment) one ball ten times the other in
weight; and let them go together from a height of 30 feet down to a plank below
. . . you will clearly perceive that the lighter will fall on the plank, not ten times
more slowly, but so equally with the other that the sound of the two in striking
will seem to come back as one single report. (Stevin, De staticae, Leyden, 1605,
quoted in Latin and translated into English by Cooper, 1935, p. 79)
Toulmin suggests that Aristotle did believe what we have called the strong dogma,
but that he was talking about terminal speeds of fall in viscous fluids:
According to Stokes, the body’s speed under those circumstances will be directly
proportional to the force moving it, and inversely proportional to the liquid’s
viscosity. (Toulmin, 1961, p. 51)
As Casper says, ‘If this interpretation is valid, Aristotle and Galileo did not really
disagree about the nature of falling objects. They were merely talking past one
another . . .’ (Casper, 1977, p. 328). However, Casper finally rejects this interpretation; and a further reason to doubt that Aristotle had terminal speeds in mind
when formulating the strong form of his dogma is that these speeds are only proportional to the weights in conditions of laminar flow, and these conditions do not
obtain for rocks of ordinary size falling a number of metres in air (see Appendix D).
Galileo limited his celebrated contradictio to a discussion of bodies of the same
specific weight (see the end of Salviati’s harangue, as cited in Sect. 2 above). This
suggests that he wanted to leave the possibility open that the rate of fall might be a
function of the specific weight, and thus of the density of the falling body. If this is
true, Galileo recognized indeed that his argument is valid only if other possible
variables that might influence the rate of fall are held constant. However, holding
the density constant is unnecessary for fall in vacuo, and it is insufficient for fall in
air. Moreover, he entertained the strange notion that a ball of wood, at the beginning of its fall, moves more quickly than a ball of iron that is released at the same
height at the same instant, but that the iron quickly overtakes the wood.8 This
8
Such a differential effect is in fact possible: at the beginning of the fall, the Stokes’ term, involving
the viscosity of air, is important, while later the aerodynamic drag term, in which the density of the air
occurs, is decisive (see Appendix D). Although Galileo could not have observed any such differential
effect, the point is that he entertained the view that different bodies can and do fall at slightly different
rates.
136
D. Atkinson, J. Peijnenburg / Stud. Hist. Phil. Sci. 35 (2004) 115–136
seems to fly once more in the face of Salviati’s triumphant conclusion that the only
way to avoid the inherent contradiction in the (weak) Aristotelian dogma is to suppose that all bodies fall at precisely the same rate.
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