Galileo and prior philosophy

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Stud. Hist. Phil. Sci. 35 (2004) 115–136

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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 sta-ted that all bodies must fall at the same rate. However, there is an empirical situation wherethe speeds of falling bodies are proportional to their weights; and even in vacuo all bodies donot fall at the same rate under terrestrial conditions. The reason for the deficiency of Galileo’sreasoning is analyzed, and various physical scenarios are described in which Aristotle’s claimis closer to the truth than is Galileo’s. The purpose is not to reinstate Aristotelian physics atthe expense of Galileo and Newton, but rather to provide evidence in support of the verdictthat 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 thatheavier 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 thoughtexperiments (Brown, 1991; Norton, 1991; Sorensen, 1992; Norton, 1996; McAllister,1996; Gendler, 1998; Brown, 2000). However, as we will show, it does not attain theimpeccable standard that is generally assumed. At first sight, it appears to refute theAristotelian paradigm in a decisive and even awe-inspiring manner. But in fact it isflawed, both in its attempted refutation of the old, as in its attempted demonstrationof the new ideas on falling bodies. One may therefore not cogently claim, as Brown

rug.nl (J. Peijnenburg).

D. Atkinson, J. Peijnenburg / Stud. Hist. Phil. Sci. 35 (2004) 115–136116

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 sys-

tem 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 contra-

dicts 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 excel-

lence: 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 sub-

sequent two sections, we reconstruct Galileo’s reasoning and analyze the defi-

ciencies 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.

117D. Atkinson, J. Peijnenburg / Stud. Hist. Phil. Sci. 35 (2004) 115–136

We close the paper with some appendices devoted to technical considerations.Their main purpose is to show that, aside from purely logical matters, Newtonianphysics has definite implications for the correctness or otherwise of Galileo’s conclu-sions. It is not relevant that Galileo lacked the physics that we possess; the point isthat empirical situations can be envisaged in which Galileo’s claims are correct, andother situations in which they are not. We can therefore state with confidence thatGalileo’s double claim (namely that Aristotle’s dogma is logically inconsistent, andthat his own dogma is necessarily true) is unfounded. In Appendix A we analyze theaccelerated fall of bodies in a uniform gravitational field, showing that in this situ-ation, 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 thiscase we show that Aristotle’s conclusion is correct. In Appendix C we return to bod-ies falling in vacuo, but now taking cognizance of the fact that the earth’s gravi-tational field is nonuniform. Here it turns out that the details of Galileo’s thoughtexperiment can be matched, step by step, but that his grand conclusion, that all bod-ies fall at the same rate (i.e. with the same acceleration) is wrong. Technical detailsare given in Appendix D of turbulent fluid motion, these being relevant to a realistictreatment of musket shot and cannon balls falling from the leaning tower of Pisa, tocite a possibly apocryphal experiment. Appendix E is devoted to questions of bothAristotelian 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 dogmaconcerning falling bodies with especial clarity:

Salviati: But, even without further experiment, it is possible to prove clearly, bymeans of a short and conclusive argument, that a heavier body does not movemore 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 twobodies whose natural speeds are different, it is clear that on uniting the two, themore rapid one will be partly retarded by the slower, and the slower will besomewhat 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 uni-ted, the system will move with a speed less than eight; but the two stones whentied together make a stone larger than that which before moved with a speed ofeight. Hence the heavier body moves with less speed than the lighter; an effectwhich is contrary to your supposition. Thus you see how, from your assumptionthat the heavier body moves more rapidly than the light one, I infer that theheavier body moves more slowly . . . We infer therefore that large and smallbodies move with the same speed, provided they are of the same specific gravity.(Galileo Galilei, 1638, p. 108)


To judge Galileo’s critique of Aristotle, let us first study what the Stagirite himselfsaid. In some places, he writes merely that heavier bodies fall more quickly thanlighter ones:

The mistake common to all those who postulate a single element only is thatthey allow for only one natural motion shared by everything . . . But in factthere 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 comparativestatement that heavier bodies fall faster than lighter ones. But in other placesAristotle 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 asgreat and more moves the same distance in a less time, the times being in inverseproportion to the weights. For instance, if one weight is twice another, it will takehalf 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 statementthat 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 shouldsubstitute for the ancient notion of natural motion, before we can reasonably con-sider the question of inconsistency. Later we will introduce two possible measuresof 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 dowith the somewhat vague term ‘natural motion’, which is quantified by the notionof ‘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 spokesmanSalviati, that the Aristotelian dogma (whether in its weak or its strong version) islogically inconsistent. To underscore the fact that Galileo is mistaken, it is suf-ficient to point to one physical situation in which Aristotle’s dogma, even in itsstrong form, is empirically correct. The case which gives Galileo the lie is that ofbodies falling in a fluid (such as air or water) at their terminal velocities in the caseof laminar fluid flow (i.e. when the fluid motion is not turbulent, see Appendices Band D for the technical details). Consider this passage:

We see that bodies which have a greater impulse either of weight or lightness, ifthey 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.


which their magnitudes bear to one another . . . In moving through plena it mustbe so; for the greater divides them faster by its force. For a moving thing cleavesthe medium either by its shape, or by the impulse which the body that is carriedalong or is projected possesses. (Aristotle, Physica 4.8, 216a14–20)

Under the restriction of laminar flow, the viscous forces on bodies of identicalsize and shape are proportional to their velocities, so the terminal rates of fall areproportional (the times of fall inversely proportional) to the weights. It is not partof our thesis that Aristotle espoused, or could have espoused this detailedinterpretation, nor that Galileo excluded, or might have excluded the particularcase of terminal fall with laminar flow. The point is simply that, since there is asituation in which Aristotle’s conclusion is correct, Galileo’s contention that it isinternally inconsistent must be wrong. Somewhere in Galileo’s argument theremust be a flaw. In Section 4 we will see precisely where the flaw lies. But first, inSection 3, we will offer three reconstructions of Galileo’s argument, with the aim offinding the weakest set of assumptions that justifies his claim that all bodies fall atthe same rate.

3. Gendler and reconstructing Galileo’s argument

Galileo’s own resolution of the imagined inconsistency in the doctrine that differ-ent bodies fall at different rates, as implied by the weak dogma, is that all bodiesmust fall at the same rate. Moreover, via the words of Salviati (‘even without fur-ther experiment’) he presents this as a truth that is accessible to reason, renderingexperiment 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 tothe claim that, if two bodies that are moving with different speeds are subsequentlytied together, the bound system will thenceforth move at an intermediate speed. Inorder 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 naturalspeeds are the same. For then ‘weight might be additive and natural speed (in avacuous 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 correlatedwith weight, would have to deny [G1] or [G2], or both. He might do this by postu-lating an essential difference in mechanical behaviour between bodies that aremerely united and bodies that are unified. Gendler even goes so far as to make, for


the Aristotelian’s intellectual delectation as it were, a mathematical model in whicha continuous function of ‘the degree of connectedness’ of two bodies determinestheir physical properties. But Gendler’s conclusion is clear: even a dyed-in-the-woolAristotelian, on being confronted with such an exotic way of saving the Master’stheory, would recant and deny any reality to a distinction between union and unifi-cation. 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 ratherthan the latter: [G3] represents the strong rather than the weak dogma. We cannevertheless also reconstruct Galileo’s argument by using merely the weakAristotelian dogma. Thus one can show that the following three conditions areinconsistent 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 for-mulation of the weak Aristotelian dogma. We have constructed [C3] with a view toexhibiting 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 increasingfunction, say U, such that the speeds of the bodies, v(1) and v(2), satisfyvð1Þ ¼ U½Wð1Þ�; and vð2Þ ¼ U½Wð2Þ�; with vð1Þ � vð2Þ. Let W(12) be the weightof the composite formed by binding the two bodies together. Then the naturalspeed of the composite body is vð12Þ ¼ U½Wð12Þ�. However, by [C2],

Wð12Þ ¼Wð1Þ þWð2Þ >Wð2Þ.

The monotonicity of U implies that U½Wð12Þ� U½Wð2Þ� U½Wð1Þ�. Hencevð12Þ vð2Þ vð1Þ, which is inconsistent with [C1], unless all natural speeds areequal (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 thesame 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 naturalspeeds are bound together, the natural speed of the composite is the same as thatof each of the two constituent bodies. We use the term here in much the same way


as when one says that temperature is intensive: two bodies at the same tempera-ture, when brought into thermal contact with one another, constitute a compositebody 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 madeabout 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 arebound 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 ofdifferent weights are combined, as it is in [C2], although it is easy to prove thatextensivity implies additivity. In [Z3] the only assumption is that there is a continu-ous function, U, such that a body of weight W has natural speed v ¼ U½W�. Sinceno assumption is made now that the continuous function U must be monotonicallyincreasing, it is clear that [Z3] is weaker than [C3]. We shall now prove that the setof 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 naturalspeed, 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 eachhave weight W/2. By [Z3], since they have equal weight the pieces have equal natu-ral speed, and by [Z1] the natural speed of the original body must be the same asthat of the pieces. Hence:

U½W� ¼ U½W=2�:However, we can now repeat the process by dividing one of the pieces into twoequal parts, thereby proving that U[W/4] is equal to U[W/2]. On iterating thereasoning 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 con-clude 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 areequal 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 hid-

2 It should be admitted that U[0], the natural speed of a falling body in the limit that the weight tendsto zero, is a theoretical construct with no direct physical relevance. It is introduced to allow the proof to

work when W/W0 is irrational.


den assumption, namely that any other parameters determinative of natural speedare 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 theweak set [Z1]–[Z3] implies the constancy of U is invalid. It is correct only if we addGalileo’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 thatthe rate of fall of a gold leaf in air is different from that of a pellet of gold of thesame weight, and therefore that the rate is dependent on the shape as well as theweight. Moreover, he recognized implicitly that his thought experiment only worksfor bodies of the same substance, or at any rate ‘provided they are of the same spe-cific gravity.’ (Galileo, 1638, p. 108) He realized that it would not do to bind twobodies of different specific weights (or densities) together, for if the natural speedwere to depend on the density, as well as on the weight (as indeed it does for fall ina viscous medium such as air), his demonstration would fail.3

Brown admits that rate of fall, even in vacuo, could logically depend on otherparameters. He cites chemical composition or colour as possibilities (Brown,1991), without however taking either of them seriously, nor does he seem toconsider any other possible empirical dependence. In spite of Brown’s dismissal,the first parameter he mentions (chemical composition) is a serious option, andindeed experiments of great sensitivity have been performed to measure the accel-erations with which different chemical substances fall in vacuo. Does a sphere oflead fall in vacuo at the same rate as a sphere of aluminium? To phrase the mat-ter more theoretically, is the ratio of gravitational mass and inertial mass thesame for all substances? The gravitational mass of a body may be defined as thecoefficient of proportionality between the body’s weight and the gravitational fieldin which it is situated. This is conceptually different from the inertial mass of thesame body, which is the coefficient of proportionality between a force acting onthe body (for example, its weight) and its resulting acceleration. That the twokinds of mass are numerically equal has been experimentally tested to high accu-racy (Eotvos et al., 1922); and the equality was built into the very foundations ofEinstein’s general theory of relativity. Because of the equality, two bodies of dif-ferent gravitational masses, if placed in the same gravitational field (with norestraining forces), will suffer the same acceleration, precisely because the ratio ofthe 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 possi-

bility 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.


masses. For our purposes, it is important to realize that this equality is an empiricalfinding, and not a logical truth. The second parameter that Brown mentioned(colour) was presumably intended jocularly; but, at the risk of ruining Brown’sjoke, we point out that if two falling bodies of different colours are exposed tovertically directed light from a laser, tuned to the frequency corresponding to thecolour of one, but not of the other body, then the light pressure experienced bythe two bodies will not be the same, and so the rates of fall of the two bodies willbe 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 theirweights. (Norton, 1996, p. 343.) The relevance of Norton’s claim can be illu-strated by reconsidering Gendler’s reconstruction of Galileo’s argument. As wehave seen, Gendler does not question the validity of [G1] and [G2]; neither woulda convinced Aristotelian or a Galilean do so. Nevertheless, the matter is moresubtle than Gendler supposes, for denying [G1] is not as bizarre as she seems toassume. 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 mediummay, or may not satisfy the mediativity condition [G1]. This can be best explainedon the basis of the set [C1]–[C3] (recall that [G1] is identical to [C1]). Forexample, two lead spheres of different weights (and therefore with differentvolumes), will have different terminal velocities. If they are tied together side-by-side, the terminal velocity of the united system will lie between the terminal velo-cities of the constituents, i.e. [C1] applies. In this case, it is [C3] that fails. If, onthe other hand, the spheres are melted and recast as one sphere of weight equal tothe sum of the weights of the two original spheres, then the terminal velocity ofthe united system will be greater than those of either of the constituents. The rea-son is that the retarding viscous force is a function of both the velocity and of thesurface area of the falling body. The smelted sphere falls more quickly than theunited spheres because the surface of the former is smaller than the combinedsurfaces of the latter. In this case [C3] applies and [C1] fails (see Appendices Band 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 terminalvelocity 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 istoo hasty in agreeing that [C1] is indubitable, and therefore that [C3] must in allcases 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 speedby the Galilean notion of acceleration. It is not even true that bodies of differentweights must fall, on earth and in vacuo, with the same acceleration, because theseaccelerations may depend on variables other than the weights. To illustrate thisfact, let us return to the tower and the falling pieces of rock, but now in the lightof Newtonian physics. Imagine that I stand at the top of the tower, holding the8 kg piece in my left hand and the 4 kg piece in my right hand. I stretch out myarms, so that the two pieces are side by side, at the same height from the ground.


If I now let them go simultaneously, they will fall with the same acceleration, tak-ing the same time to reach the ground. This is precisely the solution sketched byGalileo.4 However, if I place one piece just above its companion, but in contactwith it, it will fall with ever so little less acceleration than the other piece, becausethe gravitational field at its location is slightly smaller than that at the location ofthe 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 tiedtogether (or if they are fused into one another and thus ‘truly one’, it makes no dif-ference), then the gravitational field, averaged over the bound system, lies betweenthe fields averaged over the lower and over the upper segment separately. Conse-quently, the time of fall of the bound system will lie between the times for theupper and the lower pieces. This means that condition [C1] indeed holds non-trivially (and not, to use Gendler’s words, in a vacuous sense). [C1] is relevantbecause the position in a nonuniform gravitational field of each point of a bodyplays 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 beconsiderable. Be that as it may, the size of the effect is not at issue here. In thepresence of a homogeneous gravitational field (and in the absence of air), differentbodies would indeed fall at the same rate. However, this is not an a priori state-ment about the way bodies fall; indeed, given that the earth’s gravitational field isinhomogeneous, it is not even an accurate statement. In Appendix C we give fur-ther mathematical details, showing precisely how Galileo’s reasoning breaks downwhen the gravitational field is nonuniform.At this juncture, modern apologists for Galileo might remark that the inhomo-

geneity of the gravitational field could be seen as a disturbing factor, on a par withair friction. It requires, after all, little effort to postulate a homogeneous gravi-tational field in order to reinstate Galileo’s thought experiment in all its pristinesplendour. Moreover, aside from the effect of inhomogeneity of the gravitationalfield, we can of course cite other instances of disturbing factors. If the rocks con-tained iron, and there was a magnetic field present, the rate of fall would be influ-

4 It is generally agreed that Galileo did not in fact drop musket shot and cannon balls from the lean-

ing 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 mus-

ket 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.


enced, as it would be if the rocks were electrically charged, or if they were electri-cally conductive, and there was an electrostatic field present. The apologists wouldhave 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 allunspecified disturbing factors to be absent, so that Galileo’s law of falling bodies becorrect. However, for each putative disturbing factor that is specified, one needs atheory to be able to postulate conditions coherently under which it would beabsent. For instance, one can only identify the inhomogeneity of a gravitationalfield as a disturbing factor if one knows enough about it, within the framework ofa theory of gravitation. Galileo lacked such a theory, at least one in which gravi-tational forces (i.e. weights) drop off as the inverse square of the distance from thecentre 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 tocompare the motions of falling apples with that of the moon, as it ‘falls’ endlesslyin its month-long orbit around the earth. Since physical laws are tested by theirempirical 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 ofverities, as Brown claims. They do not stand on their own, but must be subordinatedto the theories which they inspire (and by which they are inspired). In order to gainknowledge of the world, Galileo performed, and needed to perform, real experimentswith steel balls and inclined planes.

5. Galileo and prior philosophy

A lively philosophical debate on the nature of thought experiments can be foundin the literature from about 1990. Concerning thought experiments in naturalscience, Brown, Norton and Gendler made significant contributions. As we haveseen, Brown adopts a clear stance. For him, the essence of a thought experiment isthat it teaches us new things about the world without the use of new empiricaldata. Brown’s world view is denied by Norton, for whom thought experiments aredisguised arguments. On the basis of a careful study of the epistemology ofthought experiments (as opposed to, for instance, their impact on the scientificcommunity), Norton concludes that thought experiments ‘can do no more than canordinary thinking with its standard tools of assumption and argument’ (Norton,1996, p. 366). In particular, they ‘open no new channels of access to the physicalworld’ (op. cit.).While the distinction between Brown’s view and that of Norton is fairly straight-

forward (it is the familiar dispute between rationalists and empiricists), the contrastbetween Norton and Gendler is not so easy to discern. Gendler opposes bothNorton and Brown. With Norton, and against Brown, she holds that thought experi-ments 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,


namely ones with a particularly strong persuasive power, their ‘justificatory force’.Apparently taking her inspiration from Mach, 1883, 1887), Gendler ascribes thejustificatory force to the fact that, in a successful thought experiment, instinctiveand hitherto unarticulated empirical knowledge suddenly becomes organised andmanifest. In this vein, Gendler speaks of thought experiments as ‘guided con-templations’, but otherwise underwrites the major points of Norton’s expose.Perhaps the best way to pin down the contrast between Norton and Gendler wouldbe 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 inthe background suddenly become manifest. A joke, like a thought experiment, canhowever also be explained, so that all the steps are made explicit. Whereas Nortonstresses that thought experiments can without loss of meaning be rewritten as argu-ments without imaginary particulars, Gendler emphasizes that such an approachmisses the liberating insight that is the hallmark of a good thought experiment(analogous to ‘getting’ a good joke). Gendler and Norton are merely speakingabout different aspects of a thought experiment—its subject matter, and thepsychological 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 Browncamp. Being a rationalist, Brown believes the purest thought experiments to bethose that enable us to acquire knowledge of the world without wearying ourphysical senses. Indeed, as we have seen, he cites Galileo’s thought experiment asthe paradigmatic example, for in Brown’s view this thought experiment not onlyestablished ‘the end of Aristotle’s theory’, but it also yielded a premium, ‘since theright 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 Platonicthought experiment in Brown’s sense. For the statement that all bodies fall at thesame speed is not always the right account, and of course it is not obvious unlessother conditions are satisfied. These conditions may well not be applicable to parti-cular falling bodies, and whether or not they are applicable is a matter of empiricalresearch.This may all sound plausible to empiricists’ ears. However, a towering figure like

Galileo claimed the contrary (‘even without further experiment, it is possible toprove clearly . . . that a heavier body does not move more rapidly than a lighterone’); and an astute empiricist like Gendler seems to think that Galileo’s thoughtexperiment for ever put aside Aristotle’s conclusion as erroneous (‘a simple andobvious 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 nothingof 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 Natureherself. This observation itself cannot be shown to be indubitably correct: it israther the painful lesson extracted from millennia of failures of the direct, magicalmode of apperception. The latter day Platonist is hoisted on his own petard: James


Brown’s classic Platonic thought experiment has been shown to be not only logi-cally deficient but also to fail to adequately describe the empirical world.

Acknowledgements

Special thanks are due to Pieter Sjoerd Hasper for an illuminating discussion ofsome aspects of Aristotle’s De caelo. We are grateful for the interactive contri-bution of audiences in Bertinoro, Gent, Groningen, Leusden and Rotterdam, andin particular for that of James McAllister, who brought Casper’s article to ourattention. We acknowledge also an e-mail from Jim Cushing concerning Galileo’sreasons 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 some-what vague notion of natural motion, quantified by the scarcely less precise con-cept of natural speed. In this appendix, we propose ‘acceleration’ as an interpretionof ‘natural motion’ and as a replacement for ‘natural speed’. Furthermore, we shallspeak here only of falling bodies in vacuo and in a uniform gravitational field. Theaim is to show that, under these conditions, Galileo is right.Let us investigate, within the formalism of Newtonian physics, not the con-

ditions [C1]–[C3] of Sect. 3, but rather the analogous statements:

5

to

[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 reject-ing [S3]. For bodies falling in a vacuum in a uniform gravitational field, the factsare that [S1] and [S2] are true and [S3] is false. Such, at least, is the verdict given byNewton’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. Accordingto Newton’s second law of motion, F ¼ ma, the accelerations of the bodies B(1)and B(2) are given by að1Þ ¼Wð1Þ=mð1Þ and að2Þ ¼Wð2Þ=mð2Þ. It is part ofNewton’s theory that:

[N1a] Forces, and hence in particular gravitational forces, are additive5

[N1b] Inertial masses are additive

In general, forces obey the rules of vector addition, but in the present case of forces that are parallel

one another, vector reduces to scalar addition.


According to [N1a] and [N1b], the force acting on the composite made by tying thebodies B(1) and B(2) together (its weight) is Wð1Þ þWð2Þ, while the inertial massof the composite body is mð1Þ þmð2Þ. Hence the acceleration of the composite is:

að12Þ ¼ ½Wð1Þ þWð2Þ�=½mð1Þ þmð2Þ�¼ ½mð1Það1Þ þmð2Þ að2Þ�=½mð1Þ þmð2Þ�

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 followstherefore immediately that að1Þ < að12Þ < að2Þ. In other words, Newton’s claims[N1a] and [N1b] imply [S1], i.e. accelerations are indeed mediative. This mediativityis not a mere logical possibility: it is actually realized in a nonuniform gravitationalfield, 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 afunction of a body’s gravitational mass and the local gravitational field. However,since weight is a force, [S2] is implied by [N1a]. Conclusion: Newton underwritesboth [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 areplacement for ‘natural speed’. In the present appendix, however, we shall ratherconsider bodies falling in resistive media. Here it is more useful to interpret ‘natu-ral motion’ in terms of ‘speed’ rather than acceleration; and ‘natural speed’ will bemade precise in terms of ‘terminal velocity’. The purpose is to show that, when thespeed is sufficiently slow, Aristotle’s strong dogma is an accurate description of theempirical 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 inter-ested in falling bodies in situations where fluid viscous forces are important:

. . . in the ratio which their magnitudes bear to one another . . . In movingthrough 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 theflow of air around it, the viscous retarding force is proportional to its velocity, andNewton’s second law of motion for the acceleration, a, reads:

ma ¼ mg� k vm being the mass (strictly speaking, the inertial mass on the left and the gravi-tational mass on the right), g the acceleration due to gravity (assumed constant), v


the instantaneous velocity, and k the frictional coefficient due to viscous retar-dation. This equation has the following solution for the velocity:

v ¼ ½1� exp ð�k t =mÞ� mg=kand this yields, for the terminal velocity,

vðterminalÞ ¼W=k

where W ¼ mg is the weight. Evidently, if we interpret ‘natural speed’ as terminalvelocity in a medium (in plena), then Aristotle is right that the speed of a body isproportional to its weight. But what has happened then to the Galileancontradictio? 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 sus-pended 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 otherthings, to throw further doubt on the worth of Galileo’s thought experiment. Theconclusion applies in special circumstances, namely for two bodies of the sameshape and size, but then Aristotle did write:

We see that bodies which have a greater impulse either of weight or lightness, ifthey are alike in other respects, move faster over an equal space, and in the ratiowhich 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 gravitationalfield. We argued that they all fall with the same acceleration, supporting Galileo’sconclusion. 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 wereturn to falling bodies in vacuo. However, rather than situating them in a uniformgravitational field, as we did in Appendix A, we will now place them in the inho-mogeneous 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’sthought experiment, as sketched by Salviati, will be re-enacted, but now with thetwo bodies at different distances from the earth. They will in the first instance be


considered to fall unimpeded, and in the second instance as a bound system, con-nected by a cord. It will be shown that the accelerations of the systems are media-tive, that their weights are additive, but that Galileo’s conclusion that all bodiesmust fall at the same rate is incorrect, for the two bodies, as well as the bound sys-tem, fall at different rates.The earth is a spheroid, and the weight of a body depends not only on its gravi-

tational mass, but also on how high it is above the surface of the earth. The accel-eration of a falling body at two earth-radii from the centre of the earth is only onequarter what it is when the body is close to the earth’s surface. The rate at which abody’s velocity increases is dependent on its position. Consider a body, B(1), ofmass m(1), at distance r from the centre of the earth, and another, B(2), of massm(2), at distance rþ d from the centre of the earth, directly above the first one. Ifthe 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=½rþ d�2

Suppose now that the two bodies are connected by a light, inextensible cord, oflength d. Since B(1) has a tendency to fall more quickly than B(2), the the cord willbe under tension, say T, which serves to speed up B(2) and to brake B(1). The netforce on the lower body will be mð1Þ gð1Þ � T, while that on the upper body willbe mð2Þ gð2Þ þ T. The total force on the composite system ismð1Þ gð1Þ þmð2Þ gð2Þ, the tension of the cord having dropped out of this sum.The total mass of the system is mð1Þ þmð2Þ, so its acceleration is:

gð12Þ ¼ ½mð1Þ gð1Þ þmð2Þ gð2Þ�=½mð1Þ þmð2Þ�

Since g(1) is larger than g(2), it follows trivially from the above formula thatgð1Þ > 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 þmð2Þ� gð12Þ, is indeed addi-tive, being precisely the sum of m(1) g(1) and m(2) g(2). Thus [S1] and [S2] areapplicable in this case, and so [S3] must be untrue. Indeed, the accelerations of thebodies B(1) and B(2), of weight m(1) g(1) and m(2) g(2), and that of the composite,of weight ½mð1Þ þmð2Þ� gð12Þ, are respectively g(1), g(2) and g(12), which are notin the ratio of the weights. In this case the destructive part of Galileo’s reasoning isvalid; but his conclusion, that therefore all bodies fall with the same acceleration, ispatently false. The reason is that the accelerations can, and do depend onsomething other than their weights, namely on the distances of their centres ofmass from the centre of mass of the earth.


Appendix D. laminar and turbulent flow

In Appendix B we looked at the terminal velocities of falling bodies in resistivemedia, in the domain in which the fluid motion is nonturbulent; and we showedthat the strong dogma of Aristotle is then correct. In the present appendix we willstudy falling bodies in resistive media in which turbulence does take place, and weshall demonstrate that, in this domain, Aristotle’s strong dogma is incorrect, buthis 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 theground. A cannon ball, dropped from the parapet, would take about 3.35 secondsto reach the ground, while a musket shot takes a few hundredths of a secondlonger. 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 mus-ket shot within a hundredth of a second of each other. For these, and other rea-sons, many people have relegated the story of the public experiment to the realm ofmyth.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 atspeed v in a fluid of density q and viscosity g. The Reynolds number for this sys-tem, 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, wespeak of laminar flow: under these conditions it is found that the fluid moves, nearthe sphere, in regular laminae or layers, each having its own velocity. For highervelocities, at which the Reynolds number is much larger than unity, non-steady orturbulent 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 semi-phenomenological formula of aerodynamics:

F ¼ 12Cpq r2 v2 ð3Þ

where C is called the drag coefficient—it is the inclusion of this factor that makesthe formula not a purely theoretical one. Experimentally it is found that C is about12 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 aboutR ¼ 100; 000 by:


C ¼ 24=Rþ 12

With this fit to C, the aerodynamic formula (3) reduces to:

F ¼ 6pg r v þ 1

4pq r2 v2 ð4Þ

which is Stokes’ formula (2), plus an aerodynamic term that corresponds to for-mula (3) with C ¼ 1

2. A sphere of mass m, falling in a fluid, satisfies Newton’sequation:

ma ¼ mg� 6pg r v� 14

pq r2 v2 ð5Þ

in the approximation (4). Here a is the acceleration of the sphere and g is the accel-eration 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 ðltþ uÞ � j ð6Þ

with:

j ¼ 3pg r=m k ¼ 1=4pq r2=m

and:

l2 ¼ kgþ j2 u ¼ 12log ½ðl þ jÞ=ðl � jÞ�

The distance fallen, d, can be obtained by integrating Eq. (6):

kd ¼ log cosh ðltþ uÞ � log cosh u � jt ð7Þ

For very small velocities the aerodynamic term in Eq. (4) can be neglected, and werecover the situation described in Appendix B for laminar flow, with the identifi-cation 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 ofthe leaning tower of Pisa, 55 metres from the ground. At 20

vC, the density and

viscosity of air are about q ¼ 1:2 kg=m3 and g ¼ 0:000 018 kg=ðmsÞ. Given thatthe density of iron is 7860 kg/m3, we find the radius of the small ball to ber 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 160j 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


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 sso the heavy ball arrives three hundredths of a second ahead of the light one, a dif-ference that Galileo would indeed not have been able to measure. Eq. (6) becomes,in good approximation:

v ¼ ðl=kÞ tanh ðltÞ ð8Þand the velocities of arrival of the two balls at the ground are:

vð1Þ ¼ 31:79 m=s vð2Þ ¼ 32:57 m=sAt these speeds, the Reynolds numbers can be calculated to be:

Rð1Þ ¼ 105 000 Rð2Þ ¼ 500 000so 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), andthis is very different for the two balls. However, the heavier ball achieves 95% of itsterminal velocity only after a fall of several kilometres, so one would have torepeat the experiment from a cliff several km. in height in order to find consider-able differences in the velocities of the balls. Note that these terminal velocities arenot in the proportion of their weights. The distance fallen in a given time is amonotonic decreasing function of k, see Eq. (8). Hence, for given air density, thevelocity 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, sothe distance fallen in a given time is a monotonic increasing function of r, and soof 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 thesame size but only one tenth the weight. Stokes’ formula (2) is a good approxi-mation only if the Reynolds number is not in excess of unity. For a sphere of ironof radius twenty microns (0.02 mm), the terminal velocity is 0.34 m/s, and in thiscase 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 ratioof the weights. For example, with a radius of 1 mm the air flow is turbulent andthe 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 notapplicable.


Appendix E. textual sources

The basic questions to be asked in this appendix are these: did Aristotle actuallyclaim that the rate of fall of bodies is proportional to their weight (when they movetoward their natural place), given the ambiguities in translating ancient Greek; anddid Galileo in fact state that the rate of fall of all bodies is the same in all circum-stances?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 haveintended 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 withweight and equal in bulk, which is exceeded by the other in the speed of itsnatural downward movement. (Aristotle, De caelo 4.1, 308a 31–35)

He suggests that we should construe this as an operational definition of relativelightness 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 fasterthan a lighter one (the weak dogma) would be a tautology. This interpretation issupported by Moraux (1965), who, although he translates ba´qj1 into French as‘lourd’, renders the above passage thus:

. . . nous parlons de leger relativement a autre chose et de plus leger quand, dedeux corps lourds de volume egal, l’un se porte naturellement vers le bas plusrapidement 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 meme nature sont entre eux comme leursvolumes.

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 twobodies of different material but of the same volume in terms of their relative ratesof fall. However, in 5 he shows that Aristotle also thought that, for two bodies ofthe same nature, at the same distance from the earth, one with a volume ten timesthat 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.


(correct), and the time of fall of the smaller is ten times longer than that of the lar-ger (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 sug-

gests, nor did Philoponus in antiquity (see Wolff, 1971), nor Simon Stevin at thebeginning of the seventeenth century:

. . . Aristotle . . . thinks . . . that when two similar bodies of the same density fall inair, their rate of fall is in proportion to their relative weights . . . But the experi-ment against Aristotle is like this: Take two balls of lead (as the eminent manJean Grotius . . . and I formerly did in experiment) one ball ten times the other inweight; 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 timesmore slowly, but so equally with the other that the sound of the two in strikingwill 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 directlyproportional to the force moving it, and inversely proportional to the liquid’sviscosity. (Toulmin, 1961, p. 51)

As Casper says, ‘If this interpretation is valid, Aristotle and Galileo did not reallydisagree about the nature of falling objects. They were merely talking past oneanother . . .’ (Casper, 1977, p. 328). However, Casper finally rejects this interpret-ation; and a further reason to doubt that Aristotle had terminal speeds in mindwhen formulating the strong form of his dogma is that these speeds are only pro-portional to the weights in conditions of laminar flow, and these conditions do notobtain 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). Thissuggests that he wanted to leave the possibility open that the rate of fall might be afunction of the specific weight, and thus of the density of the falling body. If this istrue, Galileo recognized indeed that his argument is valid only if other possiblevariables that might influence the rate of fall are held constant. However, holdingthe density constant is unnecessary for fall in vacuo, and it is insufficient for fall inair. Moreover, he entertained the strange notion that a ball of wood, at the begin-ning of its fall, moves more quickly than a ball of iron that is released at the sameheight 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.


seems to fly once more in the face of Salviati’s triumphant conclusion that the onlyway to avoid the inherent contradiction in the (weak) Aristotelian dogma is to sup-pose that all bodies fall at precisely the same rate.

References

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Casper, B. M. (1977). Galileo and the fall of Aristotle: A case of historical injustice? American Journal of

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Cooper, L. (1935). Aristotle, Galileo, and the tower of Pisa. Ithaca, New York: Cornell University Press.

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