Stud. Hist. Phil. Sci. 35 (2004) 115–136 www.elsevier.com/locate/shpsa
Galileo and prior philosophy David Atkinson , J...

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

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

116

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

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.

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

117

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)

118

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

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.

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

119

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

120

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

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

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

121

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.

122

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

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 (Eo¨tvo¨s 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.

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

123

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.

124

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

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.

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

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,

126

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

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 expose´. 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.

128

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

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

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

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 ð k 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

130

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

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.

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

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:

132

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

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

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

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.

134

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

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 le´ger relativement a` autre chose et de plus le´ger quand, de deux corps lourds de volume e´gal, 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 meˆme 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.

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

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. References Brown, J. R. (1991). The laboratory of the mind: Thought experiments in the natural sciences. London & New York: Routledge. Brown, J. R. (2000). Thought experiments. In W. H. Newton-Smith (Ed.), A companion to the philosophy of science. Oxford: Blackwell Publishers. Casper, B. M. (1977). Galileo and the fall of Aristotle: A case of historical injustice? American Journal of Physics, 45, 325–330. Cooper, L. (1935). Aristotle, Galileo, and the tower of Pisa. Ithaca, New York: Cornell University Press. Cushing, J. T. (1998). Philosophical concepts in physics. Cambridge: Cambridge University Press. Eo¨tvo¨s, R. von, Peka´r, D., & Fekete, E. (1922). Beitra¨ge zum Gesetze der Proportionalita¨t von Tra¨gheit und Gravita¨t. Annalen der Physik, 68, 11–66. Galileo Galilei (1954). Discorsi e dimostrazioni matematiche, intorno a` due nuove scienze (Dialogues concerning two new sciences) (H. Crew, & A. de Salvio, Trans.). New York: Dover Publications. (First published 1638) Gendler, T. S. (1998). Galileo and the indispensability of scientiﬁc thought experiment. British Journal for the Philosophy of Science, 49, 397–424. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics (2nd ed.). Oxford: Pergamon Press. McAllister, J. W. (1996). The evidential signiﬁcance of thought experiment in science. Studies in History and Philosophy of Science, 27, 233–250. Mach, E. (1960). The science of mechanics (6th ed.) (J. McCormack, Trans.). La Salle, IL.: Open Court. (First published 1883) Mach, E. (1976). On thought experiments. In E. Mach (Ed.), Knowledge and error (6th ed.) (pp. 134–147). Dordrecht: Reidel. (First published 1887) Moraux, P. (1965). Aristote, Du ciel. Paris: Les Belles Lettres, Bude´. Norton, J. D. (1991). Thought experiments in Einstein’s work. In T. Horowitz, & G. J. Massey (Eds.), Thought experiments in science and philosophy (pp. 129–148). Savage MD: Rowman & Littleﬁeld. Norton, J. D. (1996). Are thought experiments just what you thought? Canadian Journal of Philosophy, 26, 333–366. Sorensen, R. A. (1992). Thought experiments. Oxford: Oxford University Press. Toulmin, S. (1961). Foresight and understanding. New York: Harper and Row. Wolﬀ, M. (1971). Fallgesetz und Massebegriﬀ. Berlin: Walter de Gruyter & Co.

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

116

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

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.

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

117

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)

118

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

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.

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

119

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

120

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

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

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

121

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.

122

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

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 (Eo¨tvo¨s 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.

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

123

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.

124

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

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.

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

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,

126

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

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 expose´. 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.

128

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

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

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

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 ð k 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

130

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

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.

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

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:

132

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

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

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

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.

134

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

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 le´ger relativement a` autre chose et de plus le´ger quand, de deux corps lourds de volume e´gal, 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 meˆme 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.

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

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. References Brown, J. R. (1991). The laboratory of the mind: Thought experiments in the natural sciences. London & New York: Routledge. Brown, J. R. (2000). Thought experiments. In W. H. Newton-Smith (Ed.), A companion to the philosophy of science. Oxford: Blackwell Publishers. Casper, B. M. (1977). Galileo and the fall of Aristotle: A case of historical injustice? American Journal of Physics, 45, 325–330. Cooper, L. (1935). Aristotle, Galileo, and the tower of Pisa. Ithaca, New York: Cornell University Press. Cushing, J. T. (1998). Philosophical concepts in physics. Cambridge: Cambridge University Press. Eo¨tvo¨s, R. von, Peka´r, D., & Fekete, E. (1922). Beitra¨ge zum Gesetze der Proportionalita¨t von Tra¨gheit und Gravita¨t. Annalen der Physik, 68, 11–66. Galileo Galilei (1954). Discorsi e dimostrazioni matematiche, intorno a` due nuove scienze (Dialogues concerning two new sciences) (H. Crew, & A. de Salvio, Trans.). New York: Dover Publications. (First published 1638) Gendler, T. S. (1998). Galileo and the indispensability of scientiﬁc thought experiment. British Journal for the Philosophy of Science, 49, 397–424. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics (2nd ed.). Oxford: Pergamon Press. McAllister, J. W. (1996). The evidential signiﬁcance of thought experiment in science. Studies in History and Philosophy of Science, 27, 233–250. Mach, E. (1960). The science of mechanics (6th ed.) (J. McCormack, Trans.). La Salle, IL.: Open Court. (First published 1883) Mach, E. (1976). On thought experiments. In E. Mach (Ed.), Knowledge and error (6th ed.) (pp. 134–147). Dordrecht: Reidel. (First published 1887) Moraux, P. (1965). Aristote, Du ciel. Paris: Les Belles Lettres, Bude´. Norton, J. D. (1991). Thought experiments in Einstein’s work. In T. Horowitz, & G. J. Massey (Eds.), Thought experiments in science and philosophy (pp. 129–148). Savage MD: Rowman & Littleﬁeld. Norton, J. D. (1996). Are thought experiments just what you thought? Canadian Journal of Philosophy, 26, 333–366. Sorensen, R. A. (1992). Thought experiments. Oxford: Oxford University Press. Toulmin, S. (1961). Foresight and understanding. New York: Harper and Row. Wolﬀ, M. (1971). Fallgesetz und Massebegriﬀ. Berlin: Walter de Gruyter & Co.

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close