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Galileo and the Pendulum: Latching on to Time

PETER MACHAMER and BRIAN HEPBURN

University of Pittsburgh

Abstract. Galileo changed the very concepts or categories by which natural philosophy could deal with matter and motion. Central to these changes was his introduction of time as a fundamental concept. He worked with the pendulum and with the inclined plane to discover his new concept of motion. Both of these showed him that acceleration and time were important for making motion intelligible.

1. Introduction Many accounts of the work that Galileo did and why he became “the father of modern science” have been given. We will give yet another one that shows how the pendulum was crucial in Galileo’s thought. First comes a very general historical narrative outlining a new overview of Galileo’s work. Second we will present textual evidence from 1590 to 1609 that shows how Galileo used the pendulum and, by focusing on time, changed his way of thinking. Finally we will suggest what might be done with science students in ways that parallel the Galilean exemplar. 2. The Outline of the Galileo Story Galileo wanted to reconstitute the whole of natural philosophy. What Galileo accomplished was a replacement of one set of analytical concepts with another. Some researchers might phrase this claim in terms of mental models. However phrased, Galileo’s move was from the Aristotelian categories of the one celestial and four terrestrial elements and their directional natures of movement to only one element, matter. He then sought the important properties of matter and its motion trying ?rst, relative heaviness, then speci?c gravity, then momento and force of percussion, and ?nally, acceleration and time. Galileo began his critique in the 1590 manuscript, De Motu, where he argued that the balance could be used as a model for treating all problems of motion, and heaviness (weight of the object minus weight of the medium) was the characteristic of all matter. What was not worked out was the positive characterization of the replacement categories, which probably contributed to his never publishing De Motu. Later in the 1600 De Meccaniche (Galileo 1600/1960) he introduces momento and begins to look at the properties of percussion of bodies of different speci?c gravities. Still, the details of how to properly treat weight and

99 M.R. Matthews et al. (eds.), The Pendulum, 99–113. ? 2005 Springer. Printed in the Netherlands.

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movement elude him. The problem is that the Archimedian simple machines that Galileo is using as his model of intelligibility are not dynamic enough and, except for the inclined plane, time is not an aspect that one would normally attend too. The details will not fall into place until 1603–1604, when Galileo works with pendula and inclined planes. The pendulum showed Galileo that acceleration and, therefore, time is a crucial variable. The regularity of the period of a pendulum goes someway towards showing that equilibirium of times is the form of ratio that needs to be explicit in representing pendulum motion. Work on the force of percussion and inclined planes also emphasized acceleration and time. But he would not publish this until 1638 in Discourses on Two New Sciences (Galileo 1638/1954, hereafter Discorsi). In The Starry Messenger, (Sidereus Nuncius) published in 1610, he would begin his dismantling of the celestial/terrestrial distinction. But he had already laid the grounds for treating all matter as having the same nature back in 1590. The rest of Galileo’s story is well known, but we brie?y retell it in this context. With The Starry Messenger (1610) and in Letters on the Sunspots (1612), Galileo enumerated many reasons for the breakdown of the celestial/terrestrial distinction. In the latter he even went so far as saying that the new evidence supported the Copernican theory. Yet even with all these changes, two things were missing. First, there was no way to accurately describe the nature of matter in the new system. He had a start, but his developed matter theory would not come until Days One and Two of Discorsi. In the ?rst part Galileo would attempt to show mathematically how bits of matter solidify and stick together, and do so by showing how they break into bits. The ultimate explanation of the “sticking” eluded him since he felt he would have to deal with in?nitesimals to really solve this problem. The second science, Days Three and Four of Discorsi, dealt with proper principles of local motion, but this was now motion for all matter (not just sublunary stuff) and used the idea of time and acceleration as basic. The Fifth day dealt with the force of percussion, which had become an important aspect of Galileo’s thinking. The second missing part of the new natural philosophy was showing how a uni?ed theory of matter could actually be applied to a moving earth. The change here was not just the shift from a Ptolemaic, Earth centered planetary system to a Copernican solar centered one. It was also a shift from a mathematical planetary model to a physically realizable cosmography. Galileo needed to show how, on a solar centered scheme, one could intelligibly use any laws of local motion. This he did by introducing two new principles, that all natural motion was circular, in Day One of his Dialogues on the Two Chief World Systems (Galileo 1632/1967, hereafter Dialogo) in 1632, and in Day Two the famous principle of the relativity of observed motion. The joint effect of these two principles was to say that all matter shares a common motion, circular, and so only motions different from the common could be directly observed.

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This sketch provides the basis for understanding Galileo’s changes. He has a new science of matter, a new physical cosmography, and a new science of local motion. In all these he is using a mathematical mode of description based upon, though somewhat changed from, the proportional geometry of Euclid, Book VI and Archimedes. (For details on the change, see Palmieri 2002).

3. In Search Of the Replacement Categories Our historical theme is that Galileo’s law of free fall arises out of his struggle to ?nd the proper categories for his new science of motion. Galileo accepts, perhaps as early as the 1594 draft of De Meccaniche, that natural motions might be accelerated. But that accelerated motion is properly measured against time is an idea enabled only later, chie?y through his failure to ?nd any satisfactory dependence on place. Galileo must have observed that the speeds of bodies increase as they move downwards and, perhaps, do so naturally. He would have seen speeds changing particularly in the cases of the pendulum or the inclined plane. Believing the same causes were at work in free fall and projectile motion, Galileo would have been more open to acceleration there too. At this time he also begins thinking about percussive force that a body acquires during its motion and which shows upon impact. For many years he thinks the correct science of these changes should describe how speeds change according to where bodies are on their paths. Speci?cally, it seems that height is crucial. The percussive force of a dropped body is directly related to height and the motion of the pendulum seems to involve essentially equilibrium between the height of the bob at the start of the swing and its height at the end. This gives height the same role in understanding the pendulum that weight plays in understanding the balance. Of course times are also in some sort of equilibrium, each swing taking the same period (ignoring frictional losses), ?xed by the length of the pendulum. This isochrony also seems to hold regardless of the initial displacement, making the radius of the path (i.e., the pendulum length) the sole determinant of the motion. However, these times do not analogize easily with weights on a balance. The law of free fall, expressed as the distance in fall from rest being proportional to the time squared (the time-squared law), is discovered by Galileo through his inclined plane experiments. These were begun sometime before Galileo’s letter of 1604 to Paolo Sarpi in which he gives a “proof” of the time-squared law by deriving it from the assumption that velocita (a new concept that amount to degrees of speed) are proportional to the distance fallen. In this proof the overall speed, which results from the accumulation of velocita, comes out proportional to distance squared and hence to time to the fourth (see below for details). What is puzzling is that it is widely known at the time, thanks to the Mertonians and also to Oresme, that a distance-time-squared relation can easily be derived from an assumption of uniform acceleration. Galileo remains intent however, on ?nding an explanation of the time-squared law in the form of some relation between speed

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and height. The problem is that taking velocity proprotional to time might serve as a de?nition of free fall (i.e., de?ned as uniformly accelerated) but it provides no more explanation than does taking distance proportional to time squared. That this is Galileo’s attitude is even in evidence in the Discorsi. The key to his eventual con?dence in the de?nition is the mean proportional. Any proportionality between a ratio and the square of a ratio can be expressed in terms of a mean proportional, and what quantity this mean proportional represents depends on the geometric and the physical context. Galileo’s eventual de?nition of natural acceleration is an insight gained through the combination of pendulum and inclined plane “contexts” and recognition of the physical signi?cance of the mean proportional relation. What follows is a brief chronology intended to illustrate this conceptual shift. We begin with a description of how Galileo’s ?rst attempt fails in De Motu (1590), point out some key changes in De Meccaniche (1600) and ?nally we discuss the proofs of the working papers and the context that the pendulum and the inclined plane provide for the mean proportional relation. 4. The Galilean Texts An early work in Galileo’s search for the replacement categories is De Motu (1590) which ends with an incomplete attempt at describing projectile motion (Chapter 23, pp. 110–114). Addressed speci?cally is why a cannonball ?ies farther the more vertical the shot. According to the balance model Galileo just laid out for motion on inclined planes (Chapter 14, pp. 63–69), an object encounters no resistance to motion in the horizontal direction (which he calls neutral motion). But maximum range, well known from artillery, is achieved for shots of 45 degrees. The balance model must therefore be inadequate. The balance model had already been supplemented with an accidental, leaking impressed force (leaky impetus) to account for acceleration through a changing “effective” or net weight (i.e. intrinsic weight minus impressed upward force). The artillery example forces Galileo to make even greater accommodations. Galileo’s attempted reconciliation is twofold. More force will be imparted to a cannonball shot vertically, he believes, because it offers greater resistance to the motion than one shot horizontally or at any acute angle.. Secondly, an object attempting to turn downward will encounter more resistance the more directly opposed the downward and upward paths are. For any shot not directly upward, (see Figure 1).

. . . at the time when the ball begins to turn down [from the straight line], its motion is not contrary to the [original] motion in a straight line; and, therefore, the body can change over to the [new] motion without the complete disappearance of the impelling force (Galileo 1590, additions are Drabkin’s.)

The resistance offered by the original projecting force must be overcome before the ball can turn downward, and so a greater decay of the projecting force is required. The result is that the more vertical the shot, the longer the trip before it turns

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Figure 1.

downward. But “when the body moves along ae, which is almost parallel to the horizon, the body can begin to turn downward almost immediately” (Galileo 1590, p. 114). This is a coherent picture. If the body falls obliquely to the projecting force, it is plausible that it would encounter less resistance. The picture is incomplete though. Galileo has no way of determining the relation between the remaining projecting force and points on the trajectory. Near the end of De Motu Galileo mentions the pendulum and how a lead bob will oscillate longer than a lighter bob, claiming that this is due to the greater retention of the impressed force by the heavier material (cf. p.108). It is probable that in order to complete the picture Galileo turned to investigating the rates of leakage of impressed force through observing pendulums. We will pick up this thread in a moment, but ?rst we consider some other changes in Galileo’s mechanical theory. De Meccaniche (ca. 1600) is the product of a period of Galileo’s extensive work on machines. It begins with three de?nitions that are, at this stage, his proposals for the new categories of motion. The de?nitions are of heaviness (gravitas), de?ned as the “tendency to move naturally downward” (Drake 1978, p.56); of momento, also a tendency to move downward caused not only by the weight, but also compounded by speed and the geometry of the Archimedian simple machines (something like mechanical advantage); and the center of gravity. Both the early (ca. 1593–1594) and later (ca. 1601–1602) versions also conclude with a section on percussion. But the main focus is to rail against:

those people who think they can raise very great weights with a small force, as if with their machines they could cheat nature, whose instinct – nay, whose most ?rm constitution – is that no resistance can be overcome by a force that is not more powerful than it. (Drake 1978, p.56)

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Figure 2.

Through machines like the lever, resistances do seem to be overcome by less powerful forces. Galileo therefore distinguishes momento, as a force, from a body’s intrinsic weight. Galileo’s new conception is that a trade-off can occur between speed and momento. A weight can overcome a greater resistance but must do so by travelling faster than the resistance. Because the lighter object is mechanically linked to the heavier one, their motions occur in the same time. The relation of the speeds can therefore be stated, as Galileo does, in terms of the distances each object move . . .

it is seen in all other instruments that any great resistance [or weight] may be moved by any given little force (momento), provided that the space through which this force is moved shall have to the space through which the resistant shall be moved that ratio which exists between this large resistant and the small. . . . (Drake 1978, p. 62)

Galileo provides an analysis of the lever in terms of the new category of momento. He has by now abandoned a leaking impressed force and believes instead that speed and force are internal to bodies themselves (See Figure 2).

The heavy body A being placed at the point D, and the other at the point E, it will not be unreasonable that the former, falling slowly to A, raises the latter swiftly to B, restoring with its heaviness that which comes to be lost by its slowness of motion. And from this reasoning we may arrive at the knowledge that speed of motion is capable of increasing momento in the moveable body in the same ratio as that in which this speed of motion is increased. (Ibid.)

Beside the ability to overcome resistance, Galileo is also interested in percussive force, both of which are measured as a body’s momento. Speed allows a small weight to overcome a larger one due to the increased momento of the small weight. Speed also increases percussive effect. The greater the speed of the percussing object the greater its ability to overcome resistance. Speed acquired in fall then likewise contributes to momento.This commensurability, through the momento concept, between lifting force and falling force, suggests an interpretation of pendulum motion as an equilibrium between the momento or velocity gained by the weight in the downswing and the force required to overcome the weight’s own resistance and carry it back to its original height. Galileo’s further pendulum discovery was the isochrony of the swings, of which he was convinced by at least November 1602. In a letter to Guidobaldo with that date Galileo writes:

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You must excuse my importunity if I persist in trying to persuade you of the truth of the proposition that motions within the same quarter-circle are made in equal times. For this having always appeared to me remarkable, it now seems even more remarkable that you have come to regard it as false. (Drake 1978, p. 69)

Notice that Galileo is thinking of isochrony on the quarter circle. The symmetry of the pendulum path suggested equilibrium between the force gained in fall from an initial height and that required to climb back up to the same height, but this symmetry was ignored. What preoccupied Galileo was ?nding a relationship between speed and height that made the initial height not matter. Isochrony still did not suggest to Galileo that time ought to play any role in his causal account of the phenomena. Of course times mattered – speed was de?ned through distances and times. However, since for isochrony all times were the same, the features that seemed to make the difference were the path and distances (i.e. the heights of the initial displacement). In De Meccaniche Galileo was initially misled with the lever because time seems to play no part, while with the pendulum the fundamental feature is equal times. What is even more deceiving in the case of isochrony is that the equality of times is the thing to be explained. It is only natural that the explanation would be in terms of something other than time. Moreover, the determinations or explanations that Galileo sought were through ratios but the ratio of equal times is one to one. Nonetheless, his consideration of these phenomena was fundamentally new. Time itself, even if only through the phenomena of equal times, had never before been subject to mechanical investigation. Galileo’s investigation of isochrony begins with his chord theorems. The key theorem makes it into the Discorsi as:

T HEOREM V I, P ROPOSITION V I: If from the highest or lowest point in a vertical circle there be drawn any inclined planes meeting the circumference the times of descent along these chords are each equal to the other. (Galileo 1638/1954, pp. 188–189)

The connection between these theorems and the pendulum is that the circle on which the chords are transcribed represents a pendulum attached at the center (see Figure 4). The ordering of theorems in the Discorsi does not, we contend, re?ect the chronology of their discovery. The discovery of isochrony was almost certainly through observations of the pendulum. On the other hand, the chord law was probably ?rst conjectured as a step towards achieving a physical explanation of isochrony and only later, if ever, was it veri?ed by experiment. The letter to Guidobaldo suggests no distinction between the causes of the equality of times along the chords and along the arc:

Until now [that is, up to the chord theorems] I have demonstrated without transgressing the terms of mechanics; but I cannot manage to demonstrate how the arcs SI A and I A have been passed through in equal times and it is this that I am looking for.

Galileo’s investigation of the chords is thus part of his seeking the causes behind the isochrony of the pendulum.

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Figure 3.

One of the ?rst things Galileo would have done with the chords was apply his balance model and so construct a perpendicular from the ?rst point of each chord in order to ?nd the ratio of their moment (See Figure 4). He would have immediately realized that those perpendiculars all intersect at the top of the pendulum circle (Euclid Book III, props 21, 31). Since Galileo knew the times of descent along each of these chords were the same, the vertical height would have seemed a natural feature of the diagram by which to represent their common time. All the chords share the same relationship to this height, which is itself also a chord. Galileo speculated that a ball rolling down one of the inclined planes represented by a chord would take the same time as a ball freely falling along the diameter (and he later carried out experiments to verify this, see (Drake 1978, pp. 217–218) and Drake, 1989). This would have justi?ed Galileo’s investigations of free fall through inclined planes since it linked the times and distances of both. The time-squared law is discovered during Galileo’s experiments with inclined planes (see Drake, 1975). The experiments involved rolling a brass ball in a highly polished groove down an inclined plane and marking the ball’s location at successive equal time intervals. The results of the experiment were recorded on folio 107v. as a column of distances numbered 1 through 8, the numbers thus representing the times for each distance-the ?rst distance was for one time increment, the second distance was for two time increments, and so on (See Galileo 1890-1909.) At some, possibly later, point Galileo added to this folio, to the left of the times, their squares, indicating he had recognized the correct relation of distances to the square of the times. As mentioned above, a standard way for geometers of the time to deal with proportionalities to squares of ratios was through a mean proportionality. The physical signi?cance of the distance that represented the mean proportional, or even which

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Figure 4.

mean proportional he should consider, was not clear. Nor did Galileo have any understanding of what caused the time-squared relation to hold. That Galileo had not recognized the full importance of time and was not prepared to consider velocity as proportional to time, even after having worked with the mean proportionality, is indicated by his attempted explanation, given on f85v., of the time-squared relation (See Figure 5).

I assume that acceleration of bodies falling along line AL to be such that velocita grow in the ratio of the spaces traversed, so that the velocita at C is to the velocita at B as space CA is to space BA, etc. . . . But since velocita are increased successively at all points of line AE . . . therefore all these velocita [taken together] are related, one [case] to another, as all the lines [together] drawn from all points of line AE parallel to the said BM, CN, and DO. But these [parallels] are in?nitely many, and they constitute the triangle AEP ; therefore the velocita at all points of line AB are, to the velocita at all points in line AC, as is triangle ABM to triangle ACN, and so on for the others; that is, these [overall speeds through AB and AC] are in the squared ratio of lines AB and AC. But since, in the ratio of increases of [speed in] acceleration, the times in which such motions are made must be diminished, therefore the time in which the moveable goes through AB will be to the time in which it goes through AC as line AB is to that [line] which is the mean proportional between AB and AC . (Drake 1978, pp. 98–99)

This is the proof of the time-squared law that Galileo sends to Sarpi in 1604. Momento has been replaced by velocita which are proportional to distance fallen. The overall speed the body has at any point in the fall (which contributes to the momento) is a result of the accumulation of all the degrees of velocita to that point. Speed is essentially an integral of the triangle and hence proportional to distance

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Figure 5.

squared. Then, rather than state the time-squared law explicitly, Galileo gives the conclusion as a mean proportionality relation: the ratio of the times through AB and AC is in proportion to the ratio of distance AB to the mean proportional between AB and AC. Eventually Galileo will arrive at the correct de?nition of natural acceleration but this attempt to ?nd a cause for the time-squared law and the mean proportional relation is far from that. Here, since speed is proportional to distance squared and distance is proportional to time square, speed comes out proportional to time to the fourth. He has accepted acceleration but thinks a correct physical explanation must be something like a velocity-distance relation. With hindsight we know that Galileo’s search for a direct proportionality between velocity and distance would have to take him in the wrong direction. Moreover, percussion too is leading him in this way.

(f. 128) I suppose (and perhaps I shall be able to demonstrate this) that the naturally falling body goes continually increasing its velocita (speed) according as the distance increases from the point from which it parted. . . . This principle appears to me very natural, and one that corresponds to all experiences we see in instruments and machines that work by striking, where the percussent works so much the greater effect, the greater the height from which it falls. (Drake 1978, pp. 102–103)

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Figure 6.

Although Galileo may not be committed to a direct relation between speed and height, it is de?nitely height that in some way determines both speed and percussent force, the two concepts linked in De Meccaniche. Something puzzling arises in the speed-height relation when Galileo considers speeds along different inclinations (see Figure 6)

The motion along (motus per) the perpendicular AD is not perhaps quicker (velocior) than that along the inclined plane AB? It seems so; in fact, equal spaces are traversed more quickly (citius con?ciuntur) along AD than along AB; still it seems not so; in fact, drawing the horizontal BC, the time along AB is to the time along AC as AB is to AC; then, the moments of velocity (momenta velocitatis) are equal along AB and along AC. . . . (Wisan trans., p. 202)

Galileo expects the last to be true because percussion tells him the accumulated moments of velocity at B and C must be the same. Now imagine Galileo asked himself “under what conditions would both the chord law hold and percussive force be the same for all inclined planes of identical height?” In Discorsi, a proof is given that the speed of a body is the same when it traverses different inclines having the same height. We present that proof and then argue that this suggests how Galileo might have arrived at an answer to the above paradox that was physically intelligible, albeit in a new way (Figure 7). Galileo’s balance analysis of inclined planes says that the ratio of the force along the incline and along the vertical of an inclined plane are in proportion to the ratio of the length and height of the plane. Based on this he ?rst argues that:

the speed at C is to the speed at D as the distance AC is to the distance AD. . . . But, according to the de?nition of accelerated motion, the speed at B is to the speed of the same body at D as the

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Figure 7.

time required to traverse AB is to the time required for AD; and, according to the last corollary of the second proposition, the time of passing through the distance AB bears to the time of passing through AD the same ratio as the distance AC (a mean proportional between AB and AD) to AD. Accordingly the two speeds at B and C each bear to the speed at D the same ratio, namely, that of the distances AC and AD; hence they are equal. . . . (Discorsi, p.184)

Let’s restate the argument with S(AB) representing the speed through distance AB, and t (AB) the time for that distance. Therefore S(AB) = AB/t (AB). Whether S(AB) represents the instantaneous speed at the point B or average speed over the distance AB makes no difference to the ratios in this argument since we are dealing with uniform acceleration. The average speed S(AB) = 1 S(B), where S(B) is the 2 speed at B. In all reasoning with ratios of speeds the 1 drops out. 2 Notice also that, if the speeds at C and D are in inverse proportion to the distances AC and AD, then the times t (AC) and t (AD) are equal. Thus, the ?rst claim, though stated in Discorsi as resulting from the balance analysis of the forces, is also equivalent to the chord theorem, even though this theorem does not appear until later in Discorsi. The argument reads then 1. 2. 3. 4. S(C)/S(D) = AC/AD S(B)/S(D) = t (AB)/t (AD) t (AB)/t (AD) = AC/AD S(B)/S(D) = S(C)/S(D) (equivalent to the chord theorem) (de?nition of accelerated motion, see below) (mean proportionality) (from 1,2 and 3)

The third premise is a corollary of the time-squared version of uniformly accelerated motion, i.e., distance is proportional to time squared. 2. also follows because free fall is uniformly accelerated and it holds for both average speed over a distance and instantaneous speed at the end of a distance. Again, any equivocation on average and instantaneous speeds does not invalidate the argument. Our conjecture is

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that this argument is the reverse of Galileo’s actual reasoning. Rather, Galileo ?rst wondered under what conditions the speeds – and so the percussive Force – could be the same at points C and D. This would immediately imply (as we show next) the mean proportional relation. The real breakthrough would then follow when Galileo realized that this implied that velocity must change uniformly with respect to time. Time has been lurking in the background all along and now leaps to the fore as the essential measure of, in effect, all natural motion. Refer again to Figure 7, but now reverse (almost) the reasoning. By the chord law, t (AC) = t (AD). With these times equal then the speeds must be proportional to the distances, i.e., S(AC)/S(AD) = AC/AD. Now, the speeds at S(AB) and S(AC) would be the same, as implied by percussion, if S(AB)/S(AD) = S(AC)/S(AD) (two things are equal if their ratios with a third thing are proportional). Combining this with the ?rst result for the speeds, we get S(AB)/S(AD) = AC/AD. This is where the geometrical context of the pendulum lends to Galileo’s understanding of the mean proportional. The circle in Figure 7 represents a pendulum and AC, a mean proportional between AB and AD, is twice the pendulum’s length. Galileo also took double the pendulum length as characteristic of the time when investigating isochrony. This connects the speed of free fall with his broader category of natural circular motion. Our argument is not that this is how Galileo discovers the mean proportional for times. As we pointed out he had conjectured this relation earlier. He is aware ?rst of the time-squared law, probably on the basis of the inclined plane experiment, which would imply a mean proportionality. However, this gives him no physical explanation of the phenomena, and Galileo fails in his attempted explanation through accumulation of velocita. What we’ve outlined is the probable way in which t 2 , the chord law, percussive force and the mean proportional are brought together in a way that, while it still may not provide a satisfactory causal explanation of the phenomena, unites them as consequences of the simple and correct de?nition of uniform acceleration as velocities proportional to time. It is worth pointing out brie?y how our reconstruction compares with others. In ours the pendulum has played a crucial role in many ways. It exhibits acceleration, is the source of the chord law and for Galileo is subject to the same explanation that percussion is. The symmetry of the pendulum suggests to Galileo the link between speeds obtained by a body in fall and its ability to overcome resistance such as its own weight. A recent article by Renn, et al. (Renn, et al. 2002) argues that the importance of symmetry chie?y arises in Galileo’s consideration of projectile motion. In Drake’s various reconstructions (Drake 1978; Drake 1989) acceleration is accepted by Galileo only after he has overcome the hurdle of accepting instantaneous velocities. In Wisan (Wisan 1974, p.175) it is suggested that the search for the brachistochrone (the path of least time between two points) is what led Galileo to ask the right questions and that the answer to those questions turned out to be the law of free fall. Our view does not exclude any of these histories but emphasizes the pendulum: a simple machine, closely related to the balance, which beautifully

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exhibits acceleration, symmetry and the importance of the relation between path and time. 5. Science Students Discovery Brie?y, we’ll sketch how science students might “re-do” the Galilean discovery of time as crucial. This strategy accords with the idea of active learning that has dominated the science education literature in recent years. First, one might present students with a balance, an inclined plane and pendulum (and the necessary measuring instruments,) and ask them to discover and describe the basic properties of each with regard to motion. This might take the form of simply asking them to make a list, after viewing each experiment independently of one another, of nouns and verbs they could use to describe what they saw. After all three had been observed, the students would try to come up with a single, minimal set to describe the experiments. Here the teacher ought to introduce the geometry of the circle and its chords, so that students may see the geometrical relation among the machines. If acceleration is not noted by the students as a feature of the inclined plane and the pendulum it should be brought to their attention. The problem is to ?nd a way to represent time such that it can be seen as the important factor in the inclined plane and the pendulum, and why it is not necessary for balance problems. The ?nal problem is to use what was observed and measured on the inclined plane and the pendulum to transfer to the concept of a freely falling body. Discussion at this point ought to encourage students to re?ect on the basic nature of motions as observed in the world. References

Drake, S.: 1978, Galileo at Work: His Scienti?c Biography, University of Chicago Press, Chicago. Drake, S.: 1989, ‘The History of Free Fall’, S. Drake (trans.), The Two New Sciences, Wall and Emerson, Inc., Toronto. Euclid: 2002, in Dana Densmore (ed.), Elements, Thomas L. Heath (trans.), Green Lion Press, Santa Fe, New Mexico. Galileo, G.: 1590/1960, On Motion, I.E. Drabkin (trans.), University of Wisconsin Press, Wisconsin. Galileo, G.: 1600/1960, On Mechanics, S. Drake (trans.), University of Wisconsin Press, Wisconsin. Galileo, G.: 1610/1989, ‘Sidereus Nuncius or The Sideral Messenger’, A. van Helden (ed.), University of Chicago Press, Chicago. Galileo G.: 1613/1957, Letters on the Sunspots [History and Demonstrations Concerning Sunspots and Their Phenomena], selections in S. Drake (ed.), The Discoveries and Opinions of Galileo, Anchor-Doubleday, New York. Galileo, G.: 1890/1909, Galileo Galilei’s Notes on Motion, http://www.imss.?.it/ms72, Joint Project of Biblioteca Nazionale Centrale, Florence Istituto e Museo di Storia della Scienza, Florence Max Planck Institute for the History of Science, Berlin. Galileo, G.: 1632/1967, Dialogue Concerning the Two Chief World Systems, S. Drake (trans.), University of California Press, Berkeley.

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Galileo, G.: 1638/1954, Dialogues Concerning Two New Sciences, H. Crew and A. de Salvio (trans.), Dover Publications, Inc., New York. A better translation is: Galilei, Galileo. [Discourses on the] Two New Sciences [Discorsi], S. Drake (trans.), Madison, Wis., 1974. 2nd edn, Toronto 1989, [1638]. Hepburn, B.: 2003, Time as Tempo in Galileo’sUnderstanding of the Pendulum and Inclined Plane, presented at 7th International History, Philosophy and Science Teaching Conference, Winnipeg, Manitoba, mss. Palmieri, P.: 2002, ‘Proportions and Cognition in Galileo’s Early Mathematization of Nature’, mss. Renn, J., P. Damerow & S. Rieger: 2002, ’Hunting the White Elephant: When and How did Galileo Discover the Law of Fall?’, in J. Renn (ed.), Galileo in Context, Cambridge University Press, Cambridge, pp. 29–149. Wisan, W.L.: 1974, ‘The New Science of Motion: A Study of Galileo’s De motu locali’, Archive for History of Exact Sciences 13(2/3), 103–306.

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