Thomas B. Settle

GALILEO'S EXPERIMENTAL RESEARCH

Contents

Galileo's Experimental Research, an Experimental Approach (1996) 5 Appendix: Selcted Excerpts 29

The Pendulum and Galileo, Conjectures and Constructions (1965) 39

GALILEO'S EXPERIMENTAL RESEARCH,

AN EXPERIMENTAL APPROACH*

GENERAL CONSIDERATIONS

What sort of an experimenter was Galileo? What were his sources and how did he begin his experimental career? How important was experimental evidence to the formulation and reformulation of his general conclusions about certain aspects of nature? How developed was his style of work and how typical was it to what would become standard later on? These and related questions have fascinated and divided Galileo scholars over the years. Historians and philosophers have been able to find justification for quite divergent views both of Galileo and of the nature of experimental research in general. Some have concluded that Galileo did no experiments at all, in any interesting sense, that is. Others have painted simplistic "inductivist" portraits. Indeed, the purely written sources, largely those published by Antonio Favaro in the National Edition of Le Opere di Galileo Galilei, while presenting invaluable material when studied closely, leave wide margins for interpretation. And it is likely, had we no other evidence to offer, that we would have resign ourselves to the lack of further clarification of these issues.

Fortunately, there is another source of information to draw upon, nature itself, indeed several "natures". For instance, there is the nature constituted, in many variants, of our relatively untutored beliefs about the way the world works and our capacity to act successfully in that world (we can throw rocks at moving objects with some probability of hitting them). It has become a cliche, possibly a very misleading one, that Aristotelian and medieval physics provided a natural, if naive, rendering of the properties of motion. What is undeniably true is that today's children do construct, out of their everyday experiences, definite expectations about the material world, expectations that can differ markedly from the "truths" of Newtonian physics. Whether it was a distillation of such expectations that constituted the core of pre- Galilean physics is a problem to be investigated elsewhere, but we cannot ignore the possibility

* Up to now unpublished English version of Thomas B. Settle: La rete degli esperimenti Galileiani, in: P. Bozzi, C. Maccagni, C. Olivieri, T.B. Settle: Galileo e la scienza sperimentale, a cura di M. Baldo Ceolin, Padova 1995, pp. 11- 62. Thomas Settle Galileo's Experimental Research that Galileo began his own career predisposed to one or another such "natural" belief about motion. In fact, it should be part of our interests to look for traces of what Galileo may have "known", even apart from what he may have been taught in his early schooling or as a student at the University of Pisa.

Then there are our own bodily systems, which are immensely sensitive in some domains but which also have their limits. If we are tuned physiologically to the requirements of living in small groups and prospering in a hunting-gathering society, 10,000 years of cultural overlay have added both material and intellectual tools to enhance our perceptive abilities, as in the cases of the balance (allowing the making of fine distinctions in weight otherwise beyond our normal capacity) and practical geometry (permitting the precise surveying and depiction of the three dimensional world). What can we expect of Galileo's "natural" abilities? We know that he was brought up in a musical household and had both an excellent ear for distinguishing tones and harmonies and the hands of a near professional performer on the lute and organ. He was also well exposed to the Renaissance pictorial arts, including their geometrical aspects, and he must have had a keen and attentive eye. This background and training would certainly have contributed to his experimental capacities, but they may also have imposed boundaries, for example, in his ability to imagine further extensions to a particular line of investigation, or in an excessive reliance on these very abilities, unaware of their limitations. So we should also be looking for what we should be able to expect of Galileo's own system.

Thirdly, there is the "world out there". From our exposure to science courses we derive an image of how some of the principal features of the world are thought to work. But that image can present a very abstracted and sanitized version of what actually happens. We are told how bodies are supposed to move down inclined planes, for instance, and when presented with a lecture demonstration we "see" what we are expected to see, ignoring all the complications that intrude in an actual physical case, disregarding circumstances that don't fit the theoretical account. While there are distinct advantages in using this summarizing mode in the teaching of the sciences, we often make the simple mistake of assuming that Galileo saw (or ought to have seen) what we have been taught to see, forgetting that the early Galileo had never read Newton or even his own later writings. If we are to understand Galileo's own words and diagrams, we should try to learn how the physical world presented itself to him.

Finally, there is the nature of experimental research itself, or at least of Galileo's version of it. Let me introduce this most complex topic by describing the several levels of questions one might pose if given the opportunity of doing empirical studies of Galileo's work. At the very elementary level, one would hope to simulate, in equipment and procedures, the many experimental set-ups that Galileo described, sometimes in detail and sometimes only casually, in

6 Thomas Settle Galileo's Experimental Research his published and unpublished writings. An obvious example is the famous experiment of rolling balls down inclined planes and timing their motions with a water-timer, a device if not of Galileo's invention at least of his elaboration into a precision instrument. As we know, the validity of this experiment came under severe doubt two generations ago. It even came to be thought, in the history of science community, that Galileo could not have achieved the results he claimed with the equipment at his disposal [Koyré: 1939, 1943].

Then, in a contribution of my own in 1961, I showed that the inclined plane work, as described, was quite do-able [Settle: 1961]. There were two critical elements. One was the water-timer, not a clock meant to keep reliable time for several days, but a timing device capable of giving consistent measures of intervals as short as one second to about five or six seconds maximum and with a precision of at least 1/10th of a second. Galileo's device, a water container with a small tube through its bottom through which water was allowed to run during the interval to be timed, proved quite up to the task, as subsequently confirmed by many others besides myself. The other element was the experimenter, Galileo himself in the first instance. Since the experimenter has to close the flow of water in the small tube when he or she hears the ball striking an object at some distance down the plane, one had to wonder whether the human system was capable of responding in such a way as to obtain Galileo's results. Evidently, yes; the experimental results bear this out. This was not so unlikely, on reflection, in the light of the fact that trained pianists, violinists and lutanists must accomplish even more prodigious feats of ear-hand coordination in their daily exercises. This story shows, I think, that it is important to take Galileo's claims seriously and to attempt reasonable empirical simulations of them.

At a second level, beyond just deciding whether a given experimental set-up could work, we can establish the limits or boundaries of its workability. Regarding the inclined plane, Galileo himself said, in the Third Day of the Discorsi: In un regolo ... di legno, lungo 12 braccia, ... costituito che si era il detto regolo pendente, elevando sopra il piano orizontale una delle sue estremit à un braccio o due ad arbitrio, ... [GG: VIII, 212-213]. In fact, we know that the desired results obtain quite satisfactorily over a range of inclinations from about 10 degrees to about 40 or 45 degrees. Below the lower limit the motion is not reliably uniformly accelerated; in fact, with a little care we can even make the motion quasi-uniform. And above the upper limit the ball begins to skip and slide, definitely not undergoing uniform acceleration. Interestingly, this latter was also known, in a limited sense, to Marin Mersenne in the 1630s. And my own feeling is that Galileo's precise knowledge of these limits, dating back to around 1600, was behind the very carefully crafted description of the inclined plane partially cited above. In other words, our own empirical investigations promise to help us clarify the scope and intent of important written passages.

7 Thomas Settle Galileo's Experimental Research

But if they do that, they can also pose other problems. In the Discorsi Galileo used the inclined plane to justify his claim that free motions on ALL inclinations, including the vertical, were both uniformly accelerated according to his rule and related among themselves, i.e., calculable from one inclination to another. If, as I am convinced, he knew that this was not the case empirically (for the reasons I have already indicated, as well as for others beyond his ability to understand), then he must have satisfied himself that the motions that did not fit his rule of acceleration were in some sense artifacts. He had to decide that the empirical limits of the set-up did not offer strong enough contrary evidence for him to forego the general law of free fall. I am not in a position at the moment to give a complete account of how Galileo justified that decision (and Galileo himself never gave a direct justification), but it is least possible that a deeper understanding of these and related components of his experiments could suggest possibilities. Nor is this a unique case. There are many instances in which Galileo had to make decisions about fact vs. artifact as he pursued his investigations. And he made both correct and incorrect choices. Our task is to cast light on why he chose one way or another, even sometimes changing his mind after further exposure to a problem.

At yet a fourth level of approach, it is difficult to find a clear case in which Galileo completed an experimental investigation with the same understanding of a given phenomenon with which he began. To my knowledge, Galileo never simply adopted a theoretical position, deduced an empirical consequence, constructed a carefully articulated experiment to test it, and ended the matter at that point. Again in the case of the inclined plane, we know that he first used it, not in the context of confirming a theory of uniform acceleration; quite to the contrary, he used it first in the context of investigating a proposed uniform speed rule regarding natural motion in free fall, one of the three rules which were the starting points for his Pisan, uniform speed dynamics of the early, De motu antiquiora, ca. 1590 [GG: I, 243-417; Drabkin, 1960; Giacomelli, 1949; Settle, 1967]. Since neither the theoretical nor the empirical accounts of those early inclined plane trials make any mention of measured time intervals, we are probably safe in assuming that Galileo had not yet invented any precise timing mechanisms, much less the water-timer mentioned earlier. In other words, the early "inclined plane" was categorically different from the later "inclined plane". Generalizing, we could say that, over time, Galileo's understanding of inclined plane phenomena underwent periods of inception, elaboration and metamorphosis (as when he changed the theoretical context from that of uniform speed to that of uniform acceleration), then more elaboration, finally reaching the views expounded in the Discorsi. It is only this final, classic version, reduced to two paragraphs with some surrounding explanatory material, that most people know as "the inclined plane experiment". Much more interesting generally, I think, is the entire gestation of, or the developmental history behind, that final form. In this sense, Galileo's text in the Third Day of the Discorsi provides only a didactic and

8 Thomas Settle Galileo's Experimental Research very condensed presentation of final results achieved over years of real experimental research. And the phrase "Galileo's inclined plane experiment" ought to refer to his entire experience of the inclined plane from the time of the De motu antiquiora until that of the Discorsi, even the rejected early interpretations, and more besides.

I say "and more" because there is one final aspect to the complexity of Galileo's experimental research which needs introducing. Thus far I have written as though we could legitimately isolate the inclined plane, or the evolutionary story of the inclined plane, from other components of Galileo's inverstigations. In fact we simply cannot do that. For instance, in his developing mastery of inclined plane phenomena Galileo realized, more or less in the mid 1590s, that he needed a timing device, a practical way of measuring small intervals of time. He may have started by trying to use his pulse or to make use of his sense of musical time, tempo. Then, having come to appreciate the time keeping properties of swinging bodies, he probably tried to use the pendulum directly. Unfortunately, the pendulum can only mark discrete units of time and is, in any case, difficult to use. But he could use it to calibrate his water-timer, thereby giving him a means of measuring continuous intervals of time. In all likelihood, this was one of the key steps which brought him to give up the early uniform motion physics in favor of one founded on uniform acceleration.

What is important to emphasize at this point is that, besides the inclined plane, the pendulum is one of the foundation empirical devices in the Galilean sciences. It has, or certainly will prove to have, as rich and complex developmental history as that of the inclined plane itself. What is certain already is that these two stories intersected several times, beginning at least in the early 1590s, leading on the one hand to the aforementioned timing device that opened the door that allowed Galileo to move to a uniform acceleration physics. On the other hand it gave Galileo a whole new class of inclined planes to investigate, those with geometrically definable curves, in the first instance an arc of a circle (the motion of a pendulum is such a definable, non-uniform, natural acceleration). Never did any of Galileo's research lines remain isolated. They were always part of a complex network of only apparently isolated lines, each having its own story, but heavily cross-linked among themselves. Galileo simply would not have been able to advance his work on the inclined plane had he not been able to import some of the intermediate results of his work on natural oscillators, in this case the simple pendulum, and the story already promises to become very much more complicated. But it is this network of heavily cross-linked, interdependent lines of development which constitutes Galileo's experimental research.

9 Thomas Settle Galileo's Experimental Research

LIGHT BEFORE HEAVY

Let us begin our search for the network by looking at Galileo's exposition of his early uniform speed dynamics in the De motu antiquiora (ca. 1590). At several points he asserts quite plainly that, contrary to commonly accepted belief, when two bodies are dropped simultaneously the "lighter" of the two will move ahead of the "heavier", at least at first [Excerpt VII; Excerpt X]. By "light" and "heavy," he means "of lesser and greater specific gravity;" in this case, a ball of wood initially moves more quickly than a ball of iron. And he gives us to understand that he has not only seen the phenomenon in actual trials, but also that it came as a surprise to him; he had not expected it [Excerpt VI]. But how can this be? We know, and Galileo himself accepted later on, that dropped bodies of whatever weight or density should fall side by side. Did Galileo simply invent the phenomenon and deliberately mislead us by claiming that he had performed the experiments? With no other evidence than the contents of the text, we would probably have to decide that Galileo was lying or at best that he had somehow deluded himself [Schmitt: 1969]. It turns out, however, that the phenomenon is a real one. Galileo claimed as a fact something that did not obviously derive from what he then accepted as theoretically true because in all probability he actually observed it and took it seriously enough to want to explain it.

To understand what happened, let us briefly review the context. In the De motu antiquiora Galileo already had in mind a fairly thorough revision of the then "standard model", that is, what was accepted in the schools as the Aristotelian theory of natural motion. One of the tenets of that model was that bodies in free fall, natural-spontaneous motion, move with speeds (considered to be uniform) in proportion to their simple weights. Others before Galileo had already observed that this was not the case, but no one else had set out systematically to revise the account and create a consistent and comprehensive view of natural motion [Settle: 1967]. To start, Galileo adopted the supposition that all bodies in free fall move (or should move) with a characteristic uniform speed. And he made quite explicit his idea that this uniform speed was not to be understood as a "terminal velocity" (to use a modern term) caused by the resistance of the medium; instead it was to be a uniform speed that would be exhibited most exactly if we could drop heavy bodies in vacua. Then, undoubtedly impressed by the writings of Archimedes, and looking for a feature that might distinguish material bodies other than weight, he supposed that the characteristic uniform speed should be proportional to the specific gravity of the object. A gold ball, for instance, ought to fall at twice the speed of a silver one, gold having about twice the specific gravity, or density, of silver. To put it more dramatically, if the two balls were released from the top of a tall building, the gold one ought to hit the ground with the silver one only half way down.

10 Thomas Settle Galileo's Experimental Research

Galileo's problem, at this point, was that when he made the trials he found that all reasonably heavy weights (presumably using balls of iron, wood or stone) reached the ground at about the same time, contrary to his theory. He also found, as mentioned, that light bodies move ahead of heavy ones [Excerpt VI]. His first reaction was not to give up or revise his first two assumptions, but to try to find a way of "explaining away" the non-conforming experimental results. (This would seem to be a common and unremarkable practice, even if contrary to what is normally thought of as the scientific method.) For Galileo this took the form of recognizing that bodies released from rest do not jump discontinuously to their natural speeds; they undergo a period of "un-natural" motion, an "accidental" acceleration. This acceleration was un-natural in the sense that it could not, even in principle, be reduced to rational or mathematical expression. Galileo could account for it qualitatively, however, by importing one of the versions of the medieval "quality" impetus, a quality imposed on a body by an exterior agent, which would decay and eventually vanish when that agent was removed. In this case, when an object is being held by a hand, the hand transmits to it an amount of impetus, here an upward force, just sufficient to balance the object's tendency to fall by its own gravity. When the hand releases the object, the upward impetus or force cannot vanish instantaneously; it requires a period of time to drain away. For Galileo, only when the impetus had drained away completely would the object attain the speed characteristic of its specific gravity. Thus conceived, impetus allowed Galileo to argue that, since there were no buildings high enough to permit objects released from them to be free of the impressed impetus before hitting the ground, such objects were always in the phase of their un-natural, accidental acceleration, and it was not surprising that they were still falling more or less side by side [Excerpt IX].

In retrospect this was a rather ad hoc and unsatisfying way of accounting for unwanted experimental results. And we know that Galileo eventually gave up almost the entire edifice of his De motu antiquora physics. In the meantime, however, he extended the impetus solution to cover the other phenomenon he had discovered, that of the light, less dense, bodies moving ahead of the heavy ones [Excerpt VIII]. For him it was reasonable to suppose that an imposed impetus drained away from a light body more quickly than from a heavy body. The lighter body would thus acquire a greater relative preponderance of its natural gravity and thereby move more quickly downward at the beginning of its fall. But do falling bodies actually behave this way? Can we imagine and construct a set of circumstances in which they do?

A number of years ago an experimental psychologist colleague contacted me and suggested that he investigate the problem. His suspicion was that this "Galileo phenomenon" might have a physiological base. And he worked out a procedure whereby a group of subjects (25 university students, as it turned out) were to take part in a programmed "dropping" experiment [Settle: 1983]. They were not told what the real scope of the experiment was, only that there was no

11 Thomas Settle Galileo's Experimental Research correct or incorrect performance; he simply wanted to know if they actually could perform certain manual operations with both hands, in this instance holding balls of different weights, one in each hand, palms down, and releasing them at the same time. Two sets of two spheres each were provided, one set with larger diameters and one with smaller. In each set one ball was iron and one wood; the ratio of weights was about ten to one in each set. Each student was to have four "drops", two with the heavy set and two with the light, the sequence randomized in such a way as to have some beginning with the larger spheres and some with the smaller, and some beginning with the heavier ball in the "favored" hand and some with the lighter. The drops were scored by direct observation, by people who also did not know the real scope of the experiment; and they were filmed for later review.

The results were reasonably unequivocal; in 89% of the drops the lighter body visibly moved ahead of the heavier one in early part of the motion. The frames of the film clearly revealed the reason. Even though each student "felt" that he or she had released the balls simultaneously in each drop, the hands holding the lighter ball opened a fraction of a second before the others, allowing the lighter spheres a slight head-start. The reason for this has yet to be clarified precisely, but it may be because the hands and arms holding the heavier of the two balls have to work harder during the several seconds before the drop. Either the muscles or the nerves serving them become fatigued, and it takes them slightly longer to react to the signal to open the hands and release the balls. In any case, this experiment yielded a simulation of Galileo's results. The phenomenon is real, even if from the point of view of his later physics it turned out to be spurious. Initially Galileo had no obvious way of knowing that. After all, he had found a stable and repeatable phenomenon, explainable in terms of a reasonably consistent and comprehensive account of free fall. In sum, Galileo had accepted as an empirical fact a phenomenon he would later come to reject. Presumably he rejected that fact when, making the transition from a physics of uniform motion to one uniform acceleration, he realized that impetus had no place in the new explanatory scheme.

We already know part of the story of how he came to this new view. Obviously not satisfied with his inability to find uniform speeds in vertical fall, he looked for a means of systematically slowing that motion, and he hit on the idea of having spheres roll on inclined planes [Settle: 1967]. He supposed that he could find a way of calculating, for a given angle of inclination, a ratio of an expected speed on a plane to that in the vertical; and he succeeded in doing this. That same ratio would presumably give him the value of the "unsupported" component of the impetus once the sphere was freed and allowed to roll. He seemed to hope that this smaller amount of impetus would drain away quickly enough for the sphere to exhibit uniform motion before reaching the end of the plane [Excerpt VIII]. Here too he was disappointed, but the work led to an extension of his investigations. While it is difficult, if not impossible to "see" a body

12 Thomas Settle Galileo's Experimental Research accelerating in vertical fall, one can actually see it accelerating on a plane, especially at the beginning of the motion. But does it continue accelerating on the plane, or does it does it accelerate more slowly towards the end? That is less clear. Galileo did entertain the idea that the rate of acceleration would diminish (though this phrasing is somewhat anachronistic), the speed asymptotically approaching a fixed, uniform value. But it became suddenly obvious in our own experiments that one can hear an acceleration. One hears it in same sense that one can tell immediately whether an automobile engine is idling or racing. And it is more likely that a 16th century miller adjusted the speed of his millstones according to their "tone" than by what his eyes told him. Galileo never mentioned this in the early De motu, but it is unlikely to have escaped his attention. Such audio-observations would tell him that the balls accelerated most of the way down the plane; they could not tell him whether the acceleration was constant, in some as yet unknown sense, or whether it began to diminish. But at this point Galileo would have had a considerable incentive to invent a way of answering those questions.

ISOCHRONISM

How exactly he came to find a solution remains unknown, but the evidence suggests that he used some of what he had learned about another type of natural motion, that of a heavy body swinging on the end of a cord. In the De motu Galileo mentions observing swinging bodies in order to learn how long it takes them to come to rest, in other words, how long it takes for the system's impetus to drain away: in general a much longer time than it takes for an object to reach the bottom of a tower or the end of a plane. He may also have already thought of a swinging body as another experimental means of systematically constraining a free vertical motion into a more manageable, slower motion, an arc of a circle being a type of inclined plane. In all likelihood, however, the most important contribution came from his knowledge of the isochronous properties of the simple pendulum.

When precisely Galileo discovered those properties or whether he discovered them all at once or only over a period of time are questions open to debate. According to several versions of the story in the writings of Vincenzo Viviani, Galileo made them while still a student at Pisa in or around 1583 [Exerpt IV; Exerpt V]. Viviani tells us that, in the Duomo one day, Galileo's attention was caught by a swinging chandelier; he noticed that as the arcs of the swings diminished the periods of the swings seemed to remain the same, using both the beat of his pulse and his highly trained sense of musical time to check that impression. Then returning to his rooms, he and some friends devised a series of experiments to verify the original

13 Thomas Settle Galileo's Experimental Research observation and discover the remaining properties. A fanciful reconstruction of the event might read as follows:

One day, while listening to a long choral mass in the Duomo of Pisa, Galileo noticed a lamp gently swinging in one of the aisles. Evidently one of the suspending links needed an oiling; when the lamp paused momentarily before each new descent it uttered a rusty creak, and it was this that attracted his attention. As he followed the motion he realized suddenly that its rhythm coincided with that of the music. "Must be an accident of the particular arc," he mused, "as the swing diminishes, the beat will probably slow also." His eye wandered to the directing baton of the choir-master. Galileo had had a thorough musical training from his father, and he knew that the choir-master's timing was good. Right then the lamp's screech was punctuating every eighth beat of the baton. He glanced back to the lamp. Its arc had decayed to half its former extent. "Strange, it's still keeping the rhythm. But it should be out of synchronism by now." On reflecting, Galileo was not quite sure whether its beat should have slowed or speeded. While he watched, with more concentration now, the arc continued to diminish, yet the beat remained in time with the music. "Perhaps the musical time actually has been changing." He put his fingers to his pulse and counted. The lamp's creak was barely audible now; its arc was small, almost a waggle. But it did seem as if the number of pulse beats for each swing was remaining constant. While he knew that this could not be a conclusive test, it did seem an odd sort of thing. As he puzzled he could not even recall a mention of such a phenomenon in his many lecture notes or books on the various properties of motion.

A few evenings later he was at home trying to interest himself in his Galen notes. There was a heavy lamp in his room, hung by a flexible cord from the ceiling. Occasionally, when bored or otherwise uninspired, he would set it swinging and watch the shadows on the wall, letting his mind drift in speculation. This particular night he set it in motion and, as he watched the shadow of a bedpost arc back and forth, he recalled the peculiar motion of the cathedral lamp. "I wonder if it really did keep the same beat the whole time, ... ."

Fanciful as it is, this little story may capture something of Galileo's first encounter with isochronism. As we will see, it does not matter much if that encounter occurred during his student days or only later when he returned to Pisa in 1589 as a Professor of Mathematics. The next encounters probably follow the cleaned up and schematic account Galileo published much later in the First Day of Discorsi [GG: VIII, 128-131, 138-150]. First there would have been the attempt to find out if the period of a single pendulum were actually constant and independent of the length of the arc of the swing. But how could he do this? The rate of the human pulse beat is not constant, and mechanical clocks were notoriously unreliable. Though Galileo did not

14 Thomas Settle Galileo's Experimental Research actually say so, he must have realized intuitively that if the pendulum were a "natural" time- keeper, that is intrinsically a perfect time-keeper, then the only empirical way of ensuring himself of the fact was to test one pendulum against others. And however he may have reasoned, this is the picture that emerges from the passages in the Discorsi. Take two pendulums of equal length. First set them in motion on equal arcs; then on different arcs; then set one in motion and, a few seconds later, set the other in motion; and so on. In whatever sequence or configuration one can think of, the result that is most impressive is that the pendulums keep pace with one another. With a little reflection there would be no other conclusion to draw: by their inherent nature pendulums of a given length beat equal intervals of time. While much later Galileo would encounter evidence that this might not be exactly so, he remained basically convinced of the truth of this first rule of isochronism for the rest of his life. (And it is likely that this belief was reinforced by another empirical consideration which we shall come to.) Then, having taken this first step, the rest was relatively easy. By systematically substituting objects of different weight and density, he could satisfy himself that the period of a pendulum was independent of those variables. Finally, he thought to find out if there were an identifiable relationship between the lengths of different pendulums and their periods. Indeed there was: one pendulum four times the length of another will complete one swing while the other completes two; set two such pendulums in motion and they will beat a sort of syncopated harmony. If we recall that in the 1580s there was no theoretical base for any of these propositions, that the only way Galileo could have found them was through empirical research, and that our own work shows that the results obtain as described, it is no wonder that Galileo himself was very impressed. Quite probably these were the first of the many truths he discovered about the physical world which were unknown to and even unimagined by Aristotle or any subsequent natural philosopher.

And these were the discoveries, in all likelihood, that allowed him to move ahead with his investigation of motions on the inclined plane. With a means of reliably producing small intervals of time, he could try to find out if and when the rolling ball, at first accelerating, began to move with a uniform speed. He probably began by trying to use a pendulum directly, attempting to measure the distances covered on the plane during successive single swings of the weight, from the top of the swing on one side to the top on the other. But this is not easy to do, and moreover one cannot easily measure fractions of the time-interval smaller than one full swing. What Galileo apparently did was to develop the water-timer mentioned earlier, a container of water with a small tube through the bottom [Settle: 1961]. The amount of water passing through the tube during the interval measured was the index of that interval. How could Galileo assure himself that the flow of water was constant over the intervals timed? By testing the water timer against a pendulum. In my view, it was the use of this water timer, which

15 Thomas Settle Galileo's Experimental Research depended on the previous discoveries concerning isochronism, that led Galileo to the knowledge that balls continued to accelerate over the whole length of the plane, indeed that the acceleration was open and did not tend to uniform motion, and finally that one could specify an exact, mathematical rule for it: that the units of distance covered in equal time-intervals were as the odd numbers from one or, more conventionally, that the total distances covered were as the squares of the total time intervals [Lettera a P. Sarpi, 1604, GG: X, 115-116].

With this new discovery Galileo's project of creating a physics based on uniform natural motion collapsed, along with his need for impetus as originally conceived, and he redirected his efforts towards the creation of a new science of natural motions and their effects. Eventually he would want to include components on uniform acceleration, projectile motion, pendular motions (naturally oscillating motions of several types) and the mechanical consequences of motions such as percussion, and his researches in all these areas occupied him, off and on, from about the mid 1590s through and after the publication of the Discorsi in 1638.

THE PENDULUM IN A BOX

One part of this program involved trying to connect uniform acceleration with several types of curvilinear accelerations in a single general discourse. Having formulated a mathematical expression for uniform linear acceleration, and having found empirically a set of apparent rules for the isochronism of motions in arcs of circles, he thought that he ought to be able to find a mathematical demonstration leading from the first to the second. And his own work proceeded in two interlinked modes, mathematical and empirical. On the mathematical side, of course, the attempt was doomed to failure; contrary to his own conviction, simple pendulums are not, strictly speaking, isochronous; but the proof of that would only come after Galileo had passed from the scene. On the experimental side he continued to explore the phenomena. At some point, apparently, he realized that, instead of having weights swing on the ends of cords, he could have balls rolling on semicircular surfaces, and he found that they also exhibited the same properties he had found with "normal" pendulums [Lettera a G. Del Monte, GG: X, 97-100]. Our experiments show that this was a reasonable conclusion, though we should add two comments. First, the friction on the surfaces is much greater than the friction of the air, and the oscillations diminish to rest very much more quickly than with ordinary pendulums. Instead of oscillating for upwards of hundreds of swings, the rolling balls seldom reach as many as twenty. Our second observation is probably more consequential. According to our modern understanding, a ball moving through a small arc ought not to keep exact time with a ball moving through a large one, and we should see this in the test. In practice, when we release a

16 Thomas Settle Galileo's Experimental Research ball from a position fairly high on the circular surface, its arc of oscillation will diminish within two or three swings to a much smaller amplitude, small enough so that isochronism does seem to hold. So it was reasonable for Galileo to proceed to comparing linear and circular accelerations directly. He found that: 1. if two balls are released from a point on the circumference of a such a circular surface, one to descend on the circumference itself and one to descend on an inclined plane ending at the lowest point of the surface (i.e., describing a chord of the vertical circle), the one moving on the circumference will reach the bottom first, even though it follows a longer path, and 2. balls released simultaneously from any point on the circumference of the vertical circle and descending to the bottom along chords arrive at the same time. By 1602 he even seems to have found a mathematical demonstration of these effects. What he kept trying to do, without success, was to extend these arguments to a demonstration of isochronism itself.

THE INTERRUPTED PENDULUM

Meanwhile, another line of investigation led him to the conviction that several bodies descending from the same height would acquire equal speeds irrespective of the paths they traversed [GG: VIII, 205-207]. His empirical evidence came from what we call the interrupted pendulum. Galileo described the device in the Third Day of the Discorsi: a pendulum is fixed in a way so as to allow it to swing in front of a flat, vertical surface which has a horizontal line drawn on it at a convenient distance above the weight of the pendulum. If we pull the weight aside to that horizontal line and then release it, we see that it falls through its arc and then rises very nearly to the same height on the other side; then it returns through the same arc to the starting side, again rising very nearly to the previous height; and so on. (If Galileo had done this before his switch to his new physics, he would have "seen" a slow but inevitable diminishing of the arcs towards rest as the pendulum lost its initial impetus. Through the lenses of the new physics he saw the weight returning to exactly the same height except for frictional resistance.) And it returns to the same height even if we change the path. If we put a nail or a peg in the way of the cord of the pendulum, we see the weight first descending by its original arc to the bottom of its swing; when the cord hits the nail the weight swings up on a new and shorter arc, but always to the same height. And if we reverse the process, releasing the weight from the top of the shorter arc, the weight will swing down and then rise on the original, longer arc, again to the original height, In other words, the speed acquired at the bottom of the swing, through whatever arc, is always such as to make the weight rise to the same height, also by whatever arc.

17 Thomas Settle Galileo's Experimental Research

How did Galileo come to devise this apparatus in the first place? The easiest answer might be that it resulted from a series of direct attempts to discover the relationship between the height of fall and acquired velocity. We know independently that he was continuing to explore this problem in the years between 1600 and 1610. On the other hand, the device may have derived from earlier investigations. For instance, it may have resulted from Galileo's early efforts to discover all the possible properties of the pendulum and to satisfy himself that they were real. In confirming the rule relating the pendulum's period of oscillation to its length, for example, he could have thought to investigate pendulums whose lengths could be changed in mid flight. Two ordinary pendulums, one four times the length of the other, move in a kind of harmony; perhaps a "variable" pendulum might disclose other interesting patterns. Galileo never mentioned this, but the idea is consonant with the fertility of his imagination. -- Another possibility derives from an observation of our own. When the cord of the pendulum strikes the interrupting peg, the part of the cord between the peg and the point of suspension visibly vibrates. And if the pendulum's weight is heavy enough, the vibration produces a low but audible tone; and this tone will vary in pitch depending on the magnitude of the weight. If we now step back and in our mind's eye perform some simple topological transformations, we can see the device as a monochord, the teaching and experimental tool of medieval and renaissance musicians, rotated into a vertical plane. In other words, the device may have roots in Galileo's early experiments in musical acoustics. We shall return to this topic shortly.

PERCUSSION

Yet another possibility is that the interrupted pendulum may have derived from an attempt to establish a reproducible measure for percussion, a standard percussor. In the letter to Guidobaldo del Monte of 1602 Galileo alluded to a related idea without giving further details: "Al Sig.r Francesco mi farà grazia rendere il baciamano, dicendogli che con un poco d'ozio gli scriverò una esperienza, che già mi venne in fantasia, per misurare il momento della percossa". He had, after all, developed a means of producing units of time (the beat of a pendulum of such-and-such length). Here a pendulum of sufficient length, with a standard weight, and swinging down from a fixed height, would deliver a standard percussive blow. Ballistic pendulums function by the same reasoning but in reverse order. Complicating the picture for Galileo, who was still trying to sort out the important variables for his new sciences of motion, would have been the fact that the strength of the impact depended not only on the speed reached but also on the weight of the percussor, a factor with no role in isochronism.

18 Thomas Settle Galileo's Experimental Research

Galileo's first documented comments on percussion date from around 1600, in a manuscript tract on mechanics, but his initial interest probably antedated that by at least a decade [GG: II, 188-191]. The simple and convincing, if imprecise, technique of dropping stones from various heights onto stakes loosely placed in the ground could have helped persuade him that bodies in free fall do not reach uniform speed. Later he described this same procedure in his introductory remarks to "Naturally Accelerated Motion" in the Discorsi. In any case, the first sections of the ca. 1600 manuscript treat of elementary machines, from the lever through the inclined plane. In the last section, however, Galileo tried to adapt the same line of reasoning (reducing all these forms to variations on the balance) to the case of impact: how can we understand the difference in effect between merely laying a hammer on top of a nail and hitting the nail with the hammer in motion? He never did solve the problem, but he did continue looking for solutions, both empirical and rational.

Galileo described one of those experiments in a tract "Della Forza della Percossa", not published until the 18th century as the "Sesta Giornata" of the Discorsi [GG: VIII, 323-325]. From internal evidence, the trials seem to have taken place in Padua sometime between 1605 and 1610, just before his telescopic discoveries and subsequent return to Florence. Galileo initially thought that he could measure the force of percussion using an instrument very nearly like a large balance. On one end of the balance beam he hung two secchie (pails), one above the other, the upper of the two with a hole that could be closed or opened. After filling the upper secchia with water he balanced the system by hanging a convenient counterweight on the other end of the beam. The idea was to open the hole in the upper secchia, letting a column of water fall between the two. At that point he expected to see the balance-arm tip to the side of the secchie, the force of the impact of the water adding to the original static weight of the equipment: secchie, water, etc. The measure of the force of that impact would be the amount of weight he needed to add to the original counterweight to restore the balance. He was surprised, and says so, to find the balance-arm initially inclining towards the counterweights; then, as soon as the column of water began to hit the lower secchia, the balance began to return to neutral; and as the arm reached the horizontal it stopped, staying in that position while the water remaining in the upper secchia ran out. In a sense, the experiment was a failure; it did not lead to a means of measuring impact. But it did lead Galileo to a better analysis and an essentially correct understanding of what had happened. He concluded, contrary to his preliminary supposition, that the water in the falling column went out of the system and consequently did not "weigh" on the side of the secchie, hence the initial displacement, and that as soon as the column began hitting the lower secchia its "force" balanced that of the "missing" weight of water; hence the balance-arm properly returned to neutral.

19 Thomas Settle Galileo's Experimental Research

Now Galileo's apparatus does work, precisely as he described it, which ought not to be surprising. This episode does, however, give us an unusual glimpse into Galileo the researcher. While the account in the text is quite likely a very condensed version of what happened in real time, it is probably schematically correct, and it shows a Galileo quite capable of starting with certain suppositions, building equipment to explore them, attending to the results actually received (even if going counter to the initial suppositions), and then rebuilding the edifice of his understanding.

At the same time Galileo was also capable of holding on to what, in his eyes, was a compelling idea and devoting great effort in the search for experimental evidence to sustain it, especially when it seemed to have resonance in diverse phenomena and to offer the possibility of forging links among superficially unrelated realms.

MUSICAL ACOUSTICS

This general topic is one that we might call natural oscillators, and the story begins with the researches of his father, Vincenzo Galilei, into certain aspects of musical acoustics. As is well known, Vincenzo Galilei was a composer, performer, teacher and scholar of music. In all these capacities he had become interested in the theory and practice of musical harmony, including the proper tuning of instruments. For a while a student and follower of Gioseffo Zarlino, he spent the latter part of his life attacking Zarlino's theories, using to great effect the results of his own experimental investigations. And it was in this period that the young Galileo was himself maturing as a musician and, towards the end, beginning some of his own experimental researches.

Zarlino, out of his studies of ancient and medieval authors, as well as his own practical knowledge, had proposed a theory of harmony based on the existence of "sonorous numbers", the "senario". According to Zarlino, the ratios of the first six natural numbers were harmonic ratios and gave pleasing tonal combinations by their very nature; ratios of other numbers were inherently discordant. Here Zarlino was implicitly alluding to musical effects demonstrable using the monochord. The ratio of two-to-one is that of the full octave, as is obvious when one stops the string of a monochord at its half-point; if the full length of the string sounds a certain note, the half-length sounds a full octave higher. But for Zarlino, the cause of the harmony was in the ratio of the sonorous numbers and had nothing to do with the material properties of the string or the instrument or with the mechanisms of the generation, transmission, or perception of sound. The listener simply apprehended the essential "two-to-oneness" and was pleased.

20 Thomas Settle Galileo's Experimental Research

When Vincenzo Galilei began to doubt Zarlino he did so from the empirical viewpoint of the composer and performer. First examine the way singers actually modulate their voices and performers actually tune their instruments, he wrote, and then find the ratios. But the ratios of what? Not of pure numbers, but of some measurable physical characteristic. Vincenzo was led to investigate the variety of conditions in which one could produce and combine tones, looking at known musical instruments as well as other sources of identifiable notes. He tested stretched cords of ottone, acciaio and minugia (brass, steel, and gut), for instance, initially on the lute and the monochord and later simply by suspending them in the vertical with weights attached. He confirmed that the ratio of two-to-one, when referred to the lengths of stretched strings, did describe the means for producing an octave; but he also found that same musical interval could be generated by increasing the tension on the string by a factor of four. So what was the "true" or "essential" numerical ratio for the octave, two-to-one or four-to-one? And the ratio for the musical interval the fifth, commonly and correctly taken to be three-to-two when indicating lengths of strings, became nine-to-four when referring to relative tensions. Then in the case of organ pipes, if a given pipe sounded a certain note, one with all its dimensions doubled sounded an octave lower. But doubling all the dimensions meant that the ratio producing the interval was eight-to-one. In short, Vincenzo showed that these and other empirically recognized harmonic ratios could be generated "using" numbers well outside the senario, thus destroying Zarlino's attempt to unite and explain harmonic phenomena within a single system; he could provide a set of empirical rules for generating harmonies in different circumstances and could illustrate the difficulties in tuning different types of instrument and in composing for voice, but he had destroyed all previous theoretical underpinnings for the phenomena and had provided no replacement, least of all one based on the properties of the material world. In the process, however, he had cleared the way for an empirically based musical acoustics and had launched his son, Galileo, on his own career of experimental research.

What remains unclear, however, is precisely how and when Galileo entered the scene. His own apprenticeship in music and musical research would have followed closely the progress of his father's investigations, which can perhaps be divided into two phases. Vincenzo began, considerably before Galileo went to Pisa as a medical student in 1581, by looking for the empirical distinctions among the principal methods of construing musical scales and tuning instruments; and Galileo would have absorbed both his father's early results and his radically empirical attitudes while he himself was becoming an accomplished lutanist. Then, sometime after 1585, Vincenzo began looking more closely at the logic of the senario and the several ways of generating harmonic intervals mentioned above. The reports of these experiments appear prominently in his Discorso intorno all'opera di Gioseffo Zarlino ... of 1589 and his

21 Thomas Settle Galileo's Experimental Research originally unpublished manuscripts of ca. 1589/90 in the Biblioteca Nazionale Centrale di Firenze [VG (1589/90); Palisca (1989): 189-197].

If Galileo had actually discovered the laws of the isochronism of the pendulum while still a student, in about 1583 according to Vincenzo Viviani, he would have already begun his own independent research career [VV (mss); VV (1654)]. He would already have noticed that a given pendulum string, with a sufficiently heavy weight on the end of a lute string, consistently sounded the same note, and that different weights generated different notes. Did he then bring this knowledge back to Vincenzo, thus initiating the research reported by the latter? Or alternatively, did Vincenzo initiate that work independently, thus providing Galileo, by and large based in Florence between 1585 and 1589, with weights on the ends of lute strings, subsequently used in the discovery of isochronism? For the present, it is difficult to judge. But either way, at the latest by about 1590, Galileo could hardly have failed to be impressed by the fact that a single device, a weight on the end of a fine cord, seemed to yield two types of natural oscillator: the pendulum itself with the properties already described, and the vibrating string.

For Galileo, one of the essential properties of the pendulum was that its rate of oscillation did not change as the amplitude of its swing diminished. In the case of the vibrating string, the pitch of the tone remains the same both as the visible amplitude of its vibration diminishes and as its resulting strength or loudness decreases. Was it possible that the essential feature distinguishing musical tones was the rate of vibration of the sounding body and hence the medium transmitting the sound? If this were so, the dependence of pitch on a rate of vibration could provide the link re-uniting the musical phenomena left adrift when Vincenzo demolished Zarlino's theory. The suggestion had been made previously, but with little in the way of corroborating physical evidence. For Galileo, however, the analogy was compelling, even if he realized that it did not constitute unequivocal proof. What he lacked was either a direct way of counting the vibrations per unit of time of each of several lute strings tuned to identifiable notes or a way of systematically diluting the phenomena comparable to what he would achieve by rolling balls on inclined planes.

He did, however, continue to look for evidence linking the rate of vibration to pitch. Even his long attempt to derive isochronism mathematically from the general theorem describing natural motion was part of the search. Had he been able to provide such a rational link, he would have confirmed his conviction about isochronism and presumably strengthened the analogy he saw with vibrating strings. But ultimately the most convincing evidence he could offer, if not even entirely convincing to himself, came from yet another exploration.

22 Thomas Settle Galileo's Experimental Research

BICCHIERI CANTANTI

In a brief section of the Prima Giornata of the Discorsi [GG: VIII, 141-143] Galileo began a reflection on the nature of resonant phenomena by supposing that when the cord of a zither is made to vibrate, and thereby emit a tone, it also sets the air around it vibrating. These tremors or pulses in turn propagate into the space around the original source. Then if the waves or pulses strike a second object, disposed to vibrate in harmony with the first, they will begin to set that object vibrating, reinforcing the effect with each pulse and eventually causing it to produce its own audible tone. One can even make a clean, well-made goblet resonate by placing it near a sounding viola cord tuned to the goblet's natural pitch.

That a vibrating source does provoke tremors and waves in a surrounding medium can be shown with the same goblet. If we put some water in it and then rub its rim with the end of a finger, it will produce a clear tone with a definite pitch. It will also produce a pattern of wavelets on the surface of the water. And if we put the goblet into a large container filled with water almost up to the rim of the goblet and again rub its rim, we will see similar patterns of waves radiating out over the surface of the water and away from the glass. For Galileo, then, sounds in general, including musical tones, are produced by vibrating bodies and are propagated in the form of pulses or vibrations in the surrounding medium. But what was the relation between the rate of vibration of a lute string, say, and the perceived pitch of the resulting tone?

In the last portion of this section Galileo describes an effect he had produced many times. Every so often, while thus sounding a tone with a fairly large goblet almost filled with water, the pitch would jump a full octave while simultaneously the wavelets divided into two. For him this showed that the "form of the octave was the double", the inference apparently being that doubling the number of wavelets in the same space meant that the frequency of the vibrations had also doubled. Hence the ratio of two-to-one did apply to the octave after all, referring not to the senario of Zarlino, but instead to the rates of vibration of the sources of the tones.

Galileo quite probably recognized that the argument might not be entirely convincing. The example was part of a cumulation of the results of empirical investigations concerning pendulums, time-keepers, and musical acoustics which he never succeeded in arranging into a demonstrative, rationally interconnected science. Yet he must also have felt that these results could stand on their own and were worth reporting. And with one exception the results are confirmed in our own simulations of Galileo's work and do provide a sense of the effort Galileo expended while trying to give substance to his basic suppositions concerning natural oscillators.

23 Thomas Settle Galileo's Experimental Research

One can, of course, quite easily make a goblet, a "footed glass", emit a tone by rubbing a damp finger around its rim. The phenomenon is well known in our own day and apparently was also known in the 16th century; Vincenzo Galilei mentioned it twice in his writings, so we can guess how Galileo learned of it [Excerpts I & II]. With a little practice one "learns" the right pressure to apply and the right speed of motion. And one senses that both the rim of the glass and the finger itself have to be "warmed up". This probably means that the skin of the finger has to absorb a certain amount of water, has to achieve a certain degree of "morbidity", to phrase the requirement in a pseudo-medieval way. Given all this, practically any stemmed glass can be made to "sing". The pitch emitted will vary according to the height of the water in the glass; as the glass is filled the pitch will go down (possibly contrary to the a priori expectations of many). And there will be patterns of waves on the surface of the water, increasingly evident as the level of the water nears the rim.

One can also quite easily make the emitted tone jump to a higher pitch, even to more than one higher pitch. After achieving some familiarity with the basic process of rubbing the finger around the rim, one can pass to other modes of rubbing, for instance moving the finger back and forth along the circumference of the rim, more or less in one spot, or "bowing" the finger radially across the rim. In both cases one of the effects is to make the wavelets more stationary and observable. Another is to make it easier to elicit one or more of the higher tones. These are not necessarily an octave or multiples of an octave above the original; in fact, in most cases they are not. After experimenting with a multitude of glasses, however, we found that those having rims with a relatively small diameter yielded jumps of greater than an octave, while those with large diameters yielded jumps of less than an octave. It seemed reasonable to suppose, therefore, that we could find a goblet with a diameter which would yield an octave exactly, and this turned out to be the case. An ordinary trattoria wine glass with a opening of about 6.6 cm proved to give very good results both to the ears of musicians and to electronic sound analyzers. Evidently, Galileo worded his description carefully. He did not claim that all goblets yielded the octave, only that "un bicchiere assai grande", a particular glass, presumably of a determinate size.

What we have not been able to reproduce, thus far, is his doubling of the wavelets. In fact, making any sense of the patterns of the wavelets is difficult at best, and I have no absolute confidence that we are yet seeing what Galileo sought to describe. One's first impression, when rubbing one's finger around the rim of the glass, is of a general chaos of waves, rotating around the surface of the water and keeping pace with the finger. Then one perceives one or more concentrations of waves, which move as the finger moves. Unfortunately, to maintain the sounding tone one has to maintain a minimum speed of movement, and the concentrations of wavelets rotate too quickly to be examined with any precision. It is only when we change to the

24 Thomas Settle Galileo's Experimental Research alternative modes of rubbing and the waves cease to rotate that we begin to see what is happening. When we rub the rim at a single place in such a way as to elicit the base tone, we find four clusters of disturbances on the surface of the water, each one emanating from one of four equally distributed spots around the inside surface of the goblet. The curved narrow strands of the wavelets themselves splay out from those spots, and they seem too numerous and agitated to count. So one has to suppose, at least for the present, that Galileo was actually referring to the clusters of waves and not the wavelets themselves. Is it these clusters, then, that divide as the pitch of the tone jumps?

Here there is no single or easy answer. When using a finger to elicit both the base tone and the higher pitch, the results can differ from goblet to goblet. For some that we tested, including those jumping to the octave, the higher tone was accompanied by no wavelets at all and only minimal signs of disturbance very near the surface of the glass, so minimal as to preclude systematic description. On the other hand, some of the "non-octave" goblets did yield concentrations of wavelets. When the tone jumped to the higher pitch, the clusters or concentrations jumped from four to six, again evenly distributed around the inner surface of the glass. The differences could be just in the shapes of the goblets themselves. One observes, however, that when passing from a base tone to one of the higher pitches, the loudness of the tone noticeably diminishes. This might indicate that there is less energy resident in the vibrating glass and, in some cases, insufficient energy available to drive disturbances on the surface of the water.

Some further, if not final, clarification came from another line of investigation. We had wanted to measure the frequencies of the tones being emitted by the several goblets and began using an audio-analyzer which produced calibrated sine waves on an oscilloscope. This proved entirely possible but tedious when using the finger method of exciting the goblets. So we passed to using a variable frequency tone-generator with a suitable audio output, a small loudspeaker. By holding the speaker close to the goblet and varying the output from a few thousand to a few hundred hertz, we found that each goblet system (goblet plus water at a certain level) resonated at several sharply defined frequencies. As the tone was brought down from a high value, the surface of the water would become agitated at those frequencies and the glass itself could be felt to be vibrating to the touch. The frequencies of the resonant points corresponded to those elicited by finger rubbing, a further confirmation, if one needed it, of the "naturalness" of exciting more than one tone from a wine glass.

Meanwhile, the electronic equipment also gave us a way of investigating the dividing of the wavelets or their clusters. We not only had control over the frequency of the output of the audio driver; we also could vary the power or energy delivered to each goblet system. In all the cases

25 Thomas Settle Galileo's Experimental Research tried, the lowest resonant frequency (corresponding to the base tone elicited by finger rubbing) generated the four clusters of wavelets described above. At the next higher frequency all goblet systems, including those which resonated at an octave above the base level, displayed the six clusters of wavelets also mentioned above. In other words, if Galileo had meant to refer to these clusters as the entities that divided into two, he was probably mistaken. Supposing that he had found a goblet that both jumped to the octave and showed the clusters of waves dividing, those clusters probably went from four to six and not from four to eight.

For the present, our search for what Galileo may have seen rests here. In fact, we were able to make those clusters of waves go from four to eight, but only by using the audio source at a fairly high power to drive one of the yet higher resonant points of the goblets. While Galileo could easily have momentarily elicited the higher resonances with a finger, it is unlikely that he could have done so in such a way as to see and study the eight clusters. On the other hand, it could also be that we have not yet found the right goblet or the right combination of experimental circumstances to produce the doubling of the waves along with the octave jump.

What is impressive, none the less, is the amount of effort Galileo must have expended in his original explorations of these phenomena. Those few, condensed lines regarding the wavelets in the "ringing goblets" did not come automatically or easily. Galileo had no established fund of theoretical knowledge to guide him; he was himself beginning the process of laying some of the foundations for such a fund. At some point he decided to look more closely at the known but otherwise banal mode of producing tones by rubbing the rims of goblets, perhaps first seeing them as another source of controllable musical sounds. Then came the "discoveries" of the ondeggiatura (wavelets) and the tones higher than the base level. To diagnose and make sense out of all this, with what he had available to him both empirically and theoretically, required persistent work, glasses of many sizes and shapes, varied experimental conditions and, presumably, an extended period of time. He would have hoped to find good evidence for his supposition about the relationship between frequency of vibration and pitch. This would have reinforced a sense of the central importance of a weight hanging on the end of a wire as a source for two types of natural oscillator: the swinging pendulum and the vibrating string. Eventually he would have wanted to cover both in a comprehensive science of natural motion. Unfortunately nature proved uncooperative: among other things, simple pendulums are not precisely isochronous. But it was surely with swinging weights, vibrating strings and singing goblets that Galileo began his career in experimental research.

26 Thomas Settle Galileo's Experimental Research

CONCLUDING REMARKS

I began this essay by asking what sort of an experimenter Galileo was. The obvious short answer is: a very capable one, and from very early on. By now, however, it ought also to be obvious that the question has a longer and much more complex answer, even if we are only beginning to see some of its basic features.

It is important to realize that, for Galileo, experimentation was not a matter of constructing devices to illustrate truths derived from or accepted on rational grounds alone. His text in the "Third Day" of the Discorsi, leading to and including the description of the inclined plane, is more akin to a set of notes for a modern, introductory physics lecture than to a daily diary of laboratory work. It is not an account of actual research so much as a theoretical discussion of results already established, followed by a set-piece, empirical example of the main thesis. Galileo, ever the articulate expositor, wanted the reader/student to understand the central theory, including the terms which he used in expressing it, and then to have a simple way of checking it empirically. He had no primary interest in letting us know about the research, experimental or otherwise, that had led him to his conclusions in the first place.

That story began, as we have seen, with Galileo's apprenticeship in experimental physics under the tutelage of his father, Vincenzo, and followed no single path. Their attempts to elucidate the bases of musical harmonics was accompanied by Galileo's own experimental discovery and initial justification of the isochronism of the simple pendulum. From this beginning Galileo continued searching for empirical evidence to justify a hunch concerning the relationship between frequency of vibration and pitch. He also continued to look for and find other natural oscillators, including vibrating goblets and perturbed basins of water, both as a means of reinforcing what he had already learned about pendulums and as a means of "reducing" those other phenomena to the pendulum model.

Thus, when Galileo began to investigate the free fall of heavy bodies, he had already developed the habit of empirical research. Not satisfied with just proposing a new theory, he searched for the physical evidence to exemplify that theory, in this case looking for a characteristic uniform motion of free fall. He could not find it, but for a while he "discovered" that light bodies fall ahead of heavy ones in free motion. Then, because the pendulum had become a familiar instrument, he could use it both to investigate the draining of impetus and to provide a sort of diluted vertical fall. Moving to work on the inclined plane, he eventually used it to calibrate his precise water-timer, and this led him to the discovery of a new form of natural motion, uniform natural acceleration.

27 Thomas Settle Galileo's Experimental Research

Having thus clarified natural linear acceleration, Galileo wanted to extend his new science to definable types of non-linear motion. This was relatively easy with regard to projectile motion (among other things, he could show that it was parabolic), but proved intractable with regard to motion on the arc of a circle, i.e., the motion of a pendulum. His theoretical strategy was to try to construct a mathematical argument for isochronism. Empirically, he compared motions on straight inclined planes with motions on planes with circular curvatures: the pendulum in a box. The fact that pendulums are not precisely isochronous and that these experiments led to no satisfying conclusion has meant that historians have paid relatively little attention to this side of Galileo's researches; but clearly the properties of natural oscillators, pendulum-like devices, were among the major preoccupations of his career.

For Galileo, then, experimentation was in large measure exploratory in nature, a venturing into little known territory, trying to distinguish the "true" fact from the adventitious, and then revising old maps or drawing entirely new ones. Occasionally some promising natural curiosity caught his eye and he decided to pursue it. Often he started from a felt need to find a material exemplification either of his own ideas or of a previous theory. He could and did make mistakes; light-before-heavy and isochronism are examples. Some he corrected himself; some had to wait for others. But when his work led him to a convincing new "fact", he would try to incorporate into a demonstrative science. In the case of free fall, his exploratory search for uniform motion led to the inclined plane and then to the discovery of uniform acceleration. In the subsequent exposition, the inclined plane was transformed from being the immediate source of the law of free fall to being merely an illustration of it.

But Galileo's interest in the physics of motion was hardly a single-minded pursuit of the law of free fall. He studied many types of natural motion and their effects, and that law was only a small part of a much more encompassing science of motions which he had hoped to achieve. Within that larger ambit he borrowed results and instruments from one line of research to inform and aid his investigations in an another. The connections among all those components constitute an evolving network of research, both experimental and theoretical. The strength of his conclusions at any stage rested not on a single empirical example, but on the coherence of the entire net. And the coherence of the net rested on Galileo's own highly developed experimental capacities.

28 Thomas Settle Galileo's Experimental Research

SELECTED EXCERPTS

I.VG (1581): 133 imperoche il corpo co(n)cavo nel quale fusse maggiore quantità d'acqua, farebbe il suono piu grave di quello dove ne fusse meno, non altramente di quello che tutto il giorno occorre a putti: i quali mettendo un poco d'acqua in un bicchiere di vetro, di debita & accomodata proporzione, & ancora senz'acqua; & bagnandosi la sommità del dito di mezzo della destra, la vanno dolcemente in giro movendo sopra l'orlo della bocca, mentre che co(n) la sinistra mano tengano di esso bicchiere il piede, acciò stia diritto; dal quale stro picciamento esce un suavissimo & sonoro suono, simile à quello d'una corda di Viola secata dal l'arco: & quanto maggiormente si và augumentando la quantità dell'acqua, & la forza nel dito che sopra la supeerficie dell'orlo della bocca del bicchiere và in giro caminando; tanto piu si fa il suono grave & maggiore. co mezzo de quali strumenti, ò simile, si potrebbe appresso le fontane accomodare senz'alcuna spesa & poca difficultà, da udirsi un perpetuo concento.

II.VG (1589/90): f. 54v; Palisca (1989): 196

Concludesi adunque esser gl'interualli musici compresi dalla quantita discreta et non dalla continoua in modo ueruno; per quanto pero cene daua il senso dell'udito, del quale non habbiano giudice di lui piu uertierio; dal fine che de suoni musici desideriamo et dalla quantità continoua possiamo hauere maniera alcuna di modulare dal graue all'acuto, et dall'acuto al graue, non cela puo dare altro mezzo che tirare et l'allantare della corda senza tastarla; o dal uaso di uetro con l'accrescergli et con lo scemargli l'acqua girando attorno col dito sopra la superficie dell'orlo di esso uaso o bichiere che dire lo uogliamo. nelle quali maniere ancora ne fa l'udito per la sua imperfettione il giuditio medesimo che del tastare la corda nel manico della uiola si è detto.

III.GG: VIII, 141-143

Salv. Essempio che dichiara 'l mio intento non meno acconciamente di quel che questa mia premessa si accomodi a render la ragione del maraviglioso problema della corda della cetera o del cimbalo, che muove a fa realmente sonare quella non solo che all'unisono gli è concorde, ma anco all'ottava e alla quinta. Toccata, la corda comincia e continua le sue vibrazioni per tutto

29 Thomas Settle Galileo's Experimental Research

'l tempo che si sente durar la sua risonanza: queste vibrazioni fanno vibrare e tremare l'aria che gli è appresso, i cui tremori e increspamenti si distendono per grande spazio e vanno a urtare in tutte le corde del medesimo strumento, ed anco di altri vicini: la corda che è tesa all'unisono con la tocca, essendo disposta a far le sue vibrazioni sotto 'l medesimo tempo, comincia al primo impulso a muoversi un poco; e sopraggiugnendogli il secondo, il terzo, il ventesimo e pi altri, e tutti ne gli aggiustati e periodici tempi, riceve finalmente il medesimo tremore che la prima tocca, e si vede chiarissimamente andar dilatando le sue vibrazioni giusto allo spazio della sua motrice. Quest'ondeggiamento che si va distendendo per l'aria, muove e fa vibrare non solamente le corde, ma qualsivoglia altro corpo disposto a tremare e vibrarsi sotto quel tempo della tremante corda; sì che se si ficcheranno nelle sponde dello strumento diversi pezzetti di setole o di altre materie flessibili, si vedrà, nel sonare il cimbalo, tremare or questo or quel corpuscolo, secondo che verrà toccata quella corda le cui vibrazioni van sotto 'l medesimo tempo: gli altri non si muoveranno al suono di questa corda, nè quello tremerà al suono d'altra corda. Se con l'archetto si toccherà gagliardamente una corda grossa d'una viola, appressandogli un bicchiere di vetro sottile e pulito, quando il tuono della corda sia all'unisono del tuono del bicchiere, questo tremerà e sensatemente risonerà. Il diffondersi poi amplamente l'increspamento del mezzo intorno al corpo risonante, apertamente si vede nel far sonare il bicchiere, dentro 'l quale sia dell'acqua, fregando il polpastrello del dito sopra l'orlo; imperò che l'acqua contenuta con regolatissimo ordine si vede andar ondeggiando: e meglio ancora si vedrà l'istesso effetto fermando il piede del bicchiere nel fondo di qualche vaso assai largo, nel quale sia dell'acqua sin presso all'orlo del bicchiere; chè parimente, facendolo risonare con la confricazione del dito, si vedranno gl'increspamenti nell'acqua regolatissimi, e con gran velocità spargersi in gran distanza intorno al bicchiere: ed io più volte mi sono incontrato, nel fare al modo detto sonare un bicchiere assai grande e quasi pieno d'acqua, a veder prima le onde nell'acqua con estrema egualità formate, ed accadendo tal volta che 'l tuono dil bicchiere salti un'ottava più alto, nell'istesso momento ho visto ciascheduna delle dette onde dividersi in due; accidente che molto chiaramente conclude, la forma dell'ottava esser la dupla.

IV.VV (mss): 303

Questa del pendolo si è una delle più antiche invenzioni e scoperte in natura del Galileo, e fu circa l'anno 1580, quando era studente a Pisa, nel trovarsi egli un giorno in quel Duomo, dove si abbatté di vedere, lasciata in moto, una lampada pendente da una lunghissima corda. E, come quello che da giovanetto s'era anche esercitato nella Musica, sotto la disciplina di quel gran Vincenzio suo Padre, che si dottamente scrisse poi in Dialogo della Musica antica e moderna; percioch aveva impressa nell'anima l'egualità de'tempi, co'quali essa si regola, riflettendo a

30 Thomas Settle Galileo's Experimental Research quel moto, gli fu facile il giudicarlo in mente sua equitemporaneo, si nelle andate lunghe e larghe al principio del moto, come nelle strette sul fine verso la quiete. In casa poi se ne chiari in più modi replicate esperienze esattissime, trovando, coll'aiuto de'suoi compagni, che in un determinato numero di vibrazioni d'un certo pendolo, lasciato andar sempre da una distanza medesima del perpendicolo, quante ne faceva un altro pendolo delle larghe, altrettante in ciascuno ne faceva delle strette e delle strettissime. Che se il numero di queste eccedeva di qualcosa il numero di quelle, il che però si fa visibile solamente dopo un numero grandissimo delle une e delle altre, attribuiva questa piccola maggioranza al minore ostacolo, che arreca l'aria al mobile più tardo, qual'é quello del grave pendolo nel passar gli archi più piccoli, che al mobile più veloce, qual'è il medesimo nel passar gli archi grandi.

V.VV (1659): 648-649

Trovavasi il Galileo, in età di venti anni in circa, intorno all'anno 1583 nella citt di Pisa, dove per consiglio del padre s'era applicato alli studi della filosofia e della medicina; et essendo un giorno nel Duomo di quella città, come curioso ed accortissimo che egli era, caddegli in mente d'osservare dal moto d'una lampana, che era stata allontanata dal perpendicolo, se per avventura i tempi delle andate e tornate di quella, tanto per gli archi grandi che per i mediocri e per i minimi, fossero uguali, parendogli che il tempo per la maggior lunghezza dell'arco grande potesse forse restar contraccambiato dalla maggior velocità con che per esso vedeva muovere la lampana, come per linea nelle parti superiori più declive. Sovvennegli dunque, mentre questa andava quietamente movendosi, di far di quelle andate a tornate un esamine, come suol dirsi, alla grossa per mezzo delle battute del proprio polso e con l'aiuto ancora del tempo della musica, nella quale egli già con gran profitto erasi esercitato; e per allora da questi tali riscontri parvegli non aver falsamente creduto dell'ugualità di quei tempi. Ma non contento di ciò, tornato a casa pensò, per meglio accertarsene, di così fare.

Legò due palle di piombo con fili d'egualissime lunghezze, e da gli estremi di questi le fermò pendenti in modo, che potessero liberamente dondolare per l'aria (che per ciò chiamò poi tali strumenti dondoli o pendoli); e discostndole dal perpendicolo per differenti numeri di gradi, come, per essempio, l'una per 30, l'altra per 10, lasciolle poi in libertà in istesso momento di tempo: e con l'aiuto d'un compagno osservò che quando l'una per gl'archi grandi faceva un tal numero di vibrazioni, l'altra per gl'archi piccoli ne faceva appunto altrettante.

In oltre formò due simili pendoli, ma tra loro di assai differenti lunghezze; ed osservò che notando del piccolo un numero di vibrazioni, come, per essempio, 300, per i suoi archi maggiori, nel medesimo tempo il grande faceva sempre un tal istesso numero, come è a dire 40,

31 Thomas Settle Galileo's Experimental Research tanto per i suoi archi maggiori che per i piccolissimi: e replicato questo pi volte, e trovato per tutti gl'archi et in tutti i numeri sempre rispondere l'osservazioni, ne inferì ugualissima esser la durazione tra l'andate e le tornate d'un medesimo pendolo, gradissime o picolissime che elle fossero, o non iscorgersi almeno tra loro sensibile differenza, e da attribuirsi all'impedimento dell'aria, che fa più contrasto al grave mobile più veloce che al meno.

S'accorse ancora, che nè le differenti gravità assolute, nè le varie gravità in ispecie delle palle, facevano tra di lor manifeste alterazioni, ma tutte, purchè appese a fili d'uguali lunghezze i punti delle sospensioni a i lor centri, conservavano una assai costante ugualità de'lor passaggi per tutti gl'archi; se però non si fusse eletta materia leggerissima, come è il sughero, il di cui moto dal mezzo dell'aria (che al moto di tutti i gravi sempre contrasta, e con maggior proporzione a quello de'più leggieri) vien più facilmente impedito, e più presto ridotto alla quiete.

Assicuratosi dunque il Galileo di così mirabile effetto, sovvennegli per allora d'applicarlo ad uso della medicina per la misura dell'accelerazioni de'polsi, come pur tuttavia communemente si pratica.

VI.GG: I, 267, n. 1; Drabkin: 31-32, n. 12

Affinché dunque possiamo trovare questa proporzione, occorre pervenire alla causa dell'accelerazione e della decelerazione del moto. E, tanto più valido sarà anche la causa, tanto più valido sarà anche l'effetto: cosicchè, come dalla maggior gravit proverrà un maggior moto, cio più veloce; cos invece, da una minore gravità, un moto più lento. Diciamo che i mobili nei movimenti mantengono il rapporto che vi è fra le loro gravità: finchè però vengano pesati in quell stesso stesso mezzo nel quale deve avvenire il moto. Bisogna infatti fare attenzione a ciò: che per caso qualcuno non pesi in aria due palle, l'una di ferro e l'altra di legno, trovando che quella di ferro è dieci volte più grave di quella di legno; quindi, lasciasse cadere le palle in acqua, e quella di ferro si mettesse a scendere, mentre quella di legno non scendesse affatto; cosicchè il ferro si troverebbe a non mantenere nel moto nessuna proporzione con il legno.

Occorre dunque che i due mobili vengano pesati nello stesso mezzo nel quale deve avvenire il moto. Due gravi infatti non mantengono il medesimo rapporto di gravit quando si trovino indue diversi mezzi. Ma proprio qui sorge una grossissima difficoltà. Mediante l'esperienza vediamo infatti che di due palle di egual mole, delle quali l'una sia il doppio più grave dell'altra, una volta che esse vengano entrambe gettate da una torre, quella più grave non giunge a terra due volta più rapidamente. Anzi, quella più leggera all'inizio del moto precede quella più grave: quindi, per un certo tratto, si sposta con velocità maggiore.

32 Thomas Settle Galileo's Experimental Research

Questa è in effetti una obiezione, ed anche grossissima. Poichè tuttavia essa dipende da alcuni motivi non ancora spiegati, ci riserviamo di affrontarla quando verrà indicata la causa dell'accelerazione del moto naturale: laddove si dimostrerà che solo accidentalmente avviene che il moto naturale sia più lento all'inizio, e risulterà altresi che, altrettanto accidentalmente, il mobile che è il doppio più grave non scende dalla torre due volte più in fretta. E quindi si potrà anche spiegare la cause per cui, al principio del moto, il mobile più leggero si sposti più rapidamente di quello più grave. Per ora basti quanto è stato detto sulla proporzione di mobili diversi nello stesso mezzo. [Tr. L. Olivieri, III.1994]

VII.Giacomelli: 43; GG: I, 273; Drabkin: 37-38

Ma è da avvertire che qui sorge una grave difficoltà: perchè queste proporzioni, da chi ne faccia l'esperienza, si troverà che non si verificano. Infatti, prendendo due diversi mobili, che, soddisfino alle condizioni perchè l'uno debba discendere il doppio più presto dall'altro, se si lascino cadere da una torre, il primo non raggiunge certo la terra più velocemente, il doppio più presto, del secondo: che, anzi, osservando bene, quello che è più leggero all'inizio del moto precederà quello più pesante e risulterà più veloce. La quali divergenze ed anomalie, in certo modo straordinarie, donde abbiano accidentalmente origine (perchè se tratta di cose che avvengono per accidens) non è qui il luogo d'indagare, donendo prima vedere alcuni fatti che non sono stati ancora presi in esame. Occorre infatti vedere prima perchè il moto naturale sia più lento all'inizio.

VIII.Giacomelli: 56-57; GG: I, 302; Drabkin: 69

E` da avvertire come sopra si è detto per il moto verticale, che anche in questi moti sui piani accade che le proporzioni, da noi stabilite, non si verifichino, sia per le cause ora allegate (le resistenze esterne), sia anche perchè, per effetto accidentale avviene che un mobile più leggero, all'inizio del suo movimento, discenda più rapidamente di uno più grave; la ragione di che verrà dichiarata a suo luogo; giacchè la questione dipende da quella nella quale si ricerca perchè nel moto di caduta la velocità si venga intensificando.

33 Thomas Settle Galileo's Experimental Research

IX.Giacomelli: 79; GG: I, 329; Drabkin: 101

... nel qual tempo, portandosi rapidamente, discendono per grande spazio e, poichè di grandi spazi, da cui far cadere i gravi, noi non possiamo disporre, non sarà da meravigliarsi se una pietra abbandonata dalla semplice altezza di una torre, si veda accelerarsi fino a terra; dato che tale breve spazio non basta a disperdere tutta la virtù contraria.

X.Giacomelli: 81; GG: I, 334; Drabkin: 107

... l'esperienza mostra appunto il contrario. In quanto che, è bensi vero che il legno al principio del suo moto, si porta in basso più velocemente del piombo; ma pure, poco dopo, il moto del piombo si accelera tanto, da lasciarsi dietro il legno e, se lasciati cadere da un'alta torre, per grande spazio lo precede: e di ciò spesso feci esperienza. Una più seria causa da più serie ipotesi dobbiamo quindi tentare di trarre.

XI.Brunetti: Vol. 2, 727; GG: VIII, 197

E in primo luogo conviene investigare a spiegare la definizione che corrisponde esattamente al moto accelerato di cui si serve la natura. Infatti, sebbene sia lecito immaginare arbitrariamente qualche forma di moto a contemplare le proprietà che ne conseguono (cos , infatti, coloro che si immaginarono linee spirali o concoidi, originaate da certi movimenti, ne hanno lodevolmente dimostrate le proprietà argomentando ex suppositione, anche se di tali movimenti non usi la natura), tuttavia, dal momento che la natura si serve di una certa forma di accelerazione nei gravi discendenti, abbiamo stabilito di studiarne le proprietà, posto che la definizione che daremo del nostro moto accelerato abbia a corrispondere con l'essenza del moto naturalmente accelerato. Questa coincidenza crediamo di averla raggiunta finalmente, dopo lunghe riflessioni; soprattutto per il fatto che le proprietà, da noi successivamente dimostrate [dalla nostra definizione], sembrano esattamente corrispondere e concidere con ciò che gli esperimenti naturali presentano ai sensi.

34 Thomas Settle Galileo's Experimental Research

References

ORIGINAL SOURCES

Brunetti: Opere di Galileo Galilei, a cura di Franz Brunetti, 2 Voll, UTET, 1980

Caverni: Raffaello Caverni, Storia de Metodo sperimentale in Italia, 6 Voll., 1891-1900

Drabkin: "On Motion" in I.E. Drabkin & S. Drake, tr. & ed., Galileo Galilei: On Motion and On Mechanics, U. Wisconsin Pr., 1960

GG: Le Opera di Galileo Galilei: Edizione Nazionale, a cura di Antonio Favaro, 20 Voll.

Giacomelli: Raffaele Giacomelli, Galileo Galilei giovane e il suo "De motu", Pisa, Domus Galilaeana, 1949

Palisca (1989): Claude V. Palisca, The Florentine Camerata: Documentary Studies and Translations, Yale U. P.

VG (1581): Vincenzo Galilei, Dialogo della musica antica et della moderna, Firenze

VG (1589/90): Vincenzo Galilei, Discorso particolare intorno alla diversita delle forme del diapason, BNCF, Mss. Gaileiana 3: ff. 44r-54v; trascrizione e traduzione in inglese, Palisca (1989): 180-197

VV (mss): Vincenzo Viviani, BNCF, Mss. Galileiana 227, c. 60r; trascrizione Raffaello Caverni, Caverni: Vol. 1, 1891, 303 VV (1659): Vincenzo Viviani, Lettera ... al Principe Leopoldo ... intorno all'applicazione del pendolo all'orologio, GG: XIX, 647-659

SUBJECT REFERENCES

Acustica musicale; bicchiere; monocordo; Vincenzo Galilei VG (1581): 133 VG (1589/90): f. 54v; Palisca (1989): 196

Musica e pendoli: Vincenzo Viviani Mss, VV (mss): Lettera ... a Principe Leopoldo ..., VV (1659): 648-649

Bicchieri; Galileo Galilei Discorsi, Giornata terza, GG: VIII, 142-143

La caduta dei gravi; levitas, gravitas, ecc. "De motu antiquiora", GG: I, 243-417; Drabkin; Giacomelli

La caduta dei gravi Lettera a P. Sarpi, 1604.X.16, GG: X, 115-116

Piano inclinato; cronometro ad acqua Discorsi, Giornata terza, GG: VIII, 212-213

Pendoli [Oscillatori naturali] Lettera a G. del Monte, 1602.XI.29, GG: X, 97-100 Discorsi, Giornata prima, GG: VIII, 128-131, 138-50

35 Thomas Settle Galileo's Experimental Research

Pendolo interotto Discorsi, Giornata terza, GG: VIII, 205-7

Piani "curvi" [Oscillatori naturali] Lettera a G. del Monte, 1602.XI.29, GG: X, 97-100

Forza della percossa "Le meccaniche" (ca. 1600), GG: II, 188-191 Discorsi, Giornata sesta, GG: VIII, 323-325

SECONDARY SOURCES

Coelho, V., ed., 1992, Music and Science in the Age of Galileo, (University of Western Ontario series in philosophy science, v. 51) Kluwer, Dordrecht-Boston

Cohen, H. F., 1984, Quantifying Music. The Science of Music at the First Stage of the Scientific Revolution, 1580-1650, Reidel

Cohen, H. F., 1985, "Music as a Test Case", Studies in History and Philosophy of Science, 16, 351-378

Drake, S., 1970, "Renaissance Music and Experimental Science", Journal of the History of Ideas, Vol. 31, No. 4, 483-500

Drake, S., 1978, "Music and Philosophy in Early Modern Science", in Coelho, V., 1992, 3-16

Fredette, R., 1972, "Galileo's De motu antiquiora", Physis, 14, 321-348

Gozza, P., a cura di, 1989, La musica nella rivoluzione scientifica del seicento, Il Mulino: "Introduzione", esp. 13-22; 35-45; C. V. Palisca, "Empirismo scientifico nel pensiero musicale", 167-177 (tr. parziale del Palisca, 1961); "La teoria corpuscolare del suono: Isaac Beeckman", 197-206 (tr. parziale del Cohen, Quantifying Music); "Indicazioni bibliografiche", 267-273

Koyré, A., 1939, Etudes galiléennes, Paris; 1976, Studi galileiani, a cura di M. Torrini, Torino

Koyré, A., 1943, "Galileo and Plato", in Journal of the History of Ideas, Vol. 4, 400-428

Mach, E., 1960, The Science of Mechanics: A Critical and Historical Account of Its Development, Open Court: 398-402

Moyer, Ann E., 1992, Musica Scientia, Musical Scholarship in the Italian Renaissance, Cornell U. P.

Palisca, C. V., 1961, "Scientific Empiricism in Musical Thought", in H. H. Rhys, ed., Seventeenth Century Science and the Arts, Princeton, 91-137 (v: 120-137)

Palisca, C. V., 1989, The Florentine Camerata: Documentary Studies and Translations, Yale U. P.

Palisca, C. V., 1992, "Was Galileo's Father an Experimental Scientist?", in Coelho, V., 1992, 143-151

Schmitt, C. B., 1969, "Experience and Experiment: A Comparison of Zabarella's View with Galileo's in De motu", Studies in the Renaissance, Vol. 16, 80-138

Settle, T. B., 1961, "An Experiment in the History of Science", Science, Vol. 133, No. 3445, 19-23

Settle, T. B., 1967, "Galileo's Use of Experiment as a Tool of Investigation", in E. McMullin, ed., Galileo: Man of Science, Basic Books, 315-337

36 Thomas Settle Galileo's Experimental Research

Settle, T. B., 1971, "Ostilio Ricci, a Bridge Between Alberti and Galileo", in Actes, XIIe Congres International d'Historire des Sciences, Tome IIIB, 121-126

Settle, T. B., 1983, "Galileo and Early Experiment", in R. Aris, et al., Springs of Scientific Creativity, U. Minnesota Pr., 3-20

Strong, E. W., 1936, Prodecures and Metaphysics. A Study in the Philosophy of Mathematical-Physical Science in the Sixteenth and Seventeenth Centuries, Berkeley

Walker, D. P.,1978, "Vincenzo Galilei and Zarlino" and "Galileo Galilei", Studies in Musical Science in the Late Renaissance (Studies of the Warburg Institute, Vol 37), Leiden, E.J. Brill, 14-33

37

THE PENDULUM AND GALILEO

CONJECTURES AND CONSTRUCTIONS*

This outline has been drafted to serve as a basis for a discussion of some aspects of the "problem of induction". Though in general the opinions expressed here represent my current [1965] understanding of one phase of Galileo's work, I am sure that several will require a great deal more consideration. Here I have tried to draw sharp images in order to present issues and provoke discussion.

A. SETTING THE STAGE.

During his student days at Pisa Galileo attended the city's cathedral quite regularly. One day, while only half listening to a long choral mass, he noticed a lamp gently swinging in one of the aisles. Evidently one of the suspending links had needed an oiling; when the lamp paused momentarily before each new descent it uttered a rusty creak, and it was this that had attracted Galileo. As he followed the motion he realized suddenly that its rhythm coincided with that of the music. "Must be an accident of the particular arc," he mused, "as the swing diminishes the beat will probably slow also." His eyes wandered to the directing baton of the choir master. Galileo had had a thorough musical training from his father, and he knew that the choir master's timing was good. Right then the lamp's noise was punctuating every eighth beat of the baton. He glanced back to the lamp. Its arc had decayed to half its former extent. "Strange, it's still keeping the rhythm. But it should be out of synchronism by now." On reflecting Galileo was not quite sure whether its beat should have slowed or speeded. While he watched, with more concentration now, the arc continued to diminish, yet the beat remained in time with the music.

"Perhaps the musical time actually has been changing." He put his fingers to his pulse and counted. The lamp's creak was barely audible now, its arc was small, almost a mere waggle. But it did seem as if the number of pulse beats for each swing was remaining constant. While

* Up to now unpublished paper from 1965. Thomas Settle The pendulum and Galileo he knew that this could not be a conclusive test, it did seem an odd sort of thing. As he puzzled he could not even recall a mention of such a phenomenon in his many lecture notes or books on the various properties of motion.

A few evenings later he was home trying to interest himself in his Galen notes. There was a heavy lamp in his room, hung by a flexible cord from the ceiling. Occasionally, when bored or otherwise uninspired, he would set it swinging and watch the shadows on the wall, letting his mind drift in speculation. On this particular night he set it in motion and, as he watched the shadow of a bedpost arc back and forth, his thoughts were brought back to the peculiar motion of the cathedral lamp. "I wonder if it really did keep the same beat the whole time ... ."

B. TWO PROBLEMS.

B 1) Galileo's: The situation presented in A is a fanciful reconstruction, but it could represent schematically the point of departure for Galileo's investigations of the properties of the pendulum. At this stage he would not have been searching for a "proof of the isochronism of the pendulum"; he would not have been able to formulate a proposal to that effect, even if his life had depended, say, on receiving an N.S.F. grant. (Nor could he say, in another context, "I am trying to discover the law of inertia".) He only knew of a few instances of a peculiar phenomenon which seemed to have interesting characteristics; he wished to understand them; he would first have to establish the phenomenon as a genuine one. Traditional natural philosophy offered little help.

B 2) Our's: Despite the apparent sterility of the initiating circumstances, by the time Galileo had published his last and major work, the Discorsi, he had satisfied himself that: B 2 a) For any given simple pendulum the period is a definite and fixed interval of time irrespective of the amplitude of the swing. B 2 b) Similarly, the period is independent of the weight and density of the bob within wide limits, and the limits are defined by reasonable experimental factors such as the relative effect of air resistance, etc. B 2 c) For pendulums of different lengths, their periods are in the same ratio as the square roots of their lengths. Simply stated, our problem is to understand how he arrived at this knowledge and how he would have defended it. This will become more complicated, as we shall see.

40 Thomas Settle The pendulum and Galileo

C. FURTHER COMPLICATIONS.

The problems of B are heightened when we realize that Galileo had no immediately obvious way to go about discovering the time keeping properties of the pendulum. We can reasonably suppose from various types of indirect evidence that: C 1) after playing with the single lamp for a while he began to have a feeling that the time of the swing was constant as the length of the arc diminished; C 2) he even abstracted the physical problem and began investigating the properties of a lead bullet swinging on a thread or a lute string; C 3) having done so, he resolutely watched the swinging bullet; C 4) unable to discover anything by watching the lamp or the bullet alone, he compared the periods to his pulse, to the tapping out of musical intervals, and perhaps even to the oscillating escapement of a mechanical clock; C 5) he may even have tried to time each swing by measuring the amount of water flowing out of a punctured vessel.

But he would have no reason to suppose that any of these methods were guaranteed as proven, inherent, time keeping techniques. The musician senses equal intervals but cannot be certain of precision. The pulse varies. Mechanical clocks were subject to all sorts of inaccuracies. And water flow might not be constant for an extended period. Galileo apparently knew all this; at the very least, he never defended the assertion of the various properties of the pendulum with reference to any of the above types of measurements or comparisons. In fact the situation is the reverse; by 1610 he knew that the pendulum interval is somehow fundamental, and a physician friend was using a variable length pendulum to measure patients' pulses. So he had had to progress from a state in which he had had a relatively inarticulate suspicion to a state in which he was confident of a fundamental property.

In what sense had he come to consider the properties fundamental? He had no stop watches or electronic devices; he had no standard cosmic clock ticking away absolute seconds. In any case, in the end he would establish the properties as inherent to the nature of the pendulum, and he would not expect that such a position would be defended on the grounds of purely empirical comparisons with other timing devices no matter how accurate they might be. In other words he would have established what might be termed a general fact of nature or a natural principle. He would conclude finally that it was in the nature of the properties of actual free motion, or motion systematically constrained as in pendular motion, that pendular motion should behave in definite ways. He would consider that he had discovered some of those ways. Our problem will turn out to be threefold: how did he discover the properties as presented in B 2?; how did he justify

41 Thomas Settle The pendulum and Galileo to himself their acceptance after he had discovered them?; to what extent did he consider that he had established them as natural principles?

D. TOWARDS AN ANSWER TO SOME OF THE QUESTIONS.

We do not yet know the actual sequence of steps Galileo took empirically and conceptually as he was discovering the properties of the pendulum. It is not even clear that we shall ever have all the evidence we would need for a comprehensive scrutiny of the process. At least for the present, therefore, we shall have to start with an attempt to understand how he presented and defended the already discovered properties in the Discorsi. In general the remarks take the form of a discussion of various types of empirical evidence. And he obviously expects that from this evidence we shall naturally be convinced that the properties given are in fact the true ones. For our purposes we can classify the evidence into three types corresponding to the three general propositions in B 2.

D 1 a. Take two pendulums of equal length. Pull them aside from the vertical by an equal amount. Release them simultaneously. They will be observed to stay in synchronism for the whole time of their motion. This will be true no matter what the length of the pendulums, so long as both are of equal length. And it will also be true no matter how great the displacement from the vertical.

D 1 b. Take two pendulums of equal length. Pull them aside from the vertical by different amounts. Release them simultaneously. They will be observed to stay in synchronism as before. This will be true through the whole range of possible variations of the differences in initial displacement.

D 1 c. Take two pendulums of equal length. Pull them aside from the vertical by equal amounts. Release one and let its arc decay. Release the other so that their motions will be synchronous even though not of the same amplitude. They will retain their synchronism. And so on.

D 1 d. Take two pendulums of equal length. Pull them aside from the vertical by equal amounts. Release them simultaneously. When both arcs have decayed a way, carefully stop one and restart it as in D 1 c at a greater displacement. They will also retain synchronism. Conclusion: The pendulum of a given length has a fixed period of swing independent of the amplitude of swing. The pendulum is a natural time keeper.

D 2. No mention was made in D 1 a concerning the weight or density of the bob. However those experiments were carried out, repeat the entire program, systematically varying the weight

42 Thomas Settle The pendulum and Galileo and density of the bobs. Providing that the weight in each case is enough to make the suspension taught and to make relatively negligible the effect of air resistance (a bob constructed of feathers, for instance, is outlawed), the same results will be observed. Conclusion: the period of a pendulum is independent of the weight or density of the bob.

D 3 a. Take two pendulums, one four times the length of the other. Pull them aside and release them simultaneously. The longer one will complete one swing just as the shorter completes two. This "harmonic" synchronism will continue until they stop. And it will be observed no matter what the absolute length of the pendulums, so long as the ratio of their lengths is the one prescribed.

D 3 b. Take two pendulums such that their lengths are in the ratio of nine to four. Pull them aside and release them. When the longer has completed one swing, the shorter will have completed one and one half swings, i.e. in the ratio of two to three. And this harmonic synchronism will be maintained.

D 3 c. Take pendulums of any lengths so long as the ratio of their lengths are in terms of convenient square numbers and release them as above. They will also be in harmonic synchronism.

Note: the requirement that the ratio of the lengths be in terms of square numbers results from the fact that the only way we have of measuring relative periods is by observing synchronism.

Conclusion: it is clear that the ratio of the periods of pendulums of different lengths are as the square root of the ratio of those lengths.

E. FURTHER DEVELOPMENT.

The arguments outlined in D are an abstraction of the justification presented in the Discorsi. They are more organized and more systematic and formal looking than in Galileo's presentation, but I think they do no violence to the degree of rigor of the connections he construed between the relatively raw empirical observations and the conclusions which he regarded in some sense and to some degree as established natural principles. It may appear that the arguments in D do not clearly establish their conclusions. However, we must acknowledge the fact that Galileo did fully accept the conclusions. Had he been pressed for a more rigorous treatment he might (he did not, but he might) have argued as follows in establishing them:

43 Thomas Settle The pendulum and Galileo

E 1. Our object is to prove that the period of the pendulum is independent of the amplitude of the swing. We would like to be able to compare the period of the pendulum while it is swinging in a wide arc with the period of the same pendulum when its arc has decayed to a smaller amplitude. E 2. We cannot do this with one pendulum alone; we have no way of "preserving" an early time interval and matching it against a later time interval with any precision. E 3. But if as we suspect the pendulum is an inherent time keeper we ought to be able to devise a strategy, using at least two pendulums, to accomplish our purposes. E 4. We shall start by picking two pendulums of equal length, displacing them equally from the vertical, and watching them pace each other throughout their whole motion until they come to a stop. This we shall repeat under a variety of initial displacements and lengths of the two pendulums. E 5. Preliminary conclusion: these observations will show us that any two pendulums of equal length and swinging through an arc of the same length and swinging simultaneously will have the same period. And no more. E 6. Take two pendulums; mark them i and ii.

Let the period of i at its start be Pia Let the period of ii at its start be Piia Let the period of i at its half arc be Pib Let the period of ii at its half arc be Piib

Start i and ii simultaneously: Pia equals Piia - by observation Let both arcs decay to half: Pib equals Piib - by observation Carefully stop i and restart at a: Pia' equals Piib - by observation

E 7. Now: if Pia' were equal to Pia, then Piia would be equal to Piib. Which would mean that we had proved that a period at the beginning of the pendular motion is equal to one later on, i.e. that the period of a given pendulum is independent of its arc. If we can assume that each successive time we start a pendulum of a given length from a displacement of a given magnitude it beats a period of the same duration. E 8. I (Galileo Galilei) have made that assumption. It seems a perfectly reasonable one for a natural philosopher. Unless we can make it and similar assumptions we would exclude the possibility of any sort of natural knowledge. E 9. Moreover, I give it as much credibility as possible by all the variants of the experiment depicted in D 1(a-d). In general, if that assumption were not a valid one, in all the variations of observations in D 1, or even in the observations of E 4, we should have expected disconfirming

44 Thomas Settle The pendulum and Galileo results fairly rapidly. In fact I have tried to make the pendulums produce disconfirming results; short of taking the bob by hand and moving it forcibly in that way, however, the tests all produce consistent results. Therefore the assumption in E 7 seems well founded. E 10. Therefore the period of a pendulum of fixed length is independent of the amplitude of its arc. Now it is clear that if one can accept this argument the rest is easy. Galileo did not spell out any such argument as this. But it is possible, without straining the evidence, to say that some such argument as this is implicit in the exposition and that Galileo may actually have had something like it in mind when he wrote. But whatever the case, whether by arguments such as in D alone or with arguments as in E also, Galileo had satisfied himself that he had established the general facts or principles of action of the pendulum. Had Galileo completed his self appointed task? No! Though he would accept the propositions of B 2 as true natural principles, his job as a natural philosopher was not complete. For Galileo the final end of the investigations of a natural philosopher was the comprehending of several "lumps" of independently known and established general facts or natural principles within a general deductive scheme on the model of those of Euclid or Archimedes. It is helpful in this context to remember that for Galileo mathematics was not a subject that was separable in principle from physical science. He would admit the existence of a "pure" mathematics, but the "pure" would signify for him a limiting end of a continuum and not a distinct entity. (Even Newton later regarded geometry as the science of measurement.) In his investigations on motion Galileo considered himself to be doing for the properties of various natural motions what he thought Archimedes had done for the properties of flotation and Euclid had done for the properties of spatial relations. His science of motion would simply order demonstratively, in a most general and abstract way, the actual properties of the various types of motions that occur.

I am now fairly convinced that one of the motions Galileo would have liked to link demonstratively into the science presented on the Third and Fourth Days of the Discorsi is the very motion we have been discussing. He wanted to be able to connect the propositions he already accepted as factually and generally true into the deductive scheme of the Third Day. There are two important indications of this. The first is his obvious ill ease at having to use the interrupted pendulum argument only as an experimental analogue showing the plausibility of an assumption he need to make. He would have preferred a stronger connection. The second is related; the trend of the theorems and propositions he demonstrates on the Third Day is towards showing the properties of motion of a body moving in the lower half of a semi circle. He seems to want to find a demonstration to the effect that such motion will have properties like those of the pendulum he already knows about. Had he been able to do this, he would have connected

45 Thomas Settle The pendulum and Galileo by demonstration the properties of natural linear acceleration with those of bodies accelerated in a circular path and both of these to the parabolic motion of projectiles. This would have allowed him to incorporate the properties both of the interrupted pendulum and of general pendulum motion strictly into the science. As it was he could connect only linear acceleration to projectile motion. The pendulum's time keeping properties as well as its action when interrupted remained true but disconnected facts.

There is also some earlier evidence that his thoughts moved in this way, In the letter to Paolo Sarpi of 1604 he mentioned that he had discovered several things by experiment, among them the S :: T2 law for linear acceleration, and "other things". He also said that he was currently searching for a general principle from which all these things could be deduced. First, I note the general tenor of the remarks as being in the spirit of what I have discussed above. Second, I am intrigued by the "other things". He knew by that time the properties of the interrupted pendulum. He may also have known of the time keeping properties.1 He surely also knew that projectile motion was curved. He probably also knew the gunner's commonplace that the maximum range of a gun was achieved when its elevation was set at 45¯. Unfortunately there is no clear indication that he referred to any of these in the phrase "other things". But I take it as a legitimate speculation that he may have.

It would seem from the analysis I have presented that for Galileo there were at least two major categories of accepted general factual propositions: those existing in a relatively small set and conceived as an obviously distinct and demarkable type of phenomenon, e.g. the time keeping properties we have discussed in D and E, and those propositions that have been incorporated in a larger and general deductive scheme, e.g. the one concerned with linear acceleration. The two types are possibly indistinguishable in their verbiage. Galileo had accepted as true a proposition relating distance proportionally to the square of the time interval before he had worked out the deductive scheme in the Discorsi. The fact that he and others felt they had to erect such a deductive scheme to incorporate physical propositions in which they already had fairly full confidence is a commentary on the impact of Euclid and Archimedes in the period of the Scientific Revolution. For us it is more interesting to note that Galileo could place confidence in certain general facts or principles before they were thus incorporated.

1 Added note, 1996: he surely did know of the time keeping properties; see the letter to Guidobaldo del Monte of 1602.

46 Thomas Settle The pendulum and Galileo

G. COMMENTS.

Let us call the process of discovery referred to very much earlier, beginning in A, "stage 1", the level of initial justification and acceptance as in D and E "stage 2", and the final level of organization and incorporation into deductive schemes "stage 3". I have talked only briefly about stage 3. Galileo never reached it with regard to his pendulum researches, though I have inferred that he was attempting to get from stage 2 to stage 3. In any case it is clear that justification at the stage 3 level can be conceived as some combination of deductive rigor and empirical verification. It is from a reading of the stage 3 writings of such people as Galileo and Newton that the hypothetico~-deductive model of scientific method has been inferred. That this is no more than a poor approximation of the method of composing a treatise should be evident by now.

I have given a reconstruction of what I take to be a stage 2 justification. Clearly it is not deductive, at least not rigorously deductive. And though the usual sort of induction occurs it poses no problem; for Galileo the passage from "all observed A's have been B's" to "all A's are B's" is performed almost automatically, as when he accepts such basic empirical propositions as in D 1a. The reasoning that requires the effort is what one might call compositive induction or convergent induction. In it certain basic or restricted empirical universals are seen to yield a more general empirical universal by the act of their coming together. This coming together is not a summation of likes, as in the piling of silver dollars one on top of another. It is more a summation of unlikes producing a structural novelty, as the composing of a picture puzzle, or the constructing of a crystal. Carbon atoms yield different composite entities, coal, graphite, or diamond, depending on how they are put together. It is not clear that in our Galilean case the basic or restricted empirical universals could be made to yield different general universals. But it does seem that the general universal is a structural composite of the basic ones and not an enumerative composite. And further, it seems that Galileo placed confidence in a conclusion because, even if it might have had holes, the pieces he did have did fit and did interlock well. Whether or not this sort of criterion could be formulated more informatively or precisely I would not care to discuss here. But I hope this account begins to say something of how I read Galileo stage 2 justification.

Stage 1 Galileo I have hardly touched. It is the problem with which we started: how does Galileo pass from the situation in A to the point immediately before he could present such a discussion as in D. Clearly neither stage 3 nor stage 2 justification can be construed as an account of the path of research in stage 1. In the Discorsi Galileo left no obvious traces of the progress of his actual research program. We can be reasonably sure that just before stage 2, and as a direct result of the research, Galileo had become confident of propositions functionally

47 Thomas Settle The pendulum and Galileo equivalent to the ones I have formulated in B 2 though perhaps not yet reduced to sentential expression. If we were able to give an account of how he reached that state starting from the initial conditions described in A, we would have a type of "genetic" argument. It would be analogous in some way to the "argument" leading from a paleo amoeba to an ostrich or an orangutan: neither deductive nor inductive in any usual interpretation. Yet an argument, rational and proper.

H. FURTHER COMMENTS.

In retrospect some further comments are perhaps in order regarding the progress of Galileo's work. We have seen him starting, in this case, from relatively raw facts and acquiring a more and more comprehensive, rational understanding of them. This is too nice. We can be sure that it was not quite like that, not quite so simple and neat. In his investigations of linear acceleration, for instance, Galileo advanced and retreated many times. Not all the "facts" at whatever level of generality were discovered, nor did they neatly fall into place, as he needed them. Some of his most cherished ideas were ill conceived. Many of the propositions he arrived at by compositive induction he must have gone over again and again, checking the induction both empirically and conceptually, looking for debilitating weaknesses, sometimes finding them, sometimes satisfying himself that none existed. One reason for the attempt to go to ever more comprehensive deductive schemes would be to satisfy himself that no contradictions existed between propositions relating to one group of phenomena (say, the time keeping properties of pendulums) and propositions concerning other phenomena (say, the properties of linear acceleration). He would have admitted to at least one general conviction: that there could be no contradiction among true physical propositions. The more comprehensive the rational scheme, then, the more one could rely on the propositions within the scheme. But as to the historical track, the program and progress of his researches, it was not a steady climb from lowly truths to ever greater ones. Sometimes he started with a suspicion, as in the case of the pendulum work. Sometimes there was a great body of theory and observational verification which he had to understand, criticize, and then completely restructure before he could reach a stage 2 comprehension; such a process had culminated in the S :: T2 law as expressed in the Sarpi letter of 1604. But while Galileo did not always start with raw, singular empirical statements, somewhere along the line he came to terms with them. Then the process of perfecting and refining experiments went hand in hand with the process of clarifying and refining concepts, concepts which would be required for formulating propositions cast in the form of relatively basic empirical universals. Gradually he would enter the transition region

48 Thomas Settle The pendulum and Galileo between stage 1 and stage 2. Further work, presumably both empirical and conceptual, could and frequently did lead higher.

For myself, the most interesting phase of a vital science is the stage 1 research. So far as I can tell it has not been properly recognized as an important area of historical or philosophical investigation. It is where "discovery" takes place, if it can be said to take place anywhere. And it is where the investigator develops the conviction and sense of justification he will later express in relatively concise and dead propositions. I would suspect that it will be in an understanding of the processes occurring in this phase that we shall find the best account of the nature and justice of scientific knowledge.

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