Abstract
Based on the experimental record preserved in the Notes on Motion, the chapter reconstructs and discusses an experiment which, thus far, has been almost completely overlooked. Galileo timed the swinging of a pendulum as well as the rolling of a ball down along a long, gently inclined plane. From the latter measured time, he theoretically inferred the time of motion along a shorter plane, which could be inscribed into the arc of pendulum swing as a chord. Galileo initially assumed that the swinging of the pendulum along a quarter arc would be completed in the same time as fall along a chord spanning this arc and had designed the experiment to test what is here referred to as the single chord hypothesis. The hypothesis had been suggested by a challenging similarity Galileo perceived to hold between pendulum motion and naturally accelerated motion along inclined planes. Yet it was not confirmed by the experiment. In consequence, Galileo altered his conceptualization of the relation between swinging and rolling, and thus, the experiment, conducted in all likelihood in 1602, came to mark the onset of a thoroughgoing investigation of this relation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Galileo’s allusions to experiments involving pendulum motion in his published works have been scrutinized to the last detail. See, for instance, MacLachlan (1976). Drake (1975a), by changing the translation of merely a single word, for instance, revised his own earlier interpretation of a passage in Discorsi from a description of a thought experiment into the accurate description of an experiment Galileo actually conducted.
- 2.
Koyré prominently and influentially promoted the opinion that Galileo never conducted experiments with pendulums, at least not in the way he described them in the Dialogue and the Discorsi. Cf. in particular Koyré (1966). The following quote is exemplary: “It is obvious that the Galilean experiments are completely worthless: the very perfection of their results is a rigorous proof of their incorrection. (Koyré 1968, 94)” The last decades of the twentieth century saw an experimental turn in Galileo scholarship sparked not least by an influential article by Drake (1975a). Many of the experiments described by Galileo have since been recreated, and it has become commonly accepted that most of them were indeed carried out by Galileo in the way he described. A survey of Galileo’s experiments with pendulums is given in Palmieri (2009). Segre (1980) outlines the historically changing views on Galileo’s experimental work.
- 3.
Drake (1987), later republished in Drake (1990), discusses some of the entries in the Notes on Motion related to the pendulum plane experiment. Like much of Drake’s later work, the article strikes as being rather imaginative and is in parts notoriously difficult to follow. As David K. Hill put it: “Drake’s analysis suffers from specific deficiencies to numerous to list. Its general defect is a Byzantine complexity which anyone familiar with the focused simplicity of Galileo’s analytical and empirical procedures will find astonishing. (Hill 1994)” Hill himself has comprehensively discussed the experiment and my interpretation owes to Hill’s and corresponds to it in several main points. However, I share neither of its main conclusions, which Hill summarized as follows: “The analysis also demonstrates that Galileo was well aware of the non-isochronism of the pendulum (despite his published claims asserting the opposite). He was also aware of rolling inertia, and had a close estimate of its effects. (Hill 1994, 515)”
- 4.
A close relationship between the cut folio pasted onto folio 90 on the one hand and folios 115 and 189 on the other is further corroborated by an additional observation made from the original manuscript. Both folio 189 and folio 115 are folded in the middle. The cut folio pasted onto folio 90 bears the same fold. As the folio it was pasted onto is itself not folded, the fold must have been there before the cut piece was pasted, suggesting that it was kept folded together with folios 189 and 115.
- 5.
- 6.
As the folio on which Galileo had originally written down his considerations regarding the pendulum plane experiment was subsequently cut and pasted onto folio 90 to make more room for Galileo’s notes concerning projectile motion, this must thus slightly antedate the jotting down of the latter considerations. Moreover, as detailed below, the pendulum plane experiment remained inconclusive, explaining Galileo’s readiness to cut and thus, more or less, obliterate his corresponding notes. The verso side of the pasted part bears the number 90b in pencil.
- 7.
Evidence will be provided that dates the pendulum plane experiment to 1602, which suggests that Galileo drafted his first results concerning projectile motion at around this time, in contrast to the prevalent opinion, dating the onset of his considerations concerning projectile motion based on assuming natural acceleration to much later. Cf. for instance, Drake (1978).
- 8.
The entries on folio page 189 verso discussed in the present chapter were written on the page first. At some later point Galileo reused this page for noting considerations unrelated to the experiment. These entries are written around the initial entries regarding the experiment. That Galileo’s considerations pertaining to the second stage of evaluation of the experiment then had to be noted on fresh folios suggests that the second stage of evaluation succeeded the first, only after some temporal delay.
- 9.
The experimental record on page 189 verso makes no explicit mention of the length unit used to measure the lengths of the pendulum and of the inclined plane. The length given in the text is based on the assumption that Galileo specified the lengths in punti, with the punto measuring approximately 0.96 mm. This, first of all, results in very plausible dimensions for the pendulum and the plane in the experiment. That the unit in which the dimensions of the experiment were specified is indeed the punto is vindicated by an entry on folio 115 verso , where a length derived from the dimension of the experiment is specified as measured in “p[unti].” For the punto, see the discussion of folio 166 recto in the Appendix in Chap. 13.
- 10.
As detailed in the Appendix 13 in Chap. 13, Galileo determined the overall time of motion measured for the pendulum swing by adding up a column of smaller numbers. Based on these numbers, Drake believed he could infer the flow rate of the water clock he supposed Galileo had used. Given how underdetermined the problem is, this was quite plainly an exercise in overinterpretation. Whereas Drake held that the individual numbers represented individual measurements of quantities of water flowed out, Hill (1994) assumed that Galileo only measured once and that the different numbers represent the “various small weights” that he had used to counterbalance the water that had flowed out. Drake (1987), moreover, sought to identify a schematic drawing of a water clock in the Notes on Motion. However, the diagram on folio page 107 verso he was referring to has since been identified by Damerow et al. (2001) as an attempt to construct the curve of a hanging chain.
- 11.
Step V remains implicit in Galileo’s entries on page 189 verso where Galileo merely calculated the time of fall through the pendulum diameter, which, according to the law of chords, is also characteristic for motion along the pendulum chord. That Galileo perceived of the comparison as a comparison between two concrete motions connecting the same start and endpoints, the quarter swing of the pendulum and the motion along the corresponding pendulum chord, is suggested by his sketches of the experimental situation on 189 verso .
- 12.
The Florentine braccia measured between 58 and 59 cm and thus the pendulum described in the Discorsi measuring three braccia, about 175 cm, compares well in length to the one used in the pendulum plane experiment, whose length was somewhat less than 2 m. The frequency of such a rather long pendulum is sufficiently low, while at the same time the length of the pendulum is not so long as to become unmanageable.
- 13.
The inclined plane experiment has been alluded to in countless studies. A recent discussion is contained in Palmieri (2011), which provides a comprehensive bibliography on the subject.
- 14.
To this Naylor (1976, 153) remarked “[o]ne of the most controversial issues in the history of science has been the question of how far Galileo’s achievement in mechanics was dependent on the use of experiment.”
- 15.
Cf. Mach (1897, 122–126). Koyré strongly dismissed the idea that Galileo carried out experiments at all when he wrote that “experiments which Galileo, and others after him, appealed to, … were not and could never be any more than thought experiments. (Koyré 1978, 37)” In later writings he took a somewhat more modest position.
- 16.
See Settle (1961). Naylor (1974) as well replicated the experiment. He came to the conclusion that from his “observations it does not appear that the precise ratios are obtainable as suggested by Galileo, particularly in the case of the smaller inclination. (Naylor 1974, 131)” He, in particular, points to the fact that lining the groove with parchment as described by Galileo, makes the outcome worse, which he takes as an indication that Galileo’s description of the experiment cannot be taken at face value.
- 17.
Whereas Galileo may have become aware of the quadratic relation between spaces traversed and times elapsed in free fall as a heuristic assumption as early as 1592 (cf. Damerow et al. 2001), he only started to avail himself of the law as a fundamental assumption in a systematic study in his research on swinging and rolling in 1602. This would thus seem to have been the right time to seek empirical confirmation of the underlying assumption by means of an experiment.
- 18.
Drake (1975b) thought he had identified direct evidence of an experiment involving naturally accelerated motion along an inclined plane in numbers recorded on 107 verso , which he took to be recordings of time measurements of motions along an inclined plane. Yet it has since been shown that these numbers are the result of Galileo’s comparison of a hanging chain with a parabola and that, moreover, the diagram on that page, which Drake took to be a schematic representation of the water clock used in the experiment, is in fact Galileo’s construction of the center of gravity of a constellation of weights suspended on a string or rope. See Damerow et al. (2001), in particular, footnote 119.
- 19.
For a comprehensive account of the horizontal projection experiment documented on folio 116 verso , see Hahn (2002). In the experimental record, the length of the plane used is not given. Yet the maximum vertical height above the point of projection from which the motion started is given as 1000 punti. If it is assumed that Galileo used a plane of 6700 punti length as in the pendulum plane experiment, this implies an inclination of 8.6 degrees, again in astounding accordance with the 8 degree inclination used in the pendulum plane experiment. A plausible scenario would be that Galileo simply started from the exact same setup used in the pendulum plane experiment and slightly lifted the end of his plane until as desirable for the projection experiment, the vertical height amounted to the round figure of 1000 punti. The assumption that a plane comparable in length and inclination to the one of the pendulum plane experiment was likewise used in the oblique projection documented on folios 114 verso and 81 recto may provide a way to promote the deadlocked discussion concerning these experiments.
- 20.
Vergara Caffarelli (2009, 121) has rightly emphasized that fabricating this plane would have been a difficult job that would have required a good craftsman and would certainly have been rather expensive.
- 21.
- 22.
The assumption of the single chord hypothesis is by no means implausible as the example of Marci von Kronland, who in fact gave the hypothesis as a theorem, shows: “Pendulum aequali tempore mouetur per arcum Circuli & chordam eidem subtensam.” See http://archimedes.mpiwg-berlin.mpg.de/cgi-bin/toc/toc.cgi?dir=marci_figur_063_la_1648;step=thumb. Accessed 16 Feb 2017. Since Galileo never explicitly stated the single chord hypothesis in his published works, Marci must have arrived at the assumption independently, albeit based on the same conceptual prerequisites.
- 23.
According to Vincenzo Viviani, Galileo’s last and most faithful disciple and his first biographer, Galileo discovered the isochronism of the pendulum as a student while observing the swinging of a lamp in the Cathedral of Pisa. Serious doubt has meanwhile been cast on Viviani’s account. Cf., in particular, Ariotti (1971/1972). Segre (1998, 392) rightfully remarks that it “is the excellence of Viviani’s essay that makes it so difficult to interpret; even today, historians have difficulties in distinguishing between reality and myth in its pages.”
- 24.
As will become clear, Galileo’s insight into the law of fall must predate the earliest entries in the Notes on Motion. Damerow et al. (2001) have argued that Galileo most likely initially came to accept the law as a heuristic based on an experiment concerning the shape of the projectile trajectory conducted jointly with Guidobaldo del Monte, in 1592.
- 25.
Galileo’s recourse to the law of chords in its theoretical evaluation of the data was strictly speaking not necessary. He could have determined the length of a chord with the same inclination as his experimental plane but inscribed into the arc of pendulum swing geometrically and from his measured time inferred the time of motion along this pendulum chord by application of the law of fall. That he employed the law of chords to gain essentially the same result indicates that he saw no reason to be skeptical of the law of chords and took it for granted by the time the experiment was conducted.
- 26.
In principle, such a test would have been possible, using the exact same setup and evaluation of the experimental data. Galileo would simply have needed to vary the angle and/or length of the plane or the initial angle of elongation of the pendulum. Apparently, however, he did neither and made just one-time measurement for each of the two types of motion. We will of course never know if the experimental record is complete and it is in fact plausible to assume that Galileo tinkered with the experimental setup before conducting the experimental run, whose results he ended up recording on 189 verso .
- 27.
Drake (1987) referred to the ratio of these periods of the quarter swing and the time to fall through a vertical distance of twice the pendulums length as Galileo’s constant.
- 28.
Peirce, who introduced the concept of abductive inferences into modern logic, illustrated them by the following famous schema: “The surprising fact, C, is observed; But if A were true, C would be a matter of course. Hence, there is reason to suspect that A is true. (Peirce et al. 1978, V, 189)” For abductive reasoning in general, see Aliseda (2006).
- 29.
Steinle, has introduced the label explorative experiments for experiments which serve first and foremost to gain orientation in a new field where the starting positions are unclear (see Steinle 2005). Even though according to the interpretation provided in the pendulum plane experiment Galileo tested a concrete hypothesis, the label arguably still applies as this hypothesis was not a strict consequence of a well-established theory but rather a heuristic, which, should it be confirmed, would guide further investigation.
- 30.
By the law of the pendulum I refer to the assertion that the period of a pendulum varies in proportion to the square root of its lengths. From a modern perspective, it needs to be declared explicitly that the pendulum has to swing through the same angular amplitude, as the period, besides on the length of the pendulum, also depends on this angle of swing. For Galileo, who held that pendulum motion is isochronous, i.e., independent of the angle of the arc the pendulum is swinging through, the law of the pendulum held without such restriction.
- 31.
Should Galileo, against all odds, have been unaware of the law of the pendulum when he first conceived of the single chord hypothesis, the size-independence or scale invariance assumption it entailed would not have been directly implied by the underlying assumptions. That the conceived hypothesis embraced scale invariance in this case may have been the result of an unreflected choice, more or less without alternative. Indeed, based on the mathematical language of proportions, the relation between the quarter period of the pendulum and the time of motion along a corresponding pendulum chord, which can be conceived of as a ratio, cannot depend on the absolute size of the underlying geometrical configuration if it is in proportion to any of the characteristic distances defined by this configuration. Yet this is not necessarily the case, and Galileo was in fact able to deal with ratios which did change as the underlying geometry changed in size while remaining proportionally similar. In the Discorsi, Galileo thus indeed showed that the ratio between the moment of heaviness and the moment of resistance in a beam was not always the same for geometrically similar beams (EN VIII, 163–164). He was able to do so by analyzing the ratio of the ratios of the different kinds of moments for two similar cylinders instead of analyzing simply their ratio. Comparably, it was thus, at least in principle, conceivable that the relation of the period of a quarter swing to the time of motion along the corresponding pendulum arc may vary with the absolute size of the system. Galileo only commenced his investigations concerning the strength of materials in the summer of 1607, and it is not clear whether this would have been obvious to him earlier. Indeed, in the Discorsi, Galileo has Simplicio express surprise: “[t]his proposition strikes me as both new and surprising: at first glance it is very different from anything which I myself should have guessed: for since these figures are similar in all other respects, I should have certainly thought that the forces [momenti] and the resistances of these cylinders would have borne to each other the same ratio (EN VIII, 164, trans. Galilei et al. 1954, 125).” This may well portray Galileo’s position with regard to the question of the relation of the times in question in the pendulum plane experiment. Galileo, albeit at a much later time, indeed explicitly stated that one may “come principio noto” suppose that the relation of the time of motion along arc and along the chord spanning the arc must be similar for similar geometrical constellations. Cf. Chap. 9.
- 32.
Galileo never made a claim to the discovery of the law of the pendulum. Since Galileo is generally not modest about his achievements, this can be taken as an indication that he did indeed not consider himself to be the person who discovered it. As concerns his published writings, Galileo stated a qualitative pendulum law in the Dialogue, i.e., he asserted that the period of a pendulum increased monotonously with its length. Ariotti (1968, 416–417) takes the exposition in the Dialogue as an indication that Galileo is “yet unclear on the exact relationship of the length and the period.” In the Discorsi, Galileo finally introduced the law of the pendulum but gave no proof. Cf. (EN VIII, 139–140). Cf. Ariotti (1971/1972, 351–354). According to Drake (1970, 497–498), Galileo most likely witnessed his father Vincenzo Galilei’s musical experiments in 1588–1589. Drake claims that “an observer [of these experiments] can hardly escape the phenomena of the pendulum” and thus seems to imply that Galileo in this way first became aware of the law of the pendulum. In his Methodi vitandorum errorum omnium qui in arte medica contingunt published in 1602, S. Santorio introduced a device called a pulsilogium in which, by varying the length of a pendulum, the period of this pendulum is synchronized with a patients pulse. The resulting pendulum length was then interpreted as a measure of the pulse rate, which could then easily be compared, for instance, to the pulse rate at another time. Cf. Levett and Agarwal (1979). The pulsilogium does implicitly presuppose isochrone of pendulum motion. To conceive and build it, a qualitative understanding of the relation between length and period of a pendulum arguably suffices. Yet once constructed it obviously allows for qualitative observations that may suggest the law of the pendulum. In fact, in a marginal note in his private copy of the Discorsi (Gal. 79), the passage in which the law of the pendulum is introduced is annotated by the remark that “replicate esperienze” led to the law of the pendulum. Santorio and Galileo were part of the same learned circle in Venice. Cf. Recht (1931). Galileo may not necessarily have played a crucial role in the invention of the pulsilogium as commonly assumed, but he would certainly have been familiar with the device in 1602. Cf. Büttner (2008, 227–229). In the Notes on Motion, direct evidence of Galileo being aware of the law of the pendulum is provided by his considerations on folio 154 recto , where based on the law of the pendulum Galileo compared the observed time of a long pendulum to the hypothetical time calculated for a pendulum whose radius corresponded to the radius of the earth. Cf. the discussion of this folio in the Appendix in Chap. 13, where it is argued that other content on the page is related to Galileo’s construction of the ex mechanicis proof and thus most likely dates to around 1602. It is thus strongly indicated that Galileo was familiar with the law of the pendulum at that time.
References
Aliseda, A. (2006). Abductive reasoning: Logical investigations into discovery and explanation (Synthese library). Dordrecht: Springer Netherlands.
Ariotti, P.E. (1968). Galileo on the isochrony of the pendulum. ISIS, 59(4), 414–426.
Ariotti, P.E. (1971/1972). Aspects of the conception and development of the pendulum in the 17th century. Archive for History of Exact Sciences, 8, 329–410.
Brunelli Bonetti, B. (1943). Nuove ricerche intorno alla casa abitata da Galileo Galilei in via vignali. In Pubblicazioni Liviane e Galileiane a Ricordo delle Celebrazioni dell’Anno 1942. Supplemento al vol. LIX delle “Memorie” (pp. 91–102). Padua: Tip. Penads.
Büttner, J. (2008). The pendulum as a challenging object in early-modern mechanics. In R. Laird & S. Roux (Eds.), Mechanics and natural philosophy before the scientific revolution (pp. 223–237). Dordrecht: Springer.
Caverni, R. (1972). Storia del metodo sperimentale in Italia. New York: Johnson Reprint.
Damerow, P., Freudenthal, G., McLaughlin, P., & Renn, J. (1992). Exploring the limits of preclassical mechanics. New York: Springer.
Damerow, P., Renn, J., & Rieger, S. (2001). Hunting the white elephant: When and how did Galileo discover the law of fall? In J. Renn (Ed.), Galileo in context (pp. 29–150). Cambridge: Cambridge University Press.
Damerow, P., Freudenthal, G., McLaughlin, P., & Renn, J. (2004). Exploring the limits of preclassical mechanics. New York: Springer.
Drake, S. (1970). Renaissance music and experimental science. Journal of the History of Ideas, 31, 483–500.
Drake, S. (1975a). New light on a Galilean claim about pendulums. ISIS, 66(233), 92–95.
Drake, S. (1975b). The role of music in Galileo’s experiments. Scientific American, 232(6), 98–104.
Drake, S. (1978). Galileo at work: His scientific biography. Chicago: University of Chicago Press.
Drake, S. (1987). Galileo’s constant. Nuncius Ann Storia Sci, 2(2), 41–54.
Drake, S. (1990). Galileo: Pioneer scientist. Toronto: University of Toronto Press.
Galilei, G., Crew, H., & Salvio, A.D. (1954). Dialogues concerning two new sciences. New York: Dover Publications.
Hahn, A.J. (2002). The pendulum swings again: A mathematical reassessment of Galileo’s experiments with inclined planes. Archive for History of Exact Sciences, 56, 339–361.
Hill, D.K. (1994). Pendulums and planes: What Galileo didn’t publish. Nuncius Ann Storia Sci, 2(9), 499–515.
Koyré, A. (1966). Etudes Galiléennes. Paris: Hermann.
Koyré, A. (1968). Metaphysics and measurement. London: Chapman & Hall.
Koyré, A. (1978). Galileo studies (trans: Mepham J). Harvester: Brighton.
Levett, J., & Agarwal, G. (1979). The first man/machine interaction in medicine: The pulsilogium of Sanctorius. Medical Instrumentation, 13(1), 61–63.
Mach, E. (1897). Die Mechanik und ihre Entwicklung historisch-kritisch dargestellt. Leipzig: F. A. Brockhaus.
MacLachlan, J. (1976). Galileo’s experiments with pendulums: Real and imaginary. Annals of Science, 33, 173–185.
Naylor, R.H. (1974). Galileo and the problem of free fall. The British Journal for the History of Science, 7(2), 105–134.
Naylor, R.H. (1976). Galileo: The search for the parabolic trajectory. Annales of Science, 33, 153–172.
Palmieri, P. (2009). A phenomenology of Galileo’s experiments with pendulums. The British Journal for the History of Science, 42(4), 479–513.
Palmieri, P. (2011). A history of Galileo’s inclined plane experiment and its philosophical implications. Lewiston: Edwin Mellen Press.
Peirce, C.S., Hartshorne, C., Weiss, P., & Peirce, C.S. (1978). Scientific metaphysics (Collected Papers of Charles Sanders Peirce, 4th edn.). Cambridge: Belknap Press of Harvard University Press.
Recht, E. (1931). Life and work of Sanctorius. Medical Life, 38, 729–785.
Segre, M. (1980). The role of experiment in Galileo’s physics. Archive for History of Exact Sciences, 23(3), 227–252.
Segre, M. (1998). The never-ending Galileo story. In P.K. Machamer (Ed.), The Cambridge companion to Galileo. Cambridge/New York: Cambridge University Press.
Settle, T.B. (1961). An experiment in the history of science. Science, 133(3445), 19–23.
Steinle, F. (2005). Explorative Experimente: Ampère, Faraday und die Ursprünge der Elektrodynamik (Boethius, Vol. 50). Stuttgart: Franz Steiner Verlag.
Valleriani, M. (2010). Galileo engineer. (Boston Studies in the Philosophy of Science, Vol. 269). Dordrecht/London/New York: Springer.
Vergara Caffarelli, R. (2009). Galileo Galilei and motion: A reconstruction of 50 years of experiments and discoveries. Berlin/New York/Bologna: Springer, Società italiana di Fisica.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature B.V.
About this chapter
Cite this chapter
Büttner, J. (2019). Sparking the Investigation of Naturally Accelerated Motion: The Pendulum Plane Experiment. In: Swinging and Rolling. Boston Studies in the Philosophy and History of Science, vol 335. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1594-0_4
Download citation
DOI: https://doi.org/10.1007/978-94-024-1594-0_4
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-024-1592-6
Online ISBN: 978-94-024-1594-0
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)