Abstract
In November 1602, Galileo wrote a letter to his friend and patron Guidobaldo del Monte in which he outlined his current work. This letter has received particular attention as it provides the first explicit evidence that Galileo had returned to the question of the fall of heavy bodies along inclined planes, which he had already addressed in the 1590s. As is demonstrated in the succeeding chapters, the considerations explicitly referred to in the letter were in fact part of a much broader research agenda Galileo was following at the time and which has left abundant traces in the Notes on Motion. This chapter rereads and provides a fresh exegesis of the letter against the backdrop of the interpretation of Galileo’s early work on the problems of naturally accelerated motion being advanced in this book.
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- 1.
For Guidobaldo del Monte, see Gamba et al. (2013).
- 2.
Stilmann Drake, for instance, after having identified a number of folios of the Notes on Motion as containing early considerations by Galileo, claims that “these probably all belong to 1602, when it appears that Galileo, having revised and expanded his Mechanics, decided to write a new treatise on motion. Several propositions for this were neatly written out before he fully realized the importance of acceleration, and the sheets bearing these were mutilated in the course of the subsequent revisions of the projected treatise (Drake 1978, 67).”
- 3.
Humphreys (1967), for instance, speculates about Galileo’s work in 1602 without considering any of the material in the Notes on Motion at all. Galluzzi (1979, 268) more carefully restrained from speculation when he claimed concerning Galileo’s work in 1602: “È tuttavia, impossibile riconstruire il modo in cui Galileo intendeva raggiungere la dimostrazione lavarando sulla proporzione dei momenti.”
- 4.
The original letter is not extant. A copy in a later hand has been preserved. This copy was, however, not made directly from the original but from an earlier copy by Viviani, which is likewise lost today. Cf. EN X, letter 88, 97–100.
- 5.
- 6.
For Viviani’s account of how Galileo allegedly discovered the isochronism of the pendulum as a student when observing the swinging of a lamp in the dome of Pisa, see Viviani (1890–1909). For a contextualization of Galileo’s engagement, theoretical as well as practical, with pendulum motion, see Büttner (2008). Drake (1970) has suggested that Galileo’s interest in pendulum motion might well have been triggered by his father’s musical experiments. Galileo’s statements in the letter can be interpreted as based on a distinction between mere assumption and certitude. If this is the case, Galileo would be implying that it was his experiment conducted shortly before the antecedent letter that was sent that ultimately convinced him of the truth (verità) of the isochronism of the pendulum, which by then he had already held as an assumption for some time.
- 7.
See EN, VII, 474–476 and EN, VIII, 277–278. For a recent detailed assessment of these passages and Galileo’s pendulum experiments in general, see Palmieri (2009).
- 8.
Galileo had been experimenting with pendulums at least since the time of writing De Motu Antiquiora and pendulum bobs of different material figure in his very first allusion to such an experiment. In a passage of De Motu Antiquiora, Galileo writes: “The same thing is evident if two weights, one of wood, the other of lead, are suspended from two equal threads and, when they have received an impetus from an equal distance from the perpendicular, they are released; of the two, the lead will certainly be moved back and forth for a longer interval of time (EN I, 335, transl. Fredette).” See also Settle (1966, 96).
- 9.
Van Dyck (2006) convincingly shows that Galileo never genuinely got to grips with the problem of the relationship between weight, specific weight, and the naturally accelerated motion of heavy bodies.
- 10.
The reader may explore the deviation of large-amplitude pendulums from isochronous behavior virtually at http://hyperphysics.phy-astr.gsu.edu/Hbase/pendl.html. Accessed 12 Jan 2015. Lima and Arun (2006) provide a penetrating discussion of the physics of large-amplitude oscillations of pendulums.
- 11.
In his translation of the Discorsi, Drake in Galilei (1974, 226, footnote 11), for instance, claimed that Galileo’s portrayal of the results of the experiment could not correspond to actually observable facts.
- 12.
For angular amplitudes of less than 45 degrees, the period of a real pendulum deviates from that of an idealized pendulum in which the restoring force is assumed to depend linearly on the angle of elongation, which would swing truly isochronally, by less than one percent. The damping of a pendulum due to air drag forces is comprehensively treated in Nelson (1986). For large amplitudes, the rate of decay of amplitude due to damping is rather high.
- 13.
- 14.
Cf. Chap. 6.
- 15.
Much later in the Dialogue, Galileo indeed proposes to his readers to have a ball roll along a concave surface with the curvature of an upright quarter circular arc to observe that a ball’s transit to the lowest point will always be made in the same time, regardless of the starting point on the arc (EN VII, 476), just as Guidobaldo del Monte had done.
- 16.
All English quotations follow the translation given in the appendix (see Chap. 14).
- 17.
Galileo, anticipating what he would state in the next paragraph, is obviously not thinking of two arbitrary inclined planes, one measuring 100 miles and the other a span, but indeed about two chords which have these lengths and which are inscribed in the same circle sharing its nadir. Motion along such chords is completed in equal time as he will state and claim to have demonstrated a bit further down in the letter. Thus, where Guidobaldo had talked about motion along arcs, Galileo made his argument for motion along chords spanning these arcs.
- 18.
Alternatively, the “received impetus” may have been the result of an impact event, but this clearly does not seem to be what Galileo is alluding to here.
- 19.
The openness of the letter with regard to the question of how Galileo at the time conceptualized motion on inclined planes, i.e., as uniform or else as naturally accelerated, has certainly helped to nourish the incorrect understanding that in 1602 Galileo was still working under his old paradigm of uniform motion.
- 20.
There has been a tendency to assume that the proof of the law of the broken chord alluded to, just like the ex mechanicis proof of the law of chords, must have been based on Galileo’s principle of inclined plane dynamics. Damerow et al. (2001, 58) have suggested that Galileo’s remark should be understood as owing to a rhetoric strategy adopted to appease Guidobaldo del Monte, who was “skeptical with regard to studies involving motion” and who would have been reassured if “in spite of the novelty of the subject for traditional mechanics, he [Galileo ] is still adhering to the principles of this mechanics.” The proof of the law of the broken chord Galileo mentioned in the letter, as will be demonstrated here, was indeed entirely kinematical.
- 21.
In a draft of a letter composed after the publication of the Discorsi, Galileo pointed out to Giovanni Battista Baliani, who likewise had assumed a relation between pendulum motion and motion on inclined planes and founded his own approach at a new science on that assumption, that the latter had proceeded haphazardly and had not appropriately accounted for the essential differences between the two types of motions. Cf. Chap. 9.
- 22.
Settle (1996, 18) has tentatively related the impact experiment mentioned in the letter to pendulum motion when he speculates that a “pendulum of sufficient length, with a standard weight, and swinging down from a fixed height, would deliver a standard percussive blow.” There is, however, no concrete evidence that the experiment would have encompassed a pendulum.
- 23.
Some of the manuscripts of Le Meccaniche contain a chapter on the “force of percussion,” which demonstrates that Galileo had turned his attention to the problem of impact by at least 1600, if not earlier. Galileo provided an account of the impact balance experiment in a short tract entitled “Della Forza della Percossa,” which was appended to the Discorsi as an added day. See EN. VIII, 319–346. See also Favaro’s Avvertimento to the Discorsi, ibid., 11–38. According to Settle (1996, 19), this experiment “seems to have taken place in Padua sometime between 1605 and 1610”; Drake (1978, 126) dates it to 1608. Neither provides convincing evidence for their respective claims. For a more recent account, see Salvia (2014).
- 24.
See EN. II,191–192. Torricelli’s description of the experiment is comprised in his second Lezione Accademicae. Drake (1978, 72) speculates that it may have been precisely this experiment that Galileo was alluding to in his letter to Guidobaldo del Monte in 1602.
- 25.
Damerow et al. (2001, 49) have argued that during his time in Padua, Galileo “through Guidobaldo del Monte …came into close contact with the experimental techniques and the research interests of a leading engineer-scientist….” This “practical turn,” as the authors term it, included the establishment of a workshop in Galileo’s house that offered facilities for the production of instruments but allegedly also for experimentation.
- 26.
- 27.
Damerow et al. (2001) have likewise interpreted this remark as regarding matter in motion in particular.
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Büttner, J. (2019). A Glimpse at a Challenging Research Agenda: Galileo to Guidobaldo del Monte in 1602. 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_3
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