Abstract
The chapter discusses how Galileo’s attempt to approximate motion along an arc by naturally accelerated motion along paths composed of a series of inclined planes, in or around 1602—and thus, as he believed, in due course pendulum motion—bestowed him with a number of new results. These results were to have repercussions far beyond the narrower context of the investigation into the relation between swinging and rolling from which they had originally ensued. Most of the insights achieved, propositions formulated and methods developed in that period, indeed became core assets of the new science of motion and found their way into the Discorsi more than 35 years later. It is, in particular, demonstrated how, as a by-product of his attempts to construct a proof for the so-called law of the broken chord, Galileo developed a new technique for rendering the kinematics of motion combining the representation of spacial and temporal aspects of motion in just one single diagram. In this period Galileo, in particular, scrutinized the especially challenging problem of finding the geometrical configuration under which the time of motion along a path allowed to vary in a geometrically defined manner became minimal. The problem remained unsolvable, yet his attempts at a solution resulted in the formulation of a set of new propositions complete with proof that were later included, many of them almost verbatim, in the Discorsi.
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- 1.
For the history of representing motion by tracing its path and the moving body on paper and for the role this representational form played in the establishment of preclassical external ballistics, see Büttner (2017).
- 2.
Separate time representation usually embraces three components: a diagram representing the spatial geometry of the motion situation, a second, separate diagram representing its temporal aspects, and, in addition, some form of representation of the relation between the elements of the two diagrams, usually expressed in language employing the theory of proportions. Quick sketches of motion problems in the Notes on Motion often lack an explicit description of the relations between the diagrams, since those relations were immediately clear to Galileo. The diagrams representing the times of motion are, as a rule but not always, directional and monotonous, that is, the points representing later moments in time are always located to the same side of points representing earlier moments in time. Alternatively, Galileo sometimes rendered lines representing intervals of time as parallel lines.
- 3.
Galileo’s representation of times in diagrams based on integrated time representation is always conformable, i.e., it preserves the ratios that should hold between the respective times according to the laws of motion employed. It is indeed the crucial advantage of integrated time representation that ratios that hold between times according to the laws applied can be geometrically constructed based on the spacial geometry of the motion situation. Palmieri (2006) has recently emphasized the importance of integrated time representation, which he refers to as “embedding of times of descent in diagrams,” without, however, providing an account of its origin, for Galileo’s considerations.
- 4.
A limited number of propositions of the Second Book of Third Day of the Discorsi, do employ an external timeline. Not, however, to represent times of motion inferred as part of a proof argument, but rather when the ratio of times is given and a specific configuration is sought under which the times elapsed during two motions obey this given ratio. In such cases, separate lines are used to represent the given time intervals and, thus, also their ratio as, for instance, Proposition XXII of the Second Book of the Third Day of the Discorsi, the separate lines labeled A and B.
- 5.
Uninked construction lines scored into the paper and the incision marks of the compass on the pages of the Notes on Motion have thus far been virtually ignored in studies of the manuscript. The sole exception I am aware of are Damerow et al. (2001, 379), where resorting to information embraced by uninked construction marks remains restricted to the analysis of a single construction by Galileo. I owe my information concerning the uninked constructions to a group of scholars from the Max Planck Institute for the History of Science, who took photos of the manuscript under raking light in which these constructions become visible.
- 6.
Integrated time representation as argued in the text greatly facilitates chains of inferences as part of an argument and thus trivially also the putting of arguments into words. An example is provided by Proposition XIXXX of the Third Book of the Third Day of the Discorsi, in which the proposition is stated in terms of a proportion between ratios of distances and ratios of times, while in the proof itself integrated time representation is exploited and times are specified in reference to lines in the diagram.
- 7.
186 verso , T1B.
- 8.
The argument at the basis of Galileo’s diagram on folio 186 verso is discussed in detail in Chap. 5.
- 9.
186 verso , T1B.
- 10.
Ibid.
- 11.
The points s ′, t ′ and v ′ were merely marked by Galileo but not lettered.
- 12.
186 verso , T1B.
- 13.
For a detailed discussion of the proof of the law of fall on 147 recto , see Sect. 10.1.
- 14.
Throughout Galileo’s Notes on Motion, a number of diagrams can be found which embody a particular mnemonic technique for dealing with complex proportions between ratios, often, but not necessarily, between ratios of times and ratios of distances. For an example, see Fig. 7.6. In order to do so, these diagrams re-represent lines from the main diagram, ignoring their spatial orientation, focusing merely on their lengths as the relevant characteristic with regard to the proportions under investigation. Depending on the context, the appearance of an distance-time ratio diagram, can vary somewhat. These diagrams can be read like a table where usually one row contains distances covered, and the other times elapsed in traversing these distances in a given motion situation, all magnitudes being represented by lines. Times in one cell represent the times of motion along the distance represented in the cell in the same column. Neighboring columns represent valid proportions. Thus, in the example in Fig. 7.6, the first two columns represent the proportion t(ab) t(ad) ∼ ab ad, as a result of the length time proportionality for the motion situation under consideration. The second and third columns represent the proportion t(ad) t(ae) ∼ ad as, as a result of the law of fall. Such diagrams aid the construction of valid proportions. In the example provided it can, for instance, immediately be read off that the ratio between the time elapsed in traversing ab and the time elapsed in traversing ac must be in proportion to the ratio of the lines ab and a0.
- 15.
164 recto , T1.
- 16.
A more detailed discussion of Galileo’s entry on 164 recto is contained in Sect. 10.1.
- 17.
Wisan (1974, 192) was the first to recognize that Galileo’s remark on 164 recto was directly linked to the use of integrated time representation and concluded that “[t]his suggests a chronological development in which folio 164 recto represents the transitional stage between earlier proofs using a separate time line and later proofs which do not.”
- 18.
Wisan observes and correctly links the change in terminology to Galileo’s use of integrated time representation. She inaccurately claimed, however, that the entry on 164 recto provides the sole instance where tempus xy is used to refer to a time represented by line xy, instead of to the time to traverse the line xy. On folio 77 recto , for instance, Galileo states “…erit bi tempus per xa …,” and when he continues his argument, he refers to the line bi representing a time simply as “tempus bi.” Other examples can be found on 87 verso or folio 96 recto . By the time Galileo wrote these entries, he would have been so familiar with the new technique that the ambiguity entailed would no longer have been a source of confusion for him. Cf. also the discussion of folio 164 in Chap. 10.
- 19.
147 recto T5. Cf. Chap. 11
- 20.
Renn holds that the length time proportionality was initially “merely a plausible assumption which Galileo at first also believed to be a consequence of this [De Motu Antiquiora] theory (Damerow et al. 2004, 205).” With his interpretation, Renn followed Wisan who has in fact provided an account of how Galileo, still working under the assumption that motion along inclined planes was in principle uniform, may have first arrived at the insight expressed by the length time proportionality. Her speculative account strikes me as not very convincing, as it presupposes that Galileo committed a trivial error when he applied “this rule [one of the kinematic propositions] while overlooking the necessary condition that the distances be equal (Wisan 1974, 188).” She, moreover, suggests that Galileo may have successfully tested the proposition in an experiment. Wisan (1974, 200–201) attempts to supported her interpretation by making reference to an entry on folio 177 recto in the hand of Niccolò Arrighetti with no original in Galileo’s hand extant. The relevant entry is, furthermore, crossed out. Independent of the question of whether it documents a consideration by Galileo at all, the entry poses some serious interpretational difficulties. Caverni (1972) likewise located the origin of the length time proportionality in Galileo’s De Motu Antiquiora theory, attributing a similar error to Galileo, as Wisan does.
- 21.
Wisan is somewhat inconsistent in her assessment as she on the one hand asserts that the length time proportionality was an early insight achieved before Galileo’s conceptual shift to the assumption of natural acceleration, while on the other hand she maintains that the proposition was not “part of his deductive machinery” until much later.
- 22.
- 23.
As detailed in Chap. 5, the length time proportionality was employed by Galileo in all extant numerical calculations of broken chord motion situations in the context of the broken chord approach, which according to the interpretation presented, suggests that these calculations postdate Galileo’s inception of the length time proportionality as a spin-off from his search for a proof of the law of the broken chord.
- 24.
See Drake (1978, 96).
- 25.
Galileo’s construction is explained in detail in the Appendix in Chap. 13. Another instance where in a first step Galileo approached testing the validity of his assumptions, in this case the so-called theorem of equivalence, by brute force calculation of relevant cases has been analyzed in detail by Renn (1990). As discussed in Chap. 5, Galileo explored and probed the potential validity of his root-root hypothesis concerning the time of descent along bent planes in a similar fashion.
- 26.
The bent plane aec is a broken chord as the junction point e is positioned on the quarter arc connecting the points a and c.
- 27.
The hypothesis test by calculating or rather constructing the times of motion for different concrete cases could of course only show that, of the cases considered, motion along the bent plane, aec was the one traversed in the least time. What Galileo attempted to prove, theoretically, was that this was the path of absolute least time under the given constraints. Yet as argued, he was not able to achieve this.
- 28.
In Galileo’s original diagram two points are lettered q. To disambiguate, I here refer to the closest one of the two to point c as q ′.
- 29.
130 verso , T2.
- 30.
In the diagram, the positions of the points marking the mean proportionals show slight deviations with respect to their geometrically correct positions, indicating that Galileo had calculated the positions and based on the results of these calculations marked the points before. Only in a second step did he add the line x4, whose intersections with the inclined planes determined the exact positions.
- 31.
The points z, δ, k (the mean proportional on nc remained unmarked) do not lie exactly on the line from 4 to the midpoint of the horizontal cx, indicating again that their positions were first calculated by Galileo and that the general construction was added only later. The deviations from the geometrically precise positions are, thus, due to rounding errors. Folio 176 recto contains a proof of the fact that all points marking the mean proportionals lie on the line from point 4 to the midpoint of the horizontal cx. The proof is correct, yet it was marked by Galileo as “[f]alsa est.” The reason for this is not clear.
- 32.
Cf. Chap. 9.
- 33.
139 verso , T1A and T1B.
- 34.
130 verso , T1D.
- 35.
130 verso , T1F.
- 36.
Galileo wrote on 174 recto with the page turned 90∘ with respect to the modern orientation in the manuscript. Positions are specified with regard to the orientation of the page as it was used by Galileo.
- 37.
187 recto , C2 to C5 and T2.
- 38.
The time along af resulted to 54 instead of 53 time units. Thus, the time along afs amounted to 77 time units, calculated and noted by Galileo to the right of the list.
- 39.
174 recto , T2.
- 40.
That Galileo proceeded in the same way in the more extensive calculations whose results are put down underneath the diagram is indicated by the way he noted the results. In the table, the distances for ae, ad, and ed are given together with the times of motion along these distances. The respective values for fs are not contained in a separate table row, but Galileo simply wrote fs next to ed, which indicates that what he had calculated were the respective values for ed which, based on his construction, also applied to fs.
- 41.
147 recto , T1C (incorrectly transcribed in the electronic representation of Galileo’s Notes on Motion) and 89b recto T1A and T1B.
- 42.
164 recto , T1.
- 43.
47 recto , T2A.
- 44.
Galileo could have resolved this problem by splitting up what he issued as one proposition, Proposition XI, into two independent propositions. The generalized law of fall straight could then have been presented before the generalized length time proportionality from which then the generalized law of fall bent plane could have been derived.
- 45.
Wisan and Drake both claimed that Galileo formulated the least time propositions prior to his insight into the law of fall. Drake (1978, 79), referring to elaborations of two of the least time propositions on folios 140 and 127, states that they “follow immediately from Galileo’s theorem [the law of chords] without considering acceleration at all,” that is, before Galileo started to assume that falling motion is naturally accelerated. Wisan (1974, 171), who provides a thorough analysis of the propositions of the least time group, is somewhat more careful and concludes her discussion of the relevant propositions with the remark: “They [the propositions of the least time group] depend only on the law of chords which appears to be the first of the published propositions on motion, and they may precede the times-squared law.” Caverni (1972, Vol. 4, 361–363), in contrast, assumes that Galileo formulated the least time propositions after his conceptual shift toward the assumption of natural acceleration.
- 46.
A somewhat more elaborate variant of Proposition XXXII, including a lemma required for its proof, was drafted by Galileo on 168 recto , likewise, on a double folio bearing the watermark/countermark combination crown, crossbow. The facing page contains a draft of 2/12-th-12, which exactly reproduces the logic by which Galileo on 130 verso had determined the time of motion along the bent planes under consideration.
- 47.
Wisan (1981, 319) writes: “[m]ore results then quickly follow as Galileo uses these new tools to develop enough mathematical theorems on motions along inclined planes to make up a whole treatise. …Galileo now has a genuinely new ‘science’ of motion. However, it has neither an experimental nor a theoretical basis. The times-squared law was most likely confirmed by an early experiment, but he has no proof from established principles, and there remain several fundamental puzzles concerning accelerated motion. From this time on, Galileo is more and more preoccupied with the problem of clarifying and explaining accelerated motion and finding properly evident principles on which to base the growing body of mathematical theorems.” This is an excellent portrayal of Galileo’s situation around 1602. Yet I do not share her interpretation on how Galileo arrived there.
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Büttner, J. (2019). Accumulating Insights: The Problem of Motion Along Broken Chords Driving Conceptual Development. 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_7
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