Abstract
The chapter deals with Galileo’s considerations regarding motion along polygonal paths inscribed into an arc that have been preserved in the Notes on Motion. Galileo’s approach was motivated by the assumption that fall of a heavy body along a concave surface supported from below was kinematically equivalent to the swinging of a pendulum bob along the same arc supported, however, from above. As Galileo was not able to make an inference regarding motion of fall along the arc directly, he instead investigated motion along a series of adjoined inclined planes inscribed as chords into the arc. If the number of inclined planes is increased above all limits, the polygonal path exhausts the arc and in this way Galileo hoped to eventually be able to construe an argument regarding falling, and thus swinging, along the arc. Galileo’s approach yielded a number of tangible results, among them a proof for the statement that motion along a broken chord takes less time than motion along a single chord spanning the same upright arc subtending an angle of 90 degrees or less, a proof he mentioned in a letter to Guidobaldo del Monte in 1602. Based on manuscript evidence, Galileo’s extensive work on the problem is indeed dated to 1602. Despite considerable partial accomplishments Galileo’s approach was not crowned with success. By yielding indications that motion along the arc could not be isochronous, it indeed rather pointed to a problem in the underpinning conceptualization.
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Notes
- 1.
Galileo did not explicate his assumption that rolling and swinging along the same arc were kinematically equivalent. The closest he came to doing so was in a letter sent to Lorenzo Realio in June 1637. In the letter, he stated that rolling along circular channels (“canali”) was isochronous, and immediately after this, he stated that the same held true for pendulum motion, insinuating but not explicating a relation between both statements. Cf. EN XVII, letter 3496.
- 2.
As further detailed below, the swinging of a pendulum along an arc and the falling along the same arc are equivalent only if friction effects are completely ignored, i.e., in particular, when the body falling along the arc is sliding frictionlessly. In reality, due to friction, a body falling along the concave surface will usually not be slipping but rolling. As will be argued in the following chapter, Galileo was unaware of the difference between rolling and sliding with regard to the kinematics of the ensuing motion. Ultimately, this prevented him from successfully exploiting the results of his pendulum plane experiment.
- 3.
Here and in the following, I refer to an upright arc as one that has a horizontal tangent in its lower endpoint.
- 4.
If I use the word “approximate” here and in the following, I do not mean to imply that Galileo held anything that would compare to a modern concept of approximation. In particular, for him, the idea that describing a physical process as a simplified model could be used to reach a sufficiently accurate solution would be rather meaningless, not least because he had no way of demonstrating how, given a certain model, a magnitude thus calculable deviated from the magnitude approximated only within a certain limited margin of error. For Galileo, considering motion along a polygonal path was merely a step toward the solution, preferably to be established by an argument based on the method of exhaustion.
- 5.
Wisan (1974, 179) refers to the proof as being “unusually clumsy” without, however, further explicating her opinion.
- 6.
In the Discorsi, that bodies falling from the same height, here from A and D, respectively, to B acquire the same moments of velocity (“velocitatis momenta”) had been formulated as one of the fundamental principles based upon which the new science of naturally accelerated motion on inclined was founded: “Accipio, gradus velocitatis eiusdem mobilis super diversas planorum inclinationes acquisitos tunc esse aequales, cum eorumdem planorum elevationes aequales sint. (EN VIII, 205)” Clavelin (1983, 45) has suggested that initially the assumption that equal degrees of velocity were reached after fall through the same height, independent of the path taken, was based on a “general intuition that …is also the basis of statics and of the theory of simple machines.” Strictly speaking path invariance furthermore presupposes what Wisan (1974) terms the “assumption of continuity of speeds at corners.” This is, however, never made explicit by Galileo.
- 7.
In view of the fact that the majority of the proofs of the Second Book of the Third Day rely on path invariance, it is somewhat surprising that Galileo never explicitly formulated as a principle that, if naturally accelerated motion along an inclined plane is deflected to a second plane, the time elapsed in traversing this second plane is the same regardless of whether the preceding fall is made through the first plane or through any other plane of the same vertical height as the first plane.
- 8.
One can distinguish between numerical and symbolic distance time coordination. Whereas numerical distance time coordination primarily renders calculations more convenient, symbolic distance time coordination has more far-reaching consequences. Galileo’s numerical calculations of times of motion usually start by assigning a numerical value to the length of a distance traversed and a numerical value to a time elapsed during motion, usually motion along this distance. From a modern perspective, this amounts to assigning (arbitrary) units of distance and time. It is often convenient to identify the numerical value representing the measure of the length of a distance with the numerical value representing the measure of the time to traverse this distance, as this facilitates calculation. In the case of symbolic distance time coordination, two magnitudes, which are represented symbolically, that is, geometrically, with no particular numerical value being assigned to them, are identified with one another. Symbolic distance time coordination is typically established by statements of the kind: “[s]i itaque intelligatur, tempus per AC esse ipsamet AC …(EN VIII,229)” The time through AC in the example can of course not be the distance AC itself (“esse ipsamet”) in a literal sense, but AC represents the time through AC quantitatively. In diagrams making use of integrated time representation, distance time coordination amounts to the identification of one and the same line as representing a distance covered and simultaneously a time elapsed. That Galileo had a precise idea of how distance time coordination functioned as a conventional and convenient assignment of an arbitrary measure is revealed by statements such as the following taken from discussion on the Third Day: “[s]tabilite ad arbitrio nostro sotto una sola grandezza ab queste 3 misure di generi di quantità diversissimi, cioè di spazii, di tempi e di impeti …(EN VIII, 227–228).”
- 9.
In the published proof, the times of motion are represented by distances on the separate external timeline TP. Galileo’s original proof, for ease of comprehension, has been re-rendered into a form corresponding to his later argumentative technique based on integrated time representation. The original passages, whose re-interpretations are presented in the text, are provided as footnotes for means of reference and control. The sentence in Galileo’s proof that corresponds to the statement transcribed in the symbolic notation as (5.1) in the text reads: “Sit autem PS tempus quo peragitur tota.”
- 10.
In the proof: “erit tempus PR id, in quo mobile ex D peragit.”
- 11.
In the proof: “RS vero id [tempus], in quo reliquum.”
- 12.
It is worth noting that Galileo could have established the same by employing Proposition XI of the Second Book of the Third Day, the so-called generalized law of fall. As will be argued in Chap. 6, Galileo formulated this proposition only after he had drafted his proof of the law of the broken chord. Thus, the straightforward two steps argument, which leads from the law of fall to the generalized law of fall, had to be included in the proof of the law of the broken chord when it was originally devised and it remained part of the proof which, once completed was, as will be seen, never reworked but included almost verbatim in the Discorsi.
- 13.
In the proof: “erit PT tempus casus ex A.”
- 14.
According to the length time proportionality, the times of naturally accelerated fall from rest along two inclined planes of equal height are to each other as the lengths of these planes. Theorem III proposition III of the Second Book of the Third Day of the Discorsi is actually restricted to the case where one of the two inclined planes is a vertical. The more general case of motion along two inclined planes of equal height referred to here as the length time proportionality is treated in a corollary to the proposition.
- 15.
- 16.
In the Discorsi, Galileo provided a number of equivalent formulations of the law of fall, which he introduced as Proposition II of the Third Book of the Third Day of the Discorsi. The one that is used here and indeed the one that almost exclusively is brought to bear in the considerations documented by the Notes on Motion is the one Galileo formulated as the second corollary: “Colligitur, secundo, quod si a principio lationis sumantur duo spatia quaelibet, quibuslibet temporibus peracta, tempora ipsorum erunt inter se ut alterum eorum ad spatium medium proportionale inter ipsa. (EN VIII, 214)”
- 17.
In the proof: “erit PG tempus quo mobile ex A venit.”
- 18.
This conclusion could have been arrived at before by invoking the generalized law of fall which, however, as argued, had not been formulated as an independent proposition when Galileo wrote the draft of the law of the broken chord, a draft he ended up publishing almost unaltered in the Discorsi.
- 19.
In the proof: “GT vero tempus residuum motus BC consequentis post motum ex A.”
- 20.
In the proof: “ostendendum itaque est, RS maius esse quam GT.”
- 21.
Lemma 2 to Theorem XXII, Proposition XXXVI of the Second Day of the Third Book of the Discorsi reads: “Let AC be a line which is longer than DF; and let the ratio of AB to BC be greater than that of DE to EF. Then, I say, AB is greater than DE. (Galilei 1914, 235–236)”
- 22.
Apart from the final draft on folio 172, no further notes can be identified which are related to the first lemma associated with the law of the broken chord. The lemma is so trivial that it was conceivable that Galileo drafted it directly on folio 172 without any preparatory notes.
- 23.
Of the 163 folios that make up the Notes on Motion 35 bear the hand of Niccolò Arrighetti or Mario Guiducci, who were both Galileo’s disciples. For most of the entries in the hands of the disciples, an original in the hand of Galileo still exists and Favaro noted: “when we have a given fragment only in the hand of the disciples, we are allowed to surmise that they limited themselves to transcribing from a Galilean original which now is lost. (EN VIII, 34. My transl.)” All copies were made on the same type of paper not occurring elsewhere in the manuscript, i.e., no copy was made on a paper of different type, and neither does any folio of this type exist which does not contain disciple copy. The watermark is a spread-eagle with a crown above, the paper quality is thin, and white and all folios are folded in the middle. Niccolò Arrighetti and Mario Guiducci sometimes copied from the same original sheet without duplicating the work of the other, suggesting that they worked together at the same time under the supervision of Galileo. Two folios, folios 58 and 68, bear entries by both, Arrighetti and Guiducci. On folio 68 recto , the entry in the hand of Guiducci was written before the entry by Arrighetti. On folio 58, the situation is reversed. Here, the entries by Guiducci must have been made earlier, since Arrighetti added a remark referring to the content of the page that had been written by Guiducci. The entries by both disciples must hence have been made concurrently. Galileo first met Arrighetti long before his first encounter with Guiducci in 1614, which thus provides a terminus post quem for the copying. Galileo stayed in contact with both of his disciples more or less for the rest of his life and hence no terminus ante quem can be determined. Drake (1970) asserted that Guiducci must have made his copies as early as 1614 or, alternatively, between the years 1616 and 1620. Later he specified this dating to the year 1618, based on comparison of the watermark of the paper used for copying and the watermarks of letters written at this time. Cf. Drake (1978, 67, 262–263).
- 24.
186 verso , entry T1A.
- 25.
The condition which Galileo arrived at in his first attempt to solve the problem was that the line xr had to be smaller than line nm. A detailed discussion is contained in the Appendix in Chap. 13.
- 26.
186 verso , entry T1B.
- 27.
186 verso , diagram D01A.
- 28.
186 verso , entry T2.
- 29.
In the central diagram on folio 186 verso , there are two points that are lettered o. As the first sentence shows, in his note Galileo is referring to the junction point o on the arc and not the point o marking the mean proportional on the chord dc. The time of fall along the second chord oc from rest at the junction point o was, by virtue of the law of chords, equal to the time of fall over the single chord spanning the arc of the broken chord, i.e., dc. The law of the broken chord in turn stipulated that the time of motion along the single chord was longer than the time of motion along the broken chord, and this necessarily also longer than the time of motion along the first part of the broken chord do as stated in the note.
- 30.
It appears that this entry, together with entry T3, which are both written in a conspicuously black ink, had been noted on the page first, as the other content is manifestly composed around these two entries. On 131 recto , likewise, two entries are made in an ink that is noticeably darker than the ink of the other entries on the page. On 131 recto the entries in the darker ink were likewise probably the ones composed earlier. As argued in Chap. 8, the entries in darker ink on 150 and 131 were probably made at more or less exactly the same time.
- 31.
150 recto , entry T2A.
- 32.
150 recto , entries T1A to T1D.
- 33.
150 recto , entry T4.
- 34.
150 recto , entry T4.
- 35.
Most of the changes between the manuscript and the published version of the proposition are to be found in the first sentences. These changes do not concern the line of argument but for the most part serve to render more precise the geometrical construction required for the proof. The most relevant changes are the following: “mobile ex termino ferri …tempore breviori” (draft version) vs. “tempus descensus …brevius esse” (final version); “citius conficere” (draft version) vs. “citius permeare” (final version); “Quod sic ostenditur:” (final version) vs. “Quod sic demonstratur:” (draft version); “Constat insuper” (final version) vs. “Constat igitur” (draft version); and “veniens ex db ac si veniret ex ab” (draft version) vs. “ex D per DB ipsam BC, ac si venerit ex A per AB” (final version).
- 36.
An analytic solution for the calculation of the time of motion along a broken chord depending on the position of the junction point is given in Erlichson (1998). The more general solution for the time of motion along a polygonal path made up of an arbitrary number of chords of equal length and inscribed into an upright quarter circle is given in the discussion of folio 166 in the Appendix in Chap. 13.
- 37.
See Sect. 7.3
- 38.
Point r is lettered in the diagram on double folio 156 157; it is constructed but not lettered in the diagram on 148.
- 39.
In the main diagram on folio 148 recto , point b was changed to point c. Originally, the endpoints of the vertical timeline had been a and b, where the distance ab simply represented the time of fall through the distance ab. Most likely to avoid confusion, Galileo changed the lettering of these points to q and p, respectively, and made the appropriate changes in T1, which he had already written.
- 40.
148 recto , entry T1.
- 41.
In her discussion of the considerations documented on folio 131 recto , Wisan (1974, 176–177) acknowledges that these considerations may have been “directed at a search for a proof of the isochronism, rather than the brachistochrone.”
- 42.
The line tb in the diagram on 131 recto corresponds to nr marked on the radius of the big circle, na in the diagram on folio 188 recto . The mean proportional ds in the diagram on 131 recto reoccurs as no in the diagram on 188 recto .
- 43.
Folio 166 recto , table C01.
- 44.
For Galileo’s double distance rule, see Chap. 10.
- 45.
In rare cases, Galileo would also use multiples of fractions smaller than one tenth to express magnitudes smaller than the unit. Thus he had no problem in principle with expressing small fractions. Yet going over to a smaller base unit to avoid having to deal with fractions, as in the example at hand, seemed to have offered the more convenient option.
- 46.
It is striking, yet likely coincidental, that on 166 recto Galileo specified the length of the radius of the circle to measure 100, 000 units, while when illustrating the law of chords in his letter to Guidobaldo del Monte, he used the example of an arc whose radius measured 100, 000 miles.
- 47.
166 recto , entry T1C.
- 48.
166 recto , entry T3A.
- 49.
The calculations are not preserved.
- 50.
166 recto , entry T3B.
- 51.
In the Notes on Motion, Galileo does not usually use citius, the comparative of citus, synonymously with velox or velocior, respectively. Rather citius is almost always used as meaning in “in less time.” If the motions compared take place over equal distances the meaning of citius of course falls together with that of velocior. In his draft of Proposition XXXII of the Second Book of the Third Day of the Discorsi on folio 33 recto , Galileo, for instance, states: “Si in horizonte sumantur duo puncta, et ab altero ipsorum quaelibet linea versus alterum inclinetur, ex quo ad inclinatam recta linea ducatur, ex ea partem abscindens aequalem ei quae inter puncta horizontis intercipitur, casus per hanc ductam citius absolvetur quam per quascunque alias rectas ex eodem puncto ad eandem inclinatam protractas. (EN VIII, p. 253)” What is subsequently proven is that the motion claimed to be “citius absolvetur” takes less time than the motion which it is compared to without it being implied that it is also faster (velocior) according to Galileo’s understanding. On 163 recto Galileo uses “duplo citius” to mean in half the time.
- 52.
166 recto, entry T3B.
- 53.
166 recto , entry T1D.
- 54.
166 recto , entry T3A.
- 55.
Had Galileo indeed calculated the time of motion along the arc as the limiting case in the way described in the text, in the units of the base unit 100,000 approach, this would have yielded a time of 12,968 for motion on the arc. In comparison, he calculated a time of 131,078 time units for motion along the path made up of eight conjugate chords approximating the arc. From a modern perspective, the correct solution would be 130,622 time units. That the root-root hypothesis potentially applied not only to the broken chord but also to paths composed of a higher number of chords, thus allowing an inference to the limiting case of motion on the arc, however, remains merely plausible speculation.
- 56.
The hypothetical value that is calculated according to the root-root hypothesis for the time of motion along the broken chord aec is 242 time units. This overshoots the value of 236 1∕2 arrived at by doing the calculation based on the laws of motion by about as much as it is undershot if instead of the ratio of the roots of the roots, only the ratio of the roots is used for the calculation.
- 57.
According to the hypothesis, moving the junction point of a broken chord closer to the starting point of motions has the same effect as going over to an approximation of higher order, as it changes, the first incline of which the polygonal path is composed in the same manner and on which the construction of the hypothesis hinges. However, whereas the time of motion along a broken chord should get longer as the junction point moves toward the start point of motion, the time of motion along a path more closely resembling the arc should decrease. The construction underlying Galileo’s hypothesis was not capable of correctly reproducing both inverse tendencies at the same time.
- 58.
The point in question is marked with a dotted line and labeled t by Galileo. To distinguish it from the point labeled t on the timeline, it is referred to here as t′.
- 59.
In Proposition XV of the Second Book of the Third Day of the Discorsi, the first motion proceeds over a vertical and the successive one over an inclined plane. However, the proposition can easily be generalized to apply to the case of a broken chord in which the first motion likewise proceeds along an inclined plane as well. Folios 61 and 76, on which Galileo elaborated this proposition, both bear a radiant sun as a watermark, and the paper is clearly of Florentine origin. It thus appears that Galileo was only much later able to solve this particular problem that he first encountered in the context of the broken chord approach.
- 60.
The calculation 127 recto D01A, 141, 422 ∗ 239∕254 3∕5 = 132, 75[6], can be interpreted as a consistency check of the result calculated for the times of motion along the broken chord aec with respect to a base unit of 100,000 and a base unit of 180 on folio 166 recto . It must hold that: t(ac)base 100,000 ∗ t(ac)base 180∕t(aec)base 180 = t(aec)base 100,000. Aside from rounding errors, the calculated result should hence correspond to the time of motion along aec with respect to the base unit of 100,000 as it is listed on folio 183 recto , namely, 132,593.
- 61.
That the diagram on 121 recto is a modified copy of a diagram from Gilbert’s book was realized by Drake. The way in which he relates the remaining content of the folio to Galileo’s work program outlined in the letter to Guidobaldo del Monte in 1602, is tentative. Cf. Drake (1978, 67).
- 62.
Gilbert’s book is in the inventory list of Galileo’s library (Favaro 1886, 261, nr. 200). According to Drake, Galileo’s copy of the book was most likely given to him by Cremoni, who “seemed afraid to keep it on his shelves lest it infect his other books” (Drake 1978, 62–63). The discussion in the text is based on Gilbert (1958).
- 63.
The diagram is reproduced from the English edition, On the magnet, magnetick bodies also, and on the great magnet the earth: A new physiology, demonstrated by many arguments & experiments from 1900. It is reproduced from the Project Gutenberg EBook (http://www.gutenberg.org/files/33810/33810-h/33810-h.htm. Accessed 10 Dec 2016) and reused under the terms of the Project Gutenberg License.
- 64.
With regard to the degree scale, Galileo’s version differs from the original diagram by Gilbert, which has otherwise been copied more or less accurately. Galileo drew the diagram with the page turned 180 degrees with respect to the modern orientation. With regard to modern orientation of the folio, what I refer to as the apex of the circle, thus occurs as the nadir.
- 65.
For the context of Galileo’s and Sarpi’s exchange on magnetism and Gilbert in particular, see Heilbron (2010, 97).
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Büttner, J. (2019). Squaring the Pendulum’s Arc: Motion Along Broken Chords. 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_5
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