Abstract
In a letter to Guidobaldo del Monte written in November 1602, Galileo mentioned having found a proof for the so-called law of chords, i.e. the statement that fall along any chord inscribed in the same circle and sharing either its apex or nadir is completed in equal time. The proof alluded to in the letter is identified with one of the proofs Galileo provided for this proposition in the Discorsi. It is set apart from all other proofs regarding naturally accelerated motion contained in the Discorsi in that it makes a dynamical argument. This chapter reconstructs and discusses in great detail the sequence of steps by which in 1602 Galileo initially established the dynamical argument forming the basis of this proof. Contrary to the widely accepted opinion that Galileo constructed this argument while still holding the assumption characteristic of his older De Motu Antiquiora that motion along inclined planes is in principle uniform, it is thus shown that the argument was constructed only after he had come to accept that falling motion was naturally accelerated. It is, moreover, demonstrated that Galileo was already aware of the law of chords when he first constructed a dynamical argument in support of it. The new proof could be considered to rest on fundamental principles and Galileo indeed earmarked it as being intended to replace a proof of the same proposition he had constructed earlier. The way in which Galileo may have come to accept the law of chords as a heuristic in the first place is discussed.
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Notes
- 1.
In a series of articles together with different co-authors, Souffrin has analyzed Galileo’s concept of velocity. Cf. Chap. 2. He comments upon Galileo’s application of one of the kinematic propositions as a premise in the ex mechanicis proof as follows: “It is generally assumed that Galileo’s mechanical demonstration of theorem VI on accelerated motion in the Discorsi relies on an illegitimate use of a theorem valid only for uniform motions. I will show that this view depends on an anachronistic interpretation of the terminology of theoretical kinematics, and propose an alternative interpretation which conforms more to kinematics at the time of Galileo and which leads to a demonstration that is devoid of the alleged error (Souffrin and Gautero 1992, 269).”
- 2.
“Sin qui ho dimostrato senza trasgredire i termini mecanici (EN X, 99)” Cf. Chap. 3.
- 3.
Theorem VI Proposition VI of the Second Book of the Third Day of the Discorsi compared motion along a chord to motion along the vertical diameter of the circle. The more general case stating that motion along two arbitrary chords inscribed in the same circle and comprising either the apex or nadir was made in the same time, i.e., the statement referred to here as the law of chords, is actually established in a corollary to the proposition.
- 4.
“Idem aliter demonstratur ex mechanicis (EN VIII, 221)”.
- 5.
EN, VIII, 221–223.
- 6.
According to the wording, Galileo’s sentence could also be interpreted as implying that if the time of motion along any distance on CA was the same as the time of motion along any other distance on DA then the spaces traversed must be in the ratio of BE to DF. This is, however, quite clearly not what was intended, and it is assumed here that Galileo was referring to motions from rest in C and D, respectively, only.
- 7.
In drawing this conclusion, with regard to the premise expressed in the previous step, Galileo seems to have reversed the direction of the inference. Yet, as we will see, the relation expressed in the previous step was for Galileo indeed a logical equivalence and thus an inference from equal times of motion to a ratio of distances traversed was for him as legitimate as the inference from a ratio of distances to the equality of the times of motion.
- 8.
Cf. Chap. 2.
- 9.
- 10.
Galileo introduced the propositions advanced in the First Book of the Second Day of the Discorsi as follows: “…in prima parte consideramus ea quae spectant ad motum aequabilem, seu uniformem (EN VIII, 19).”
- 11.
Renn has already remarked that neither the statement nor the proof of Proposition II of the First Day are directly restricted to the case of uniform motion and concluded that “[t]heorem II, corresponding to one of the Aristotelian proportions, is in fact also valid for non-uniform motion …(Damerow et al. 2004, 260).”
- 12.
If the ex mechanicis proof had indeed been conceived under the old conceptualization of motion of fall, it could be argued that Galileo may at some point have become convinced that it was likewise valid under his new conceptualization, not least because the premises employed were ultimately indifferent to the type of motion. Such a position was taken, for instance, by Souffrin. When issuing a version of the same proposition Torricelli, possibly because he realized the problem entailed by Galileo’s reference to Proposition II of the First Book, explicitly stated that it was valid for accelerated motion. Cf. Torricelli (1919, 112).
- 13.
The same happens once more in the Discorsi, when in the proof of Proposition III of the Second Book of the Third Day of the Discorsi, i.e., the length time proportionality, Galileo shows that the velocities of the motions along two inclined planes of equal height are equal and then concludes “Sed demonstratum est, quod si duo spatia conficiantur a mobili quod iisdem velocitatis gradibus feratur, rationem habent ipsa spatia, eamdem habent tempora lationum (EN VIII, 216–217).” This is an explicit reference to Proposition I of the First Book, where thus again a proposition proved formally only for uniform motion is invoked in a conclusion about naturally accelerated motion.
- 14.
It is tempting to attribute the slight inconsistency in the deductive structure to the unfavorable circumstances under which the Discorsi were composed. Cf. Drake (1978), in particular Chap. 20.
- 15.
- 16.
In a number of additional entries Galileo argued for motions along inclined planes completed in equal time in a manner comparable to the ex mechanicis proof. However, in contrast to the law of chords, in these considerations Galileo did not specify the planes under consideration as chords inscribed in a circle but through a different geometrical constraint. These considerations will be discussed in Chap. 10 where it will be argued that these considerations must postdate the construction of the ex mechanicis proof.
- 17.
The ex mechanicis proof was eventually copied by Mario Guiducci from folio 172 recto to 47 recto introducing only some minor corrections that may have been directly dictated by Galileo to his disciple.
- 18.
131 recto , T1 and T3.
- 19.
The different inks used on folio 131 recto are suspiciously similar to the different types of inks used on folio 150 recto . As argued in Chap. 5, the entries on the latter page should have been drafted at about the same time as those on folio 131 recto .
- 20.
131 recto , T2.
- 21.
Not only did Wisan claim that here Galileo was already referring to a body moving continuously but she has even proposed that Galileo may, in this way, have inferred that the descent of a body on an arc was accelerated. Cf. Wisan (1974, 161–162). Galluzzi, in contrast, emphasized that Galileo does not talk explicitly about a continuous descent, and concluded that “la sua analissi si mantiene, cioè, ancora sostanzialmente << statica>> . (Galluzzi 1979, 217)” He consigns, however, that Galileo probably would have asked himself how the diminishing static moment on the arc could be reconciled with the accelerated motion of a descending pendulum. Galluzzi indeed suggested that it may have been due to precisely this problem that Galileo decided to present a purely static analysis and avoided discussing any dynamic implication in Le Meccaniche. Cf. Galluzzi (1979, 216–217 and 269).
- 22.
131 recto , T4.
- 23.
A diagram on folio 154 recto compares to that on 131 recto in that it contains a plane tangent to a circle together with a construction that would allow for an immediate application of the argument of the bent lever proof, as well as a parallel transposed plane inscribed as a chord in a circle. Just as based on the construction on 131 recto , likewise based on the construction on 154 recto , the ratio of the mechanical moment on this chord and on the vertical chord could immediately be inferred. The diagram on 154 recto is unfortunately not associated with any textual entry, and it cannot, thus, unanimously be concluded that the diagram served such a consideration even though this appears to be rather likely. Folio 154 recto otherwise contains considerations concerning the period of a pendulum whose length corresponds to the radius of the earth. Cf. Büttner (2009).
- 24.
151 recto , T1A, T1B and D01A as well as D02A.
- 25.
That the construction on folio 151 recto is closely related to Galileo’s bent lever proof has already been noted by Galluzzi. He argued that the method and terminology used on folio 151 recto , in particular Galileo’s use of the expression “momentum super” instead of “gravitas in plano”, places the sketch of the ex mechanicis proof on the page conceptually closer to Le Meccaniche than to De Motu Antiquiora. Cf. Galluzzi (1979, 267).
- 26.
In their own interpretations of folio 151 recto , Wisan and Galluzzi have likewise assumed that the argumentative steps, which remained implicit, must be the same as those of the published ex mechanicis proof, i.e., the application of the basic principle of inclined plane dynamics and of the second Aristotelian kinematic proposition. Galluzzi states “[t]uttavia, la proporzionalità implicita, ma evidentissima, momento-velocità, indica che lo scienziato pisano è tornato a lavorare sull’ipotesi dinamica utilizzata nel De motu. (Galluzzi 1979, 267). Wisan remarked “[t]he conclusion follows from the converse of the mediaeval assumption, which we have seen in the Liber Karistonis and in BRADWARDINE, that in equal time intervals, speeds are proportional to distances traversed. Speed is not distinguished from momentum (Wisan 1974, 164).”
- 27.
Galluzzi (1979, 266) has observed that the proof on folio 151 recto may “conservarci la dimostration [della quale Galileo dichiarava a Guidobalo nel novembre del 1602] esattamente come Galileo la concepì al tempo della lettera a Guidobaldo.” He explicitly rejects making a decision on whether the entry was drafted before or after Galileo’s conceptual shift but clearly leans toward the second alternative when he states “[n]on esiste elemento preciso che ci consenta di stabilire se Galileo stia pensando, in questi theoremi, a moti accelerati o uniformi; anche se la ricomparsa del teorema del De motu, precedentemente abbandonato, potrebbe far supporre che Galileo abbia voluto verificare la validità nell’ipotesi discese accelerate. (Galluzzi 1979, 267)” Wisan (1974, 163), in her discussion of the proof on folio 151 recto , carefully circumnavigates the question as to whether Galileo was already working under the assumption that motion on inclined planes was naturally accelerated. Yet, only some pages later, she states that a number of propositions and proofs, among them the ex mechanicis proof, “may precede the times-squared law. (Wisan 1974, 171)” In her periodization of the content of the Notes on Motion presented in the form of a list, she finally assigns the ex mechanicis proof to the period antedating Galileo’s conceptual shift. Cf. Wisan (1974, 163, 171 and 296).
- 28.
151 recto , C01.
- 29.
The first sketch of the argument on 151 recto , the draft on 160 recto and naturally its copy on 172 recto , all regard the motion situation where motions along the chords under consideration end at the lowest point of the circle. An earlier proof of the law of chords contained on folio 172 recto , in contrast, considered the situation where the motions along the chord all started from the highest point of circle. Both cases are of course eventually equivalent yet regarding motions starting from the same point seems to have been somewhat more natural for Galileo than regarding motion ending at the same point. The focus put on motion along chords ending at the same point in the construction of his ex mechanicis proof argument was apparently inherited from his considerations regarding the relation between swinging and rolling that had prompted the proof idea for the ex mechanicis proof.
- 30.
As detailed in Chap. 10, it is somewhat unclear whether the watermark of folio 151 is in fact a small star or, alternatively, a large star. These different marks may in fact merely be different manifestations of what is, in principle, the same watermark, suggesting that all folios bearing these marks are affiliated.
- 31.
The average height of those folios sharing the large star watermark with 160 is 281.5 mm; what remains of folio 160 measures exactly 140 mm in height; hence the folio was apparently cut exactly in half.
- 32.
172 recto , D02A.
- 33.
The fact that the mechanical moment of a body on an inclined plane depends merely on the inclination of the plane is reflected in Galileo’s use of expressions such as “momentum ponderis super plano secundum lineam abc elevato …”.
- 34.
Statements of the law of the inclined plane have been preserved on the cut sheets 179a verso for an inclined plane and its vertical height and on 179b verso for two inclined planes of equal height. In his argument on folio 160 recto , Galileo employed the law of the inclined plane as formulated on 179a verso and inferring the ratio of mechanical moments for the more general case of two inclined planes of equal height takes up most of the entry on 160 recto . Handwriting and layout suggest that the content of the second pasted sheet, 179b may indeed have been added at a later date and that, at the same time the first sentence on 179a verso containing the general statement was added. Cf. also Chap. 10.
- 35.
Guisti has pointed out that whereas in his argument on 151 recto Galileo inferred that the motions under considerations were completed in equal times, in the argument on 160 recto , he presupposed the times of motion to be equal, inferring the ratio of the distances that would be covered to find that this corresponded to the ratio between the chords. He claimed that “There is no reason to compare spaces traversed in equal times, unless one knows that the mechanical method can be only applied to the comparison of motions taking place in equal times” and takes this as a “sign that Galileo has become aware of the fact that the mechanical method has a limited validity…(Giusti 2004, 125)” However, as will be argued in Chap. 10, Galileo’s insight into the restricted applicability of his mechanical method almost certainly postdated his entries on 160 recto .
- 36.
That the entry from 160 recto was copied to 172 recto and not vice versa is entirely clear as in the former the phrase “extensam dg usque ad circumferentiam,” which does not occur in the latter, is written out in the former and then deleted. Otherwise only two spelling errors have been corrected, the punctuation was changed and “demonstratur” substituted for “probatur” when copying. The virtual identity of the two entries indicates that writing down the new entry was not engendered by the intention to introduce a change in the argument but rather served the insertion of the proof in its correct intended position in the collection of material being prepared at this time. Cf. Chap. 10.
- 37.
The leaps in the sequence of individual steps that, as reconstructed here, led Galileo to establish the ex mechanicis proof, are so small that it is hardly conceivable that he wrote more down in this respect than has been preserved. In other words, the ex mechanicis proof seems to provide one of the rare cases where Galileo’s considerations have been preserved in the manuscript without gaps. It is, thus, entirely clear that Galileo is not recapitulating or exploring an existing argument here but that he is developing it.
- 38.
172 verso , T1C. Cf. Chap. 10.
- 39.
Laird (2001, 267), for instance, states: “The conclusions of Galileo’s new science of motion may have been purely kinematic and modelled after the statics of Archimedes, but they are ultimately founded upon …the principle of the balance …”. As the ex mechanicis proof is indeed the only one of Galileo’s arguments which directly relates the principle of balance to the kinematics of naturally accelerated motion, this is a perfect example of how repeatedly, the ex mechanicis proof has been assigned a crucial role in Galileo’s new science with rather broad-brushstroke claims. Such claims, as a rule, are not the result of a careful analysis of the Notes on Motion but rather appear to be based on an understanding drawn mainly from Galileo’s correspondence.
- 40.
Numerous studies have touched upon the questions related to the ex mechanicis proof and its genesis and only those that have examined the proof and its relation to the general framework of the new science of naturally accelerated motion in some detail are briefly surveyed here. Wisan refers to the law of chords as the oldest of the propositions that Galileo published in the Discorsi and claims that the ex mechanicis proof was the first proof Galileo had found. In her discussion of the proof, she hesitates to decide whether it was conceived before or after Galileo’s conceptual shift but at a later point of her analysis assumes the former. She argues somewhat circularly when she claims the least time propositions must have been drafted early, because they exclusively employ the law of chords as a premise, just to conclude some pages later that the law of chords must be the earliest proposition exactly because it is used as a premise in the proofs of these propositions. According to Wisan the ex mechanicis proof must stem from the “quite early” Paduan period, and she expresses little doubt that it is, in fact, datable to 1602, based on the letter to Guidobaldo del Monte. Cf. Wisan (1974), in particular pages 162–163. Galluzzi (1979, 266) likewise leaves the question of whether the ex mechanicis proof was conceived before or after Galileo’s paradigm shift open but leans toward the second alternative. Drake (1978, 67) held that the ex mechanicis proof was originally anchored in the framework of the De Motu Antiquiora: “Given Galileo’s two erroneous conclusions in De motu, that acceleration may be neglected and that their ‘speeds’ of descent (regarded as constant) along two different incline planes of equal height are inverse to the lengths of the planes, any Euclidean geometer could easily reach Galileo’s theorem [the law of chords ] by inspection of the diagram.” Renn follows the standard interpretation, when he states, for instance, that the law of chords is “a theorem on motion along inclined planes which directly follows from his theory in De Motu…” Some pages later he reiterates that “Galileo had obtained the first presupposition, The Isochronism of Chords [law of chords ], as a direct consequence of the theory of motion along inclined planes in De Motu …” Naylor (2003, 156–159), when addressing the “remarkable developments” revealed by Galileo’s letter to Guidobaldo del Monte in 1602, expresses his belief that the ex mechanicis proof must hence have succeeded Galileo’s conceptual shift toward the assumption of natural acceleration but states that there is a “complete lack of direct evidence” for these developments in the sources. (Humphreys 1967, 234), without any consultation of the manuscript sources, speculates about a way in which the law of fall may have been inferred as an “out-growth of his [Galileo’s ] juvenile speculations on dynamics” and states with regard to the law of chords that it is “plausible to assume that the proof he had was the dynamical one given in Two New Sciences and based on the work done around 1598 for the book On Mechanics.” Hooper (1998), who in his Ph.D. thesis has quite carefully examined parts of the Notes on Motion, essentially follows Humphreys’ account. Hooper (1992) has included a discussion of the manuscript sources in his analysis of the conceptual underpinnings of the ex mechanicis proof, but remains conspicuously silent about whether Galileo, when he first came up with his ex mechanicis proof, still regarded motion on inclined planes as essentially uniform or already as naturally accelerated. Souffrin (2001) finally, despite having convincingly argued that the assumption that led people to believe that the law of chords was conceptually anchored in the De Motu Antiquiora theory in the first place was erroneous, sticks to the assumption that the ex mechanicis proof predated Galileo’s conceptual shift toward assuming natural acceleration.
- 41.
Put anachronistically, for two uniformly accelerated motions from rest completed in the same time the average velocities are in the same ratio as the accelerations. Hence distances traversed in naturally accelerated motions will likewise be traversed in the same time by two uniform motions, provided their velocities are in the same ratio as the constant accelerations of the accelerated motions.
- 42.
Cf. Chap. 2.
- 43.
Drake (1978, 67).
- 44.
That the standard dating of the discovery of the law of fall to the year 1604 rests primarily on Galileo’s remarks in the letter sent to Paolo Sarpi in October 1604, and not on a careful analysis of other sources such as the Notes on Motion has been argued by Damerow et al. (2001). Resorting to the work of Drake as an example, they state: “But even after Drake had extensively studied Galileo’s working papers, and after repeatedly changing his views on Galileo’s discoveries, he rather accommodated the dating of Galileo’s manuscripts to the standard dating than the other way around (Damerow et al. 2001, 304–305).” Analyzing the material related to Galileo’s insight into the parabolic trajectory of projectiles the authors state that “according to common historiographic criteria, Galileo must be credited with having made this discovery already as early as 1592” and that the law of fall “was merely a trivial consequence of this discovery (Damerow et al. 2001, 300).” The authors question the notion of discovery and call instead for a careful reconstruction of the “network of interdependent activities which only as a whole make an individual step understandable as a meaningful ‘discovery’.” Their stipulation has greatly influenced the current work.
- 45.
Assuming the onset of Galileo’s investigations into naturally accelerated motion to be 1604 has forced a number of authors to adopt rather unorthodox assumptions in their exegesis of the Notes on Motion. They acknowledge that some of the considerations documented in the manuscript conform well to Galileo’s announcements in the letter of 1602 and indeed date these considerations to this time. Based on the understanding that Galileo began to conceive of natural acceleration only in 1604, they are thus forced to assume that such considerations must be anchored in the understanding that motion along inclined planes is essentially uniform. Confronted with indications of the contrary, the additional assumption is usually invoked that years later Galileo must have returned to his older considerations and reassessed them from the perspective of his new conceptualization, adding new entries to older ones. A good example of this is Wisan. As has been done here, she as well has located the point of departure for Galileo’s construction of the ex mechanicis proof in the central diagram on folio 131 recto . She too dates this work to approximately 1602 but assumes that at this time Galileo still assumed motion along inclined planes to be characteristically uniform. At the same time she of course realized that the diagram on the page in the form of the points v and o, which mark mean proportionals, embodies an application of the law of fall. Thus in order to salvage her interpretation she simply but somewhat artificially claimed that these points were added much later in an alleged second attempt at the same problem after 1604: “It may be conjectured then that the central part of this fragment exhibits a very early stage in the investigation of the brachistochrone and that the first sentence (together with the smaller circle and the points O and V ) is a later addition stemming from a new attack that makes use of the corollary on mean proportionals [i.e., the law of fall ] (Wisan 1974, 177)”.
- 46.
The points reached from a given starting position in the same time of course form a surface. However, due to the rotational symmetry of Galileo’s problems it suffices to study the problem in two dimensions, i.e., to seek isochrone curves.
- 47.
Once Galileo had become convinced that the spacio-temporal behavior of the motions he had treated as uniform earlier was correctly specified by the law of fall, what would have been missing for a more comprehensive treatment of naturally accelerated would be an argument allowing to relate naturally accelerated motions along inclined planes of different inclination. Such an argument is of course exactly what an investigation of the isochrone curve of naturally accelerated motion can potentially provide.
- 48.
In the seventeenth century a circle was recognized as a geometrical curve, whereas the spiral was seen as belonging to the class of the mechanical curves. Geometrical curves could be determined from one motion and produced with the conventional tools, compass and ruler. The mechanical curves were defined by two or more motions and hence required more complex instruments to produce them. Descartes (2001) vehemently attacked the distinction. Cf. Büttner et al. (2003, 5–6).
- 49.
The experiment is described in Mersenne (1963, 111) and analyzed in detail in Settle (1966) and Naylor (2003). Naylor (1974, 121) claims that “it is almost certain that from 1602 onward Galileo tested this statement [the law of chords ].” Using an inclined plane he had built to recreate Galileo’s projection experiments, he also conducted an experimental test of the law of chords. His results compare to the ones given by Mersenne, and “it seems credible to allow that Galileo could have obtained observations similar to these (Naylor 1974, 122).” As the difference between rolling and free falling shows in the results, these observations, however, do not unreservedly support the law of chords, at least as long as a free falling motion is compared to motion along an inclined plane.
- 50.
Alternatively, it has been claimed that Galileo’s emphasis on having proven everything thus far without transgressing the terms of mechanics was in fact a reaction to Guidobaldo del Monte’s position that mechanical principles “did not only comprise the theory as it is exposed in the ancient texts but also a strict correspondence between theory and practical experience,” an attitude which “led him to be skeptical with regard to studies involving motion.” Hence Galileo needed to convince Guidobaldo that “in spite of the novelty of the subject for traditional mechanics, he is still adhering to the principles of this mechanics which had been the starting point of their exchange (Damerow et al. 2001, 358).”
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Büttner, J. (2019). Toward a Foundation: The Ex Mechanicis Proof of the Law of 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_8
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