Abstract
The chapter discusses Galileo’s attempt to provide his new results concerning the pheno-kinematics of naturally accelerated motion—his new propositions concerning the regular relations between spaces traversed and the corresponding times elapsed in motions of this type—with a foundation. Providing his new propositions with a deductive structure rooted in fundamental principles or assumptions would allow Galileo to publish them as a new science. He, in particular, systematically explored the question of which of his new statements could serve as a minimal yet strong enough set from which all remaining propositions could be derived. This search for an axiomatic foundation disclosed that to root the deductive tree, at least two of his propositions needed to be equipped with a proof from basic principles. Further considerations deprived the only such fundamental proof he had found so far, and which was based on a dynamical argument, of its explanatory power. This brought Galileo’s search for a foundation temporarily to a halt. It is discussed how in 1604, as evidenced by a letter written to Paolo Sarpi, Galileo picked up work on the problem of a foundation once again, availing himself of a new approach based on representing accelerated motion as characterized by the change of degrees of velocity.
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Notes
- 1.
According to the Aristotelian conception, a demonstrative science has to be based on principles “true, primary, immediately better known than, prior to and grounds of the conclusion” (Posterior Analytics, I.2 71b20-22). For a discussion of the Aristotelian notion of basic principles, see McKirahan (1992, 21–50). In the contemporary philosophical debates, the thought movement from an effect to the existence of cause of this effect followed by an inference from this very cause to the effect was referred to as the regressus method. It was considered to be the foundation of scientific reasoning, yet usually not discussed in relation to physico-mathematical sciences such as Galileo’s new science of motion. For the regressus method in general, see, for instance, the corresponding entry in the Routledge Companion to Sixteenth Century Philosophy (Palmieri 2017). For the question to what extend Galileo can be said to have followed this method, see in particular Wallace (1992) and Jardine (1976).
- 2.
147 recto , D01A.
- 3.
147 recto , T2.
- 4.
147 recto , D02A.
- 5.
147 recto , T3, D03A and D03B.
- 6.
147 recto , T4 and T5.
- 7.
147 recto , T1A and T1B.
- 8.
147 recto , T1C.
- 9.
147 recto , D01A.
- 10.
147 recto , T2.
- 11.
Part of Galileo’s consideration concerning motion in media of De Motu Antiquiora was published in the Discorso intorno alle cose che stanno in sull’acqua in 1612. It was there that Galileo first introduced his assumption that all bodies would show the same kinematical behavior in a vacuum.
- 12.
The second entry in the Notes on Motion, which relates directly to motion in media, is contained on folio 66 recto . It is written in the hand of Mario Guiducci. In the entry, the change of the natural heaviness of a body on an inclined plane depending on the inclination is compared to the change of the heaviness of a body depending, according to the principle of buoyancy, on the density of the medium in which it is immersed. It is insinuated that the kinematics of falling motion should be comparable in both cases. After the consideration documented by the mirandum paradox, Galileo would have perceived the argument as problematic which may speak in favor of Mario Guiducci being the author. Indeed, no original draft in Galileo’s hand is extant from which the entry could have been copied.
- 13.
147 recto T1A. “It should be considered, that, just as all heavy [bodies] rest on the horizontal, be they very big or very small, so they are moved on inclined lines with the same velocity, just as along the perpendicular itself; it will be beneficial to prove that by saying that if a heavier [body] [be] faster it would follow that a heavier [body] [be] slower, having combined differently heavy [bodies], etc. (trans. Damerow et al. 2004, 203).”
- 14.
The wording indicates that the hollow paradox was written before the mirandum paradox. After the considerations documented by the latter, it would not have made sense for Galileo to speak of the equal velocity of two naturally accelerated motions without further specification, as he does in the hollow paradox.
- 15.
In the hollow paradox, Galileo uses the expressions “celeritas” and “velocitas” synonymously. Morphologies of the noun celeritas or the adjective celer occur 16 times on 7 folio pages in Galileo’s Notes on Motion, with one of the occurrences being a duplication by copying. Most of the occurrences congregate on one text contained on folio 91 verso , which contains early considerations pertaining to projectile motion. Just as celeritas is used by Galileo synonymously with velocitas, so momentum celeritatis is, albeit rarely, used synonymously with momentum velocitatis. The disappearance of the former expression from Galileo’s vocabulary seems to owe to his attempt to unify terminology.
- 16.
“Moreover, not only homogeneous and unequal heavy bodies would move at the same speed, but also heterogeneous ones such as wood and lead. Since as it was shown before that large and small homogeneous bodies move equally, you argue: Let b be a wooden sphere and a be one of lead so big that, although it has a hollow for b in the middle, it is nevertheless heavier than a solid wood sphere equal [in volume] to a, so that for the adversary it should move faster than b; therefore if b were to be put into the hollow i, a would move slower than when it was lighter; which is absurd. (trans. Damerow et al. 2004, 204).”
- 17.
For the concept of specific weight in Galileo, see Van Dyck (2006) (Weighing falling bodies. Galileo’s thought experiment in the development of his dynamical thinking, unpublished. Available at http://www.sarton.ugent.be/index.php).
- 18.
Cf. Damerow et al. (2004, 202–204).
- 19.
If the air included in the hollow sphere is taken into consideration when forming the composite body, the argument breaks down because then the example can be constructed such that the specific weight of this composite body lies between that of the wooden sphere b and the hollow lead sphere a. Cf. Damerow et al. (2004, 204, footnote 126).
- 20.
In the Discorsi the statement that the velocity of fall is independent of weight is directly adjoined to a discussion of the independence of pendulum frequency from the weight of the pendulum bob:
…and if Simplicio is satisfied to understand and admit that the gravity inherent [interna gravità] in various falling bodies has nothing to do with the difference of speed observed among them, and that all bodies, in so far as their speeds depend upon it, would move with the same velocity, pray tell us, Salviati, how you explain the appreciable and evident inequality of motion; (EN VIII, 131, trans. Galilei et al. 1954, 87.)
This accords well with the conjecture that Galileo had originally conceived the independence of the kinematics of motion along inclined planes on the weight of the falling body, which he argued for in the hollow paradox, as a consequence of the independence of the period of the pendulum on the weight of the bob together with his conceptualization of the relation between swinging and rolling.
- 21.
Galileo’s construction of a proof of the law of fall on folio 147 bears striking resemblance to his elaboration on folio 189 verso of a proposition that was later to be published as Proposition V in the Discorsi. In both cases, for instance, Galileo employs distance-time ratio diagrams and uses different types of lines to distinguish between lines representing times of motion and such lines representing distances covered by motion, lending additional plausibility to the assumption that work on the two folios was conducted roughly at the same time.
- 22.
Based on his discussion of the proof of the law of fall on 147 recto , Renn concludes: “this argument for the law of fall cannot be considered a proof on the basis of the De Motu theory, since one of the two premises on which it is based, the Length-Time-Proportionality, actually cannot be justified on the basis of this theory.” Yet when discussing the length time proportionality, Renn claims that it “was merely a plausible assumption, which Galileo at first also believed to be a consequence of his [De Motu Antiquiora ] theory (Damerow et al. 2004, 205).”
- 23.
Galileo’s use of the indicative future II has been overlooked by the authors who have discussed this entry. Renn, for instance, translates “After it has been demonstrated that the times through ab and ac are equals …” and states “At the beginning of his proof Galileo formulates the Isochronism of Chords as a theorem which he had already shown (Damerow et al. 2004, 205).”
- 24.
The assessment that the geometrical lemma invoked in the proof on 147 recto is the one on 172 verso is shared by Caverni (1895, 347).
- 25.
147 verso , D04A.
- 26.
147 verso , T2.
- 27.
147 verso, D02A.
- 28.
When examining the original manuscript, I missed the opportunity to inspect the page for incisions of a compass or uninked lines. Comparison to the related diagram on folio 155 recto suggests that such marks could be present.
- 29.
Hooper (1992, 345), for instance, lists a number of alternative interpretations and states that “[f]olio 147 may be the record of a historically important moment in Galileo’s real conceptual growth, or it may simply be an exercise of something he had already achieved, or it may have been a rehearsal for some public or private demonstration.” He, however, strongly suggests that the content of the folio was composed before Galileo’s conceptual shift toward the assumption of natural acceleration. Wisan (1974, 174–175), summarizing her detailed analysis of the entries on the page carefully surmised: “In this stage GALILEO may still regard motion as ‘naturally’ uniform” just to drop her caution and to more strongly conclude just a few sentences later: “[a]s has been remarked there is nothing yet to suggest a line of thought leading to the times-squared theorem, or even to a general law of fall.”
- 30.
In the copy that Arrighetti made of the advertas note on folio 57 verso , a second diagram with identical keying, but depicting the more general case of motion on two inclined planes, was in fact added.
- 31.
147 verso , T1A.
- 32.
As argued in the previous chapter the qualification of “ut demonstratur” by “infra” indicates that Galileo does not merely relate to a logical dependency between statements but is considering these statements as ordered as a consequence of their sequential exposition in a treatise or book. With the expression “ut infar demonstratur,” he may thus have been referring to the law of the inclined plane drafted on a folio bearing the small star watermark and later cut and pasted onto folio 179. This was, however, formulated in terms of mechanical moment and not in terms of impetus.
- 33.
Hooper, as well as Drake, assumes a reverse relationship between the respective entries on 147 and 180. According to Drake (1978, 63) the results of folio 180 were “recast …in terms of impetus rather than in terms of moment of heaviness” on folio 147. The assumption that the tidily written draft of a proposition should predate its elaboration on a page whose content clearly has provisional character is, however, rather preposterous. For Hooper’s position, cf. Hooper (1992, 345).
- 34.
The considerations on 147 verso start from considering forces along planes of equal length but different heights, whereas in the ex mechanicis proof, Galileo started by considering the moments on inclined planes of equal height and different lengths. The latter corresponds to the formulation of the law of the inclined plane drafted on the sheets pasted to folio 179.
- 35.
147 verso , 3A, 3B and 3C.
- 36.
Wisan has commented upon this seeming duplication as follows: “GALILEO could have given a proof from the law of chords …and it is somewhat odd that it should be proven as above [i.e., without using the law of chords as a premise]. A note …suggests the reason why. …The question is, why does this relation hold? Perhaps it is for the answer to this that GALILEO goes back to his dynamical principle (Wisan 1974, 172).”
- 37.
The proof of the law of fall from 147 recto was copied by Arrighetti to 50 recto , the isochronism considerations to 57 recto and 57 verso . That almost the entire content of folio 147 was copied shows that Galileo considered it to be of crucial importance. This is not surprising in view of the fact that, as argued here, the folio was part of his earliest attempt to provide his new insights with a foundation and to, thus, erect a new science of motion.
- 38.
Arrighetti copied the draft from 180 recto to 177 verso .
- 39.
57 recto , T3.
- 40.
An elaboration of the first of the two proofs which Galileo provided for Proposition IX in the Discorsi is contained on folio 194 recto . The entry remained incomplete and breaks off in the middle of a sentence. It is written rather thoroughly and the text floats around a blank space intended to hold the proof diagram which was, however, never added. The handwriting differs considerably from most of the entries in the Notes on Motion but resembles that of an entry in the lower part of folio 182A recto. According to Favaro, the entry is in Galileo’s hand.
- 41.
I had the opportunity to examine the watermarks in the Notes on Motion cursorily by means of a light-sheet device courteously provided by the Bibliotheca Nazionale Centrale di Firenze. A systematic campaign to analyze the watermarks on the paper used by Galileo is a desideratum.
- 42.
Drake has identified the watermark of folio 148 as type C18 of his typology. Unfortunately, this type is missing from the table in which he provided detailed descriptions. The watermark of folio 147 is specified by him as “Mountains” 20 × 17 mm. See Drake (1979, XXXIII). I have measured a width and height of 18 mm for the watermarks on folio 147 and 148. Both folios, moreover, share the exact same height of 305 mm.
- 43.
Arguments on the relationship between folios based on terms with low frequencies are, as a rule, only meaningful if combined with an hypothesis concerning the reason why a particular term is rarely used. This may, for instance, simply be due to the fact that the term in question pertains to a topic treated only marginally, in which case it is, of course, not suggested that two entries where the term occurs belong to the same period of work. In the case of the Latin “gravia,” it can be argued, however, that the low frequency is due to the fact that over time it was replaced by the term “mobilia” to get rid of the dynamical connotations of the former. It is then, indeed, suggested that entries which employ the term “gravia” stem from the same early period of work.
- 44.
Without a comprehensive linguistic analysis, arguments based on the frequency of words such as the one made in the text need to be taken with a pinch of salt.
- 45.
The results of Galileo’s considerations regarding the iso-temporal surfaces of different types of motion were presented as part of a longer Scholium appended to Proposition XXIII of the Second Book of the Third Day of the Discorsi. Proposition XXIII shows how to construct a plane which, appended to a vertical, is traversed after fall through the vertical in the same time. Cf. Chap. 8.
- 46.
According to the Aristotelian understanding, a science “explains them[facts] by displaying their priority relations (APo. 78a22/28). That is, science explains what is less well known by what is better known and more fundamental, and what is explanatorily anemic by what is explanatorily fruitful (Shields 2016).” A statement that is not proven or demonstrated but considered either self-evident or subject to necessary decision is commonly referred to as an axiom, whereas I refer to this, in accordance with Galileo’s own parlance, as a principle or basic principle. When speaking about axioms or more precisely about Galileo’s fundamental or axiomatic propositions, I refer to his statements concerning the spacio-temporal behavior of accelerated motion which served as a starting point for inferring all other propositions. The use of axiomatic in the text is thus not fully compliant with its use in contemporary logic.
- 47.
The results of the pendulum plane experiment did not fully comply with Galileo’s theoretical expectation but could, as argued, nevertheless be interpreted by him as partially corroborating his exposition of the pheno-kinematics of naturally accelerated motion. Moreover, as discussed in Chap. 6, it is very likely Galileo had performed additional experiments, in particular, an experiment to validate the law of fall as he later described it in the Discorsi.
- 48.
We are thus now in a position to answer in the affirmative the question posed by Settle (1966, 99): “[a]re we to infer that much of the substance of the Third Day was already in hand in 1602?” to which he, based on the limited sources available to him, could then merely state, “there is warrant for entertaining the idea.”
- 49.
The only page of the Notes on Motion in which Galileo uses the expression “momentum” to refer to the force on a inclined plane which has not been discussed here is 173 recto . Based on material criteria, the order in the manuscript, and the diagram on the verso side, it is plausible to assume that the page pertains to the period of work discussed in the present chapter. The notes are too sketchy to reliably reconstruct Galileo’s underlying considerations. The diagram in the upper left depicts the motion situation of the law of chords, i.e., of two inclined planes of different inclination traversed in equal times. The diagram below and in the lower right could indicate that Galileo was considering how the dynamical argument could be transferred to the motion situation of the length time proportionality, i.e., to motions over inclined planes of equal height or even to the most general case of motion along inclined planes of different inclination, height, and length.
- 50.
Galileo had all of the isochronism considerations from folio 147 copied. The copying must have been preceded by a rather systematic process in which he selected the material to be copied. The duplication entailed by isochronism considerations with regard to the ex mechanicis proof of the law of chords, if not when writing them down, would most likely have been realized then. That he had the isochronism considerations copied nevertheless suggests that they originally served a somewhat different intellectual goal, which even though eventually not reached, they still represented for Galileo.
- 51.
Cf. Galluzzi (1979).
- 52.
For the idea of exploring the limits of a conceptual system, see Damerow et al. (2004).
- 53.
Galileo’s proof of the double distance rule already rested on the assumption that the degrees of velocity increased in proportion to the distance fallen, i.e., the Sarpi letter principle. Yet, for the time being, his argument rested on showing that the kinematic velocity of successive motion was double that of the preceding motion, which seemed reconcilable with both the Sarpi letter principle and the later correct principle. The conflict entailed which led Galileo to abandon the erroneous principle and accept the correct principle was realized only after systematic elaboration. See the discussion in Damerow et al. (2004, 180–188). From a modern perspective, the double distance rule, indeed, implies the law of fall, but for this to become manifest, an appropriate argument has to be constructed. Galileo, however, was never quite able to integrate the double distance rule into his new science, and it is indeed not formally presented as a proposition in the Discorsi.
- 54.
Letter and translation are provided in Chap. 14.
- 55.
An exception is Renn, who maintained that a deeper investigation of naturally accelerated motion must have antedated the letter. Thus, for instance, he assumes, as is done here, that the proof of the law of fall on 147 was drafted before the letter. Cf. Damerow et al. (2004, 205–208).
- 56.
The letter sent by Galileo to Paolo Sarpi on the 16th of October 1604 was a direct reply to an antecedent letter by Sarpi, dated 9 October 1604 (EN X, 114, a translation is contained in Drake 1969, 340). The letter by Sarpi indicates that the two men had recently discussed problems of motion in person. Drake assumes that the considerations alluded to in Galileo’s letter were inspired by this conversation.
- 57.
I follow the interpretation of folios 128 and 85 established by Renn in Damerow et al. (2004).
- 58.
As was seen, Galileo had already employed the Sarpi letter principle in 1602. The emphasis in his claim is, thus, not on the principle itself but on the practicability of its use as an axiom (“assioma”), i.e., a basic principle in the foundation of the new science. Wallace (1992, 266) has rightly emphasized that to “serve as a principle for a demonstrative science there would have to be independent evidence of its truth, either as per se nota in its own right or as demonstrated on other grounds” and that in “1604 Galileo was optimistic that he could produce such a demonstration.”
- 59.
EN X, 115, letter 105. As the remaining part of the sentence shows, Galileo explicitly distinguished the “other conclusions” from conclusions concerning projectile motion and must, thus, here indeed be referring to his conclusions concerning naturally accelerated motion on inclined planes.
- 60.
EN X, 116, letter 105.
- 61.
The proof on 179 recto rests on a proposition drafted on folio 138 verso . As part of his argument, Galileo considered a step-by-step increase of the velocity which then remains constant over a minimal distance of fall. An, according to my opinion, essentially correct reconstruction of Galileo’s reasoning in the proof on 179 recto has been given by Wisan (1974, 216–219). I, however, disagree with most of the conclusions she draws on the basis of her reconstruction, in particular, her dating of the entry to 1609.
- 62.
In the original draft, Galileo stated that the degrees grew according to the Sarpi letter principle and this is also represented in the associated diagram. Yet the assumption was not invoked as a premise in the argument which is, in fact, indifferent to the precise way in which the degrees of velocity increase with the distance fallen as long as this increase is related in particular manner for the different motions compared, which facilitated its reworking.
- 63.
That the underlined passages were marked for deletion and replacement was first noted by Wisan (1974, 218). The content of folio 179 recto was copied by Arrighetti to folio 88 recto without, however, deleting or otherwise altering the underlined passages. Based on this observation, Wisan (1974, 218) has claimed that if these passages were in fact marked for deletion, then this must have happened after the proof was copied by Niccolò Arrighetti which, however, does not appear to make much sense in view of the fact that the passages in question are underlined in the original entry on 179 recto as well.
- 64.
Wisan denies a close relation between the law of fall on 128 and the length time proportionality on 179; in fact she dates the former to 1604 and the latter to 1608 or 1609. According to her, the content of 179 reveals “an attempt by GALILEO to rework his foundations using ARCHIMEDEAN methods rather than those of the medievals, which were used to establish the times-squared theorem on folio 128 (Wisan 1974, 216).”
- 65.
That the systematic resort to the doctrine of the configurations of motion in 1604 was stimulated by Galileo’s “having reached an impasse in his work on the theory of motion” as a result of which “Galileo may have sought and began seriously to consider some of the writings of the fourteenth century” has congenially been anticipated by Settle (1966, 199).
- 66.
How Galileo realized that the Sarpi letter principle was not compliant with the law of fall has convincingly been reconstructed by Damerow et al. (2004, 188–197), when this happened exactly is not entirely clear. Direct evidence that Galileo had abandoned the Sarpi letter principle by 1611 is offered by a letter that Daniello Antonini sent to Galileo on 9 April 1611. “Ho pensato alcuna volta a quella sua propositione: Mobile secundum proportionem distantie, a termino a quo movetur velocitatem acquirens, in instanti movetur …(EN XI, letter 512).” Just as he would later argue in the Discorsi, Galileo had obviously communicated to Antonini before that if the Sarpi letter principle held motion would be necessarily instantaneous. As Galileo categorically rejected the possibility of instantaneous motion, this implies that before 1611 Galileo renounced the Sarpi letter principle and thus most likely adopted the correct principle of acceleration.
- 67.
In 1610 in a letter to the Grand Duke’s Secretary of State Belisario Vinta (EN X, 348–353, Galileo felt confident enough to announce he had completed three books on local motion in such a way that he could justifiably call it new science established from its first principles (“onde io la posso ragionevolissimamente chiamare scienza nuova et ritrovata da me sin da i suoi primi principii (EN X, 352).” I side with Wisan (1981, 330) who stated, with regard to new science as finally laid out in the Discorsi: “In fact, Galileo never completely solved the problem of foundations for the new science, but his treatise demonstrated the potential inherent in his new mathematical approach to motion.”
- 68.
The ex mechanicis proof explicitly compares two motions made by one and the same falling body. One could claim that, if the same schema of argument were applied to bodies of different absolute weights, this would immediately show that the kinematics of naturally accelerated motion should indeed depend on the absolute weight of the body. Yet there is now evidence that Galileo engaged in such a consideration. Van Dyck (2006, 210) has rightly remarked that essentially the same inconsistency can already be pinpointed in De Motu Antiquiora and that it represents “an instance of the dynamical conundrum that threatens the whole of De motu.” The problem of the dynamical framework of De Motu Antiquiora is that “[s]pecific weight appears impotent to cause any effects (Van Dyck 2006, 211).”
- 69.
In the Discorsi, the principle of acceleration according to which the degrees of speed increase in proportion to the time elapsed, which replaced the Sarpi letter principle, is justified by ruling out the Sarpi letter principle as the only sensible alternative. Galileo’s refutation of the Sarpi letter principle in the Discorsi has recently been interpreted by Norton and Roberts (2012) with a critical response by Palmerino (2012) and Laird (2012). For the velocity proportional to vertical height principle, a plausibility argument is provided in the Discorsi which makes recourse to pendulum motion.
- 70.
The belief that Galileo must have arrived at his results in a way comparable to the way he argued for the results in the Discorsi in part rests on insufficiently distinguishing “two types of historical processes, namely, the process of discovery and the process of justification of this discovery (Hoyningen-Huene 1987, 504).”
- 71.
With regard to the Discorsi, Clavelin (1983, 36–37) remarked: “the definition of naturally accelerated motion does not occur in any of the theorems or in any of the constructions which will achieve the geometrisation [of naturally accelerated motion]. The conceptual analysis could only remain exterior to the mathematical analysis, incapable of supporting it or of being supported by it …Whoever makes the effort of reading closely the theorems and propositions will find that the demonstrative apparatus is principally based on two relations of proportionality established independently one from the other [propositions II and III, the law of fall and the length time proportionality].” This state of affairs in the Discorsi, so acutely observed by Clavelin, thus finds its explanation in the concrete historical process by which it was brought about.
- 72.
One cannot help but get the impression that for Galileo the work on the conceptual underpinnings was important primarily in that it was a requirement for establishing a foundation for his new science. Harriot, in contrast, seems to have considered it a theoretical question in its own right and, thus, for instance, diligently investigated the implications of the fact that the Sarpi letter principle and the correct principle of acceleration were irreconcilable. Cf. Schemmel (2008).
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Büttner, J. (2019). Toward a New Science: Axiomatization and a New Foundation. 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_11
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