Abstract
Confronted with overwhelming problems, in or shortly after 1602, Galileo abandoned his ardent investigation into the relation of swinging and rolling without, however, entirely giving up on his approach. The fundamental idea of his early research, namely, to approximate motion along an arc by motion along polygonal paths made up of a series of inclined planes, still resonates in the Discorsi, albeit in but one argument. By means of this argument, Galileo intended to demonstrate that the brachistochrone, i.e., the curve between two points along which motion is completed in least time, was an upright arc. It is argued that this was a late attempt by Galileo to make his earlier considerations regarding swinging and rolling bear fruit and that he had reason to surmise that his argument was flawed. The chapter, moreover, discusses how, stimulated by his perusal of Giovanni Battista Baliani’s De motu naturali gravium solidorum published in 1638, Galileo briefly resumed his work on swinging and rolling. Based on the understanding he himself had gained in his own earlier investigation into the same problem, Galileo drafted a critique of Baliani’s approach as part of which, for the first time, he devised a proof of the law of the pendulum. Galileo had thus accomplished in part what he had sought to achieve some 35 years earlier, and the new proof was indeed earmarked for inclusion in a second edition of the Discorsi.
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Notes
- 1.
At one point in the discussion in the First Day of the Discorsi it is first asked whether pendulums do in fact swing isochronously. From this, Salviati immediately jumps to recounting that Galileo had demonstrated that motion along appropriate chords subtending arcs of the same circle was isochronous, i.e., the law of chords. After this, he goes on to state that the isochronism of the pendulum is confirmed by experience (“mostra …l’esperienza”), motion along chord and motion along the arc spanned are directly compared, and it is emphasized that, counterintuitively, motion along the arc, despite being made over the longer distance, is completed in shorter time (“E quanto al primo dubbio, che è, se veramente e puntualissimamente l’istesso pendolo fa tutte le sue vibrazioni, massime, mediocri e minime, sotto tempi precisamente eguali, io mi rimetto a quello che intesi già dal nostro Accademico; il quale dimostra bene, che ’l mobile che descendesse per le corde suttese a qualsivoglia arco, le passerebbe necessariamente tutte in tempi eguali, tanto la suttesa sotto cent’ ottanta gradi (cioè tutto il diametro), quanto le suttese di cento, di sessanta, di dieci, di due, di mezzo e di quattro minuti, intendendo che tutte vadano a terminar nell’infimo punto, toccante il piano orizontale. Circa poi i descendenti per gli archi delle medesime corde elevati sopra l’orizonte, e che non siano maggiori d’una quarta, cioè di novanta gradi, mostra parimente l’esperienza, passarsi tutti in tempi eguali, ma però più brevi de i tempi de’ passaggi per le corde; effetto che in tanto ha del maraviglioso, in quanto nella prima apprensione par che dovrebbe seguire il contrario: imperò che, sendo comuni i termini del principio e del fine del moto, ed essendo la linea retta la brevissima che tra i medesimi termini si comprende, par ragionevole che il moto fatto per lei s’ avesse a spedire nel più breve tempo; il che poi non è, ma il tempo brevissimo, ed in consequenza il moto velocissimo, è quello che si fa per l’arco del quale essa linea retta è corda. (EN VIII, 19–20)”). Besides the brachistochrone argument, nowhere in the Discorsi does Galileo’s early research agenda on swinging and rolling resonate so clearly as in this passage.
- 2.
See Baliani (1638).
- 3.
We briefly touched upon this episode in Büttner et al. (2004, 111–112).
- 4.
EN VIII, 263, trans. Galilei et al. (1954).
- 5.
Cf. EN VII, 476.
- 6.
In her study of the Notes on Motion, Wisan (1974) somewhat reservedly claimed: “Possibly then it was a traditional problem, at least among some mechanicians of the time, but one receiving little attention in written works. In fact it is possible that it was the brachistochrone which originally turned Galileo’s attention to motion on inclined planes.” Yet in a later paper she lost her reservations and firmly and explicitly stated: “To solve the brachistochrone, however, Galileo had to create a new science, one based on new and fundamental propositions (Wisan 1984, 270).” More recently Palmieri has followed Wisan’s lead in assuming that the search for brachistochrone played a central role in Galileo’s considerations documented in the Notes on Motion: “Galileo sought to prove his conjecture [that the arc of the circle is the brachistochrone], as the many remnants of related theorems and problems surviving in the folios of Manuscript 72 suggest. (Palmieri 2009, 47)”.
- 7.
The analysis given leans on an analysis of the argument in Ariotti (1971/1972).
- 8.
For the use of verisimile (or veri simile) in the sense of probable or plausible, at least in classical Latin, cf. Fuhrer (1993).
- 9.
In 1696 John Bernoulli challenged the mathematicians of his day to solve the brachistochrone problem. He himself gave two solutions. His brother James sent a solution as well as Leibniz, and another solution was submitted anonymously by Newton. Cf. Goldstine (1980).
- 10.
That Galileo’s argument proves only a weaker statement than the one he actually makes was already realized by Dijksterhuis who, however, did not mention the second, more serious problem with Galileo’s argument. Cf. Dijksterhuis and Maier-Leibnitz (2002, 385)
- 11.
Galileo’s argument suggests that under slight variation of the path, the time of motion along the arc is indeed minimal. It can be argued, based on the understanding that nature acts in a simple manner, that it would not be very plausible to assume that this was a local minimum and that for a path substantially different from the arc the time of motion is minimized absolutely.
- 12.
With respect to the central diagram on 166 recto , Galileo calculated for motion along gc from rest at a a time of motion of 19,896, corresponding almost precisely to the recalculated value of 19,895. For motion along g8c, likewise from rest at a, Galileo calculated a smaller time of motion of 19,821 which should be 19,894 according to my recalculation. Compare Table 13.1.
- 13.
There is no indication that Galileo ever calculated the time of motion composed of 16 conjugate chords and inscribed into a quarter arc. Yet, he at least replaced the lowest chord of his eight-chord approximation of the arc on 166 recto , the chord 8c, by the broken chord 89c. The times of motion along the chords 89 and 9c, both motions after fall from rest in a are not listed in the table of results on 183 recto , yet the time of motion along 9c is given as 9821 in the diagram on 166 recto . The number is written next to the number 19,605 representing the time Galileo had calculated for motion along 8c made from rest in a.
- 14.
EN. VI, 627–647. The letter was sent to Raffaello Staccoli in January 1631. For a discussion of the broader context in which this letter was sent, see Westfall (1989). Galileo’s brachistochrone argument in the letter starts on p.643. His diagram shows a quarter arc spanned by a chord EC, a broken chord EFC, and a polygonal line with all junction points equally distributed on the arc EGFNC: ‘…che, posta l’istessa pendenza tra due luoghi tra i quali si abbia a far passare un mobile, affermo, la più spedita strada e quella che in più breve tempo si passa non esser la retta, ben che brevissima sopra tutte, ma esservene delle curve, ed anco delle composte di più linee rette, le quali con maggior velocità ed in più breve tempo si passano. E per dichiarazione di quanto dico, segniamo un piano orizontale secondo la linea AB, sopra ’l quale intendasi elevata una parte di cerchio non maggiore di un quadrante, e sia CFED, sì che la parte del diametro DC, che termina nel toccamento C, sia perpendicolare, o vogliamo dire a squadra, sopra la orizontale AB; e nella circonferenza CFE prendasi qualsivoglia punto F: dico adesso, che posto che E fusse il luogo sublime di dove si avesse a partire un mobile, e che C fusse il termine basso al quale avesse a pervenire, la strada più spedita e che in più breve tempo si passasse non sarebbe per la linea o vogliàn dire per il canale brevissimo EC, ma preso qualsivoglia punto nella circonferenza F, segnando i 2 canali diritti EFC, in più breve tempo si passeranno questi che il solo EC; e se di nuovo ne gli archi EF, FC si noteranno in qualsivoglia modo 2 altri punti G, N, e si porranno 4 canali diritti EGFNC, questi ancora si passeranno in tempo più breve che li 2 EFC; e continuando di descrivere dentro alla medesima porzion di cerchio un condotto composto di più e più canali retti, sempre il passaggio per essi sarà più veloce, e finalmente velocissimo sopra tutti sarebbe quando il canale fusse curvo secondo la circonferenza del cerchio EGFNC. Ecco dunque trovati canali che hanno la medesima pendenza (essendo compresi tra i medesimi termini E, C), e che son di differenti lunghezze, ne i quali i tempi de’ passaggi sono (al contrario di quello che comunemente si stimerebbe) sempre più brevi ne i più lunghi che ne i più corti, e finalmente lunghissimo è il tempo nel brevissimo, e brevissimo nel canale lunghissimo.”
- 15.
Erlichson (1998) has come to the same conclusion. “We are inclined to hypothesize that Galileo knew full well that he did not have a complete proof, and that he also knew that it would be quite difficult to prove his ‘yet it seems that …’ (Erlichson 1998, 347).” He shows that the argument that Galileo provided in his proof of the law of the broken chord cannot simply be extended to the case where motion along the broken chord does not start from rest and challenges his readers to “try to prove the unproven assumption of Galileo …preferably by methods available to Galileo (ibid.).” As we have seen, however, the assumption does not hold and hence the challenge can be closed.
- 16.
A number of considerations in the Notes on Motion could potentially be related to the brachistochrone argument. In a construction on folio 190 recto , Galileo exploited an approach developed on 130 verso where he had sought the bent plane traversed in least time connecting two points whose vertical and horizontal distances were the same. On 190 recto Galileo applied the same construction to two points whose vertical distance was bigger than their horizontal distance. Apparently he was inquiring if, in this case, the junction point defining the bent plane traversed in least time still lay on the arc connecting start and endpoint of motion, as he (falsely) believed to be the case in the situation under scrutiny on 130 verso . Should this have been found to be the case the insight could have potentially been used to flesh out the brachistochrone argument. Yet it seems very unlikely that this is what motivated the construction on 190 recto and related constructions on 191 verso .
- 17.
Wisan (1974, 176).
- 18.
The Acta Eruditorum of 1697 contain Johann Bernoulli’s solution to the brachistochrone problem on pp.206–211. The quote is on p.210. Quoted from www.nlb-hannover.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/III7A.pdf. Accessed 28 Nov 2016.
- 19.
Yet based on studying the Discorsi perspicaciously, Settle (1966, 99) surmised that “the exposition of the Third Day is such that it seems that Galileo was looking for a demonstrative link between linear and circular natural motions.”
- 20.
By the time Giovanni Battista Baliani’s book came out Galileo’s manuscript had been sent off to Leiden and he was expecting the return of the printed copies. For a detailed account of how the Discorsi came to be printed, see Raphael (2012).
- 21.
For Giovanni Battista Baliani’s work on the problems of motion, see above all Moscovici (1967).
- 22.
On 20 February 1627, Baliani wrote to Benedetto Castelli: “…Io altre volte feci un trattato de’ moti dei solidi, e della loro maggiore o minore velocità ne’ piani più o meno declinanti: volli poi far quello de’ liquidi, e lasciai l’opera imperfetta, perchè mi si accrebbero le difficolà. La causa principale è la seguente. Facendo il trattato de’ solidi che ho detto, avvenne che, senza cercarla, mi riuscì, a parer mio, ben dimostrata una proposizione per una via molto stravagante, la quale già il Sig. Galileo m’avea detta per vera senza però addurmene la dimostrazione; ed è, che i corpi di moto naturale vanno aumentando le velocità loro con la proporzione di 1, 3, 5, 7, ec., e così in infinito: me ne addusse però una ragione probabile, che solo in questa proporzione più o meno spazi servano sempre l’istessa proporzione. Non mi dichiaro maggiormente, perchè so che parlo con chi intende. Però io l’ ho dimostrata con principi molto diversi; …(EN XIII, 348–349; letter 1806)” This shows first that Baliani had been working on his treatise for quite a while before 1638; second, that Galileo had communicated results to him; and third, that Baliani had a clear awareness that his approach differed fundamentally from that of Galileo with respect to the principles on which it was based. Capecchi (2014, 177), in contrast, assumes that “Baliani’s claims …according to which all heavy bodies fall with the same temporal law and the periods of the pendulums are proportional to the square roots of their lengths, were independent of the results obtained by Galileo.”
- 23.
With regard to De motu naturali gravium solidorum, Richard S. Westfall has, for instance, remarked that “[t]he level of discussion in Baliani does not begin to approach Galileo’s, so that issues of plagiary have inevitably arisen.” See http://galileo.rice.edu/Catalog/NewFiles/baliani.html. Accessed 4 Sep 2016.
- 24.
Pendulum motion was clearly among the topics that Galileo and Baliani discussed. A letter (EN XIV, 342–344; letter 2258) from 23 April 1632, sent by Baliani to Galileo, for instance, testifies that the two men debated the use of a pendulum for measuring seconds.
- 25.
These and all following quotes from De motu naturali gravium solidorum are made according to Baliani et al. (1998); translations, where given, are mine.
- 26.
For the role played by the observation that lines from weights at different positions at a balance or lever to the center of the world cannot be parallel in discussions of mechanics in Galileo’s day, see Damerow and Renn (2012).
- 27.
Baliani (1638, 14).
- 28.
In 1615, Baliani traveled to Florence where he visited Galileo and also met with Benedetto Castelli. Based on a proposal by Galileo, Giovanni Battista Baliani became a member of the Accademia Lincei in the following year. Correspondence between Baliani and Galileo continued sporadically for many years. The letter sent in December 1638 reads: “Havendo io risoluto di mandar fuori un’operetta del moto naturale de’ corpi gravi mi parrebbe far mancamento se non la mandassi subito a V. S., pregandola che a tanti favori fattimi voglia aggionger questo di legerla e corregerla e dirmene il suo parere. Son sicuro che, se non per altro, la stimerà almeno degna di comparirle dinanti per conoscer la fattura di autore che, ancorchè da lontano, si ingegna di seguir le sue pedate; et io in tanto starò con desiderio di veder uscir in luce le opere di V. S., in cui spero di vedere ridotto a perfettione ciò che io ho abbozzato così alla grossa. E pregandola conservarmi nella sua buona gratia, le baccio per fine le mani, e priego dal Signor ogni vero contento (EN XVII, 413–414; letter 3824).” On 7 January 1639, Galileo wrote back indicating that he had received the book and read it, respectively, had it read to him due to his worsening eyesight, as he noted in a letter to Renieri on 28 March 1639. Cf EN XVIII, letter 3829 and 3858.
- 29.
The echo the publication of Baliani’s book provoked in Galileo’s circle in the years 1638 to 1639 was characterized by “dramatic tones of bewilderment and sometimes outrage. (Istituto della Enciclopedia italiana (Roma) 1963)” Galileo himself does not appear to have been too bothered. He kept up his correspondence with Baliani, retaining a polite tone. Occasionally, he interspersed remarks in the conversation to the effect that the Discorsi was submitted to the printer much earlier than Baliani’s book and that what was demonstrated therein goes beyond what Baliani had demonstrated. Cf. EN XVIII, letters 3829, 3897 and 3912.
- 30.
The draft is preserved in the Biblioteca Nazionale Centrale in Florence as part of Ms. Gal. 74, folios 35v -38v. Favaro did not include the document in the EN as he did not consider Galileo to be the author. Cf. EN VII, 36–37. It was first published by Caverni, who attributed the authorship to Galileo. See Caverni (1972, IV, 313–314). The document is likewise discussed in Moscovici (1967) who, without providing further evidence, assumes that the text was dictated by Galileo to Viviani. Capecchi (2014, 180) even states that the document reproduces “a letter of Galileo to Vincenzo Renieri (1606–1647) which is lost but was reported by Vincenzo Viviani.” He too, however, provides no direct evidence for his claim.
- 31.
Bound as Gal. 79, an annotated copy of the Discorsi is preserved in the Biblioteca Nazionale Centrale di Firenze. A leaf bound with the book refers to it as a copy with corrections notes an addenda by Viviani. Damerow et al. (2001, 314) refer to it as Galileo’s own copy, and this assumption is shared here. Gal. 79 is one of the texts Guisti used as the basis for his edition (cf. Galilei and Giusti 1990) of the Discorsi. Therein the handwritten annotations of Gal. 79 are given as footnotes.
- 32.
On a separated leaf (Gal. 79 folio 58) inserted into the copy of the Discorsi, the proof idea cursorily added in the lower margin is expanded upon. A full transcription of this page is contained in Galilei and Giusti (1990, 105–106). The marginal note directly underneath the thumbnail figure and which is to be appended after “troverò la lunghezza della corda” in the main text as indicated by an insertion mark reads: “perché facendo come il quadrato del piccol numero delle vibrazioni del lungo pendulao al quadrato del gran numero dell vibrazioni del corto, così la lunghezza nota di questo ad un’altra, essa sarà l’ignota lunghezza del lungo (Galilei and Giusti 1990, 108)”.
- 33.
Capecchi (2014, 187) similarly remarks that Baliani’s approach “evidences an understanding of infinitesimal analysis at least on an intuitive level. It is quite acceptable for a modern, but it was certainly not such for Baliani’s contemporaries. Although mathematics was evolving in the direction of the infinitesimal calculus, this assumption of Baliani still seems very daring and interesting.”
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Büttner, J. (2019). Whatever Happened to Swinging and Rolling: Faint Echoes and a Late Insight. 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_9
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