Abstract
The role of Galileo in the history of modern science has been and will always be subject to debate. If it is not true that he invented the scientific method – it is of Hellenistic origin – based on the comparison between theory and experiment, it is true that he made a fundamental contribution to its clarification and dissemination. If Galileo was unclear about some of his scientific results and also on epistemological aspects, they were all solved by his colleagues and students. Castelli, Torricelli, Cavalieri, and Viviani stand out among them. Cavalieri and Torricelli generalized the uncertain Galilean principle of inertia bringing it to its modern form, which was only implicit in Galileo. Viviani in his biography attributed to his master a purely empirical method, charging it with experimental activities in many sectors; in addition to falling bodies, he also took care of thermology and magnetism experiments. Although the description of Viviani was most probably not faithful, it represents a sign that Galileo had transmitted to his heirs a method in which the role of experiment was crucial.
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Notes
- 1.
vol. 6, p. 232.
- 2.
vol. 6, pp. 347–348.
- 3.
vol. 7, p. 60.
- 4.
vol. 6, p. 296.
- 5.
vol. 1, p. 285.
- 6.
vol. 8, p. 175.
- 7.
vol. 8, p. 202. Translation in [73].
- 8.
vol. 6, p. 237. Translation [42].
- 9.
p. 165.
- 10.
pp. 151–188.
- 11.
p. 162.
- 12.
pp. 58–59.
- 13.
vol. 2, pp. 261–266.
- 14.
vol. 8, p. 197.
- 15.
vol. 8, p. 197. Translation in [73].
- 16.
vol. 8, p. 198.
- 17.
vol. 8, p. 202.
- 18.
pp. 8–9.
- 19.
vol. 2, pp. 267–276.
- 20.
pp. 267–276. Even the modern infinitesimal analysis, from the proportionality between speed (v) and space (s) leads to an absurd, although in a different manner. Indeed the proportionality between v and s in a modern notation reads as \(v = ks \), k being a constant of proportionality. This relation can be written as a differential equation in the form: \(ds/dt = ks\). It, after integration, leads to the relation \(s = S_0e^{kt}\), which if one assumes \(s = 0\) for \(t = 0\), implies \(S_0 = 0\) and then \(s = 0\) for each value of the time, leading to an absurd result: one assumes a downward motion and comes to the conclusion that this motion does not exist; thus it is not possible that the speed increases proportionally to the space of fall, at least if one starts from rest.
- 21.
vol. 8, p. 197.
- 22.
vol. 8, p. 273.
- 23.
vol. 17, pp. 90–91.
- 24.
vol. 18, pp. 12–13.
- 25.
p. 205.
- 26.
The mathematical physicist of the nineteenth century, before introduced definitions that, in fact, inspired by physics, were completely detached from it officially. Then he drew inspiration from these definitions to develop mathematical theories (different from the traditional ones, at that time the Euclidean geometry and calculus). Mathematics then gave its formal system to develop arguments and completely rigorous “histories” that did not strictly concern the purely mathematical world. Only in the twentieth century, with the clarification of the axiomatic and hypothetical deductive methods, was this function absolved by formal logic.
- 27.
pp. 324–327.
- 28.
vol. 2, pp. 211–212. Translation in [49].
- 29.
vol. 4, pp. 108–109.
- 30.
p. 145.
- 31.
p. 98.
- 32.
p. 102.
- 33.
vol. 5, pp. 349–370.
- 34.
p. 235.
- 35.
vol. 5, p. 351.
- 36.
p. 274.
- 37.
p. 8.
- 38.
vol. 4, pp. 13–16.
- 39.
vol. 4, p. 521.
- 40.
pp. 408, 410.
- 41.
pp. 32–35.
- 42.
p. 43.
- 43.
p. 22.
- 44.
p. 199.
- 45.
p. 66; p. 501.
- 46.
p. 606.
- 47.
pp. 115–116.
- 48.
pp. 212–213.
- 49.
p. 158.
- 50.
pp. 144–145.
- 51.
pp. 142–143.
- 52.
p. 30.
- 53.
p. 271.
- 54.
pp. 240–265.
- 55.
p. 28.
- 56.
vol. 10, p. 115.
- 57.
In the following I consider the two laws equivalent, and for simplicity I always speak about the fall as the law of odd numbers.
- 58.
This is the same argument adduced by Torricelli in Sect. 4.5.2.1.
- 59.
vol. 13, p. 348.
- 60.
vol. 7, p. 75. Translation in [74].
- 61.
p. 115.
- 62.
pp. 348–349.
- 63.
p. 203.
- 64.
pp. 212–213. Translation in [73].
- 65.
p. 154.
- 66.
The law is consistent with the increase of speed with time, as can be easily proved.
- 67.
p. 208, Theorem 1, Proposition 1.
- 68.
p. 213. Translation in Galileo 1656.
- 69.
p. 524.
- 70.
p. 350.
- 71.
pp. 158–159.
- 72.
p. 173. Translation in [68].
- 73.
vol. 8, pp. 371–372. Translation in [49].
- 74.
p. 147r.
- 75.
p. 4.
- 76.
pp. 180–181.
- 77.
vol. 4, p. 275.
- 78.
p. 379. Translation [62].
- 79.
vol. 1, p. 383.
- 80.
vol. 4, p. 67.
- 81.
vol. 4, p. 69.
- 82.
p. 159.
- 83.
For the causal role attributed to mathematics in the Renaissance see, for example, [11].
- 84.
p. 187.
- 85.
p. 140.
- 86.
p. 140.
- 87.
pp. 140–141.
- 88.
p. 142.
- 89.
p. 142.
- 90.
pp. 106, 152.
- 91.
p. 237.
- 92.
vol. 7, pp. 260–261. Translation in [74].
- 93.
p. 51.
- 94.
p. 55.
- 95.
p. 59.
- 96.
p. 59.
- 97.
For a history of the concept of horror vacui, see [83], pp. 329–355.
- 98.
Galileo was never explicit about the role that the vacuum played in his atomistic conception. Hero’s theory is attributed to him that vacuum existed in nature only as minute elements of separation between the atoms (see [12], pp. 433–435; [7], pp. 91–164; p. 152). But perhaps the matter is more complicated [111], p. XX).
- 99.
vol. 8, p. 60.
- 100.
p. 60.
- 101.
p. 65.
- 102.
pp. 121–126.
- 103.
p. 61.
- 104.
p. 67.
- 105.
- 106.
The prudence of Galileo in speaking of atoms could be sought in the political climate of the time. According to many religious, believing in atoms involved denial of the dogma of the Eucharist as it was accepted by the Council of Trent (See [115]).
- 107.
vol. 4, pp. 63–142.
- 108.
vol. 6, pp. 197–372.
- 109.
The two terms, non-quanti and indivisible seem to be used interchangeably by Galileo , which still favors the first.
- 110.
p. 96.
- 111.
p. 96. Translation in [73].
- 112.
p. 66.
- 113.
The measure of this resistance is carried out considering a column of water, in which the adhesion among the parties is motivated only by the effect of a macroscopic horror vacui; it is quantified in 18 braccia (about 10 m). In the case of copper Galileo estimated the resistance offered by the macroscopic vacuum with respect to the total one in the ratio of 2 : 5000 or so, then practically negligible [66], pp. 65–66.
- 114.
pp. 153–154.
- 115.
The concept of “remote actions” was not new; it could be derived by Philo of Byzantium according to [111], p. 125.
- 116.
p. 9.
- 117.
vol. 4, p. 112.
- 118.
vol. 4, p. 89.
- 119.
vol. 7, p. 443.
- 120.
vol. 7, p. 443.
- 121.
vol. 7, pp. 446–447. Translation in [74].
- 122.
vol. 7, p. 445.
- 123.
vol. 7, p. 450.
- 124.
vol. 7, p. 456.
- 125.
vol. 7, p. 451. Translation in [74].
- 126.
vol. 7, p. 474. Translation in [74].
- 127.
vol. 7, p. 454.
- 128.
p. 152.
- 129.
p. 161.
- 130.
p. 154.
- 131.
pp. 154–157.
- 132.
p. 156.
- 133.
pp. 157–158.
- 134.
p. 160.
- 135.
p. 165. Actually Cavalieri declared it differed insensibly from a parabola, because he knew that the lines of fall were not exactly parallel to each other, but converging toward the center of the earth.
- 136.
pp. 168–169.
- 137.
p. 4.
- 138.
pp. 171–172.
- 139.
p. 48.
- 140.
pp. 49–50.
- 141.
p. 178.
- 142.
p. 97.
- 143.
Chapter 7.
- 144.
p. 98.
- 145.
p. 140.
- 146.
pp. 180–181.
- 147.
p. 205.
- 148.
p. 99.
- 149.
vol. 8, pp. 214–218.
- 150.
vol 3, pp. 461–462.
- 151.
vol. 13, p. 348. Letter of Baliani to Benedetto Castelli. February 20th, 1627.
- 152.
pp. 219–221.
- 153.
vol. 3, p. 479.
- 154.
p. 191.
- 155.
vol. 3, p. 80.
- 156.
pp. 192–193.
- 157.
vol. 3, p. 276.
- 158.
pp. 62–71.
- 159.
Lezione 2, p. 6.
- 160.
- 161.
Lezione 2, p. 7.
- 162.
Lezione 3, pp. 13–14.
- 163.
Lezione 2, pp. 9–10.
- 164.
The summation of moments over time, though suggestive is in fact wrong according to modern mechanics [19], p. 176.
- 165.
Lezione 2, p. 12. See Fig. 4.10.
- 166.
Lezione 7, pp. 48–49.
- 167.
Lezione 7, p. 49.
- 168.
Lezione 6, p. 37.
- 169.
p. 37.
- 170.
p. 191.
- 171.
Castelli is referring to the first proposition of the second part of his treatise.
- 172.
p. 7.
- 173.
p. 11.
- 174.
p. 27.
- 175.
p. 33.
- 176.
pp. 46–47.
- 177.
p. 47.
- 178.
p. 52.
- 179.
p. 53.
- 180.
p. 48.
- 181.
pp. 76–98.
- 182.
pp. 99–184.
- 183.
p. 78.
- 184.
p. 79.
- 185.
p. 82.
- 186.
p. 83.
- 187.
pp. 77–78.
- 188.
p. 77.
- 189.
pp. 194–199. The precise reference is to pages 197–198. Actually the speed distribution along the height of a fluid in a section of an open channel is not linear.
The real profile has the pattern shown in Fig. 4.14 ([40], p. 334). The distribution is thus closer to the constant than to the linear. Perhaps Castelli realized this fact and declined to accept the suggestion of Cavalieri .
- 190.
p. 88.
- 191.
In channels with rectangular sections, according to modern theories of hydraulics, the flow rate is proportional to \(h^{5 / 3} \sim h^2\) [40], p. 316.
- 192.
pp. 94–95.
- 193.
vol. 19, pp. 599–632. See also [119].
- 194.
- 195.
The monograph was reprinted in Galileo works of 1842–1856 as vol. 14 [70].
- 196.
vol. 3, pp. 193–305. The Florentine edition is the second Italian edition of the works of Galileo after that of Bologna and was edited by Grandi , Tommaso Bonaventuri, and Benedetto Bresciani. The writings of Viviani on the strength of materials that do not appear in the National Edition by Favaro , were given the title: Trattato delle resistenze principiato da Vincenzo Viviani per illustrare le opere di Galileo .
- 197.
The definition continues with the text “perpendicular to the horizon”, which has been omitted because it is misleading for a modern reader.
- 198.
This is the Galilean absolute resistance, that is, the resistance to pure traction.
- 199.
vol. 14, pp. 3–6.
- 200.
Today it is known that such a center does not exist for a bent beam.
- 201.
vol. 14, p. 6.
- 202.
vol. 14, pp. 7–10.
- 203.
vol. 14, p. 9.
- 204.
pp. 17–45.
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Capecchi, D. (2018). Galilean Epistemology. In: The Path to Post-Galilean Epistemology. History of Mechanism and Machine Science, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-58310-5_4
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