1 Galileo to Guido Ubaldo dal Monte, Padua, 29 November 1602, in Galileo, Le opere di Galileo Galilei: Edizione Nazionale (ed. A. Favaro), 20 vols., Florence, 1890–1909, x, 97–100, translated in P. Palmieri, Reenacting Galileo's Experiments: Rediscovering the Techniques of Seventeenth-Century Science, Lewiston, NY, 2008, 257–60. ‘Isochronism’ is the property of certain physical systems to oscillate at constant frequency regardless of the oscillations' amplitude. We now know that simple pendulums are not isochronous. ‘Isochronism’ is not a Galilean word. As the reader will see, Galileo uses other expressions to refer to this property. I will retain ‘isochronism’ since it has become common in the literature.

2 Koyré, A., ‘An experiment in measurement’, Proceedings of the American Philosophical Society (1953), 97, 222–37Google Scholar; P. Ariotti, ‘Galileo on the isochrony of the pendulum’, Isis (1968), 59, 414–26; idem, ‘Aspects of the conception and development of the pendulum in the 17th century’, Archive for History of Exact Sciences (1972), 8, 329–410; Drake, S., ‘New light on a Galilean claim about pendulums’, Isis (1975), 66, 92–5CrossRefGoogle Scholar, reprinted in idem, Essays on Galileo and the History and Philosophy of Science, 3 vols., Toronto, 1999, ii, 316–20; idem, Galileo: Pioneer Scientist, Toronto, 1990, 9 ff.; idem, Galileo at Work: His Scientific Biography, New York, 1995 (1st edn Chicago, 1978), 69–70; Naylor, R., ‘Galileo's simple pendulum’, Physis (1974), 16, 23–46Google Scholar; idem, ‘Galileo's need for precision: the point of the Fourth Day pendulum experiment’, Isis (1977), 68, 97–103; idem, ‘Galileo, Copernicanism and the origins of the new science of motion’, BJHS (2003), 36, 151–81; MacLachlan, J., ‘Galileo's experiments with pendulums: real and imaginary’, Annals of Science (1976), 33, 173–85CrossRefGoogle Scholar; Costabel, P., ‘Isochronisme et accélération 1638–1687’, Archives internationales d'histoire des sciences (1978), 28, 3–20Google Scholar; Hill, D., ‘Pendulums and planes: what Galileo didn't publish’, Nuncius (1994), 9, 499–515CrossRefGoogle Scholar; T. Settle, ‘La rete degli esperimenti Galileiani’, in Galileo e la scienza sperimentale (ed. M. Baldo Ceolin), Padua, 1995, 11–62; Machamer, P. and Hepburn, B., ‘Galileo and the pendulum: latching on to time’, Science & Education (2004), 13, 333–47CrossRefGoogle Scholar; Matthews, M., ‘Idealisation and Galileo's pendulum discoveries: Historical, philosophical and pedagogical considerations’, Science & Education (2004), 13, 689–715.CrossRefGoogle Scholar

3 Naylor, ‘Galileo's simple pendulum’, op. cit. (2), 23.

4 Naylor, ‘Galileo's simple pendulum’, op. cit. (2), 23. The Discorsi referred to by Naylor are the Two New Sciences.

5 Hill, op. cit. (2), 513. Hill's conclusion is untenable: I have suggested a counterargument based on a repetition of Galileo's calculations concerning the so-called brachistochrone, on which Hill's indictment hinges, in Palmieri, op. cit. (1), 248–55.

6 MacLachlan, op. cit. (2), 173.

7 A. Koyré, Études Galiléennes, Paris, 1966 (1st edn 3 vols., Paris, 1939).

8 Costabel, op. cit. (2), 6. Costabel doubts that Galileo could have done the experiments with cork and lead pendulums since in his view Galileo's reasoning about the experiment is erroneous. Costabel's argument only shows his misunderstanding of what Galileo's experiments with cork and lead pendulums are really about and their function as evidence (see below, ‘A crucial experiment in the historiography of isochronism’).

9 Naylor, ‘Galileo's simple pendulum’, op. cit. (2). Settle's groundbreaking reconstruction of the inclined plane experiment takes a different approach and carefully avoids the trap of the matching problem. Settle, T., ‘An experiment in the history of science’, Science (1961), 133, 19–23.CrossRefGoogle ScholarPubMed

10 Belloni, L., ‘The repetition of experiments and observations: its value in studying the history of medicine (and science)’, Journal for the History of Medicine and Allied Sciences (1970), 25, 158–67CrossRefGoogle Scholar; and Belloni's commentaries in M. Malpighi, Opere scelte (ed. L. Belloni), Turin, 1967. In recent years, more scholars have embraced an experimental approach to the history and philosophy of this science: for instance, Settle, op. cit. (9); Wilson, C., ‘Re-doing Newton's experiment for establishing the proportionality of mass and weight’, St John's Review (1999), 45, 64–73Google Scholar; J. Renn and P. Damerow, ‘The hanging chain: a forgotten “discovery” buried in Galileo's notes on motion’, in Reworking the Bench: Research Notebooks in the History of Science (ed. F. Holmes, J. Renn and H.-J. Rheinberger), New York, 2003, 1–24.

11 Naylor, ‘Galileo's simple pendulum’, op. cit. (2).

12 Naylor, ‘Galileo's simple pendulum’, op. cit. (2), 38–9.

13 See Palmieri, op. cit. (1), 217, for the mathematical details.

14 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60; Galileo, Dialogue on the Two Chief World Systems (1632), Edizione Nazionale, vii, 256–7, 474–6, and Palmieri, op. cit. (1), 260–2, 262–3; Two New Sciences (1638), Galileo, Edizione Nazionale, viii, 128–9, 139–40, 277–8, and Palmieri, op. cit. (1), 263–4, 264–5, 265–8.

15 Hemp is a material that would have been easily available in Galileo's time. Linen would also have been available. Galileo says spago or spaghetto, i.e. thin string, which implies that the string's material would probably have been hemp or linen. See the ‘Supporting document’ for further details.

16 The choice of the lead balls is somewhat problematic. Galileo does not specify the size of the lead balls he uses, but the words he chooses seem to indicate that he used very small balls, roughly the size of musket balls, or little more. Musket balls at that time would have been in the range of one to two ounces (between about twenty-eight and fifty-six grams). See ‘Supporting document’ for further details.

17 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60; Galileo, Two New Sciences (1638), Edizione Nazionale, viii, 128–9, 277–8, and Palmieri, op. cit. (1), 263–4, 265–8. I will consistently use ‘oscillation’ to indicate a complete swing of the pendulum, back and forth from starting point to maximum height on the other side with respect to the perpendicular and return. Galileo's language is not always clear when referring to pendulum swings, at times suggesting oscillations, other times perhaps half oscillations.

18 This is how I performed the adjustment of the lengths of the pendulums. It is, of course, possible to think up better fine-tuning methods, but they are trial-and-error procedures, since the lengths of the pendulums will always look the same all the time (obviously within the limits of technologies that would have been available to Galileo).

19 See the ‘Supporting document’; and also Palmieri, op. cit. (1), Appendix 1.

20 Galileo, Dialogue on the Two Chief World Systems (1632), Edizione Nazionale, op. cit. (1), vii, 256–7, and Palmieri, op. cit. (1), 260–2.

21 For a description of the thin string see Galileo, Two New Sciences (1638), Edizione Nazionale, op. cit. (1), viii, 128–9, and Palmieri, op. cit. (1), 263–4.

22 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60; Galileo, Two New Sciences (1638), Edizione Nazionale, op. cit. (1), viii, 128–9, and Palmieri, 263–4.

23 Braun, Cf. M., ‘On some properties of the multiple pendulum’, Archive of Applied Mechanics (2003), 72, 899–910Google Scholar, for a discussion of a multiple-mass pendulum model. I have adopted Braun's linear approximation for my hundred-mass model of the chain pendulum.

24 Galileo, Dialogue on the Two Chief World Systems (1632), Edizione Nazionale, op. cit. (1), vii, 474–6, and Palmieri, op. cit. (1), 262–3.

25 Galileo, Dialogue on the Two Chief World Systems (1632), Edizione Nazionale, op. cit. (1), vii, 256–7 and Palmieri, op. cit. (1), 260–2.

26 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60.

27 Galileo, Edizione Nazionale, op. cit. (1), viii, 205–8.

28 It is rather easy to cast lead balls since lead's liquefying temperature is not too high. It is therefore possible that Galileo would have cast his own lead balls, in which case he would have been free to cast balls of different weights than musket balls (see the ‘Supporting document’ for more on lead balls and other materials).

29 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60.

30 At one point in the course of the experiments, the impression emerged that the two pendulums might somehow interfere with each other. The possible effects of interference are discussed later in this section. The two pendulums did not interfere in any appreciable way. I tested this conclusion by leaving one of the two pendulums at rest and operating the other to see if the motion of one might excite some movement in the other. Video 8 (‘The stability of the pendulum’) shows that the pendulum left at rest remains at rest while the other oscillates for a long period of time, well beyond the four to five minutes of the time window allowed.

31 Galileo to Guido Ubaldo dal Monte, 29 November 1602, Edizione Nazionale, op. cit. (1), x, 97–100, and Palmieri, op. cit. (1), 257–60; Galileo, Two New Sciences (1638), Edizione Nazionale, viii, 277–8, and Palmieri, op. cit. (1), 265–8.

32 Galileo, Two New Sciences (1638), Edizione Nazionale, op. cit. (1), viii, 277–8, and Palmieri, op. cit. (1), 265–8.

33 In Two New Sciences (1638), Edizione Nazionale, op. cit. (1), viii, 277–8, Galileo seems to argue that the count would disagree not only by not even one oscillation, but also by not even a fraction of one oscillation, and for angles up to more than eighty degrees. I believe that this is an exaggeration, since the discrepancy produces that kind of difference of a fraction of an oscillation.

34 Galileo, Dialogue on the Two Chief World Systems (1632), Edizione Nazionale, op. cit. (1), vii, 474–6, and Palmieri, op. cit. (1), 262–3.

35 Galileo, Edizione Nazionale, op. cit. (1), i, 323–8, is the most elaborate version of the battery of counterarguments levelled by Galileo at the theory of the punctum reflexionis in the whole of De motu.

36 The argument reconstructed by Galileo is rather obscure. Galileo's original Latin is as follows. ‘Quod movetur ad aliquod punctum accedendo et ab eodem recedendo, ac ut fine et principio utendo, non recedet nisi in eo constiterit: at quod ad extremum lineae punctum movetur et ab eodem reflectitur, utitur eo ut fine et principio: inter accessum, ergo, et recessum ut stet, est necessarium’. Galileo, Edizione Nazionale, op. cit. (1), i, 323–4.

37 Galileo, Edizione Nazionale, op. cit. (1), i, 326–8.

38 Experience does not teach causes, says Galileo: ‘quaerimus enim effectuum causas, quae ab experientia non traduntur’. Galileo, Edizione Nazionale, op. cit. (1), i, 263.

39 Galileo, Two New Sciences (1638), Edizione Nazionale, op. cit. (1), viii, 128–9, and Palmieri, op. cit. (1), 263–4.

40 Palmieri, P., ‘Galileo's construction of idealized fall in the void’, History of Science (2005), 43, 343–89CrossRefGoogle Scholar.

41 In order to get closer to the 1:100 ratio, I repeated the tests with a heavier lead ball, as shown in Video 26 (‘Cork lead 4lb interference’), and Videos 27 to 29 (‘cork lead 4lb discrepancy’). The lead ball was 1812 grams and the cork ball 18.5 grams, very close to the exact ratio claimed by Galileo. As Video 26 shows, the heavy lead ball causes some interference that somehow affects the results. The origin of this interference is mechanical, as will be explained below in this section. Unfortunately, the experiments with the heavy lead ball are affected by an interference that precludes further conclusions.

42 Palmieri, op. cit. (40); idem, ‘“… spuntar lo scoglio più duro”: did Galileo ever think the most beautiful thought experiment in the history of science?’, Studies in History and Philosophy of Science (2005), 36, 223–40; and idem, ‘The cognitive development of Galileo's theory of buoyancy’, Archive for History of Exact Sciences (2005), 50, 189–222.

43 Galileo, Edizione Nazionale, op. cit. (1), i, 273. Galileo is clear and honest. The proportions of motions calculated according to the Archimedean rules of the specific gravities do not pass the test of experience. Galileo does not say more about the tests.

44 An intriguing answer might be given by the tests made by Thomas Settle and Donald Miklich, even though the experiments only aimed at determining the plausibility of the empirical basis underlying another theory espoused by Galileo in De motu, namely the theory according to which light bodies fall faster than heavy bodies at the beginning of a free fall. See T. Settle, ‘Galileo and early experimentation’, in Springs of Scientific Creativity: Essays on Founders of Modern Science (ed. R. Aris, H. Ted Davis and Roger H. Stuewer), Minneapolis, 1983, 3–20.

45 See the ‘Supporting document’ for reasons why this arrangement is unconvincing.

46 ‘Pathway’ in the sense articulated by F. L. Holmes, Investigative Pathways: Patterns and Stages in the Careers of Experimental Scientists, New Haven and London, 2004.