Galileo's theory of motion: Processes of conceptual change in the period 1604–1610

References

• Drake , Stillman . 1972 . On the probable order of Galileo's notes on motion . Physis , 14 : 55 – 68 . and his ‘Galileo's discovery of the law of free fall’, Scientific American, 228, No. 5 (1973), 84–92 (see esp. p. 87).
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• Wisan , Winifred . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 103 – 306 . (see esp. p. 127).
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• Drake . 1972 . Galileo's discovery . Physis , 14 : 87 – 87 .
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• Galilei , Galileo . 1890–1909 . Opere Edited by: Favaro , A. Vol. 8 , 363 – 436 . Florence 20 vols.
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 230 – 230 .
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• Drake . 1972 . Galileo's discovery . Physis , 14 : 89 – 91 .
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 210 – 214 .
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 209 – 209 .
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• Reproduced in Drake Galileo's discovery Physis 1972 14 84 84
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• Drake . 1972 . Galileo's discovery . Physis , 14 : 89 – 89 .
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• Drake . 1972 . Galileo's discovery . Physis , 14 italics added
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• Opere , 1 268 – 268 . 301. See also Galileo Galilei, On motion and on mechanics (translated by I. E. Drabkin and S. Drake: Madison, 1960), 32, 68.
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• Drake . 1972 . Galileo's discovery . Physis , 14 : 90 – 90 .
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• Drake , Stillman . 1974 . Galileo's work on free fall in 1604 . Physis , 16 : 309 – 322 . This paper includes reproductions of folios 107v and 152r of volume 72 of the Galilean manuscripts on pages 312 and 315 respectively. It will be seen that there is no sign of any experimental arrangement on folio 107v, nor are there any words or phrases providing clues to such arrangements or procedures.
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• Drake , Stillman . 1973 . Galileo's experimental confirmation of horizontal inertia: unpublished manuscripts . Isis , 64 : 291 – 305 . see pp. 293–296.
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• Naylor , Ronald . 1975 . An aspect of Galileo's study of the parabolic trajectory . Isis , 66 : 394 – 396 . An important aspect of the task of reconstructing the use of a particular manuscript emerges in this case. It revealed clearly that only when it has been possible to demonstrate a precise correspondence between a proposed explanation and the manuscript data is it possible to place any confidence in a proposed reconstruction. The analysis of the data on folio 117r demonstrated that the figures on it were the result of a mathematical analysis which yielded the series of numbers 100, 121.3, 155.3 and 195.4.
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• A reproduction of folio 116v appears Drake Galileo's experimental confirmation of horizontal inertia: unpublished manuscripts Isis 1973 64 297 297 Details of the means of reproducing data similar to that on folio 116v are provided in R. H. Naylor, ‘Galileo and the problem of free fall’, British journal for the history of science, 7 (1974), 105–134 (pp. 107–115).
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• The reference is to experiments capable of producing data of the form and accuracy seen on folio 107v rather than in the sense ‘might possibly’ used by Drake in advancing his hypothesis for folios 107v, 117r and 116v. Even assuming that an inclined plane was involved, there are experiments that evidently could yield data similar to that on folio 107v. Consideration of reconstructions provided in my Drake Galileo's experimental confirmation of horizontal inertia: unpublished manuscripts Isis 118 120 suggest this, in particular the data of table 7. I hope to return to this at a later date. However, I think it is clear that the evidence available on folio 107v is of such a limited nature that any interpretation will remain speculative. For this reason folio 107v appears incapable of playing more than a secondary role as a source of evidence of experimentation. A close resemblance between the data of any reconstructed experiment and that on folio 107v is clearly a prior requirement for any worthwhile discussion.
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• Opere , 8 213 – 213 .
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• It is reasonably certain that in the Discorsi Galileo described those experimental arrangements which he believed to be the best means of displaying the evidence most relevant to his theory. We also know from the study of folios 116 and 114 that Galileo carried out experiments on planes inclined at between 15° and 40° to the horizontal: these experiments gave measurements agreeing with s∞t2 to within ±2% (see Naylor An aspect of Galileo's study of the parabolic trajectory Isis 1975 66 107 117 Evidence obtained from electrically timing the rolling of a sphere down differing distances on an incline of 6·5° gave deviations of ±2% from the s∞t 2 law (ibid., 130). This agreed well with Galileo's data on folios 116 and 114. Similar measurements on an incline of 3·2° revealed derivations in time of ±4·5% from s∞t 2 (ibid.). This evidence, combined with Galileo's statements in the Discorsi that inclines between 5° and 10° were those used to confirm s∞t 2, can only indicate that Galileo's recommendations were not without basis. Galileo obviously exploited the fact that the smaller the inclination of the plane the greater the time taken for a sphere to roll down its length. Even so, he appears to have appreciated that there were limits to how far one could go in reducing the inclination before the effects of friction, air resistance, and so on, would interfere with the motion of the sphere and so offset the increased accuracy gained by extending the time to be measured. The effects of friction, etc., on a steel ball-bearing moving in a polished wooden groove of small inclination are examined in Naylor (footnote 15), 117–118. It so happens that the incline used was of the length suggested by Drake for his reconstruction of folio 107v. The measurements conducted on an incline of 220 cms revealed that as the incline decreased in size the effects of friction became more noticeable (ibid., table 5). Thus, comparing runs over 220 cms on an incline of 8·3° with runs at 4·1°, those on the incline of 4·1° took 4% longer than predicted by the law. Compared with runs down the 8·3° incline, runs down inclines of 3·2° took 6% longer and down inclines of 2° took 7·5% longer, while down an incline of 1·7° they took 10% longer. Electrical timing reveals that deviations from s∞t 2 for distances travelled down a particular incline also increases with decreasing slope. The variations of the times of runs from s∞t 2 on inclines of 1·7° are more than 6%. This indicates clearly that it would be impossible to confirm s∞t 2 to within ± 1·5% on an incline of 1·7° as on this slope the law no longer holds to this accuracy. In the examination of this issue it was essential not to confuse errors of measurement with deviations from the law. The errors of measurement involved in using electrical timing equipment are very small and are fixed (0·5–1·0%). Obviously the errors involved in this means of timing would be the same at inclines of 6·5° as at inclines of 3·2°. The deviations from the law are greater and change with inclination (ibid., table 5).
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• The striking feature of Drake's subsequent paper illustrating the idea of his experiment is that it seems to provide a convincing demonstration of its impracticability. The photograph provided goes no way to making credible Drake's view that Galileo would have been able to measure the successive distances to the nearest point (0·95 millimetres) and still think it worthwhile adding + or — to indicate over or under reading. See Drake S. The role of music in Galileo's experiments Scientific American 1975 232 6 98 104
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• Drake . 1973 . Galileo's experimental confirmation of horizontal inertia: unpublished manuscripts . Isis , 64 : 296 – 296 . 299, 300
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• Naylor . 1974 . Galileo and the problem of free fall . British journal for the history of science , 7 : 109 – 115 .
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• Naylor . 1974 . Galileo and the problem of free fall . British journal for the history of science , 7 : 110 – 110 . Drake indicated his withdrawal in ‘Galileo's discovery of the parabolic trajectory’, Scientific American, 232, No. 3 (1975), 102–110.
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• The sole evidence advanced to date has been the photograph in Drake The role of music in Galileo's experiments Scientific American 1975 232 6 98 104 No details of specific measurements for this, or other proposed reconstructions have as yet been published by Drake.
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• Opere , 8 213 – 213 .
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• Opere , 8 213 – 213 .
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• Naylor . 1975 . An aspect of Galileo's study of the parabolic trajectory . Isis , 66 : 113 – 113 .
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• This is my conclusion based on the study of the manuscript in Florence. The details of the watermarking on the letters held in the National Library, Florence, is provided in Drake On the probable order of Galileo's notes on motion Physis 1972 14
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 210 – 210 .
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 213 – 213 .
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 214 – 214 .
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• Drake . 1972 . On the probable order of Galileo's notes on motion . Physis , 14 : 86 – 86 .
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• Opere , 10 115 – 116 .
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• Opere , 8 373 – 374 .
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• Drake . 1974 . Galileo's work on free fall in 1604 . Physis , 16 : 319 – 319 .
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• Drake . 1974 . Galileo's work on free fall in 1604 . Physis , 16 : 214 – 214 . footnote 9
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• Drake . 1974 . Galileo's work on free fall in 1604 . Physis , 16 : 210 – 210 .
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• Drake . 1974 . Galileo's work on free fall in 1604 . Physis , 16 : 207 – 207 .
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• As translated in Drake Galileo's 1604 fragment on falling bodies British journal for the history of science 1969 4 342 343 The ‘[have]’ near the end of the quotation has square brackets because it is Drake's addition to the text.
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 235 – 235 . A discussion of the manuscript material is given on pp. 229–236.
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 233 – 233 . Galileo's letter to Valerio which dates from 1609 is lost. Wisan provides an important analysis on pp. 222–226 of the material in vol. 72 that appears to be related, and offers a reconstruction of Galileo's letter on the basis of the manuscript and Valerio's reply to Galileo.
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• It appears in the Discorsi Opere 8 242 242 and in the form of a scholium. This informal presentation indicates Galileo's dissatisfaction with it as a demonstration, as does the fact that he could not bring himself to complete the final steps in the proof involving infinite magnitudes. Wisan discusses this (ibid., 235). To explain the use of two proofs Wisan suggests that Galileo's aim may have been to establish that the first kind of proof involving the argument from limits and infinite magnitudes was acceptable in that it was justified by its consistency with accepted procedures involving medieval proof techniques. This is evidently a viable interpretation and not one that suggests any fundamental differences in the interpretation of the Discorsi evidence from that advanced here. However, I suspect that Galileo's original aim was to establish the double distance rule securely by a means other than the proof technique. Galileo's attitude may have been the reverse of that suggested by Wisan in that he really felt that the proof technique could only be justified by its agreement with an independent demonstration based on quite different principles.
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• It appears in the Discorsi Opere 8 277 295
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• It appears in the Discorsi Opere 8 277 277 Wisan's arguments for a redating of the ‘Liber secundus’ draft are convincing. She suggests a new date for this manuscript as ‘some time not long before 1630’. Other evidence relating to this draft, as yet unpublished, confirms the proposal for a new date. Koyré's view was that the manuscript could not date from as early as 1604 (as Favaro had proposed) as it contained a discussion of the correct law of fall. In view of this, Koyré suggested a date of 1609. Wisan's proposal, which has been independently corroborated, reveals later changes in Galileo's plans for the foundation of the treatise than had ever previously been suspected.
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• It appears in the Discorsi Opere 8 227 229
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• Opere , 8 427 – 427 .
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• Drake . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 92 – 92 . Wisan (footnote 2), 206, 211, suggest that Galileo had obtained observational knowledge of the form of the projectile trajectory before developing his explanation. In R. H. Naylor, ‘Galileo: the search for the parabolic trajectory’, Annals of science, 33 (1976), 153–172, a reconstruction of a Galilean experiment on folio 81r of the manuscript revealed how the investigation of the projectile trajectory proceeded. From this experiment, which provided evidence of the parabolic form of the trajectory, it seems likely that Galileo proceeded to experiments of the form seen on folio 116v. Though folio 81r throws added light on the interpretation of the manuscripts advanced here, as do many other manuscripts not discussed, detailed reference to it is omitted. As the main features of the relationship between experiment and theory can be established by the examination of folio 116v discussion is restricted to that document. (In a personal communication Dr. Wisan informed me of her acceptance of the reconstruction of the use of folio 81r referred to above.)
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• Drake . 1973 . Galileo's experimental confirmation of horizontal inertia: unpublished manuscripts . Isis , 64 a photograph appears on p. 297. See also pp. 294 and 301 for photographs of folio 117r and folio 114v referred to above.
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• See Drake On the probable order of Galileo's notes on motion Physis 1972 14 55 68
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• Opere , 10 229 – 229 . A letter to Antonio de' Medici dated 11 February 1609 announces that all horizontal shots from a cannon will have a common time of flight no matter what the powder charge, provided they are fired over a horizontal plane. This corresponds precisely to the arrangement in the folio 116v experiment.
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• Naylor . 1974 . Galileo and the problem of free fall . British journal for the history of science , 7 : 107 – 115 .
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• Wisan . 1974 . The new science of motion: a study of Galileo's De motu locali . Archive for history of exact sciences , 13 : 206 – 206 . 211; and Raymond Fredette, ‘Les de motu “plus anciens” de Galileo Galilei: prolégomènes’ (Ph.D. thesis, Université de Montréal, 1969), 147–163.
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• Discussed by Fredette Les de motu “plus anciens” de Galileo Galilei: prolégomènes Ph.D. thesis Université de Montréal 1969 206 206
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• See Wisan The new science of motion: a study of Galileo's De motu locali Archive for history of exact sciences 1974 13 249 258 for a discussion of the evidence and the technique.
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• See Wisan The new science of motion: a study of Galileo's De motu locali Archive for history of exact sciences 1974 13 206 206
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• Added in proofs , Since completing this paper I have examined folio 152r again in the company of Dr. Thomas Settle. He pointed out to me then that the mark ‘i?’, which has been the source of such varied speculation (sub-section 2.2 above), is almost certainly an abbreviated form of ‘in’. Galileo's use of this abbreviation can be seen quite clearly elsewhere in the folio, Thus, at the very bottom of the manuscript there is ‘d’ i peto’, twice in succeeding lines, while most clearly in the very last line we have ‘i un'hora et 16 i 2 hore’. Settle has thus solved the puzzle, beyond any reasonable doubt, that is.
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