¹ Galileo, , Dialogues concerning Two New Sciences, trans. Crew, Henry and de Salvio, Alfonso (New York: Macmillan, 1914), 166 [202]Google Scholar. Cited hereafter as Two New Sciences. The bracketed numerals represent the pagination of the Italian original as given by Crew and de Salvio.

² Galileo, , On Motion and On Mechanics, trans. Drabkin, I. E. and Drake, Stillman (Madison, Wisc., 1960), 16–17.Google Scholar Cited herefter as On Motion and On Mechanics. Concerning Benedetti see Mach, Ernst, The Science of Mechanics, trans. McCormack, Thomas (LaSalle, Ill., 1960), 156Google Scholar, and Drabkin, I. E., “G. B. Benedetti and Galileo's De Motu”, Proceedings of the 10th International Congress on the History of Sciences (Ithaca, 1962), 627–630.Google Scholar

³ This experiment consists in thinking of a body being cut into two unequal pieces as it falls freely, or of two pieces of the same material being joined while falling. In either case, there is no reason to believe that the velocities of fall will be changed.

⁴ On Motion and On Mechanics, 107.Google Scholar

⁵ Ibid., 107–109. See also Galileo's, Dialogue concerning the Two Chief World Systems, trans. Drake, Stillman (Berkeley, 1953), 151–152Google Scholar: Salviati asks Simplicio whether a cotton or lead pendulum will remain in motion longer and Simplicio chooses the latter. “Salv. So that whatever the cause of that impetus and mobility it is conserved longer in the heavy material than in the light.” This work will be cited hereafter simply as the Dialogue.

⁶ On Motion and On Mechanics, 88–90, 110–114.Google Scholar

⁷ Ibid., 63–69.

⁸ Ibid., 65.

⁹ Ibid., 68: “Now the question is what ratio the speed [of a body falling or rolling] on ca bears to the speed on ad. And since line da is to line ab as the slowness on ad is to the slowness on ab, as has been shown above; and since the slowness on ab is to the slowness on ac as line ab is to line ac, it follows, ex aequali, that line da will be to line ac as the slowness on ac. And therefore, line da will be to line ac also as the speed on ac is to the speed on ad.” The ratio of slownesses here apparently is the inverse of the ratio of speeds.

¹⁰ Ibid., 64.

¹¹ Koyré, Alexandre, Études galiléennes (published as nos. 852–854 of Actualités Scientifiques et Industrielles, Paris, 1939), fasc, ii, 4.Google Scholar But see also the English translation of Boas (Hall), Marie, The Scientific Renaissance (New York, 1962), 224Google Scholar. Koyré's use of the expression “la proportion double” is dubious.

¹² An English translation is given in the Appendix below. I am indebted to Professor A. R. Hall for having checked the translation (based on Duhem's French) against the original Italian.

¹³ See Hall, A. R.. Note on “Another Galilean Error”, Isis, 1 (1959), 261–262.CrossRefGoogle Scholar

¹⁴ Professor Cohen's remark was made in discussing another of Galileo's errors. But if it is relevant there it is even more relevant here. The point is that the remark is not relevant in either case. See Cohen, I. B., “Galileo's rejection of the Possibility of Velocity Changing Uniformly with Respect to Distance”. Isis, xlvii (1956), 235Google Scholar. See also the discussion by Hall, Cohen A. R. and Drake, Stillman in Isis, xlix (1958), 342–346.CrossRefGoogle Scholar

¹⁵ But see Dijksterhuis, E. J., Mechanization of the World Picture, trans. Dikshoorn, C. (Oxford, 1961), 339Google Scholar. Dijksterhuis seems to intimate that Galileo might have got the law of squares from some Aristotelians or other. But he is neither sufficiently explicit nor definite about this for the reader to know what evidence he might have for the contention.

¹⁶ Galileo, , Opere, Edizione Nazionale (Florence, 1934), x, 97–100.Google Scholar

¹⁷ Loc. cit., 451.Google Scholar

¹⁸ Two New Sciences, 188–190 [221–222], 237–239 [261–263].Google Scholar

¹⁹ Ibid., 180 [214].

²⁰ Ibid., 189–190 [221–222]. The dynamical proof is the second one given by Galileo.

²¹ Two New Sciences, 180–185 [214–219]Google Scholar; On Motion and On Mechanics, 170–175.Google Scholar

²² On Motion and On Mechanics, 135.Google Scholar

²³ Dijksterhuis, , op. cit. (15), 340–341Google Scholar: “By extending the axiom of the proportionality between velocity and distance to the motion of bodies rolling down smooth inclined planes, so that the distance travelled is replaced by the vertical distance from the starting point, Galileo was able to deduce from his false premise the correct proposition that the final velocity of a body rolling down an inclined plane depends solely on the amount of the vertical descent of the moving body and not on the angle of inclination of the plane. Moreover, he deduced the proposition which was to be of importance for his later work, namely that particles, when released simultaneously at points of a vertical circle, move to the lowest point … of this circle along chords, and arrive there simultaneously. This proof, however, calls for dynamical reasoning.” According to the order of Dijksterhuis's exposition this demonstration follows the letter to Sarpi but no explicit date is given.

²⁴ Two New Sciences, 186 [218–219].Google Scholar

²⁵ Loc. cit., 170 [205].Google Scholar

²⁶ Hall, A. R., “Galileo and the Science of Motion”, British Journal for the History of Science, ii (1965), 197.Google Scholar

²⁷ This first portion of the proof is an abbreviated version of Galileo's argument in Theorem III, Proposition III of Two New Sciences. Loc. cit., 185–186 [215–218].Google Scholar

²⁸ It is likely that Galileo never completely cleared up in his own mind the distinctions between force and momentum though, as we know, he completely succeeded in analysing the correlative kinematical concepts of acceleration and velocity. That Galileo was concerned about this matter late in life is evidenced by the rather obscure “elucidation” on momentum which he had Viviani draw up for inclusion in later editions of Two New Sciences. (See pp. 180–185 [214–219] of the Crew-de Salvio translation.) This undoubtedly has some bearing on the fact noted by Professor Hall in his recent paper (op. cit. (26), 192) that correspondence with Luca Valerio in May of 1609 seems to involve residual confusion on Galileo's part about the speeds of bodies on inclined planes. Although Galileo's own contribution to the correspondence is lost it would not be surprising to find that even at this late date he was still drawing incorrect inferences from his earlier dynamical studies.

²⁹ Dialogue, 23.Google Scholar

³⁰ Ibid., 23–24.

³¹ Ibid., 24–25.

³² Ibid., 26.

³³ Which is not to say, of course, that he had become clear about all particulars of accelerated motion or that he was actually capable of resolving the apparent contradiction troublinġ Sagredo. Galileo's difficulties with the ideas of speed, velocity and momentum very likely did not end immediately with the discovery of the correct definition of uniform acceleration but only resolved themselves as he developed more and more of the theory of motion. See footnote (28) Supra.

³⁴ Boas (Hall), Marie, op. cit. (11), 224.Google Scholar

³⁵ Two New Sciences, 208 [237]Google Scholar. Corollary to Problem VI, Proposition XIX: “… if the time of fall from rest through any given distance is represented by that distance itself, then the time offall, after the given distance has been increased by a certain amount, will be represented by the excess of the mean proportional between the increased distance and the original distance over the mean proportional between the original distance and the increment.”

³⁶ Two New Sciences, 169–170 [205].Google Scholar

³⁷ Ibid., 171–172 [207].

³⁸ Hall, , “Galileo and the Science of Motion”, op. cit. (26), 195–196.Google Scholar

³⁹ Ibid., 190.

⁴⁰ Ibid., 195.