Proofs and Paradoxes: Free Fall and Projectile Motion in Galileo’s Physics

  • Peter Damerow
  • Gideon Freudenthal
  • Peter Mclaughlin
  • Jürgen Renn
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)


According to a well established view the work of Galileo marks the beginning of classical mechanics.1 His work does not yet represent the full fledged classical theory as it emerged in the contributions of Newton and others, but, following this widespread interpretation, Galileo did take the first decisive steps: he criticized and overcame the traditional Aristotelean world picture, he introduced the experimental method, he concentrated on a systematic and concise description of single phenomena rather than searching for their causes and elaborating an overarching philosophy of nature, and he succeeded in the mathematical analysis of some of the key problems of classical mechanics 2


Incline Plane Free Fall Uniform Motion Oblique Projection Horizontal Projection 
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  1. 1.
    See, e.g., Clavelin 1974, p. 271: “While the defense of the Copernican doctrine enabled Galileo to outline some of the most significant ideas and methods of classical mechanics, his Discourses must be considered an integral part of that science.”Google Scholar
  2. 2.
    See, e.g., Drake’s characterization of Galileo’s pioneering contribution: “Physics began to depart from the Aristotelian conception with Galileo’s investigations of motion in bodies free to descend, whether unrestrained, or supported from below, or suspended from above as with pendulums.… The historical evolution of modern physics will not be thoroughly understood without the attention to the labors of men who put aside the ambition to explain everything in the universe and sought to find sure particular limited rules by careful measurements. So far as we know, Galileo was the first to do this for actual motions.” (Drake 1979, p. VII)Google Scholar
  3. 5.
    See, e.g., Drake 1964, pp. 602–603: “But if Galileo never stated the law [of inertia] in its general form, it was implicit in his derivation of the parabolic trajectory, and it was clearly stated in a restricted form for motion in the horizontal plane many times in his works.”Google Scholar
  4. 6.
    This view is expressed drastically in Szabô 1977, p. 60: The value of Aristotle’s and Descartes’ natural philosophies for natural science may be seen at most in having stimulated Galileo and Newton to refute their errors.Google Scholar
  5. 7.
    According to Drake (1978, p. 387) Descartes’ reaction “throws light on the reception of Galileo’s science outside the circle of his friends and former pupils, illustrating the conception of science most opposed by Galileo and the ineffectiveness of his own work in altering traditional goals.”Google Scholar
  6. 8.
    Letter to Mersenne, June 29, 1638; AT II, 194.Google Scholar
  7. 9.
    Letter to Mersenne, August 23, 1638; AT II, 336.Google Scholar
  8. 10.
    See the recent review of various interpretations given in Ariew 1986. Emil Wohlwill (1884, pp. 128–129) presents a remarkable exception claiming in fact that the essential points of Descartes’ criticism are correct, even though Descartes failed to recognize Galileo’s great achievements. For a similar view, see also Dijksterhuis 1924, pp. 301ff.Google Scholar
  9. 11.
    AT II, 380; translation adapted from Drake, 1978, pp. 387–388. See document 5.3.1.Google Scholar
  10. 12.
    Letter to Mersenne, Oct 11, 1638. “It is to be noted that he takes the converse of his proposition without proving or explaining it, that is, if the shot fired horizontally from B toward E follows the parabola BD, the shot fired obliquely following the line DE must follow the same parabola DB, which indeed follows from his assumptions. But he seems not to have dared to explain it for fear that their falsity would appear too evident.” (AT II, 387; translation, Drake 1978, p. 391) It has been claimed (e.g., Shea, 1978, pp. 150–151) that this criticism is insub-stantial, but we shall see below that Descartes actually pointed to a serious problem in Galileo’s treatment of projectile motion. (See document 5.3.1)Google Scholar
  11. 13.
    For instance Drake (1978, p. 31) maintains that, in his early unpublished writings on motion, Galileo had exhausted the traditional preoccupation with the search for causes as such.“Google Scholar
  12. 14.
    Galileo’s scholastic notebooks on physical questions were published in EN I and are now also available in a translation by Wallace (1977).Google Scholar
  13. 15.
    According to Wallace (1984, p. 230), Galileo’s earliest writings on motion were the direct continuation of his scholastic studies of logical and physical questions.Google Scholar
  14. 16.
    De Motu was first published in EN I, 251–340; it will be cited in Drabkin’s translation (Galileo 1960b). For a thorough philological study of the treatise, including earlier versions and later reworkings, see Fredette 1969, partly summarized in Fredette 1972.Google Scholar
  15. 18.
    Earlier versions and later reworkings of De Motu, including a dialogue version, are published in EN I, 341–408. For an English translation of some of these, see Drake and Drabkin 1969, pp. 115–131. Fredette (1972, p. 321) claims that Galileo first wrote the dialogue version and then two complete treatise versions of De Motu before he abandoned it. Fredette’s ordering of the manuscript material was challenged by Drake (1976b), but Drake’s criticism was rejected by Galluzzi (1979, p. 168, note 58). Wallace (1984, p. 231) follows Fredette’s ordering, while Drake further develops his argument in Drake 1986. The most recent account of this debate is Wallace 1990.Google Scholar
  16. 19.
    This comment, dating from 1674, is quoted from Fredette 1972, p. 326.Google Scholar
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    They are preserved today at the Biblioteca Nazionale in Florence; many are transcribed in EN VIII, and many are also reproduced in facsimile in Drake 1979. Here, the manuscripts will be referred to by their folio numbers as in MS and, if transcribed in EN, by the page number of the transcription. For a more detailed description of this collection of manuscripts, see also Renn 1988, and the references given there. Comprehensive studies of Galileo’s science incorporating an analysis of his unpublished manuscripts are Caverni 1895, vol. IV, Wohlwill 1883–1884 and 1909, DijksterhuisGoogle Scholar
  18. 21.
    Some such attempts are documented in the English translations of 16th century works on motion and mechanics collected in Drake and Drabkin 1969. For a discussion of Tartaglia’s role in making the works of Archimedes accessible, see the Introduction to this volume. The significance of Tartaglia’s contribution, in particular, and that of the engineering tradition, in general, to early modem science is strongly emphasized in Olschki 1927. On their importance for Galileo’s early science, see Drake 1986, pp. 438–439, and Settle 1987.Google Scholar
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    See Benedetti 1585 (English translation in Drake and Drabkin 1969).Google Scholar
  20. 23.
    This, at least, is the understanding of Aristotle’s theory found in works of early modem science. For reconstructions of Aristotle’s original thinking on falling bodies, see Young 1967 and Casper 1977; for a discussion of its later reception, see Sorabji 1988.Google Scholar
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    For a discussion of a similar criticism of Aristotle by Benedetti and its relationship to Galileo’s, see Drake 1986, pp. 438–439.Google Scholar
  22. 32.
    See, e.g., DM, 31; EN I, 267. The systematic treatment of all other cases actually presupposes a conceptualization of the notion of specific weight in terms of compounded proportions that was not yet available to Galileo at the time he composed De Motu. For a comprehensive study of the development of this notion in Galileo’s later work and the role played by compounded proportions in it, see Napolitani 1988.Google Scholar
  23. 39.
    For the evidence of these reworkings, see note 18. Several commentators on De Motu have pointed out the existence of internal contradictions in Galileo’s treatise. Besides Fredette 1969, the most comprehensive study of De Motu, including an analysis of such contradictions, is Galluzzi 1979, pp. 166–197. Among the puzzles of Galileo’s early theory of motion that become evident from his reworkings of De Motu, is the problem whether or not the upward motion of a light body can properly be classified as a natural motion since its “Archimedean” explanation by extrusion brings an external force into the play and hence is in conflict with the Aristotelian understanding of natural motion as an intrinsic property of the moving body. Another puzzle arises from Galileo’s use of statics in his theory of motions along inclined planes in De Motu, which suggests a dependence of the speed of motion on the weight of the moving body, contrary to Galileo’s conclusions discussed in the present section.Google Scholar
  24. 40.
    DM, 37–38; EN I, 273. For a discussion of Galileo’s early experimentation and its historical context, see Settle 1983.Google Scholar
  25. 46.
    DM, 78–79. According to Duhem (1906–1913, Vol. 1, p. 110) it was common, in the Middle Ages as well as in early modern times, to conceive the impressed force generating the motion as a quality comparable to heat or cold and to make assertions about its varying degrees. For a discussion of the antique and medieval background of Galileo’s understanding of impressed forces, see Moody 1951.Google Scholar
  26. 55.
    Galileo’s description of the fall of different bodies in a medium is examined from the point of view of classical mechanics in Coulter and Adler 1979 and from the point of view of a 16th century experimentalist tradition in Settle 1983. Settle and Miklich (see Settle 1983) have repeated Galileo’s experiment, confirming his description, and provide an explanation for the paradoxical results based on psychological and physiological differences in perception of simultaneity when dropping heavy and light bodies.Google Scholar
  27. 58.
    DM, 69; EN I, 302. This statement refers specifically to motion along inclined planes.Google Scholar
  28. 60.
    Tartaglia 1537, for an English translation, see Drake and Drabkin 1969. A later treatment by Tartaglia of projectile motion is found in Tartaglia 1546. For historical surveys of projectile motion in the time between Tartaglia and Galileo, see Hall 1952 and Barbin and Cholière 1987.Google Scholar
  29. 61.
    Tartaglia’s trajectory also became the basis for measuring instruments used by gunners; see Wunderlich 1977. Modern interpretations of the history of ballistics often overlook this practical context and judge progress in this field exclusively by the similarity of the trajectories found in the historical sources with that countenanced by classical mechanics; see, e.g., Barbin and Cholière 1987, pp. 63–66. As Tartaglia’s explanation (Tartaglia 1564) and the description of the trajectory in Cardano (1550, Liber primus, p. 394) show, early modern engineers and natural philosophers were better aware of the actual shape of the trajectory than the widespread use of Tartaglia’s construction of the trajectory, taken by itself, would suggest.Google Scholar
  30. 62.
    Naylor (1974c, pp. 325ff) reports that the transcriptions of the diagram in EN I and DM do not correspond to the original De Motu manuscript; our diagram has been altered accordingly.Google Scholar
  31. 63.
    See Tartaglia 1564. Benedetti criticized Tartaglia’s approach but used the same conceptual tools as Tartaglia in his own treatment of problems of projectile motion. Trying to avoid problematical consequences resulting from Tartaglia’s comparison of the cannon with a balance, he compared it instead with an inclined plane (see Drake and Drabkin 1969, pp. 224ff).Google Scholar
  32. 71.
    Galileo’s derivation of the Isochronism of Chords within the conceptual framework of the De Motu theory is documented by several of his manuscripts, see, e.g., MS, fol. 151r (EN VIII, 378 and document 5.3.2). The relevant manuscript evidence is extensively documented and analyzed in Wisan 1974, pp. 162–171. For an analysis of the conceptual background of Galileo’s proof, see Galluzzi 1979, pp. 266–267, Souffrin 1988, and Souffrin and Gautero 1989.Google Scholar
  33. 72.
    MS, fol. 107; both sides of this folio contain diagrams, numbers, and calculations but no text. For a plausible reconstruction of Galileo’s experiment see Drake 1978, pp. 86–90. Drake’s reconstruction was first published in 1974 and followed an earlier, successful attempt to reconstruct Galileo’s inclined plane experiment by Settle (an attempt which was not, however, based on manuscript evidence); see Settle 1961. The reconstruction of the fol. 107v experiment, as well as its role for the development for Galileo’s understanding of motion, has been extensively discussed by historians. Drake’s reconstruction has been challenged by Naylor who reconstructs the figures in this manuscript on the basis of an experiment on projectile motion; see Naylor 1977a, pp. 373–377, and Naylor 1980a, p. 554.Google Scholar
  34. 73.
    For the reconstruction of Galileo’s diagram as a catenary, see Naylor 1980a, p. 554.Google Scholar
  35. 74.
    del Monte (ms). The manuscript is quoted in Libri 1838–1841, Vol. IV (1841), pp. 397–398. It is now preserved in the Bibliothèque Nationale, Paris (supplément latin 10246, p. 236). We should like to thank Gad Freudenthal for arranging the reproduction of the manuscript and updating Libri’s reference. Guidobaldo’s manuscript was discussed by Fredette (1969, pp. 154–155), who quotes most of this section of it. An English translation was made by Naylor (1974c, p. 327). See document 5.3.4 for transcription and translation of the text. Guidobaldo’s experiment is discussed extensively in Fredette 1969 and Naylor 1974c. Galileo describes the experiment in the Discorsi (EN VIII, 185f/142). Fredette (1969, pp. 148–163) provides a convincing argument that Galileo must have been familiar with Guidobaldo’s experiment before 1601. Here are the main points of his argument: In a letter to Cesare Marsili of Sept. 11, 1632 (EN XIV, 386), Galileo claims to have found the shape of the trajectory 40 years earlier, and also, that this discovery had been a primary goal of his studies of motion. In a letter to Galileo of Sept. 21, 1632 (EN XIV, 395), also concerning the discovery of the shape of the trajectory, Cavalieri reports that he had learned from Muzio Oddi 10 years before that Galileo had performed experiments with Guidobaldo del Monte precisely on this subject. Oddi had received his education in Pesaro from Guidobaldo, who was likely to have told him about his collaboration with Galileo. But since Oddi went to prison from 1601 to 1610, he could not have obtained — at least from Guidobaldo — any information about experiments jointly performed by Guidobaldo and Galileo during that time, nor after he was freed since Guidobaldo died in 1607. Hence, Oddi must have heard from Guidobaldo about the experiments with Galileo before 1601, which is — cum grano salis — in agreement with Galileo’s own dating of his discovery of the shape of the trajectory.Google Scholar
  36. 75.
    The translation is adapted from Naylor 1974c, p. 327; 1980a, p. 551.Google Scholar
  37. 76.
    This interpretation differs from that of Naylor (1974c, p. 333), who claims that Galileo would have required evidence less ambiguous than that provided by the Guidobaldo experiment before embarking on an explanation of either a parabolic or a hyperbolic trajectory. But it is quite conceivable that Galileo engaged in a speculative reconstruction of the trajectory on the basis of evidence that was even less conclusive. In fact, as Hill (1988, p. 667) convincingly argues, Galileo’s familiarity with Apollonius and Archimedes equipped him to analyze a number of ordinary events in terms of mathematical curves. On the other hand, his later, more sophisticated experiments on projectile motion seem to presuppose a knowledge of the parabolic form of the trajectory.Google Scholar
  38. 77.
    This reconstruction was suggested by Settle (1967, pp. 320–335).Google Scholar
  39. 78.
    It is precisely the close relationship between these two elementary properties of projectile motion and the motion of fall that has made it so difficult for historians to establish which was discovered first, in spite of the relatively rich manuscript material documenting Galileo’s thinking on motion. Drake has recently suggested another speculative reconstruction of the discovery of the times squared relationship, based on manuscript evidence; see Drake 1989b, pp. 35–49, and Drake 1990, Chapter 1. According to Drake, Galileo first measured the speeds of a motion of fall along an inclined plane to grow as the series of odd numbers, but he did not recognize the times squared relationship. Galileo then set out to relate fall to pendulum oscillations, first discovering the pendulum law and then the law of fall, both in the form of a mean proportional relationship (on proportions see section 1.2.1 above). Only then did he return to his inclined plane measurements, finally discovering the times squared law. According to Drake, Galileo discovered the parabolic shape of the trajectory only several years later as the result of a new set of careful measurements. While Drake’s reconstruction makes Galileo a sophisticated experimentalist, it presupposes that he is an incompetent mathematician, unable immediately to recognize the elementary relationships between the sequence of odd numbers and that of the square numbers, between the expression of a relationship in terms of mean proportionals and in terms of double proportions or between a sequence of square numbers and a parabola.Google Scholar
  40. 81.
    Most historians of science have failed to realize that this question was a serious puzzle in Galileo’s early theory of motion. In Wolff’s comprehensive analysis of Galileo’s conception of neutral motions and its historical background, for instance, he identifies Galileo’s assertion that a “neutral” motion can be caused by the smallest force with a disposition in favor of neutral motions, and ultimately with Galileo’s statements about the continuation of uniform motion along a horizontal (or spherical) plane; see Wolff 1987, pp. 247–248. Drake explains Galileo’s hesitation in a similar context to make a clear pronouncement on inertial motion by the scruples of a careful experimentalist; with reference to a later comment on the continuation of motion by Galileo’s disciple Castelli, Drake writes: “In a way it is a pity that Galileo never published his inertia idea in as general a form as his pupil thus ascribed it to him, though in another sense it is a great credit to Galileo as a physicist that he refused to go so far beyond his data to no purpose. Descartes did, being a less cautious physicist than Galileo; and being a more ingenious theologian than Castelli, he managed to derive the general law of inertia from the immovability of God” (Galileo 1969a, p. 171, note 26).Google Scholar
  41. 82.
    Except for the emphasis on the role of the logic of contraries as the traditional conceptual background of Galileo’s argument, a similar reconstruction of his discovery of the continuation of uniform motion along the horizontal was given in Wertheimer (1945, pp. 160–167) and by Chalmers and Nicholas (1983, p. 335). Chalmers and Nicholas stress in their concise historical reconstruction that Galileo’s argument neither presupposes nor, contrary to Wertheimer, implies the principle of inertia.Google Scholar
  42. 83.
    MS, fol. 147r (EN VIII, 380). See document 5.3.5. Without referring to this particular manuscript, a similar derivation has been suggested as a possible root of the discovery of the law of fall; see Humphreys 1967. In view of the more direct ways to discover this relationship described in the previous section, this suggestion is not very plausible.Google Scholar
  43. 90.
    The translation is adapted from Drake 1978, p. 95. In his interpretation of the memorandum (Drake 1978, pp. 95–96), Drake overlooks the fact that the second part does not specifically refer to motion along inclined planes but rather refers to motion of fall in general. Drake calls Galileo’s argument an “unsuccessful gambit.” But he fails to note the crucial difference between the argument given in the memorandum and the original De Motu argument, as is clear from his assertion that the equal speed of fall of bodies differing in weight already follows from the De Motu argument. But, in fact, this conclusion is in flat contradiction to the De Motu theory, as we have seen in the previous section.Google Scholar
  44. 91.
    Underneath this memorandum, Galileo wrote, at a later time, the word “Paralogismus” indicating that he had discovered a flaw in his argument. One could indeed object that, in his argument, Galileo had not properly determined the volume of the composite body because, when considering the specific weight of the composite body, he did not take into account the volume of the air (or whatever other medium the two bodies are moving in) which originally filled the hollow of the leaden sphere. One could argue that, if the wooden body is placed into the hollow of the leaden sphere so as to form the composite body, the volume of the air originally filling the hollow has to be counted as part of the volume of the composite body, because otherwise the composite body would have a volume smaller than the sum of the volumes of the single bodies. In other words, a certain volume of the contiguous air, equal to the volume of the hollow, has to be considered as part of the composite body. Although it may seem plausible to take into account this volume, from the point of view of hydrostatics - on which the whole argument is based - an arbitrary volume of the contiguous medium can in fact be added to the composite body without changing anything. Hence, this objection to Galileo’s argument, based as it is on a simple addition of volumes, by no means refutes the argument, but rather serves to strengthen it by pointing to the crucial weakness of a theory of fall based on hydrostatic conceptions, i.e., the determination of the volume of the body falling in a medium, which is, to a certain extent, arbitrary. Nevertheless, in view of Galileo’s later work on hydrostatics, it seems possible that the objection sketched above could have appeared convincing to him. Galileo’s discussion of the floating of thin lamina of materials having greater specific weight than water is indeed based on an erroneous argument according to which the floating object has to be conceived as a composition of the contiguous air and of the floating thin lamina, whereas the modern understanding of this phenomenon is based on thesurface tension of the water. (See Galilei 1612, Theorem VI, and the discussion preceding it.) It is in fact conspicuous that Galileo’s argument in fol. 147r does not appear in his published works and that, in the above mentioned work on hydrostatics, he still refers to the dependency of the velocity of fall on the specific weight; see Galileo 1663, p. 70.Google Scholar
  45. 92.
    Galileo Galilei to Paolo Sarpi, October 16, 1604; EN X, 115–116. (Translation, adapted from Drake 1969.) See document 5.3.3.Google Scholar
  46. 93.
    EN I, 119; Wallace 1977, p. 172. It has been claimed that Galileo’s work on free fall is independent of the medieval traditions going back to the Mertonians and to Oresme, because one particular application of the conceptual tools developed in this tradition, the Merton Rule, is not documented, at least in its original form, by Galileo’s working papers; see, e.g., Drake 1969, p. 350. But this argument is inconclusive because it focuses on one particular application of the conceptual tools in question, while, as we shall see, other applications can in fact be identified among Galileo’s manuscripts. It has, on the other hand, been suggested that concepts and arguments associated with these medieval traditions found their way in several stages into Galileo’s work, and that Galileo supposedly turned directly to medieval sources precisely when confronted with particular problems of motion. See Wisan 1974, p. 288, note 18, pp. 296–297, and also Galluzzi 1979, pp. 273–274, note 36, for a brief review of the relevant arguments by various other authors. However, although, as argued in section 1.2.3 above, the early modern tradition of the configuration of qualities did not include its use as a calculational tool, it cannot be doubted that some of its basic concepts were part of common knowledge, and that hence no particular medieval text has to be assumed to be the priviledged source of Galileo’s familiarity with this tradition. In fact, the notion of “degree of velocity,” for instance, was generally adopted in lectures at the Collegio Romano, from which Galileo drew much of his knowledge about scholastic traditions; see Wallace 1984, p. 268. For a recent study of the role of the Mertonian tradition in Galileo’s work, also emphasizing conceptual similarity rather than dependence on technical results, see Sylla 1986.Google Scholar
  47. 94.
    MS, fol. 107v. One of the two diagrams may suggest that Galileo took the possibility of a discontinuous change of motion into consideration. It seems to be misleading, however, to assert, as Drake (1990, p. 39) does, a supposed “passage from a quantum-concept of speeds to the concept of continuous change” as an essentially new element in Galileo’s mechanics. In fact, according to Wallace (1984, p. 267), Galileo adopted a scholastic view when claiming that velocity varies continuously. There is, however, manuscript evidence showing that Galileo did consider discontinuous changes of speeds; see MS, fol. 182r (EN VIII, 425–426), which Drake (1979, p. LII) dates as late as 1618 (see document 5.3.6). But this manuscript evidence does not support Drake’s suggestion of a deep conceptual gap between the consideration of a gradual and a “quantum” change of speeds, but rather suggests that the choice between these considerations seems to be linked to the mathematical technique applied to a given problem. It shows that an attempt by Galileo to use arithmetic techniques in the study of accelerated motion led him into difficulties. In this manuscript, he attempts to construct the motion of upward projection by subtracting the naturally accelerated motion downward from the violent motion upward. In this construction, he divides the motion of upward projection into a certain number of equal parts and then uses simple arithmetic to determine the result of a step-wise combination of the violent and the natural motion. But his result remains inconclusive because it depends on the number of parts into which the motion was originally divided. For a critique of Drake’s understanding of the role of discontinuity in the medieval and early modern analysis of motion, see also Franklin 1977.Google Scholar
  48. 97.
    For a comprehensive study of Galileo’s use of the concept of moment of velocity, see Galluzzi 1979, in particular Chapter III; for his discussion of fol. 179v, see pp. 285–286. The significance of the relationship between the infinitesimal meaning of the concept of moment and its meaning in Galileo’s statics, thoroughly analyzed by Galluzzi, lies beyond the scope of this chapter.Google Scholar
  49. 98.
    Drake claims that Galileo may have left the argument in fol. 179v unfinished because he discovered a problem with the erroneous principle mentioned in the letter to Sarpi, but his argument is based on an entirely fictitious mathematical difficulty which Galileo is supposed to have discovered in his argument in fol. 179v; see Drake 1970a, p. 23, and Drake 1978, p. 116. Wisan (1974, pp. 220–221) suggests that the argument is left unfinished because Galileo discovered an “elementary error” in his attempt to prove the law of fall from the erroneous principle mentioned in the letter to Sarpi. But, as we shall see in the following, this attempted derivation did not involve an elementary error, at least not from the point of view of Galileo’s knowledge.Google Scholar
  50. 99.
    MS, fol. 85v (EN VIII, 383). According to Favaro this manuscript is in the hand of Galileo’s assistant Mario Guiducci. The passage is crossed out in the manuscript. We follow Wisan in interpreting this argument as an early version of the somewhat more polished argument in fol. 128, to be discussed in the following; see Wisan 1974, pp. 207–209. Contrary to Wisan, we interpret the line marked “S” next to the triangular diagram as representing space and not as being a geometrical mnemonic. The translation is adapted from Drake 1978, pp. 98–99. See document 5.3.9.Google Scholar
  51. 100.
    This interpretation follows Sylla (1986, pp. 73–74), who solves a much debated puzzle. Sylla’s reconstruction is supported by the treatment of this mathematical technique in a standard textbook of the time, Tartaglia 1556b, pp. 111, 123, and 128. There is evidence that Galileo actually learned mathematics from this textbook; see Renn 1988, Appendix B.Google Scholar
  52. 101.
    MS, fol. 128 (EN VIII, 373–374). The following English translation is adapted from Drake 1978, pp. 102–103. See document 5.3.7.Google Scholar
  53. 103.
    See his early unpublished work on mechanics (EN II, 147–191; translated in Galileo 1960c). A close relationship between the principle of acceleration mentioned in the letter to Sarpi, and Galileo’s earlier work on statics has been convincingly argued by Galluzzi (1979, p. 272).Google Scholar
  54. 109.
    The same conclusion is the core of Galileo’s derivation of the Isochronism of Chords, see note 71. The interpretation given here essentially follows Wisan 1974, pp. 204–207.Google Scholar
  55. 110.
    This is disputed by some interpreters of Galileo’s argument, who reinterpret it in the conceptual framework of classical mechanics; see, e.g., Sylla 1986, p. 84, note 90, where the concept of “average velocity” is used in the sense it has in classical mechanics in order to criticize Galileo’s argument.Google Scholar
  56. 112.
    MS, fol. 91v. EN VIII, 280, 281–282, and 427. The manuscript contains three texts (see document 5.3.13, and Plate IV). For a corrected transcription of the second of these texts, which will be discussed in this section, see Wisan 1974, p. 227. The first and the third text will be discussed in section 3.5.3. MS, fol. 152r (partly transcribed in EN VIII, 426–427; see also Fig. 3.21).Google Scholar
  57. 113.
    The argument in the second passage of fol. 91v was first identified as a derivation of the proportionality between degrees of velocity and times of fall by Wisan (1974, pp. 227–229). The relationship between this text and the notes in fol. 152r which document the same line of reasoning has not been recognized before. Galileo’s discovery of the proportionality between the degrees of velocity and the times of fall has been the subject of numerous studies. Naylor (1980a, pp. 562–566) claims that Galileo encountered a conflict with the principle mentioned in the letter to Sarpi when studying the composition of velocities in projectile motion. This reconstruction cannot be correct, since, as we shall see in section 3.5.3, Galileo had not yet mastered the composition of impetuses in horizontal projection at a time when he already used the correct proportionality between degrees of velocity and times of fall. Wisan (1984, in particular pp. 276ff) has claimed that the discovery of this proportionality is due to a crucial experiment by which Galileo supposedly tested the principle mentioned in the letter to Sarpi. Her reconstruction of the supposed crucial experiment on the basis of a manuscript (MS, fol. 116v) has been convincingly refuted by Hill (1986, pp. 284–288). In his own reconstruction of Galileo’s discovery Hill argues that it depends on the prior discovery of the proportionality between the velocities and the square roots of the distances traversed in a motion of fall, a proportionality that is in fact equivalent to the proportionality between degrees and times, if the law of fall is taken for granted and if velocities are understood in the sense of degrees of velocity. Hill claims that the first proportionality is necessarily entailed by the Isochronism of Chords, but this is problematical. In fact, in his speculative reconstruction of Galileo’s discovery, which is not supported by direct manuscript evidence, Hill argues that the Isochronism of Chords implies that the velocities along the chords are in the same proportion as the square roots of the vertical descents, obviously referring to velocity as an overall characteristic of the motion. He then assumes that the velocities along the chords are the same as the velocities along the corresponding vertical descents, a highly problematical assumption, as will be shown in section 3.4.2. On the basis of this problematical assumption, Hill finally draws the conclusion that the velocities of the motion of fall along the verticals must be in the same proportion as the square roots of the lengths of these verticals, now apparently referring to velocities in the sense of degrees of velocity, since he claims that Galileo tested this conclusion in the experiment on projectile motion also referred to by Wisan, an experiment which, in fact, involves instantaneous velocities. Hill’s failure properly to distinguish between overall velocity and degree of velocity, together with the use of a problematical assumption, makes his reconstruction unconvincing; see Hill 1986. The interpretation of Galileo’s refutation of the proportionality between the degrees of velocity and the distances traversed given in Sylla 1986, pp. 79–82, is a speculative reconstruction not based on manuscript evidence, in which use is made of an argument similar to the one later published in the Discorsi (see section 3.6.1). In his most recent account of Galileo’s rejection of the Sarpi principle, Drake (1989b, p. 55, and 1990, pp. 104–105) claims that Galileo abandoned this principle after recognizing a paradox concerning motion along inclined planes, but he does not provide a detailed reconstruction. For a discussion of this paradox, see the following section.Google Scholar
  58. 114.
    MS, fol 91v (EN VIII, 281–282 and Wisan 1974, p. 227). This passage consists of one long paragraph which has been divided up here for better understanding. See document 5.3.13 and Plate IV.Google Scholar
  59. 117.
    According to Drake (1990, p. 101) the notes in fol. 152r (reconstructed in the following) refer to a study by Galileo of accretions of impetus to the natural tendency downward, following what Drake describes somewhat vaguely as the “medieval impetus theory as mathematized by Albert of Saxony.” In Drake 1978, p. 93, he claims that the result of this study appeared to vindicate the argument from which, in fol. 163v, the Double Distance Rule was derived. In the following, however, we shall argue that the argument documented in fol. 152r presupposes the Double Distance Rule and obtains a conclusion that is incompatible with the Sarpi principle, which in turn was used as a premise in the derivation of the Double Distance Rule in fol. 163v.Google Scholar
  60. 119.
    Most interpretations of fol. 152r have concentrated on this part of the manuscript. Drake (1973b) was apparently the first systematic interpretation. At that time, he saw fol. 152r as “the starting point of the modern era of physics” (p. 90). According to this interpretation, Galileo discovered the law of fall in this manuscript by accident in a search for consistent ratios between speeds, distances, and times (p. 89). This interpretation was, however, based on an erroneous transcription; it has been severely criticized by Wisan (1974, p. 214, note 9), and Naylor (1977a, pp. 367371). In later publications Drake (1978, pp. 91–93, 1990, pp. 100–102) has corrected his transcription but not changed his interpretation. Since all relationships between distances and times in fol. 152r indicate, contrary to Drake, that Galileo presupposed the law of fall, all other interpretations of this manuscript relate it to a period in which Galileo was attempting to prove the law. The alternative interpretation suggested by Wisan (1974, pp. 210–214) is, however, based on the assumption that Galileo performed erroneous calculations based on “some vaguely recalled medieval formulas for local motion.” Her interpretation has been convincingly refuted by Naylor (1977a, pp. 377–380), who (pp. 381–386) has succeeded in reconstructing the figures 4 and 131/2 in the upper right-hand corner (see Fig. 3.21) as resulting from a calculation of overall velocities based on the traditional represesentation of velocity by an area. The partial interpretation of fol. 152r by Sylla (1986, p. 76), as well as the complete one given here, are indebted to Naylor for stressing this point. In several interpretations, such as those by Naylor (1977a, p. 386), Drake (1978, pp. 135136), and Romo Feito (1985, p. 107), the significance of the difference between the proportionality between the degrees of velocity and the times of fall, on the one hand, and the proportionality between the degrees and the square roots of the distances, on the other hand, is emphasized. Romo Feito even claims that in fol. 152r Galileo used the latter and attempted to avoid the former. In view of the fact that these two proportions are mathematically equivalent if the law of fall is taken for granted, this emphasis seems to be misleading.Google Scholar
  61. 120.
    The actual distances corresponding to ab and ac in the diagram of the upper part of fol. 152r are approximately 4 and 8, and not 4 and 9, as indicated by Galileo’s table. This feature of the diagram suggests that it may initially have been drawn with the intention of representing subsequent equal distances traversed in a motion of fall, as in the diagrams accompanying Galileo’s attempted proofs discussed in the previous section.Google Scholar
  62. 123.
    The kinematic rule used in this reconstruction had actually also been applied by Galileo to accelerated motion, as for instance in the derivation of the Isochronism of Chords; for references, see note 71.Google Scholar
  63. 127.
    With respect to these two memoranda we do not wish to establish any particular chronology. The reconstruction of the arguments documented in these memoranda aims exclusively at revealing another structural difficulty in Galileo’s reasoning about accelerated motion, a difficulty that he indeed became aware of as his subsequent research makes evident.Google Scholar
  64. 134.
    The possibility of this inference proves Hall (1965, p. 187) incorrect when he claims that it is inconceivable that anyone could have formulated a proposition such as the Isochronism of Projectile Motion who had not transcended the limits of medieval and 16th century mechanics.Google Scholar
  65. 135.
    MS, fol. 106v (EN VIII, 433). This manuscript also contains a table of contents for a book on projectile motion similar to the one later published in the Discorsi. Two other manuscripts (MS, fol. 81r and MS fol. 114v) which deal with oblique projection have been discussed extensively in the literature. See Hill 1988 for a recent review.Google Scholar
  66. 139.
    MS, fol. 116v. This manuscript was first published by Drake (1973a) and has since been the object of numerous controversies; for a recent critical review, see Hill 1988. Most interpretations relate the experiment documented by this manuscript not to a technical result of Galileo’s theory of projectile motion, as is suggested here, but to more fundamental insights he supposedly achieved by this experiment. Drake (1973a, p. 303; 1978, pp. 127–132; 1985, p. 10; 1989b, p. 58) relates the experiment to a test of what he claims is a principle of horizontal inertia in Galileo’s physics or, alternatively, to what he sees as a principle of composition in Galileo’s mechanics, a test in the course of which Galileo accidentally hit on the parabolic trajectory. Naylor (1974a, p. 116) relates fol. 116v to a confirmation of the law of fall (see also Naylor 1977a, pp. 389–391), while Wisan (1984, pp. 276ff) relates it to a discovery of the proportionality between the degrees of velocity and the times in free fall. Hill (1986 and 1988) criticizes Wisan’s interpretation but also relates the experiment to a test of a fundamental principle related to the law of fall. In spite of obvious indications that the drawing in the manuscript is in fact related to an experimental situation, even this general character of the manuscript has been doubted; see Costabel 1975, where, however, no convincing alternative is offered.Google Scholar
  67. 140.
    MS, fol. 175v, MS, fol. 171v. Folio 175v was first published in Drake 1973a and has been discussed in the literature; fol. 171v has not been dealt with. Both manuscripts contain diagrams but no text. Based on the watermarks, Drake (1979, pp. li, lxi, and lxvi) dates fol. 175v to about 1609 and fol. 171v to a much later period (after ca. 1626).Google Scholar
  68. 141.
    This interpretation was suggested by Drake (1973a, p. 296), and by Drake and McLachlan (1975a, p. 104). According to a conjecture by Drake (1985, p. 12) this device was used by Galileo when he also tested the Isochronism of Projectile Motion for oblique projection.Google Scholar
  69. 142.
    These features were first identified by Naylor (1980a, pp. 557–561). According to Naylor’s interpretation of this manuscript, it represents a theoretical analysis by Galileo of the trajectory for oblique projection. The interpretation given here follows this general idea but reconstructs the reasoning that guided Galileo in his analysis in a different way. Naylor interprets the construction in fol. 175v as an examination of the conservation of horizontal inertia and of the principle of superposition, an examination leading to a confirmation of these principles because they alone were capable of accounting for the experimentally determined form of the trajectory. As we have seen, however, the physical properties Naylor refers to by these principles were, for horizontal projection, unproblematic consequences of Galileo’s physics, while Naylor does not provide any evidence that Galileo ever used both principles to derive the trajectory for oblique projection. Drake (1990, p. 121) claims that the marks identified by Naylor actually document traces of an experiment and, contrary to Naylor, that the trajectory shown in this manuscript is parabolic, which is certainly not the case.Google Scholar
  70. 143.
    The transcription is based on the one given in Naylor 1980a, but has been confirmed and improved by a comparison with the original in Florence.Google Scholar
  71. 151.
    For a more detailed analysis of the role played by this purported general property of projectile motion in Galileo’s physics, see Renn 1990a.Google Scholar
  72. 153.
    For a systematic analysis of Galileo’s use of the concepts moment, degree of velocity and impetus in his theory of projectile motion, see Galluzzi 1979, pp. 372383. On p. 383, note 36, Galluzzi remarks that in Galileo’s work impetus almost always refers to the physical effect of velocity. This acute observation supports the interpretation that Galileo’s central propositions on projectile motion are pronounced for oblique projection but proven for horizontal projection because of the problems he encountered in deriving the trajectory for oblique projection. In fact, this transition from oblique to horizontal projection transforms impetus conceived as a cause of horizontal projection into impetus conceived as the effect of oblique projection. Of course, this interpretation does not preclude that the understanding of impetus as the effect of projection was also part of its traditional meaning.Google Scholar
  73. 154.
    Most interpretations of Galileo’s science of motion — with the remarkable exception of Dijksterhuis (1924, p. 275) — have confounded this composition rule with the vector addition of velocities in classical mechanics, ignoring the fact that the geometrical representative of Galileo’s compounded impetus does not actually lie along the tangent to the parabola as does the vector sum of two instantaneous velocities in a given point of the trajectory; see, e.g., Drake 1978, p. 135; see also Hill 1979, p. 270, and Naylor 1980a, p. 565.Google Scholar
  74. 156.
    MS, fol. 117r. This manuscript was first published by Drake (1973a, pp. 293–294) and interpreted as documenting the results of a measurement of the deceleration of horizontal motion by friction; see also Drake and McLachlan 1975a, p. 104. In spite of Naylor’s identification of the “measurements” as the results of a geometrical construction, Drake (1985, p. 11) proposed this interpretation once again. The construction in this manucript that is the object of our subsequent discussion was first identified in Naylor 1975; see also Naylor 1980a, pp. 562–566. Several other geometrical features in this manuscript were identified in Hill 1979. Naylor (1980a, p. 562) interprets fol. 117r as an attempt by Galileo to establish “the way in which the average velocity of the projectile was changing in relation to time.” According to Naylor, Galileo’s study of the increase of this “average velocity” (a notion that Galileo in fact did not possess) led him to the insight that something was wrong with the principle of acceleration mentioned in the letter to Sarpi, i.e., the proportionality of the degrees of velocity and the distances traversed. In fact, however, Naylor’s interpretation is made very implausible by the existence of Galileo’s Scalar Addition Rule, documented in fol. 90ar, since this rule shows that Galileo already adhered to the correct proportionality of degrees of velocity and times in free fall, at a time when he did not yet compound velocities or impetuses in projectile motion by the Pythagorean addition (extracting the square root of a sum of squares) applied in the Vectorial Composition of Scalar Impetus as well as in fol. 117r.Google Scholar
  75. 157.
    See Naylor 1975 and 1980a, pp. 562–566. Our interpretation that these calculations are related to a study of the composition of impetuses is confirmed by the existence of similar calculations in MS, fol. 115v, which also contains an application of the Scalar Addition Rule.Google Scholar
  76. 158.
    See fols. 83v and 86v (EN VII, 427–428). Although in the course of his studies on the Theorem of Equivalence, Galileo discovered that it was incorrect, he did succeed in proving that his Vectorial Composition of Scalar Impetus is indeed compatible with the assumption underlying his derivation of the Equal Amplitude Theorem; see Renn 1990a and for the complete manuscript evidence Renn 1984 and Renn 1988.Google Scholar
  77. 162.
    The problem of the relationship between Proposition I and the Double Distance Rule has been much discussed by historians of science precisely because it is related to the question of whether or not Galileo had knowledge of the traditional Merton Rule; for a recent account see Sylla 1986, p. 89. Sylla tries to find an answer to the question why Galileo did not formulate Prop. I as referring to the mean degree, i.e., in precise analogy to the Merton Rule. But the close similarity in the formulations of Prop. I and the Double Distance Rule (both refer to the maximal degree), the ample documentation of the crucial role played by the Double Distance Rule in Galileo’s studies of motion, and the fact that Galileo actually could derive Prop. I from it, make the answer to the question obvious, independent of the whether or not he knew about the original Merton Rule.Google Scholar
  78. 163.
    There have been numerous attempts to reconstruct the meaning of the proof of Prop. I, and there seems to be no general agreement whether or not this proof is still rooted in the same conceptual framework as Galileo’s earlier attempts to prove the law of fall. Drake (1970a, 1972a) has emphasized the difference between this proof and earlier arguments including the Merton Rule, claiming that the Discorsi proof is not based on a comparison of areas but on a one-to-one comparison of degrees of velocity. Although Drake’s emphasis on the conceptual aspects of the difference between this proof and the traditional proofs is, as we shall see, problematical, his acute identification of the existence of a difference was an important starting point for the analysis given here. Sylla follows Drake’s interpretation of the proof of Prop. I (Sylla 1986, pp. 85–89) and also reconstructs it as a reaction to difficulties Galileo had previously encountered (Sylla 1986, p. 77, note 79). Different interpretations of the proof of Theorem I, which link this proof to Galileo’s earlier proofs or to the Merton Rule have been proposed, among many others, by Clavelin and Ogawa. But in view of the results by Drake and Sylla, Clavelin’s (1974, pp. 298ff) claim that, in Theorem I the area is treated as a sum of lines, and the degrees of velocities are summed to an overall velocity as in Galileo’s earlier proofs seems to be just as unacceptable as Ogawa’s (1989, p. 48) anachronistic treatment of Theorem I and the Merton Rule on the same footing. Settle 1966, pp. 172–183, presents an ingenious reconstruction of Galileo’s argument in the proof of Prop. I on the basis of infinitesimal considerations, but this reconstruction presupposes that Galileo’s problem was, as Settle puts it, to prove that an area can represent a distance (p. 166), whereas the analysis by Drake indicates that areas do not crucially enter Galileo’s argument. Nardi (1988, pp. 49, and 51–52) even argues that Galileo may have used s a vt in his proof. In support of this anachronistic interpretation he appeals to the role of the concept of moment in Galileo’s proof and to the implausible claim of a similarity between the diagram to Prop. I and a balance.Google Scholar
  79. 164.
    The formulation of the proposition and the final conclusion of the proof of Prop. I are actually different; for an interpretation of this fact, see section 3.7.1. This reconstruction disagrees with Wisan (1974, p. 220), who claims that the proof of Prop. I does not presuppose kinematic proportions which (in classical mechanics) are restricted to uniform motion.Google Scholar
  80. 165.
    This identification of the key problem of Galileo’s proof essentially agrees with Galluzzi 1979, p. 354 and Blay 1990, p. 3. Galluzzi supports his interpretation by a careful analysis of Galileo’s infinitesimal considerations presented in the First Day of the Discorsi and by an examination of Galileo’s correspondence, in particular with Cavalieri. Blay confirms this identification of the problem in Galileo’s proof by referring to the contemporary reception of Galileo’s proof by Torricelli and Varignon.Google Scholar
  81. 166.
    The connection between Galileo’s proof of Prop. I and the infinitesimal considerations of the First Day of the Discorsi was suggested in Settle 1966, Chapter IV; for a comprehensive discussion, see Galluzzi 1979, Chapter V.Google Scholar
  82. 167.
    There is no general agreement among historians of science on the question of whether or not Galileo’s proof is correct, either as a proof in classical mechanics, or at least within Galileo’s conceptual system. But contrary to the position defended here, most interpretations tend to represent this proof as an actual solution to Galileo’s earlier problems, even if they reconstruct it in radically different terms. While according to Wisan (1974, p. 214) in Theorem I Galileo has finally resolved the problem of foundations, Drake (1978, p. 371) goes as far as to claim that “Proposition One is perhaps the only theorem capable of rigorous proof relating the law of fall to the definition of uniform acceleration.” Similarly, according to Giusti (1981, p. 39), in his proof of Prop. I Galileo has definitely overcome his previous difficulties. The extreme opposite view is held by Clavelin (1983, p. 47), who does not even grant the derivations of Galileo’s theorems the status of a proof but just that of “simple ordered recapitulations of the main reasons for their acceptance.”Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Peter Damerow
    • 1
  • Gideon Freudenthal
    • 2
  • Peter Mclaughlin
    • 3
  • Jürgen Renn
    • 4
  1. 1.Max-Planck-Institut für BildungsforschungBerlin 33Germany
  2. 2.Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityRamat-Aviv, Tel AvivIsrael
  3. 3.Fachgruppe PhilosophieUniversität KonstanzKonstanz 1Germany
  4. 4.The Collected Papers of Albert EinsteinBoston UniversityBostonUSA

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