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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)

Abstract

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 Aristotelian 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

Keywords

Incline Plane Free Fall Uniform Motion Oblique Projection Horizontal Projection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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    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
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    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 privileged 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 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
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    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
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    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
  54. Ms. Gal. 72, 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. The fact that Galileo had this page copied by a disciple as late as his Florentine period has been taken as the basis for the claim that, even at that time, he still clung to the assumption that the velocity increases in the ratio of the space traversed (see Takahashi 1993a, 1993b). This interpretation is plausible if Galileo first had the text copied and later crossed it out once he recognized the fallacy of space-proportionality. However, our analysis of the ink used by Galileo has shown that the text and the lines by which it is crossed out were written in the same ink, which suggests that the copy was actually made from a (now lost) original itself containing an already crossed-out text. See document 5.3.10.Google Scholar
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  59. Ms. Gal. 72, fol. 163v (EN VIII, 384–385), discussed in Wisan 1974, pp. 215–216. See Plate II. But he was also able to derive this theorem directly from the principle proposed to Sarpi so that he could now demonstrate the Isochronism of Chords on the basis of the law of fall and of the Length-Time-Proportionality. This derivation of the Isochronism of Chords within the conceptual framework of the De Motu theory is documented by several manuscripts; see, e.g., Ms. Gal. 72, fol. 172r (EN VIII, 39293) and fol. 151r (EN VIII, 378 and document 5.3.2). The relevant manuscript evi- dence is extensively documented and analyzed in Wisan 1974, pp. 162–171 and 190191. For an analysis of the conceptual background of Galileo’s proof, see Galluzzi 1979, pp. 266–267, Souffrin 1992, and Souffrin and Gautero 1989.Google Scholar
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    Ms. Gal. 72, fol. 179r (EN VIII, 388). For studies of this proof, see Wisan 1974, pp. 217–220, Drake 1978, pp. 113–115, Galluzzi 1979, pp. 294–297, and Souffrin 1986.Google Scholar
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    Ms. Gal. 72, fol. 85v, 88v, 128r, 179r.Google Scholar
  62. 98.
    Wisan has suggested that this derivation may have actually been the motive for Galileo’s search for such a relationship; see Wisan 1974, p. 206.Google Scholar
  63. 99.
    Ms. Gal. 72, fol. 163v (EN VIII, 383–384); discussed in Wisan 1974, pp. 204–207.Google Scholar
  64. 100.
    The same conclusion is the core of Galileo’s derivation of the Isochronism of Chords, see note 95. The interpretation given here essentially follows Wisan 1974, pp. 204–207.Google Scholar
  65. 101.
    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
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  67. 103.
    Ms. Gal. 72, fol. 91v. EN VIII, 280, 281–282, and 427. The manuscript contains three texts (see document 5.3.15, and Plate III). 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. Gal. 72, fol. 152r (partly transcribed in EN VIII, 426–427; see also Fig. 3. 21 ).Google Scholar
  68. 104.
    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. Gal. 72, 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 one of his last accounts 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
  69. 105.
    Ms. Gal. 72, 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 5.3.15 and Plate III.Google Scholar
  70. 106.
    See sections 1.2.1 and 1.2.2.Google Scholar
  71. 107.
    Galileo’s use of the notion of moment as referring to a velocity to which the Aristotelian proportions can be applied is documented in Ms. Gal. 72, fol. 151r (EN VIII, 378; document 5.3.2); see also note 95.Google Scholar
  72. 108.
    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
  73. 109.
    A similar refutation of the proportionality between the degrees of velocity and distances has been identified by Drake (1970a, pp. 39–41) in a later published work: Giovanni Andres, (1779) Raccolta di opuscoli scientifici, e letterarj, Ferrara, p. 64.Google Scholar
  74. 110.
    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. 367–371). 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. 377380), 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 representation 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. 135–136), 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
  75. 111.
    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
  76. 113.
    The point f’ is not marked or labeled in Galileo’s original diagram; it has been added to the transcription in order to make the following calculation more transparent.Google Scholar
  77. 114.
    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 95.Google Scholar
  78. 115.
    See section 1.2.1Google Scholar
  79. 116.
    See Tartaglia 1556a, in particular Chapter 16, pp. 240ff. For evidence of the role played by this book in Galileo’s mathematics education, see Renn 1988, Appendix B.Google Scholar
  80. 117.
    EN VIII, 614, Ms. Gal. 72, part. 6, vol. 3, fol. 56475v (according to Favaro this is a copy of Galileo’s original in Viviani’s hand); and Ms. Gal. 72, fol. 164v (EN VIII, 375). The first has not been discussed before in recent literature, while the second was first extensively discussed (under the designation given here) in Wisan 1974, pp. 201–204. The other two memoranda are found on Ms. Gal. 72, fol. 147r (EN VIII, 380); see document 5. 3. 6.Google Scholar
  81. 118.
    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
  82. 119.
    Ms. Gal. 72, part. 6, vol. 3, (EN VIII, 614); this note contains no diagram.Google Scholar
  83. 120.
    See, for instance, the derivation of the Isochronism of Chords; for references, see note 95.Google Scholar
  84. Ms. Gal. 72, fol. 164v (EN VIII, 375). The interpretation given in the following (as well as the translation) essentially follows Wisan (1974, pp. 201–204). For Drake (1978, pp. 124–125) on the other hand, the Mirandum Paradox is not a paradox at all but it is precisely this argument, in connection with the correct proportionality between moments of velocity and square roots of distances, which shows that Galileo “had reached complete clarity on puzzles of accelerated motion that had long plagued him.” Accordingly, Drake translates “Mirandum” (“it has to be seen”) as “Remarkable!” (1978, p. 125) and even as “Marvelous!” (1989b, p. 55). However, he leaves it unclear how Galileo’s supposed resolution of this “seeming paradox” could have made use of the correct proportionality between moments of velocity and square roots of distances; see Drake 1990, p. 105. Galileo’s reiteration of this paradox in his published works (see section 3.7.1) does not support this interpretation. 22For an analysis of the conceptualF backgound of Galileo’s argument, and, in particular, of the role of the concept of moment in this argument, see Galluzzi 1979, pp. 292–293; see also Souffrin 1992.Google Scholar
  85. 123.
    See DM, Chapter 14, pp. 63–69 (EN I, 296–302).Google Scholar
  86. 125.
    The translation is adapted from Drake 1978, p. 95. In his interpretation of the memorandum Drake (1978, pp. 95–96) 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
  87. 126.
    Underneath this memorandum, Galileo wrote, at a later time, the word “Paralogismus” indicating that he had discovered the 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 the surface tension of the water. (See Galileo 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 specific weight; see Galileo 1612, pp. 66–67; English in Galileo 1663, p. 70. 127Ms. Gal. 72, fol. 147r (EN VIII, 380). See document 5.3.6. 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
  88. 128.
    See note 95.Google Scholar
  89. 129.
    This is suggested by an argument in Ms. Gal. 72, fol. 177r (EN VIII, 386), translated in Wisan 1974, p. 200. Our interpretation follows Wisan 1974, pp. 188, and 200–201.Google Scholar
  90. 130.
    Ms. Gal. 72, fol. 147r (EN VIII, 380). Later, in the Discorsi, this form of the law of fall is expressed as follows: “… if at the beginning of motion there are taken any two spaces whatever, run through in any [two] times, the times will be to each other as either of these two spaces is to the mean proportional space between the two given spaces.” (Discorsi, pp. 170–171; EN VIII, 214). The following translation has been adapted from Drake 1978, p. 94.Google Scholar
  91. 131.
    On the doctrine of proportions see section 1.2.1.Google Scholar
  92. 132.
    Ms. Gal. 72, fol. 58r, as pointed out by Drake 1978, p. 94.Google Scholar
  93. 133.
    For a detailed reconstruction of this manuscript page, see Renn et al. 2000; and for a reconstruction of Galileo’s diagram as a catenary, see Naylor 1980a, p. 554.Google Scholar
  94. 134.
    EN X, 229–230; translation adapted from Drake 1973a, pp. 303–304.Google Scholar
  95. 135.
    In view of the context of patronage in which this letter has to be read, it remains uncertain how new this result really was for Galileo. For an extensive discussion of this context of Galileo’s science, see Biagioli 1993.Google Scholar
  96. 136.
    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
  97. 137.
    Ms. Gal. 72, 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. Gal. 72, fol. 81r and Ms. Gal. 72 fol. 114v) which deal with oblique projection have been discussed extensively in the literature. See Hill 1988 for a review.Google Scholar
  98. Ms. Gal. 72, fol. 87v (EN VIII, 428–429). This manuscript has not received much attention from historians of science, who have focused their attention on Galileo’s discovery of the shape of the trajectory and neglected the question of the conceptual tools required to determine its size. Galileo’s two results are later published in the Discorsi as Prop. V and its Corollary in the book on projectile motion, see Discorsi, p. 243; EN VIII, 294–295.Google Scholar
  99. 141.
    Ms. Gal. 72, fol. 116v. This manuscript was first published by Drake (1973a) and has since been the object of numerous controversies; for a 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
  100. 142.
    Ms. Gal. 72, fol. 175v, Ms. Gal. 72, 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
  101. 143.
    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
  102. 144.
    These features were first identified by Naylor (1980a, pp. 557–561). According to Naylor’s interpretation, this manuscript 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, which is certainly not the case. A completely analogous procedure for constructing the trajectory of oblique projection was followed by Thomas Harriot, see Schemmel 2002.Google Scholar
  103. 145.
    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
  104. Fol. 90r was discussed by Caverni (1895, pp. 534–537) and Wisan (1974, p. 269) without mention of Galileo’s Scalar Addition Rule. This was first discussed in Renn 1990a. Other manuscripts containing applications of the Scalar Addition Rule are Ms. Gal. 72, fol. 110v and Ms. Gal. 72, fol. 115r.Google Scholar
  105. 150.
    Ms. Gal. 72, fol. 80r (EN VIII, 433, table III).Google Scholar
  106. 151.
    Ms. Gal. 72, fol. 80r (EN VIII, 433, table III); the division into paragraphs has been added.Google Scholar
  107. 153.
    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
  108. 154.
    Ms. Gal. 72, fol. 91v; see document 5.3.15 and plate III. The first text corresponds closely to Prop. II of the book on projectile motion in the Discorsi (EN VIII, 280/229); the translation is adapted from Drake’s translation of the Discorsi. The third text is transcribed in EN VIII, 427.Google Scholar
  109. 155.
    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
  110. 156.
    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
  111. 157.
    Ms. Gal. 72, fol. 110v (EN VIII, 428). The corresponding diagram, as identified by Drake (1979, p. XXXVI), is found in fol. 87v. Wisan (1974, pp. 271–272) claims that in fol. 110v Galileo is searching for a proposition relating the impetus in projectile motion to the impetus acquired in free fall, but in fact there is no indication of such a search in this manuscript.Google Scholar
  112. See fol. 83v and fol. 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
  113. 161.
    See, e.g., the letter to Elia Diodati, Dec. 6, 1636 (EN XVI, 524) where Galileo writes: “… I experience how much old age takes away from the mind’s vitality and quickness, when I have a hard time understanding not a few of the things discovered and demonstrated by me in a fresher stage of life.” There is an early draft of the beginning of the book on accelerated motion in the Discorsi entitled “Liber secundus: in quo agitur de motu accelerato” (EN II, 261–266). This draft, which can be dated about 1630, was bound together with Galileo’s early treatise De Motu. Fredette (1972, pp. 329–330) argues that it was filed by Galileo himself together with this treatise. If Fredette’s reconstruction is correct, it suggests that Galileo kept the manuscript of his first treatise in order to exploit its results and to integrate them into his late treatise on motion. For a detailed study of the phase in which Galileo completed the Discorsi, see Wisan 1974, section 8.Google Scholar
  114. 164.
    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 an 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
  115. 165.
    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
  116. 166.
    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
  117. 167.
    This identification of the key problem of Galileo’s proof essentially agrees with Galluzzi 1979, p. 354 and Blay 1998, pp. 73–75. 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
  118. 168.
    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
  119. 169.
    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
  120. 170.
    Galileo’s refutation of the proportionality between the degrees of velocity and the distances traversed was often criticized as a fallacious argument (see, e.g., Hall 1958), until Drake (1970a, pp. 28–36) proposed a reconstruction of this argument that makes it at least plausible. Drake’s reconstruction, on which the interpretation given in the following is based, is similar to the earlier interpretation of Dijksterhuis (1924, pp. 246–250); this interpretation makes use of a one-to-one correspondence between aggregates of infinite velocities. Drake supports this reconstruction citing an interpretation of the argument along the same lines by a contemporary of Galileo’s (Tenneur 1649, p. 8). A criticism of Galileo’s argument was reproposed by Finocchiaro (1972) and convincingly refuted by Drake (1973c). In the Discorsi (EN VIII, 203/160, see below) Galileo characterizes the principle he had earlier suggested to Sarpi by a quotation from Vergil (Aeneid, IV, 175): “vires acquirat eundo.” For Descartes’ use of the same quotation, see document 5. 1. 3.Google Scholar
  121. 171.
    Ms. Gal. 72, fol. 128, EN VIII, 323; see also section 3.3. 2. 1.Google Scholar
  122. 174.
    Wisan (1974, p. 220) claims that the proof of Theorem III is based on a rule for uniform motion, but according to the interpretation proposed here this was not an error but justified by the relationship Galileo had previously established between accelerated and uniform motion. Galluzzi (1979, pp. 359–362) reconstructs the proof of Theorem III on the basis of the claim that Galileo did not consider a division of the line into an infinite number of points but just into an arbitrary number; however, in his proof Galileo in fact refers explicitly to the “parallels from all points of the line AB.” In another reconstruction of the proof of Theorem III, Souffrin (1986) refers to a manuscript (Ms. Gal. 72, fol. 138v; EN VIII, 372) documenting an attempt by Galileo to generalize the Archimedean proposition on uniform motion to accelerated motion. It is, however, implausible to assume that, in a published text, Galileo referred to an unpublished manuscript.Google Scholar
  123. 175.
    lndeed, in one of Torricelli’s manuscripts, an argument concerning a comparison between a nonuniformly accelerated motion and a uniform motion is found, similar to the hypothetical argument reconstructed above; see Blay 1999, pp. 76–77. In Torricelli’s argument, too, areas do not represent distances traversed but are related to aggregates of degrees of velocity. In order to avoid paradoxical consequences of this argument such as the ones discussed above, Torricelli introduces an additional ad hoc assumption. Cavalieri’s first reaction to the Discorsi indicates that he, too, was aware of the problematical character of Galileo’s proof technique, because he criticizes Galileo for not having emphasized that the indivisibles have to be taken as equidistant (see Bonaventura Cavalieri to Galileo, June 28, 1639, EN XVIII, 67); for a discussion of Galileo’s contemporary correspondence with Cavalieri, as well as for other reactions to Galileo’s treatment of indivisibles, see Galluzzi 1979, Chapter V.Google Scholar
  124. 176.
    This lack of a proof of the trajectory for oblique projection was noticed and extensively discussed by Wohlwill (1884, pp. 111–116) and Dijksterhuis (1924, pp. 264284) but has been ignored by many later interpretations of Galileo’s science. Wohlwill, whose interpretation is the starting point for the one given here, explained this gap in the deductive structure of Galileo’s theory of motion by the absence of a general principle of inertia and a general principle of superposition from the conceptual foundations of the theory.Google Scholar
  125. 177.
    In the definition quoted above, the notion of moment may seem to be close to the modern notion of an infinitesimal increment of velocity, while in remarks immediately preceding the definition Galileo provides an explanation of the degrees of velocity in accelerated motion that makes them appear rather close to the modern concept of instantaneous velocity. In fact, in these remarks Galileo draws the consequences of his earlier insight — related to the Double Distance Rule — into the relationship between a degree of velocity in uniformly accelerated motion and uniform motion: “… if the moveable were to continue its motion at the degree or moment of speed acquired in the first little part of time, and were to extend its motion successively and equably with that degree, this movement would be twice as slow as [that] at the degree of speed obtained in two little parts of time.” (Discorsi, EN VIII, 198/154) Galileo’s use of “degree” or “moment” of speed as equivalent in certain contexts makes it however clear that neither of these can be identified with the modern notion of an infinitesimal increment of velocity or with that of an instantaneous velocity, but that both are still part of a traditional conceptual system.Google Scholar
  126. 178.
    This reversal was noticed, for instance, by Wisan (1974, pp. 289–290).Google Scholar
  127. 179.
    This conclusion is in disagreement with Kuhn’s interpretation of this passage from the Dialogo as well as with the conclusions he draws from it for an understanding of cognitive development. With respect to Galileo’s “thought experiment” as well as with respect to a related experiment in child psychology by Piaget, Kuhn (1977, p. 264) writes: “Full confusion, however, came only in the thought-experimental situation, and then it came as a prelude to its cure. By transforming felt anomaly to concrete contradiction, the thought experiment informed our subjects what was wrong. That first clear view of the misfit between experience and implicit expectation provided the clues necessary to set the situation right.” For a discussion of the problems of cognitive development in Galileo’s science, see also Renn 1989, and 1990b. For references to the historical literature on this problem, see section 3. 4. 2.Google Scholar
  128. 180.
    This theory has been extensively studied by Wisan (1974, pp. 281–286), Clavelin (1983), Giusti (1986), Souffrin (1992), and Napolitani ( 1988 ). The interpretation presented here is indebted, in particular, to Giusti for the clear exposition of the deductive structure of this treatise, and to Souffrin for his insight into the role of the Aristotelian concept of velocity in Galileo’s theory. Napolitani’s paper contains an important analysis of the parallel between Galileo’s treatment of velocity and his treatment of specific weight.Google Scholar
  129. 181.
    The present text has often been interpreted as providing evidence for Galileo’s recognition of the principle of inertia; for a classic interpretation see Mach 1942, pp. 168–169, and pp. 330–337. In fact, in this passage, Galileo does not use the terminology “natural” and “forced,” but characterizes the motion along the horizontal plane by the absence of “external causes of acceleration or retardation” without explaining what these external causes are; see Mittelstrass 1970, pp. 268–282. However, not only is Galileo’s formulation in this passage an isolated case in his writings, as has been stressed by Wohlwill (1884, pp. 126–134), but also the crucial point is that the justification of what some historians see as a principle of inertia actually still follows the pattern of an argument derived from the De Motu theory; see section 3.3.1. An interpretation similar to that given here, also following Wohlwill, was proposed by Dijksterhuis ( 1924, pp. 264–271 ).Google Scholar
  130. 182.
    The interpretation given here is indebted to the analysis of the discussion between Salviati, Sagredo, and Simplicio given in Dijksterhuis (1924, pp. 271–277), Wisan (1974, pp. 261–263), and Chalmers and Nicholas (1983, pp. 329–333). A passage in the First Day of the Dialogue (EN VII, 43/19) shows that Galileo’s cosmological views indeed made it impossible for him to accept rectilinear uniform motion as a principle of natural motion, i.e., as a motion which is proper to bodies constituting an ordered universe. According to the argument given in this passage, straight motion is incompatible with the notion of an ordered universe, first, because the terminus ad quem is different from the terminus a quo so that, if the world was perfect in the beginning, it will no longer be perfect after the motion has taken place, and second, because, properly speaking, there is no terminus ad quem for an infinite straight motion so that it cannot be a natural motion aiming for a natural place. This argument illustrates that Galileo’s cosmology with its emphasis on circular motion was not only influenced by Platonic philosophy or by his early insight that circular motion is neither natural nor forced, and hence perpetual but also that it was shaped by such aspects of the Aristotelian concept of motion as the determination of a rectilinear motion by its terminal points. There are, however, interpretations that claim that Galileo’s cosmological statements should not be taken that seriously; see Coffa 1968, p. 280.Google Scholar
  131. 183.
    EN VIII, 446–447. The manuscript is in the handwriting of Viviani; the text was to be inserted before Prop. VIII of the 4th Day of the Discorsi. See section 3.6.2 and document 5. 3. 21.Google Scholar
  132. 184.
    For a detailed discussion of Galileo’s plans for the Fifth Day see Renn et al. 2000.Google Scholar
  133. 186.
    For reconstructions of Galileo’s theory of fall in media presented in the Discorsi, see Dijksterhuis 1924, pp. 227–234, 1961, p. 336, and Clavelin 1974, p. 333. Dijksterhuis’ early interpretation also stresses the similarity between the De Motu and the Discorsi theories.Google Scholar
  134. 187.
    Galileo’s skeptical discussion of the causal explanations of acceleration has led many historians to assume that Galileo’s physics in the Discorsi is pure kinematics, abjuring all attempts of a causal explanation (see, e.g., Settle 1966, p. 152, Barbin and Cholière 1987, p. 94, Drake 1990, p. 68 ). First of all, they have overlooked, the fact that Galileo did highlight one particular explanation as a plausible one and, second, that this explanation is not just a reiteration of the initial explanation of acceleration given in De Motu. The Discorsi explanation can in fact be considered as a correction of the De Motu explanation, and, according to the interpretation presented here, it is precisely this correction that makes it problematical for Galileo.Google Scholar
  135. 188.
    EN X, 116; see section 3.3.2.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Peter Damerow
    • 1
  • Gideon Freudenthal
    • 2
  • Peter McLaughlin
    • 3
  • Jürgen Renn
    • 1
  1. 1.Max Planck Institute for the History of ScienceBerlinGermany
  2. 2.The Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityTel AvivIsrael
  3. 3.Philosophisches SeminarUniversität HeidelbergHeidelbergGermany

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