Abstract
By way of introduction, the chapter considers aspects of Galileo’s considerations regarding motion and mechanics prior to his conceptual shift toward the assumption that motion of fall is naturally accelerated. The aspects discussed have been selected for the relevance they assumed when, from 1602 onward, Galileo started to engage in the investigations which eventually led to the establishment of his new science of motion. In particular, his understanding of the free fall of heavy bodies and of acceleration is being presented as it is reflected in a compilation of early manuscripts, which have come to be referred to as De Motu Antiquiora. One of these manuscripts, commonly designated as On Motion, contains a chapter in which Galileo investigated the dynamics of motion along inclined planes and, in particular, provided a proof for the law of the inclined plane relating the inclination of the plane to the force along this plane experienced by a body placed upon it. Galileo’s arguments in this chapter are discussed in some detail.
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- 1.
Cf. Damerow et al. (2004).
- 2.
Cf. Favaro’s Avvertimento to De Motu Antiquiora in EN I, 245–246. Favaro himself referred to the collection as De Motu. According to Vincenzo Viviani and Giovanni Battista Nelli, Galileo himself, on a cover lost today, used the title De Motu Antiquiora to refer to these texts. Cf. Camerota and Helbing (2000).
- 3.
The content of the manuscript can be accessed online through the digital collection of the Biblioteca Nazionale Centrale at: http://www.bncf.firenze.sbn.it. Accessed 10 Feb 2017.
- 4.
The different parts of De Motu Antiquiora have been addressed by different names. Sometimes different scholars have even used the same name to refer to different parts. For disambiguation I provide the following table which gives the names I use for the parts, the page numbers in Volume I of the EN, and the shelf mark and folio numbers of the corresponding manuscript.
EN reference
Coll. of Galilean manuscripts
On motion
I,251–340
Ms. Gal. 71, 61v–124v
First two chapters reworked
I,341–343
Ms. Gal. 71, 133r–134v
Treatise in 10 chapters
I,344–366
Ms. Gal. 71, 43r–60v
Dialogue
I,367–408
Ms. Gal. 71, 4r–35r
Memoranda
I,409–417
Ms. Gal.46, 102–129
Work plan
I,418–419
Ms. Gal. 71, 3v
- 5.
Fredette (1972) has compellingly made the case that On Motion was conceived of by Galileo as being composed of two books and that the treatise in 10 chapters represents a reworking of the first one.
- 6.
- 7.
Throughout this work a great number of Galileo’s (and other’s) statements have been rendered into modern symbolic notation. Such an approach has been criticized by Palmieri (2003, 230): “the translation of verbal mathematical arguments into a symbolic notation inevitably carries with it the risk of missing important aspects that are inextricably linked with the original language.” Even though such critique has its justification, I believe that this technique can have great benefit, but only if it is applied with an awareness of its limitations. The conventions used are more or less self-explanatory and, where they are not, will be introduced explicitly.
- 8.
In his own text (Souffrin 1992, 237) even puts the word between slashes and writes /velocitas/ to “représenter toutes les formes du latin velocita.”
- 9.
From a modern perspective the difference between velocity and speed amounts to the difference between a vector quantity and a scalar quantity. Thus, for instance, the velocity of a body that has returned to its point of departure is zero as the displacement vector is zero. For Galileo and his contemporaries, it was distance traversed rather than displacement that mattered, and this seems to be the reason why many authors who have discussed the same material I discuss here have preferred speed over velocity in translations but also refer to Galileo’s concept in general. However, the early modern concept is as incommensurable with our modern concept of speed or average speed as it is with our concept of velocity or average velocity, and thus the reason to prefer the term speed over the term velocity is revealed to be void. My choice to use velocity is based simply on the fact that it is linguistically closer to Galileo’s own terminology.
- 10.
My exposition follows closely the one given by Souffrin (1992), who provides examples of late medieval and early modern authors including Galileo, to support the reconstruction given.
- 11.
The holistic character needs to be emphasized mainly because in the classical framework velocity is defined as the derivative of position with respect to time and is thus an instantaneous magnitude. Average speed, which applies to distance traversed and time elapsed and which is, in this respect, somewhat closer to the preclassical concept, is understood to be a derived concept. Notwithstanding some resemblances to the modern average speed, the preclassical kinematic velocity is distinct from it in important ways. Thus what holds for the ratio of two kinematic velocities in the preclassical framework holds for the ratio of the average speed of the same motions in the classical framework. On the other hand, the definition of a magnitude as a distance traveled divided by the time elapsed is categorically ruled out in the preclassical framework.
- 12.
Throughout the book, primarily for ease of apprehension, I use modern symbolic notation to transcribe verbal statements. Such an approach is criticized by Palmieri (2003, 230) who states that “the translation of verbal mathematical arguments into a symbolic notation inevitably carries with it the risk of missing important aspects that are inextricably linked with the original language.” This is true. Yet I believe it should not lead us to abandon such use of symbolic notation but to use it carefully and with an awareness of its limitations.
- 13.
Cf. Clagett (1979).
- 14.
Proposition 1 of On Spirals reads: “If a point moves at a uniform rate along any line, and two lengths be taken on it, they will be proportional to the times of describing it. (Archimedes and Heath 2002, 155)”.
- 15.
Cf. Damerow et al. (2004, 16).
- 16.
For the relation between the definition of velocities as equal and specification of motion as uniform, see Souffrin (1992, 238–239).
- 17.
- 18.
Dynamics refers to the study of motion that happens under the influence of forces. It embraces on the one hand kinematics and on the other hand kinetics, which is concerned with the effect of forces and torques on motions. For this categorization, see, for example, Burton (1890). Even though I believe this terminology to be adequate, I follow the terminology more commonly used in studies on early modern natural philosophy and distinguish dynamics and kinematics according to the distinction between the study of causes and the study of space-time effects of motion.
- 19.
- 20.
For the theory of proportions as laid out in Book V of Euclid as Galileo’s principal mathematical tool, see, in particular, Giusti (1992, 1993). Also Palmieri has recurrently and rightly emphasized the importance of interpreting Galileo’s considerations against the background of this mathematical means. Cf., e.g., Palmieri (2005b, 345).
- 21.
Such a position is prevalent in many assessments of Galileo’s new science. It has recently been forcefully explicated by Henry (2011).
- 22.
Cf. Jung (2011).
- 23.
- 24.
Cf. in particular Schemmel (2014).
- 25.
In his early writings Galileo explicitly refers to the “Doctores Parisienses,” and he clearly “knew of their teaching from the Jesuit lecture notes (Galilei and Wallace 1992, 219).” What exactly Galileo owed to the doctrine of the configurations of motion and where he went beyond its traditional scope has been intensely debated in the history of science. On this question see, e.g., Wallace (1981) or Lewis (1980). Extreme opposite positions in this debate were taken by Duhem and Koyré. Wallace (1990, 239) pointedly represents Koyré’s position: “[t]he medieval and Renaissance development that had been traced in such detail by Duhem might be of antiquarian interest, but it was not at all necessary for Koyré’s understanding of Galileo and the ‘new science’ he had brought into being.” Wallace holds that an important result of his own studies “is their vindication of Duhem, as contrasted with Koyré.” As I will demonstrate, the doctrine of the configurations of motion was crucial for Galileo’s attempt to provide what had been achieved independently with a conceptual underpinning.
- 26.
- 27.
The Merton rule is often imprecisely referred to as the mean speed theorem, yet the latter more specifically refers to the application of the former to the case of local motion proceeding with a uniformly growing degree of velocity. The mean speed theorem is already found in the works of the Calculators, for instance, in Heytesbury but also in the works of Oresme. Cf. Boccaletti (2016, 32–38).
- 28.
For an excellent rendition of the difference between the preclassical and the classical concepts of velocity, see Souffrin (1990), in particular the schematic table he provides therein to illustrate the history of the concept.
- 29.
The few entries in the Notes on Motion which are written in Italian and in which the expression occurs always use the equivalent to the Latin “gradus velocitatis,” i.e., “grado di velocità.”
- 30.
According to Wallace (1984), Galileo’s use of “gradus velocitatis” traces back to the Collegio Romano. Referring to Galluzzi, Wallace (1984, 268) furthermore claimed that the early use of “gradus velocitatis” was replaced by “momento velocitatis” from about 1605. In my analysis I was unable to confirm this alleged trend. On the contrary, Galileo already used “momentum velocitatis” in entries which must date from before 1604, for instance, on 163 recto .
- 31.
Cf. Schemmel (2014, 18).
- 32.
The identification of aggregate velocity with kinematic velocity would eventually force Galileo to give up on the idea that velocity was represented by the area in an Oresme diagram. This was replaced by an argument published as Proposition I of the Third Book of the Third Day of the Discorsi, in which equality of aggregate velocity and thus also of kinematic velocity is established arguing via pairwise correspondence of degrees of velocity instead.
- 33.
According to Galluzzi (1979, 182), it needs not even to be particularly stressed that “la teoria galileiana del De motu, la ‘dinamica pisana’, come è stata definita, è una dinamica dei moti uniformi.”
- 34.
See Gregory (2001, 3).
- 35.
Cf. Camerota and Helbing (2000, 346). An important contribution was made by Buridan, who had given an explanation of acceleration in terms of the concept impetus, i.e., impressed force. Alternatively, “antiperistaltic” explanations of the acceleration of natural motion were advanced. For the discussions on the cause of acceleration in the natural philosophical debates of the Late Middle Ages, see Grant (2010, 207 ff.) or Guerrini (2014).
- 36.
Cf. Palmieri (2005a). Benedetti had advanced a comparable account of motion through media based on Archimedean hydrostatics before. The question to what extent, if any, this influenced Galileo is still open. Grant (1966) meticulously reconstructs possible common origins of Benedetti’s and Galileo’s approach also pointing out the differences.
- 37.
- 38.
The full sentence from which the quotations in the text are taken reads: “In utroque motu ex eadem causa pendere tarditatem et celeritatem, nempe ex maiori vel minori gravitate mediorum et mobilium, mox demonstrabimus (EN I, 260).”
- 39.
A complete English translation of De Motu Antiquiora by Raymond Fredette is available online in the cultural heritage collection ECHO at http://echo.mpiwg-berlin.mpg.de/content/scientific_revolution/galileo. Accessed 15 Jan 2017. Where not noted otherwise, this and all following English quotations from De Motu Antiquiora are taken from his translation. For the relation of Galileo’s stance regarding acceleration to the position taken by Girolamo Borro and Francesco Buonamici, his teachers at the university at Pisa, cf. Camerota and Helbing (2000).
- 40.
Galileo’s discussion of acceleration reoccurs in similar form in the Dialogue of De Motu Antiquiora, which is now generally accepted to have been written before On Motion. Cf. EN I, 405–408.
- 41.
- 42.
See EN I, 318. Transl. Fredette.
- 43.
Drabkin in Galilei (1960, 90) traces Galileo’s remark concerning Hipparchus to Simplicius Commentary on Aristotle’s De Caelo.
- 44.
According to Galileo’s account, the accelerated part of the downward motion is, strictly speaking, a mix of violent and natural motion and would thus, according to orthodox Aristotelian understanding, have to be classified as violent. Galileo, indeed, concedes that there is no essential difference between the decelerated forced upward motion of his first example and the accelerated downward part of motion and states “this alternative motion, while the mobile is moved from lightness to heaviness, is a single and continuous motion (EN I, 322, trans. Fredette).” In the Memoranda, Galileo is even more explicit and states in an entry “a violent motion precedes every case of natural motion, as we have made clear (EN I, 411, trans. Fredette).” As Wisan (1978, 9) puts it: “Galileo’s acceleration is unnatural (or violent) motion.”
- 45.
In the Dialogue of De Motu Antiquiora, Galileo even advances an argument why it is that, under certain conditions, bodies falling at a uniform rate are falsely perceived to accelerate. Cf. EN. I 407.
- 46.
Cf. Camerota and Helbing (2000). In trials in which persons were asked to drop objects of same shape but different weight at the same time, Thomas Settle and Donald Miklich showed that the heavier object is consistently dropped with a short delay even though the actors perceive dropping the objects simultaneously. Settle (1983) relates this phenomenon, which can be explained physiologically, to the early modern debates.
- 47.
According to Fredette (1972, 347–348), it was the irreconcilability of his account of acceleration with the revised conceptualization of the dynamics of motion through media, as elaborated in the treatise in 10 chapters, which led Galileo to abandon the project of De Motu Antiquiora.
- 48.
Cf. Palmieri (2005b).
- 49.
The dating of the text is controversial. Favaro and Wohlwill have both dated it to Galileo’s Paduan period, and a possible dating “to well before 1604, quite possibly to the late 1590s” has recently been proposed by Palmieri (2005b). Drake et al. (1999, 216) dates the text to 1630 or 1631. Based on comparison with material in the Notes on Motion, I tend to agree to a dating after Galileo’s move to Florence.
- 50.
“Idem est mobile, idem principium movens: cur non eadem quoque reliqua? Dices: eadem quoque velocitas. Minime: iam enim re ipsa constat, velocitatem eandem non esse, nec motum esse aequabilem: oportet igitur, identitatem, seu dicas uniformitatem, ac simplicitatem, non in velocitate, sed in velocitatis additamentis, hoc est in acceleratione, reperire atque reponere (EN II, 262).”
- 51.
Settle (1966, 148) too emphasizes that “[i]t would seem, then, that two things had to occur …before …Galileo could establish the foundations of a new science: a shift in primary focus from uniform motion to acceleration as the essential mode of natural motion, and the discovery of a mathematical description of that acceleration. And it would seem that we must credit systematic experimentation by Galileo with a key role in both.”
- 52.
- 53.
For an account of how Galileo may have come to accept the law of chords, see Chap. 8.
- 54.
The experimental setup of the pendulum plane experiment which Galileo conducted in 1602, as argued in Chap. 4, is strikingly similar to the setup of the famous inclined plane experiment described by Galileo in the Discorsi suggesting that he indeed may have tested the law of fall before or in 1602.
- 55.
“Constat ergo, eiusdem mobilis in diversis inclinationibus celeritates esse inter se permutatim sicut obliquorum descensuum, aequales rectos descensus compraehendentium, longitudines.”
- 56.
In the Ps. Aristotelian Mechanical Problems, which famously define mechanical devices as those in which “the less master the greater, and things possessing little weight move heavy weights (Aristotle and Hett 1980, 331),” the inclined plane is not investigated explicitly. Yet in Problems 17 and 19, the force altering effect of the wedge, treated by the later tradition as an instance of an inclined plane, is explored.
- 57.
Cf. Schiefsky (2008).
- 58.
For Guidobaldo del Monte’s Mechanicorum liber, see Del Monte et al. (2010).
- 59.
Galileo’s treatment of the problem of the inclined plane in De Motu Antiquiora has been discussed in detail and been thoroughly contextualized in a history of mechanics by Festa and Roux (2008).
- 60.
In a footnote to this chapter, Drabkin remarks: “[e]ven on the specific question of the ratio of speeds Galileo can hardly claim priority (Galilei 1960, 63).” To me this remark seems to lack justification. An exception is Giovanni Marliani, who treated motion along inclined planes in his Quaestio de proportione motuum in velocitate of 1482. For Marliani’s work, see Clagett (1941). Palmieri (2017) opposes the idea that Marliani conducted experiments involving the falling of bodies down along inclined planes.
- 61.
Festa and Roux (2008, 210) likewise highlight this difference when they state: “it is not a matter of using the law of the inclined plane to explain the function of the screw as it will be in the Mecaniche, but rather to answer two questions concerning the movement of a body along an inclined plane.”
- 62.
Quaestio, quam nunc explicaturi sumus, a philosophis nullis, quod sciam, pertractata est: attamen, cum de motu sit, necessario examinanda videtur illis, qui de motu non mancam tractationem tradere profitentur.
- 63.
From an orthodox Aristotelian perspective, natural motion, of course, includes the upward motion of light bodies. Yet Galileo began to question and eventually went on to relinquish the distinction between heavy and light in De Motu Antiquiora. Cf. Fredette (2001, 173).
- 64.
…mobile, nullam extrinsecam habens resistentiam, in plano sub horizonte …naturaliter descendet, nulla adhibita vi extrinseca;”
- 65.
For Aristotle, natural motion is not simply downward motion but the motion of bodies to their proper place, i.e., the center of the cosmos, which is incidentally the same as the center of the world. Cf. Machamer (1978). On an inclined plane a body will not descend, however, in a straight line, to its proper place.
- 66.
The challenge accruing to the traditional framework from the inclusion of free motion along inclined planes among natural motion starts with this very chapter of De Motu Antiquiora. Galileo will argue that a “mobile in plano quantulumcunque super horizontem erecto non nisi violenter ascendit: ergo restat, quod in ipso horizonte nec naturaliter nec violenter moveatur. Quod si non violenter movetur, ergo a vi omnium minima moveri poterit (EN I, 299).” Thus his identification of the motion of a heavy body down along an inclined plane as natural and that up along an inclined plane as violent leads Galileo to conclude that motion along the horizontal partakes in neither category and from this to conclude that a vanishingly small force would suffice to set a body in motion, paving the way to his principle of circular inertia. Cf. Miller (2014, chap. 5) and Damerow (2006, 15). Galileo retained the assumption (expressed by Cardano before him) that a body will be moved by a vanishingly small force on the horizontal, from De Motu Antiquiora onward. In the Aristotelian framework, a finite force depending on the weight of the body was required to set the body in motion along a horizontal. Laird (2001, 263), for instance, thus writes “[i]n a brilliant insight he grasped, contrary to Pappus and Guidobaldo, that ideally a body on a horizontal plane can be set in motion by any force, however small, so that the power to sustain a weight and the power to move it are effectively equal.” Heron had assumed the same in the first book of his mechanics (Heron 1976, 54); however, his text was, according to present knowledge, not known in the Latin West in Galileo’s day. Cf. Schiefsky (2008).
- 67.
“Quaeritur enim cur idem mobile grave, naturaliter descendens per plana ad planum horizontis inclinata, in illis facilius et celerius movetur quae cum horizonte angulos recto propinquiores continebunt; et, insuper, petitur proportio talium motuum in diversis inclinationibus factorum.”
- 68.
“…manifestum est, grave deorsum ferri tanta vi, quanta esset necessaria ad illud sursum trahendum; hoc est, fertur deorsum tanta vi, quanta resistit ne ascendat.”
- 69.
“Qui è manifesto, tanto essere l’impeto del descendere d’un grave, quanta è la resistenza o forza minima che basta per proibirlo e fermarlo: per tal forza e resistenza, e sua misura, mi voglio servire della gravità d’un altro mobile. Intendasi ora, sopra il piano FA posare il mobile G, legato con un filo che, cavalcando sopra l’F, abbia attaccato un peso H….” This passage is part of what is known as Viviani’s Scholium. It was dictated by Galileo to Viviani who put it into dialogue form and inserted it into the second edition of the Discorsi published in 1655. Cf. Halbwachs and Torunczyk (1985).
- 70.
In his resort to the compensation model, Galileo is not unique. The same compensation model is at least implicit in Jordanus’ treatment of the inclined plane, and Festa and Roux (2008, 207) wonder if manuscripts of Jordanus’ text may have “contained illustrations representing such an [compensation] arrangement.” Stevin’s proof of the law of the inclined plane of course rests fundamentally on a particular compensation model in which weights on inclined planes connected by a rope keep each other in equilibrium. Cf. Dijksterhuis (1943)
- 71.
- 72.
“…quanto maior vis requiritur ad sursum impellendum mobile per lineam bd quam per be …”
- 73.
In the first edition of Guidobaldo’s Mechanicorum Liber of 1577, Pappus’ proof of the principle of the inclined plane was merely referred to. See Del Monte (1969). In Pigafetta’s 1581 Italian translation of the work, the proof was added as a commentary. See Del Monte (1581). For Jordanus’ proof of the law of the inclined plane, see Tartaglia (1565). All three works can be accessed through the cultural heritage online collection ECHO at http://echo.mpiwg-berlin.mpg.de/home. Accessed 15 Jan 2017.
- 74.
Festa and Roux (2008, 214) rightly stress that “details of his [Galileo’s ] demonstration prevent us from concluding, as did Caverni and Duhem, that he owes his demonstration of the law of the inclined plane to this reading [of De ratione ponderis].”
- 75.
My notion of the balance-lever model corresponds to what Festa and Roux (2008, 214) outline as follows: “In the following discussion, we take ‘model of the balance’ to mean the idea that all mechanical systems (and in particular the inclined plane) can be understood from the starting point of weights balanced on a balance; we note that this idea does not prejudge the manner in which this balance is itself explained.” Since antiquity, the balance-lever model had indeed remained almost without alternatives for devising both qualitative and quantitative explanations of mechanical phenomena. Cf. Renn et al. (2003). The only thinker until Galileo’s time to have attended to the problem of the inclined plane without making direct recourse to the balance-lever model is Stevin (1586, 41–42).
- 76.
The one aspect Galileo criticized explicitly with regard to Pappus’ approach in his treatment of the problem of the inclined plane in Le Meccaniche was that the latter had tried to determine the force to move a body up along an inclined plane from the force required to move the same body on the horizontal. According to Galileo, this force is vanishingly small and, from his perspective, Pappus’ approach is indeed rendered inappropriate. Jordanus’ proof may well have been known to Galileo, yet the notion of positional heaviness on which it is fundamentally based is found, for example, also in the works of Cardano, Scaliger, Tartaglia, and Benedetti. Cf. Festa and Roux (2008). For the concept of positional weight or heaviness, see Damerow and Renn (2012).
- 77.
Issued as Proposition 8 in Book I of De ratione ponderis, Jordanus’ treatment of the bent lever replaced two propositions on the same subject in the earlier Liber de ponderibus. The proposition reads: “If the arms of a balance are unequal, and form an angle at the axis of support, then, if their ends are equidistant from the vertical line passing through the axis of support, equal weights suspended from them will, as so placed, be of equal heaviness (Gillispie 1981).” On the question of the attribution of various medieval mechanical manuscripts to Jordanus and for a reconstruction of their historical succession, see Clagett (1979). For Jordanus’, Benedetti’s, and Guidobaldo del Monte’s approaches to the bent lever, see Damerow and Renn (2012).
- 78.
- 79.
Galileo himself refers to the contrivance in his argument as a balance (libra). In the text, I address it by the more familiar term lever and bent lever, respectively. In the abstract case of a beam not extended in space that figures in Galileo’s argument, the difference between a lever supported from below and the balance supported from above vanishes.
- 80.
“Rursus, quando mobile erit in puncto s, in primo puncto s suus descensus erit veluti per lineam gh; quare mobilis per lineam gh motus erit secundum gravitatem quam habet mobile in puncto s.”
- 81.
Gatto dates Le Meccaniche to 1598 based on the assumption that a course Galileo gave in the academic year 1598/1599 at the University of Padua was based on the content of Le Mecaniche. Cf. Galilei (2002, LV). For a synopsis of Le Meccaniche, see the editor’s introduction in Galilei (2002). Le Mecaniche was circulated only in manuscript copies. It was published in French translation by Mersenne (Mersenne and Galilei 1635). For the bent lever proof in Le Meccaniche, see EN II, 180–183. In Le Meccaniche the law of the inclined plane derived by the proof is applied in the treatment of the screw. Galileo had already issued the law of the inclined plane in the earlier Delle Machine, which is sometimes also referred to as the short version of Le Meccaniche. However, due to its different style, length, focus, as well as date of composition, it should be treated as an independent text. R. Gatto in Galilei (2002) dates the text to 1593 based on the assumption that it presents the lecture notes for a lecture held by Galileo at the University of Padua. Valleriani (2010), in contrast, assumes that it was used by Galileo to teach his private pupils. Delle Machine, of which four copies are still extant, was not included by Favaro in the EN. Today, the text has been edited and published by Gatto (Galilei 2002).
- 82.
In Le Meccaniche, instead of “gravitas in plano,” Galileo uses the term mechanical moment to refer to the force along the inclined plane by which quite generally he expressed the varying effect of a weight depending on the “arrangement which different bodies have among themselves (EN II, 159, transl. Galilei 1960, 151).” Galluzzi (1979), in his eminent study on the subject, refers to “momento” as the link between statics and dynamics. For a clarification of the relation between mechanical moment and “momento” as in “momentum velocitatis,” see Chap. 10, Sect. 10.1. With regard to the bent lever proof, the substitution of “gravitas in plano” by “momento” appears to me to be less significant than has been assumed by, for instance, Festa and Roux (2008, 217), who state that “De motu brought together under the name of gravity two distinct quantities, weight and the effectiveness of weight; the Mecaniche distinguishes them by calling gravità the weight, and momento the effectiveness of the weight, that is to say the force required to support it or move it.” It can be objected that in De Motu Antiquiora Galileo was well able to distinguish weight and the effectiveness of weight not only conceptually but also on a linguistic level, when he uses, for instance, “gravitas” as clearly distinguished and distinct from “gravitas super plano.” With regard to this conceptual nexus, Clavelin (1983, 26) states succinctly that through the inclined plane “the tendency to descend, that is gravity as a motive force, is actually modified. A distinction thus becomes necessary between the weight and the natural motive force or, better still, between a properly gravific function and a motor function of weight.”
- 83.
Being based on model equivalence rather than a physical instantiation of the balance-lever model, Galileo’s argument seems to have been too abstract for some of his contemporaries. Davide Imperiali, for instance, devised his own version of a proof of the law of the inclined plane in which he blended Galileo’s approach with that of Pappus. Imperiali identified the abstract bent lever of Galileo’s construction within a body of circular cross section on the inclined plane, thus essentially annihilating the advantages of Galileo’s proof in favor of being able to provide an argument based on the idea of a physical instantiation of the balance-lever model. Imperiali himself claimed that his proof in no way fell short of Galileo’s. Cf. Gatto et al. (1996, 38 and 88).
- 84.
…est autem tanto gravius in puncto d quam in s, quanto longior est linea da quam linea ap;
- 85.
constat igitur, tanto minori vi trahi sursum idem pondus per inclinatum ascensum quam per rectum, quanto rectus ascensus minor est obliquo; et, consequenter, tanto maiori vi descendere idem grave per rectum descensum quam per inclinatum, quanto maior est inclinatus descensus quam rectus.
- 86.
From a modern perspective, the restriction to planes of either equal height or length when stating the law of the inclined plane simply allows avoiding invoking the sine. The bent lever proof itself, in a sense, offered an alternative formulation of the principle in which indeed merely the inclinations, but not specific length or heights of the planes considered, need to be specified, and it was, as argued in Chap. 8, exactly for this reason that the argument and construction of the bent lever proof could provide the gateway for the construction of the so-called ex mechanicis proof of the law of chords.
- 87.
Unde causetur celeritas et tarditas motus naturalis.
- 88.
- 89.
“…proportiones consequenter velocitatum, gravitatum proportiones, sequ[u]ntur.” Wisan (1974, 151) comments: “this implicitly assumes the Aristotelian dynamic principle that a constant force generates a constant speed. Galileo initially assumes this relation, thinking of natural fall as uniform.” The sentence quoted refers to motion through media and not to motion along inclined planes. Yet, as Souffrin (2001) has convincingly demonstrated in De Motu Antiquiora, exactly the same dynamical scheme forms the basis of Galileo’s treatment of motion on inclined planes and his treatment of motion in media. For Aristotelian dynamics, see, for instance, Clagett (1979, 425–433). Aristotelian dynamics was subject to debate from the Middle Ages to Galileo’s time, and alternatives were developed. These are discussed by Maier (1949). Van Dyck, 2006, Weighing falling bodies, Galileo’s thought experiment in the development of his dynamical thinking, unpublished, available at http://www.sarton.ugent.be/index.php rightly comments that “the only model that he [Galileo] possesses for understanding forces is the balance which measures absolute weights; and all his dynamical thinking is based on the idea that speeds are caused by such forces.”
- 90.
See Souffrin (2001) my emphasis. It needs to be added, however, that as regards natural motion, from an Aristotelian perspective, heaviness cannot be perceived as the potentia.
- 91.
Festa and Roux (2008, 24) claim “that this result does not take into account the trivial observation that a body accelerates when it descends an inclined plane.” However, at the end of the chapter, Galileo refers the reader forward to the chapter in which he goes on to give his account of accidental acceleration.
- 92.
Cf. Chap. 10.
- 93.
Vergara Caffarelli (2009, 101) referring to this remark by Galileo thus, for instance, claims that “Galileo wanted to tell us that he had found other applications of the theorem of the inclined plane (probably the theorem of chords).”
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Büttner, J. (2019). Before Natural Acceleration. In: Swinging and Rolling. Boston Studies in the Philosophy and History of Science, vol 335. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1594-0_2
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