Phys. Perspect. 21 (2019) 194–221 Ó 2019 Springer Nature Switzerland AG 1422-6944/19/030194-28 https://doi.org/10.1007/s00016-019-00243-y Physics in Perspective

What the Middle-Aged Galileo Told the Elderly Galileo: Galileo’s Search for the Laws of Fall

Penha Maria Cardozo Dias, Mariana Faria Brito Francisquini, Carlos Eduardo Aguiar and Marta Feijo´ Barroso*

Recent historiographic results in Galilean studies disclose the use of proportions, graphical representation of the kinematic variables (distance, time, speed), and the medieval double distance rule in Galileo’s reasoning; these have been characterized as Galileo’s ‘‘tools for thinking.’’ We assess the import of these ‘‘tools’’ in Galileo’s reasoning leading to the laws of fall (v2 / D and v / t). To this effect, a reconstruction of folio 152r shows that Galileo built proportions involving distance, time, and speed in uniform motions, and applied to them the double distance rule to obtain uniformly accelerated motions; the folio indicates that he tried to fit proportions in a graph. Analogously, an argument in Two New Sciences to the effect that an earlier proof of the law of fall started from an incorrect hypothesis (v µ D) can be recast in the language of proportions, using only the proof that v µ t and the hypothesis.

Key words: Galileo Galilei; uniformly accelerated fall; double distance rule; theorem of the mean speed; kinematic proportions in uniform motions.

Introduction Galileo Galilei stated the proof of the laws of fall in their finished form in the Two New Sciences,1 published in 1638, only a few years before his death in 1642, and shortly before turning 78. After the 1974 paper by Winifred Lovell Wisan on Galileo’s folio 91v,2 and Stillman Drake’s reconstruction of Galileo’s folio 152r,3 it has become accepted that the first accomplished proofs of the two laws of the uniformly accelerated fall (v2 / D and v / t) appear respectively in folios 152r and 91v. These folios are both placed in Galileo’s Paduan period and are dated sometime between 1609 and 1610,4 when Galileo was a middle-aged man (about 45–46).

* Penha Maria Cardozo Dias researches on the foundations of classical physics, using the history of physics to clarify meanings of concepts and laws. Mariana Francisquini teaches at the Instituto Federal do Rio de Janeiro. Her main interests are history of physics and physics education. Carlos Eduardo Aguiar is associate professor of physics at the Universidade Federal do Rio de Janeiro. His main research interests are nuclear physics and physics education. Marta Feijo´ Barroso is professor of physics and the Universidade Federal do Rio de Janeiro and researches on physics education.

194 Vol. 21 (2019) Middle-Aged Galileo 195

Results in Galilean studies by Matthias Schemmel, Peter Damerow et al., and Carla Rita Palmerino recognize in Galileo’s folios ‘‘tools for thinking’’ (an expression borrowed from Schemmel).5 The first is the description of uniform motion from three proportions involving ratios among distance, speed, and time in uniform motion. The second is the use of the medieval ‘‘double distance rule;’’ this is a very early formulation of William Heytesbury mean speed theorem.6 The third is ‘‘representations of the change of qualities and motions by geometric figures, or diagrams.’’7 Palmerino makes the claim that: ‘‘For those who, like Galileo, regarded mathematics as the language in which the book of nature was written, diagrams were not just a conventional representation of the laws of motion, but could in fact assist in the discovery of these laws.’’8 This historiography leaves an open question. It still remains to assess the use- fulness of those ‘‘tools’’ and how Galileo put them together; that is to say, it still has to be shown how Galileo reasoned from graphs, proportions, and the double distance rule to a proof of the laws of fall. By following Galileo’s reasoning in the folios, it is possible to find the import of the ‘‘tools’’ to the cognition of the laws of motion. To this end, we reconstruct folio 152r (see the ‘‘Graphical Representa- tions’’ section below). Galileo’s problem can be stated as the problem of finding laws for a uniformly accelerated motion from proportions involving distance, speed, and time for uni- form motions. Galileo conceived of two uniform motions and a third auxiliary uniform motion, conveniently defined. The double distance rule transforms dis- tances in uniform motions into distances in uniformly accelerated motions. The uniform motions and the associated uniformly accelerated motions are displayed in figure 3 and in table 1 (‘‘The Mathematics of Proportions’’). From the table, there immediately follows proportions comparing the distances moved in two different instants of time during a fall with the distance traveled in the auxiliary uniformly accelerated motion (equation 4); from the definition of meanp propor-ffiffiffiffiffi tional, the proportions are respectively transformed into the laws of fall, D µ v (equation 5) and v µ t (equation 6); equation 5 and equation 6 are also compact representations of the proofs of the two laws, as they respectively appear in folios 152r (appendix A) and 91v (appendix B). The table also points to meanings the numbers scattered all over folio 152r (figure 4) must have if they are to make any sense at all: they are attempts to fill table 1 with numbers. Eventually, Galileo hit upon a good choice of numbers (table 2, in the section ‘‘Discovery of the Para- bola’’); this choice could have been found by trial and error, without assuming a ‘‘dynamic,’’ as proposed by Drake and by Damerow et al.9 (see the section ‘‘Text 1,’’ below). Otherwise, we follow Damerow et al. and Wisan in interpreting the numbers in the folio as attempts to build proportions describing uniformly accelerated motion from uniform motion.10 However, neither Damerow et al. nor Wisan explore the drawing of the triangle that occupies most of the folio; the drawing suggests that the motivation in starting the folio was to make a diagram of distances and speeds consistent with the times of fall, once a choice of numbers 196 P. M. C. Dias et al. Phys. Perspect. that fits table 1 was found (see the section ‘‘Discovery of the Parabola’’). The numbers in table 2 give the points a, e, and f in figure 4 and in figure 5; actually, these points are in the parabola v2 =29 50 9 D. The general proof of the parabola v2 µ D is displayed in the folio: it involves only the proportions in table 1, which is shown by mere reproduction of the proof (appendix A). Galileo first found the odd numbers sequence (1, 3, 5, 7, …) for the distances moved in equal times (which is the same as D µ t2). It is not exactly known how Galileo found it; Drake reconstructs numbers in folio 107v, and proposes that they result from experiments,11 a conclusion challenged by R. H. Naylor.12 In a letter to Paolo Sarpi, dated October 16, 1604,13 Galileo claimed to have proved that the square of the time was proportional to the distance. Galileo sought to prove the sequence from a ‘‘more natural’’ principle;14 the principle is the hypothesis that, in the fall, the speed is proportional to the distance. It is believed that the proof mentioned to Sarpi is in folio 128;15 the proof is based on v µ D and on the double distance rule. The proof is wrong: besides starting from a false premise, Galileo seems to have made an error in calculating, or else made another conceptual error; as Wisan interprets it: ‘‘perhaps the most charitable conclusion is that GALILEO was too deeply engrossed in the larger problem of adapting traditional methods to an entirely new treatment of motion. His effort to provide elegant mathematical proofs in the style of ARCHIMEDES to matter previously dealt with in largely rhetorical terms had led him into new territory where no trail breakers had pre- ceded him.’’16 The incorrect hypothesis (v µ D) poses a new problem. According to Damerow et al.,17 Galileo understood that the hypothesis was wrong when he realized that the numbers in table 4 were inconsistent with the v µ D. Also, in Palmerino’s opinion: ‘‘I am inclined to believe that it was precisely this fact—that the principle of the space/speed proportionality did not yield a diagrammatic representation of the odd number law—that helped Galileo to realize the falsity of this principle. The manuscript notes clearly reveal that Galileo did struggle to produce a diagram into which the three parameters of acceleration, speed, time and space, could be made to fit.’’18 The claim is plausible, from the point of view of a reconstruction of folio 152r, and the evidence therein. In fact, when plotted in a graph of speed versus distance, speeds and distances must form the sides of similar triangles, if the hypothesis v / D is true, but they do not (figure 5): the speeds in two instants in the fall (points e and f ) are not on a same line drawn from the beginning of the motion ðÞa . What emerges from table 1 is D / v2 (equation 5) and D2 / t (equation 6), which Galileo could have noticed; but he marks point f in figure 4, and there are indi- cations of an attempt to plot. However, as appealing as the claim might be, Galileo’s known argument to the effect that the hypothesis v / D is wrong seems to follow a different line of thought. In the Two New Sciences, Salviati, Galileo’s avatar, gives an argument to the effect that in the (uniformly accelerated) fall, the speed cannot be proportional to Vol. 21 (2019) Middle-Aged Galileo 197 the distance, as ‘‘The Author’’ (Galileo himself) previously believed.19 The argu- ment consists in showing that the hypothesis leads to an absurdity. This argument was analyzed by many authors in the seventeenth and eighteenth centuries,20 by Ernst Mach, and more recently by John D. Norton and Bryan W. Roberts;21 the interpretations were also subjected to criticisms by Isaac Bernard Cohen, A. Rupert Hall, Drake, Palmerino, and W. R. Laird.22 Perhaps the argument in Two New Sciences is not different from the reasoning in both folios. We show (see the section ‘‘The Argument’’) that the argument can be recast in terms of the proportions in table 1; in fact, the absurd result follows by algebraic manipulation, when putting together equation 6 and the incorrect hypothesis. In the same way that Galileo, having already obtained the sequence of odd numbers, searched for a proof of the sequence from ‘‘more natural’’ principles, the argument in Two New Sciences could be a proof of a result he found in trying to fit proportions in a graph, now based only on the proportions in table 1. Whether a principle is ‘‘natural’’ might have to do with Galileo’s Platonism; Noel Swerdlow defines the meaning of ‘‘Platonism’’ in Galileo’s thinking: ‘‘what is often called ‘Platonism’ in Galileo, his appeal to mathematics and idealized conditions, is in fact the abstract mathematical analysis of mechanics, and he came to regard anything outside mechanics, anything not subject to mathematical analysis, any- thing invoking hidden causes, as not ‘within the limits of nature.’’’23 Analyzing Galileo’s dictum that the book of nature is written in the language of mathematics, Palmerino argues: Galileo’s claim that nature is written in the language of mathematics, far from being a rhetorical statement or an unwarranted metaphysical conviction, is grounded in coherent ontological and epistemological arguments. In his works Galileo repeatedly argues that mathematical entities are ontologically inde- pendent from us and that the physical world has a mathematical structure. This structure is, however, too complex to be fully grasped by our finite intellect, which is why we need to simplify physical phenomena in order to be able to deal with them mathematically.24

The Mathematics of Proportions and the Double Distance Rule Uniform motion is defined in Two New Sciences in the language of proportions, and can be written in modern notation (propositions I, II, and III in Two New Sciences, respectively): 198 P. M. C. Dias et al. Phys. Perspect.

s1 t1 when v1 ¼ v2 then ¼ s2 t2 v1 s1 when t1 ¼ t2 then ¼ v2 s2 v1 t2 when s1 ¼ s2 then ¼ v2 t1 The first proportion is in Archimedes’s On Spirals.25 The other two are based on Aristotle’s definition of ‘‘quicker,’’ which is given in Physica.26 Damerow et al. observe that even when d, v, and t were different, the comparison of two motions was still made by taking together the above proportions, as follows.27 Considering a third motion that moves with a speed equal to the speed of the first ðÞv3 ¼ v1 , and time as in the second motion ðÞt3 ¼ t2 : s t t 1 ¼ 1 ¼ 1 ð1Þ s3 t3 t2 s v v 3 ¼ 3 ¼ 1 ð2Þ s2 v2 v2 then (which is allowed by theorem IV in Two New Sciences): s v t 1 ¼ 1 Â 1 : ð3Þ s2 v2 t2 Wisan notes that the latter might have been used by Galileo in the folios.28 The definition of uniform motions by these proportions is also found in the later Two New Sciences. Medieval thinkers already knew that this mathematics of propor- tions did not hold for accelerated motion, and Galileo and his predecessors faced a problem: ‘‘one of the key problems of preclassical mechanics was to derive con- sequences for nonuniform, accelerated motions from Aristotle’s definition of velocity.’’29 Many historians recognize Galileo’s debt to the medieval.30 According to Damerow et al., Galileo’s terminology, such as ‘‘degree of velocity,’’ points to the Mertonian theory of increase and remission of qualities: ‘‘one can consider degrees through which it [that is, a quality] is constituted hot or cold, etc., and so can be more or less perfect; on this account the quality has a certain latitude over which there are a number of degrees that do not vary the essence.’’31 The ‘‘intension’’ or ‘‘degree of motion’’ is the instantaneous speed; but the interpretation of the ‘‘extension’’ or ‘‘latitude’’ of a motion was a debated problem in the Middle Ages.32 Sometimes, the extension was taken as the time, as with Richard Swineshead, John Dumbleton, and Nicole Oresme; sometimes it was the distance, as with Albert of Saxony.33 Furthermore, in Marshal Clagett’s dictionary of Mertonian concepts, the ‘‘quantity of motion’’ or ‘‘total velocity’’ is ‘‘measured by the distance traversed in that time,’’ and is given by the area in a graph distance versus time; in Oresme’s geometry of graphs, the area represented the distance.34 Vol. 21 (2019) Middle-Aged Galileo 199

According to Clagett, Galileo probably learned the proof of the mean speed theorem ‘‘from [Giovanni di] Casali, or Blasius [of Parma], or even from the 1494 edition of [William] Heytesbury’s Regule solvendi sophismata.’’ 35 Casali illustrates the ‘‘extension’’ (or ‘‘latitude’’) taking as example the variation of heat, as does Galileo in the above quotation. In particular, Casali’s treatment of the mean speed theorem is close to Heytesbury’s, according to Clagett.36 Heytesbury states the double distance rule, not the full theorem: ‘‘From the foregoing it follows that when any mobile body is uniformly accelerated from rest to some given degree [of velocity], it will in that time traverse one-half the distance that it would traverse if, in that same time, it were moved uniformly at the degree [of velocity] terminating that latitude.’’37 The double distance rule is found in folio 163v. Figure 1 and figure 2 below are reconstructions of the figures in the folio. The first paragraph in the folio explains the double distance rule in figure 1: ‘‘Let the motion from a to b be made in natural acceleration: I say, if the velocity in all points ab were the same as that found in the point b, the space ab would be traversed twice as fast; because all velocities in the single points of the line ab have the same ratio to all the velocities each of which is equal to the velocity bc as the triangle abe has to the rectangle abed.’’38 Literally, from figure 1: Area of Mabc 1 ðÞab  bc ¼ 2 ; Area of (abcd ab  bc whatever the interpretation of coordinates and areas. If now the vertical repre- sents distance, and the horizontal represents speed, this can be written: Area of Mabc ðÞab  v ¼ b ; Area of (abcd ðÞÂ2ab vb therefore, in the time of the fall D ¼ ab, the distance in a uniform motion made with speed vb is d ¼ 2D. But this is also Area of Mabc ab  v ¼ b ; Area of (abcd ab Âð2vbÞ therefore, if the distance d ¼ ab is traversed with a uniform motion in the same time that an equal distance D ¼ ab is traversed in the fall, the uniform speed is 2vb. Wisan instead proposes that ‘‘twice as fast’’ is best understood as ‘‘in half the time;’’39 this is true, when the distance in the uniform motion is d ¼ D ¼ ab and t the speed is vb, because then the time in the uniform motion is 2. Figure 2 illus- trates the application of the rule: by construction, the time of the motion on the incline is t, and the speed of the uniform motion is vb; the distance is then d ¼ bc ¼ 2ab ¼ 2D. 200 P. M. C. Dias et al. Phys. Perspect.

Fig. 1. Double distance rule. The vertical represents distance of fall; the horizontal represents instantaneous speed. The area of the triangle abc is half the area of the rectangle abcd

The second paragraph in the folio shows how the double distance rule is used to evaluate distances in uniformly accelerated motions in figure 2: ‘‘From this it follows, that if there were a plane ba inclined to the horizontal line cd, and be being double ba, then the moving body would come from a to b and successively from b to c in equal times: for, after it was in b, it will be moved along the remaining bc with uniform velocity and with the same with which [it is moved] in this very terminal point b after fall through ab. Furthermore, it is obvious that the 1 38 whole time through abe is 1 2 the time [sesquialterum] for ab.’’ Vol. 21 (2019) Middle-Aged Galileo 201

Fig. 2. Application of the double distance rule. The horizontal bc is the distance in a uniform motion with the speed at the bottom of the incline, accomplished in the same time of fall through ab : bc ¼ 2ab

In order to use the double distance rule, and compare two distances of fall, say D1 and D2, two uniform motions, d1 and d2, are then introduced, which are respectively made with the instantaneous speeds at the end of the distances of fall, as in figure 3. A third uniform motion is then introduced, which is made with speed vb, and in which the distance is the mean proportional between cd and be; it is easy to see that bl ¼ 2as, where as is the mean proportional between ab and ac.* Palmerino interprets figure 3 as follows: ‘‘As I mentioned above, in folio 91v Galileo makes use of the Double Distance Rule to demonstrate that the speed of fall must increase in proportion to the time elapsed, but produces a hybrid diagram (figure 7 [figure 3 here]) which conflates the representation of speed with that of space.’’18 The conflation of coordinates suggested by Palmerino is possible only by con- sidering uniform motions, as in table 1: Clearly, from table 1: D t D v mp ¼ 1 and 1 ¼ 1 ; ð4Þ D2 t2 Dmp v2 so that ab and as,orcd and bl also measure time, and ac and as,orcd and bl also measure speed. But this implies:

* The third motion does not belong in the same fall with the other two: neither the speed nor the time of fall when the body falls a distance equal to the mean proportional are equal to the speed and the time in the third motion. Furthermore, the third motion seems to be introduced as a computational trick; the mean proportional is only a way to find square roots. 202 P. M. C. Dias et al. Phys. Perspect.

Fig. 3. The three uniform motions. The vertical represents a fall. The horizontal lines cd ¼ 2ac pandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibc ¼ 2ab representpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi distances in uniform motions respectively made with vc and vb; bl ¼ cd  be ¼ 2as ¼ ab  ac is the distance in a third uniform motion made with vc

Table 1. The three motions. Uniform motions obey equation 1 and equation 2, and accelerated motions obey the double distance rule

Distance in the uniform motion Speed in the uniform motion Duration of the uniform motion d1  cd  2D1  2ac v1 t1 d2  be ¼ 2D2  2ab v2 t2 pffiffiffiffiffiffiffiffiffi dmp ¼ d1d2  bl  2Dmp  2as v2 t1

rffiffiffiffiffiffi D D v 1  1 ¼ 1 ; ð5Þ Dmp D2 v2 which is a compact notation of the steps of the proof that vb and vc fall on the same parabola (section ‘‘Graphical Representations,’’ below, and the appendix A). Also: rffiffiffiffiffiffi t D D v 1 ¼ mp ¼ 1 ¼ 1 ; ð6Þ t2 D2 D2 v2 which is a compact notation of the steps of the proof of the straight line v / t (see the section ‘‘The Argument,’’ below, and the appendix B). Vol. 21 (2019) Middle-Aged Galileo 203

Graphical Representation of Proportions The parabola v2 / D first appears in folio 152r. Drake was the first to recognize the importance of the folio, and places it before October 1604.40 However, many historians have criticized his reconstruction of the folio.41 According to Drake, the discovery results from a chance choice of numbers, a medieval addition of impetus,

Fig. 4. Folio 152r (su concessione del Ministero per i beni e le attivita` culturali/Biblioteca Nazionale Centrale. Firenze—further reproduction or duplication of this image by any means are prohibited without the permission of Biblioteca Nazionale Centrale). The folio contains many texts with numbers. At the center, occupying most of the paper, is the diagram speed versus distance 204 P. M. C. Dias et al. Phys. Perspect.

2 0 Fig. 5. The parabola v ¼ 2 Â 50 Â D. The horizontal lines be ¼ va ¼ 20, dd ¼ vc ¼ 30 and cX ¼ 45 fall on the hypotenuse aX, but not cf ¼ vc ¼ 30. The points a, ebeðÞ¼ va , YbYðÞ¼ vd and fcfðÞ¼ vc fall on a parabola and owes nothing to the medieval theorem of mean speed or to the double dis- tance rule. This section presents a reconstruction of folio 152r. Any reconstruction of the folio has some amount of speculation, as Wisan would recognize: ‘‘One of the main difficulties lies perhaps in Galileo’s failure to provide a systematic account of his views. Brief comments on philosophy and method are scattered throughout his writings and occasionally one finds a paragraph or so of sustained discussion. But these passages are often obscure, usually fragmentary, and sometimes contradict one another. Consequently, they have provided the basis for radically different interpretations covering virtually the entire spectrum of philosophical stances and methodological precepts.’’42

Discovery of the Parabola v2 / D This subsection proposes the following reconstruction of folio 152r. Galileo first gives numerical values to the uniform motions in table 1, which results in table 2, below; then he plots these numbers in a graph speed versus distance. The numbers in table 2 do no fit the proportionality between v and D when graphically Vol. 21 (2019) Middle-Aged Galileo 205 expressed; in other words, speeds and distances of the fall expressed by the numbers in table 2 do not form similar triangles. Then Galileo proves that the curve consistent with the proportions in table 1 is a parabola (the proof is algebraic, not geometric, and is in the appendix A). Folio 152r is in figure 4. There are texts with numbers scattered all over the folio. In the interpretation given here, the texts seem to be attempts to build motions as in table 1, which is shown in section ‘‘The texts in folio 152r,’’ below. These texts have been commented by Wisan, Damerow et al., and Drake.43 Wisan’s interpretation is similar to ours; although Damerow et al. recognize the use of the kinematic relations, their interpretation differs from ours,44 which is discussed in section ‘‘Text 1’’. The drawing on the bottom, with a vertical, and horizontal lines issuing from it resembles figure 3, which Damerow et al. also notice. In particular, the following set of numbers is a perfect fit for table 1:

Time through ab4 ab4 ad6 be20 ac6 ad9 cf 30

In fact, this is shown in table 2: Suppose now that these numbers are plotted in a graph speed versus distance (figure 5). From table 2: D ab 4 2 v be 20 2 D v 2 1 b 1 b ;  ¼ ¼ and  0 ¼ ¼ or ¼ ¼ Dmp ad 6 3 vc dd 30 3 Dmp vd 3 then points a, e, d0, X, c, d, and b fall on sides of the same triangle, aXc; fur- thermore, triangles abe, add0 and acX are similar:* ab be ac 2 ¼ ¼ ¼ : ad dd0 cX 3 However the point f does not fall on the line aed0X (the hypotenuse), as it should, if the speeds and distances are in a unique proportion; in fact the third speed is cf  vc ¼ 30, but the point on the hypotenuse is cX ¼ 45. At this point, if Galileo was indeed trying to plot the numbers in table 2 into a graph speed versus distance, he could not have failed to realize that the hypothesis

 d 20 30 cX * tan Xac ¼ 4 ¼ 6 ¼ 9 ¼ 5 so that cX ¼ 45: 206 P. M. C. Dias et al. Phys. Perspect.

Table 2. Table 1 with numerical values. A choice of numbers that obey the correct proportions

Distance in the Distance in the uniform Speed in the uniform Duration of the uniform fall motion motion motion

Dab  ab ¼ 4 d1  2ab ¼ 8 vb  be ¼ 20 tab ¼ 4

Dac  ac ¼ 9 d2  2ac ¼ 18 vc  cf ¼ 30 tac ¼ 6 0 Dmp  ad ¼ 6 dmp  2ad ¼ 12 vc  dd ¼ 30 tab ¼ 4 v / D was incompatible with table 2. But in table 2 the double distance rule is used to evaluate distances in a uniform accelerated fall from distances in uniform motions (where d / v), then v / D is incompatible with the double distance rule. The role of the double distance rule in finding that it cannot be v / D is commented by Palmerino,45 and by Damerow et al.: ‘‘As a consequence [of the double distance rule] he [Galileo] discovered that the degree of velocity in accelerated motion cannot be proportional to space but must be proportional to time, if the law of fall is supposed to hold. But since, up to this point, Galileo had attempted to derive the law of fall from the proportionality between velocities and spaces, this consequence leads to an open contradiction. Consequently, he is driven to new attempts at reconciling his premises, which in turn result in new paradoxes.’’46 Galileo did not draw the parabola in figure 4. But the vertical in figure 4 has marks that indicate that Galileo might have been trying to represent the numbers in table 2 into a graph; there is a horizontal line from d to the hypotenuse, f and c are clearly on the same horizontal, and there is a vertical line through f from the point where the horizontal from d intersects the hypotenuse. Galileo did not have to actually draw the parabola to realize that v is not proportional to D; it suffices to notice that the horizontal line cf does not fall on the hypotenuse. The proof of the parabola overlaps the graph, squeezed into a corner, suggesting that there is not left much room in the folio, so that the proof must have been the last thing to be made; furthermore the fact that the triangular diagram occupies most of the folio suggests that Galileo’s initial intention might have been a representation of the motion in a graph involving the three kinematic variables, as claimed by Palmerino. An important set of numbers is found near the graph: 8:18; 27:8. Wisan and Damerow et al. interpret these numbers as the ratios:47  Area of (abeE area of Mabe v  ab 20  4 8 4 2 2 c ; 0 ¼ 0 ¼ ¼ ¼ ¼ ¼ ð7Þ Area of (add D area of Madd vd  ad 30  6 18 9 3  Area of (acfD area of Macf v  ac 30  9 27 13 1 2 2 ¼ ¼ c ¼ ¼ ¼ 2 6¼ : ð8Þ Area of (abeE area of Mabe vb  ab 20  4 8 4 3 Vol. 21 (2019) Middle-Aged Galileo 207

ÀÁ 3 4 The inequality above should give the ratio 2 ,ifv / D, which happens only if f is brought into X:

The Texts in Folio 152r

ÀÁTo build table 2, Galileo had to find three numbers in geometric proportion 4 6 2 6 ¼ 9 ; he was shrewd to choose the numbers at the extremities (4 ¼ 2 and 9 ¼ 32) to be square numbers. From equation 4, the speeds and the times are in 2 the ratio 3; an immediate choice is v1 ¼ 20 and v2 ¼ 30; the times could have been 20 and 30, but Galileo made the ‘‘next’’ choice 4 and 6 (actually the correct values are t1 ¼ 0:4 and t2 ¼ 0:6, but it makes no difference, since Galileo considers ratios). This subsection shows that the various texts scattered throughout folio 152r can be interpreted as unsuccessful attempts to fill table 1 with numbers.48 This interpretation seems cogent, because table 1 gives the prescription to concoct motions: the proportions in equation 5 and equation 6 are already the laws v2 / D and v / t; as mentioned before, the equations also summarize the steps in the proofs, which can be verified by following the proofs verbatim, as shown in appendix A and in appendix B.

Text 1 With one degree of impetus it makes 2 miles per hour: With 4 degrees of impetus it will make 8 miles in one hour and 16 in 2 hours: Text 1 is at the bottom left of the folio, close to a drawing similar to figure 3, and seems to refer to the drawing. Marks and letters in the drawing indicate that Galileo might be trying to do some sort of plotting. Furthermore, the drawing indicates that the distance moved in one hour is D ¼ ab ¼ 1. From table 3:

Table 3. Text 1. Attempt to find nice numbers to fill in Table 1

Distance in the Distance in the uniform Speed in the uniform Duration of the uniform fall motion motion motion

D1 ¼ 1 d1 ¼ 2 v1 ¼ 1 t1 ¼ 1

D2 ¼ 8 d2 ¼ 16 v2 ¼ 4 t2 ¼ 2 pffiffiffi pffiffiffi pffiffiffi pffiffiffi Dmp ¼ 8 ¼ 2 2 dmp ¼ 2 8 ¼ 4 2 v2 ¼ 4 t1 ¼ 1 208 P. M. C. Dias et al. Phys. Perspect.

Table 4. Text 1 interpreted by Damerow et al. The second line results from the law of odd numbers and the proportion v µ D

Distance in the Distance in the uniform Speed in the uniform Duration of the uniform fall motion motion motion

D1 ¼ 1 d1 ¼ 2 v1 ¼ 1 t1 ¼ 1 0 0 D 2 ¼ 4 d2 ¼ 8 v2 ¼ 4 t2 ¼ 2 D2 ¼ 8 d2 ¼ 16 v2 ¼ 4 t2 ¼ 2

Dmp ¼ 4 dmp ¼ 8 v2 ¼ 4 t1 ¼ 1

pffiffiffi 2 d v 1 ¼ 1 6¼ 1 ¼ 4 dmp v2 4 pffiffiffi 2 d t 1 ¼ mp 6¼ 1 ¼ ; 4 d2 t2 2 which shows that Galileo has not yet obtained good numbers. To Damerow et al., text 1 is the ‘‘when and how’’ Galileo realizes that the hypothesis v / D is wrong.49 Their interpretation and the interpretation in this paper differ. Table 4 summarizes their interpretation. Damerow et al. reason as follows: For, if the degree of impetus were to grow in proportion to distance, a falling body would acquire the 4 degrees of impetus mentioned in the text after falling through the 4 units of distance that are marked in Galileo’s diagram. According to the law of fall, the body would take 2 hours in order to traverse this distance, since 1 hour was needed to fall through a distance of 1 mile. The Double Distance Rule then implies that, if the body is deflected into the horizontal after these 2 hours, it will traverse 8 miles in 2 hours in a uniform motion. Therefore, 4 degrees of impetus are characterized by a uniform motion that traverses 8 miles in 2 hours.50

The first line in table 4 is equal to the first line in table 3: D1 ¼ 1, v1 ¼ 1, t1 ¼ 1. 0 From the ‘‘law of fall’’ (odd numbers sequence?), the falling body makes D2 ¼ 3 þ 1 ¼ 4int2 ¼ 2 hours, and from v / D, it follows that the (instantaneous) speed at the end of D ¼ 4isv2 ¼ 4, which gives the second line in table 4. They continue: ‘‘But, according to the kinematic proportion that led to the second proposition of this text, 4 degrees of impetus correspond to 8 miles traversed in 1 hour or in other words (for uniform motion) 16 miles traversed in 2 hours. Hence, the assumption of a proportionality between the degree of impetus and distance leads to a contradiction.’’51 The only difference between table 4 and table 3 is the introduction of a line (the second line) that shows how Galileo might have found the speed at D ¼ 4. Vol. 21 (2019) Middle-Aged Galileo 209

Table 5. Text 2. An attempt to find nice numbers to fill in table 1

Distance in the Distance in the uniform Speed in the uniform Duration of the uniform fall motion motion motion

D1 ¼ 2 d1 ¼ 4 v1 ¼ 15 t1 ¼ 4

D2 ¼ 4 d2 ¼ 8 v2 t2 ¼ 8 pffiffiffi pffiffiffi Dmp ¼ 2 2 dmp ¼ 4 2 v2 t1 ¼ 4

But Galileo does not need it if he uses the proportions in equation 4. Furthermore, the choice of D2 in the other set of numbers does not seem to be an application of the odd number sequence: in text 1 and in text 2, where t2 ¼ 2t1, Galileo does not make D2 ¼ D1 þ 3D1 ¼ 4D1; in text 3, t2 is left blank, and D2 seems to be chosen rather to get a workable mean proportional. It seems that Galileo only uses the kinematic relation in table 1.

Text 2 4 miles with 15 of velocity in 4 8 miles in 8 From table 5: pffiffiffi 2 d v 15 ¼ 1 6¼ 1 ¼ 2 dmp v2 v2 pffiffiffi 2 d t 4 1 ¼ mp 6¼ 1 ¼ ¼ : 2 d1 t2 8 2

v1 t1 If v2 ¼ 30, then ¼ , according to table 1, but the distances do not fit. v2 t2

Text 3 4 miles with 10 of velocity in 4 hours 9 miles with 15 of velocity inÀÀhours From table 6: 2 d v 10 ¼ 1 ¼ 1 ¼ 3 dmp v2 15 2 d t 4 ¼ mp ¼ 1 ¼ : 3 d2 t2 t2 210 P. M. C. Dias et al. Phys. Perspect.

Table 6. Text 3. An attempt to find nice numbers to fill in table 1

Distance in the Distance in the uniform Speed in the uniform Duration of the uniform fall motion motion motion

D1 ¼ 2 d1 ¼ 4 v1 ¼ 10 t1 ¼ 4 9 D2 ¼ 2 d2 ¼ 9 v2 ¼ 15 t2 ¼ ?

Dmp ¼ 3 dmp ¼ 6 v2 ¼ 15 t1 ¼ 4

If t2 ¼ 6, the numbers fit table 1. Then table 2 is obtained from table 6 by 9 multiplying the distances and speeds by 2, which eliminates the fraction 2, and keeps the proportions. In the folio (figure 4), there is a 6 above the number 4, and an unreadable scribble in place of the ‘‘—.’’ Drake and Wisan give different interpretations: Drake makes it ‘‘5 hours,’’ because of his way of adding impetus gained per hour;52 1 1 Wisan makes it 13 2 by visual inspection, corroborated by the number 13 2 in text 4, below.53 However, since it is impossible to read the scribble, it is possible that the 6 above the 4 is intended to be the value that fills the ‘‘—,’’ and makes the numbers in table 6 to be in the correct proportion.

Text 4 through ab velo - city as 4 through ac velo - 1 city as 13 2 It seems that the text refers to the ‘‘total velocity,’’ that is, to areas: the ratio 4 8 131 ¼ 27 appears in the graph in figure 5, as already discussed. Then text 4 seems to refer2 to areas obtained with the numbers in text 3:

d1 Â v1 4 Â 10 4 ¼ ¼ 1 d2 Â v2 9 Â 15 13 2

Text 5 through ab velocity as 10 through ac as 15 Probably another failed attempt. Vol. 21 (2019) Middle-Aged Galileo 211

The Argument in Two New Sciences Recast from Folio 91v The purpose of this section is to show that Galileo’s argument in Two New Sci- ences to the effect that v / D leads to an absurdity is obtained from the proof of t / v in folio 91v, by making v / D (or from equation 6, which follows from table 1). The sequence of odd numbers is a consequence of D / t2. Once it is proved that D / v2, it suffices to prove that v / t to obtain D / t2. This proof is in folio 91v, first noted by Drake.54 Wisan places folio 91v in 1610 at the latest.53 It has already been commented that equation 6 is a compact form of the steps of the proof. Recalling equation 6: v D D t 1 ¼ 1 ¼ mp ¼ 1 ; v2 Dmp D2 t2 where the first equality follows from steps 1 and 2 below, the second from the definition of mean proportional, and the third from steps 1 and 2. In fact, using the notation in figure 3 and table 1, the steps in the proof are: d (1) Step 1. From table 1 and figure 3: vc ¼ dc  d1 and tac ¼ bl  mp vb bl dmp tab be d2 (2) Step 2. Using the double distance rule: vc ¼ dc  d1 ¼ 2Dac ¼ Dac and vb bl dmp 2Dmp Dmp tac bl dmp 2Dmp Dmp t ¼ be  d ¼ 2D ¼ D . ab 2 ab ab D (3) Step 3. From the definition of mean proportional, Dac  ca ¼ as  mp. Dmp as ba Dab (4) Step 4. Use step 1 and step 2 to obtain vc ¼ tac . vb tab The argument in Two New Sciences is as follows.55 The first part is: ‘‘if therefore the speeds with which the falling body passed the space of four braccia were the doubles of the speeds with which it passed the first two braccia, as one space is double the other space, then the times of those passages are equal.’’ The second part is: ‘‘but for the same moveable to pass the four braccia and the two in the same time cannot take place except in instantaneous motion.’’ Finally, the undisputed third part seems to be observational: ‘‘But we see that the falling heavy body makes its motion in time, and passes the two braccia in less [time] than the four; therefore it is false that its speed increases as the space.’’ As already noted, this argument has been long debated. There are two points of dispute. The first point is about the first part: it is true for uniform motions, and it seems to be valid for uniformly accelerated motions, if either Galileo refers to the average speed or to the double distance rule. The second point is about the solution (that the times are equal for different distances): the solution is not unique, and infinite times is also a possibility,56 as well ‘‘no distance at all.’’57 212 P. M. C. Dias et al. Phys. Perspect.

Fig. 6. The argument by Andres. The figure is similar to figure 3

From step 2 above: rffiffiffiffiffiffiffiffi v D D c ¼ ac ¼ ac ð9Þ vb Dmp Dab rffiffiffiffiffiffiffiffi t D D ac ¼ mp ¼ ac ð10Þ tab Dab Dab

vc Dac If v ¼ cD, then vb ¼ cDab and vc ¼ cDac, then ¼ ; with this result, equa- vb Dab tion 9 becomes: rffiffiffiffiffiffiffiffi v D D c  ac ¼ ac; v D D qffiffiffiffiffiffiq ffiffiffiffiffiffi b ab ab qffiffiffiffiffiffi qffiffiffiffiffiffi or Dac Dac À 1 ¼ 0, which for finite, non-null Dac implies Dac ¼ 1. Taking Dab Dab Dab Dab this result in equation 10: rffiffiffiffiffiffiffiffi t D ac ¼ ac ¼ 1: tab Dab

This is the conclusion in Two New Sciences. It holds whatever the distances Dac and Dab, which is an absurdity, unless, as Galileo recognizes, tab ¼ tac ¼ 0, or ‘‘in instantaneous motions.’’* This reconstruction is similar to a proof by Giovanni Andres, dated 1779 (figure 6).   * Mathematically, tac ¼ 1 does not preclude the solutions 1 ¼ lim tac and 1 ¼ lim tac ; the tab tab t tab first solution means ‘‘instantaneousqffiffiffiffiffiffi motion;’’ the second meanst! infinite0 time, as!1 Norton and Robertsqffiffiffiffiffiffi claim. Furthermore, Dac ¼ 0 is also a solution, and likewise, it does not preclude Dab lim Dac ¼ 0, or ‘‘no distance at all,’’ as claimed by Mach. D!0 Dab Vol. 21 (2019) Middle-Aged Galileo 213

Andres’s argument can be paraphrased as follows.* In an equal time t, body A moves AC and B moves BD. By construction, let it be AC ¼ 2BD.Ifv / D, then vAðÞ¼C 2vBðÞD where vAðÞC is the speed of body A at point C, and vBðÞD is the speed of B at D. The motions on the horizontal lines from C and D are uniform, and are made with the speeds at C and D, respectively. In time t, the distances are: vA dA ¼ vAðÞC t, then dB ¼ vBðÞD t ¼ 2 t,ordA ¼ 2dB;ifdA ¼ 4, then dB ¼ 2. dA 4 Applying the double distance rule: AC ¼ 2 ¼ 2 ¼ 2 in the time of fall tAC from A dB 2 to C, and BD ¼ 2 ¼ 2 ¼ 1 in the time of fall tBD from B to D. But tAC ¼ tBD ¼ t,so AC that in t, B falls the same as A falls in both AE ¼ BD ¼ 2 and AC. Therefore ‘‘EC was passed over without time.’’

Conclusion The span of about thirty years between the discovery of proofs of the laws (circa 1605–1610), and their publication in Two New Sciences (published in 1638), might have to do with Galileo’s involvement with astronomy, researches in other fields, and new researches in mechanics.59 According to Swerdlow, the treatment of the accelerated fall was first published in Two Chief Systems (1632);60 this treatment is along the same line of though in folios 152r and 91v. Salviati argues that the speed increases continually, and ends with the statement that ‘‘[the falling body] would pass with uniform motion during the same time through double the space which it passed with the accelerated motion.’’61 This is the double distance rule in the earlier folios. In the much earlier De Motu, written during Galileo’s Pisan period (1589–1592), he follows an entirely different approach, in which ‘‘heaviness’’ and ‘‘lightness’’ is associated with specific weight.62 In Two New Sciences, Galileo proves the laws using the Mertonian theorem of mean speed (which is a devel- opment of the double distance rule) and the mathematics of proportions, and also the motion of fall is represented in a graph speed versus time, which are also in the folios. The folios written decades before Galileo’s last work consist in what the middle-aged Galileo told the elderly Galileo.

*‘‘A at C has double the velocity of B at D; therefore A has such velocity as to run in uniform motion four braccia in the time that B in uniform motion runs two braccia. The time in which B at D runs two braccia is the time in which it has run through BD; therefore, the time in which A at C will run four braccia in uniform motion is equal to that in which B, accelerated, has run through BD. Now, the time in which A at C runs four braccia uniformly is equal to that in which it has run through AC with accelerated motion; therefore, the time of motion of A through AC is equal to the time of motion of B through BD. But the time of motion of B through BD is equal to the time in which A has passed through AE; therefore, the time of motion of A through AC, which is seen to be the time of B through BD, is equal to that of the motion of A through AE. Therefore EC was passed over without time.’’ 214 P. M. C. Dias et al. Phys. Perspect.

Postscript It may be worth a general comment, in order to attract those that should be the most interested in the history of physics—the physicists. This paper is not on historiography, but uses good historiography written by knowledgeable historians to discuss the foundations of physics. A concern of the foundations of physics is to justify a specific law of nature, a challenge met by Galileo. In the folios, Galileo’s effort to find convincing arguments for the law of fall is organized in an intellectual context of problems and ‘‘tools,’’ the same context in Galileo’s ultimate and last work, Two New Sciences. Focus on the folios thus discloses relevant points in the arguments, and makes possible their analysis. We have chosen not to broaden the scope of the paper exploring the ins and outs in Galileo’s working process, as this would lead the paper in too many directions, loss of focus, and a failure to fix the context. Formulating old arguments in modern mathematical language can enrich his- toriography by clarifying it. In this paper, mathematics allows the organization of numbers in the folios in tables, analogously to the periodic table of elements, or to the standard model of particle physics; continuing the analogy, this places the numbers in the folios in a determined position, which clarifies their meanings, and pins down interpretations. Furthermore, the mathematics shows that the argument in Two New Sciences discussed in this paper fits in a same package with the proportions, graphs, medieval kinematics, and the tables. Incidentally, this modern mathematical language brought a new result: various interpretations of the argu- ment given by different authors are consistent with the way it is derived using the modern mathematical language; the result is presented in a footnote (footnote c), because these interpretations involve the concept of limit, a concept that is quite fuzzy in many passages in Galileo’s book.

Appendix A: The Proof of the Parabola v2 ¼ 2gD (Folio 152r)

63 In this appendix, the proof in folio 152r is reproduced, and recast in the notation in figure 5 and table 2.

First Paragraph Let ba be to ad as da to ac and let be be the gradus velocitatis in b.

 ba da ¼ and be ¼ v ad ac b In the paragraph, ad is defined as the mean proportional between the distances Dac ¼ ac and Dab ¼ ab: Vol. 21 (2019) Middle-Aged Galileo 215

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ad  Dmp ¼ D1D2  ba  ca

That be ¼ vb follows from the construction of the motions in table 1 or table 2, and the double distance rule.

Second Paragraph And let be be to cf as ba to ad, then cf will be the gradus velocitatis in c:

 be ba ¼ then cf ¼ v cf ad c With the notation in table 2, this is:

vb Dab ¼ then cf ¼ vc ð11Þ cf Dmp The result follows from the construction in table 2, and the double distance rule.

Third Paragraph Since therefore as ca is to ad, cf is to be, also (ac will be to (ad as (cf to (be:

 ca cf ac2 cf 2 ¼ then ¼ ad be ad2 be2 With the notation in table 2, this is:

2 2 Dca vc Dca vc ¼ then 2 ¼ 2 ð12Þ Dmp vb Dmp vb

The result follows, making vc ¼ cf (equation 11).

Fourth Paragraph, Part 1 But as (ca to (ad, ca is to ab;

 ca2 ca ¼ : ad2 ab The result follows from the definition of mean proportional: 216 P. M. C. Dias et al. Phys. Perspect.

2 Dca Dac 2 ¼ ð13Þ Dmp Dab

Fourth Paragraph, Part 2 But therefore as ca to ab, (cf is to (be. Therefore, the points e, f are on a parabola

 ca cf 2 ¼ ab be2 From equation 12 and equation 13:

2 Dca vc ¼ 2 ð14Þ Dab vb

Appendix B: The Proof of the Straight Line v = gt (Folio 91v)

64 In this appendix, the proof in folio 91v is reproduced, and recast in the notation in figure 3 and table 1. The folio starts with a proof for the composition of velocities. Then Galileo proposes a problem: In motion from rest the moment of velocity and thetime of this motion are intensified in the same ratio  v t to prove that : c ¼ ac vb tac For let there be a motion through ab from rest in a, and let an arbitrary point c be assumed; and let it be posited that ac is the time of fall through ac, and the moment of the acquired speed in c is also as ac,

½Štac ¼ ac vc / ac

[Explanation: in uniform motion, for fixed vc,itisd / t. Then d1  cd ¼ 2Dac (vc constant on cd), the time is tac / d1 ¼ 2ac. This is the ‘‘conflation’’ observed by Palmerino.] and assume again any point b: I say that the time of fall through ab to the time through ac will be as the moment of velocity in b to the moment in c  t v ab ¼ b tac vc Let as be the mean [proportional] between ba and ac; Vol. 21 (2019) Middle-Aged Galileo 217

hipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as ¼ ba  ac and since the time of fall through ac was set to be ac, as will be the time through ab: it thus has to be shown that the moment of speed in c to the moment of speed in b is as ac to as: [Explanation:qffiffiffiffi because it is tac ¼ d1, it has to be proved that pffiffiffiffi tab d2 ac ¼ ac ¼ d1 ¼ vb :] as ab d2 vc

The Proof First Paragraph Assume the horizontal [line] cd to be double ca, but be to be double ba:it follows from what has been shown, that the [body] falling through ac, deflected into the horizontal cd, will traverse cd in uniform motion in an equal time as it also traversed ac in naturally accelerated motion; and, similarly, it follows that be is traversed in the same time as ab: but the time of ab itself is as: therefore, the horizontal [line] be is traversed in the time as ½Šcd ¼ 2ca; be ¼ 2ba

Second Paragraph Let eb be to bl as the time sa is to the time ac; and since the motion through be is uniform, the space bl will be traversed in the time ac according to the moment of speed in b: but according to the moment of speed in c, in the same time ac the space cd will be traversed; but the moments of speed are to one another as the spaces, which according to these moments are traversed in the same time: therefore the moment of speed in c is to the moment of speed in b as dc to bl. The text states relations among the uniform motions that follow from table 7: cd v bl t ¼ c and ¼ ac : ð15Þ bl vb be tab

Table 7. Double distance rule in the accelerated motion in figure 3

Distance in the uniform motion Speed in the Duration of the uniform motion uniform motion d1  cd  2ca  2Dca vc tac d  be ¼ 2ba  2D v t ¼ as 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiba pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ab dmp  bl ¼ ðÞÂcd ðÞbe ¼ 2 ðÞÂca ðÞba ¼ 2as  2Dmp vb tac 218 P. M. C. Dias et al. Phys. Perspect.

When Galileo refers to ac and as also as time, he is using what Palmerino calls ‘‘conflation’’ of coordinates. This shows that Galileo requires a separated vertical line to represent time. In the Two New Sciences, he uses the coordinates speed versus time, and an independent line for distance.

Third Paragraph, Part 1 But as dc to be, so are their halves, i.e., ca to ab :  dc 2ca ¼ : be 2ab

The uniform motions are associated with uniformly accelerated motions, using the double distance rule: D ac dc ac ¼ ¼ ð16Þ Dab ab be

Third Paragraph, Part 2 But as eb to bl,soba to as:

 eb ba t ba ¼ that is ab ¼ : bl as tac as This result uses the double distance rule: d2 ¼ 2Dab ; from equation 16 and dmp 2Dmp table 7 rffiffiffiffiffiffiffiffi D D t ba  ba ¼ ab : ð17Þ Dmp Dca tac

Third Paragraph, Part 3 Therefore, by the same [ex aequali], as dc to bl,soca to as: that is, as the moment of speed in c to the moment of speed in b,soca to as,  dc ca v ca ¼ that is c ¼ : bl as vb as

This is analogous to paragraph 3, part 2: from the double distance rule: d1 ¼ 2D1 ; dmp 2Dmp from equation 16 and table 7 Vol. 21 (2019) Middle-Aged Galileo 219

rffiffiffiffiffiffiffiffi D D v ca  ba ¼ c : ð18Þ Dmp Dca vb

Third Paragraph, Part 4 That is, the time through ca to the time through ab. Quod erat demonstrandum

 v t c ¼ c : vb tb This follows from equation 17 and equation 18.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References 1 Galileo Galilei, Discourse Concerning Two New Sciences, trans. Stillman Drake (Toronto: Wall & Thomson, 1989). 2 Winifred L. Wisan, ‘‘The New Science of Motion: A Study of Galileo’s De motu locali,’’ Archive for History of Exact Sciences 13, no. 2–3 (1974), 103–306. 3 Stillman Drake, ‘‘Galileo’s Discovery of the Law of Free Fall,’’ Scientific American 228, no. 5 (1973), 84–92. 4 Drake, Galileo at Work (Chicago: University of Chicago Press, 1978); Wisan, ‘‘New Science of Motion’’ (ref. 2), 227. 5 Matthias Schemmel, ‘‘Medieval Representations of Change and Their Early Modern-Applica- tion,’’ Foundations of Science 19, no. 1 (2014), 11–34; Peter Damerow, Gideon Freudenthal, Peter McLaughlin, and Ju¨ rgen Renn, Exploring the Limits of Preclassical Mechanics (Dordrecht: Springer, 2004), ch. 3; Carla Rita Palmerino, ‘‘The Geometrization of Motion: Galileo’s Triangle of Speed and its Various Transformations,’’ Early Science and Medicine 15, no. 4–5 (2010), 410–47. 6 Marshal Clagett, The Science of Mechanics in the Middle Ages (Madison: University of Wis- consin Press, 1959), 414. 7 Schemmel, ‘‘Medieval Representations’’ (ref. 5), 12. 8 Palmerino, ‘‘The Geometrization of Motion’’ (ref. 5), 414. 9 Drake, ‘‘Galileo’s Discovery’’ (ref. 3); Damerow et al., Exploring the Limits (ref. 5). 10. Damerow et al., Exploring the Limits (ref. 5); Wisan, ‘‘The New Science of Motion’’ (ref. 2). 11 Drake, Galileo at Work (ref. 4), 86–90. 12 R. H. Naylor, ‘‘Galileo’s Theory of Motion: Processes of Conceptual Change in the Period 1604–10,’’ Annals of Science 34, no. 4 (1977), 365–402. 13 ‘‘Galileo Galilei,’’ in A. Favaro, ed., Le Opere di Galileo Galilei, vol. 10, 115–16 (Florence: Barbe`ra, 1900). Also in Damerow et al., Exploring the Limits (ref. 5), 354–55. 220 P. M. C. Dias et al. Phys. Perspect.

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38 Galileo, cited in Damerow et al., Exploring the Limits (ref. 5), 361. 39 Wisan, ‘‘The New Science of Motion’’ (ref. 2), 206. 40 Drake, ‘‘Galileo’s Discovery’’ (ref. 3), 85. 41 Cohen, The Birth of a New Physics (ref. 30); Naylor, ‘‘Galileo’s Theory of Motion’’ (ref. 12). Palmerino, ‘‘The Geometrization of Motion’’ (ref. 5) 42 Wisan, ‘‘Galileo’s Scientific Method: a Reexamination,’’ in New Perspectives on Galileo, ed. R. E. Butts and J. C. Pitt (Dordrecht: D. Reidel, 1978), 1–58. 43 Wisan, ‘‘The New Science of Motion’’ (ref. 2), 211–12; Damerow et al., Exploring the Limits (ref. 7), 187; Drake, ‘‘Galileo’s Discovery’’ (ref. 3). 44 Damerow et al., Exploring the Limits (ref. 5), 182–88; Wisan, ‘‘The New Science of Motion’’ (ref. 2), 185–87. 45 Palmerino, ‘‘The Geometrization of Motion’’ (ref. 5). 46 Damerow et al., Exploring the Limits (ref. 5), 180. 47 Wisan, ‘‘The New Science of Motion’’ (ref. 2), 212; Damerow et al., Exploring the Limits (ref. 5), 192. 48 The texts are quoted from Damerow et al., Exploring the Limits (ref. 5), 365. 49 Damerow et al., Exploring the Limits (ref. 5), 184–85. 50 Damerow et al., Exploring the Limits (ref. 5), 185–87. 51 Damerow et al., Exploring the Limits (ref. 5), 187. 52 Drake, ‘‘Galileo’s Discovery’’ (ref. 3), 86. 53 Wisan, ‘‘The New Science of Motion’’ (ref. 2). 54 Drake, ‘‘Galileo’s Discovery’’ (ref. 3). 55 Galileo, Two New Sciences (ref. 1), 203, 160. 56 Norton and Roberts, ‘‘Galileo’s Refutation’’ (ref. 21). 57 Mach, The Science of Mechanics (ref. 21). 58 Giovanni Andres, cited in Drake, ‘‘Uniform Acceleration’’ (ref. 20), 41–42. 59 Drake, ‘‘Uniform Acceleration’’ (ref. 20), 27–28. 60 Swerdlow, ‘‘Galileo’s Mechanics’’ (ref. 32), 36; Galileo, Dialogue Concerning the Two Chief World Systems – Ptolemaic & Copernican, trans. Stillman Drake (Berkeley: University of Cali- fornia Press, 1967). 61 Galileo, Dialogue (ref. 60), 228–29. 62 Swerdlow, ‘‘Galileo’s Mechanics’’ (ref. 23). 63 Damerow et al., Exploring the Limits (ref. 5), 365. 64 Damerow et al., Exploring the Limits (ref. 5), 366.

Instituto de Fı´sica, Universidade Federal do Rio de Janeiro Rio de Janeiro, Brazil

Instituto Federal de Educac¸a˜o, Cieˆncia e Tecnologia do Rio de Janeiro, Estrada Washington Luı´s Sapeˆ Nitero´ i, RJ 24315-375, Brazil e-mail: [email protected]