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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.

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