The Geometrization of Motion: Galileo’s Triangle of Speed and its Various Transformations

Abstract

is article analyzes Galileo's mathematization of motion, focusing in particular on his use of geometrical diagrams. It argues that Galileo regarded his diagrams of acceleration not just as a complement to his mathematical demonstrations, but as a powerful heuristic tool. Galileo probably abandoned the wrong assumption of the proportionality between the degree of velocity and the space traversed in accelerated motion when he realized that it was impossible, on the basis of that hypothesis, to build a diagram of the law of fall. e article also shows how Galileo's discussion of the paradoxes of infinity in the First Day of the Two New Sciences is meant to provide a visual solution to problems linked to the theory of acceleration presented in Day ree of the work. Finally, it explores the reasons why Cavalieri and Gassendi, although endorsing Galileo's law of free fall, replaced Galileo's diagrams of acceleration with alternative ones.

Aggregating Speeds and Scaling Motions: A Response to Norton and Roberts

“…In the lines just quoted Galileo explicitly states that a finite interval of time is composed in actu of an infinite number of instants without duration and that in each of these instants a new degree of speed is acquired. As I have tried to show elsewhere (Palmerino, 2001(Palmerino, , 2010, the analysis of the paradox of the wheel and of other paradoxes of infinity offered by Galileo in the First Day of the Two New Sciences (to which Norton and Roberts refer in footnote 16) is precisely to provide an ontological foundation for the theory of acceleration put forward in the Third Day, by demonstrating that all physical magnitudes are actually composed of an infinite number of non-extended atoms (atomi non quanti ). In the First Day of the work, Salviati also provides an answer to the question of how if there are two aggregates of infinite points one can be bigger:…”

“…I am personally convinced that Galileo abandoned the hypothesis of the speed‐distance proportionality, when he understood that it was impossible, on the basis of that hypothesis, to build a diagram representing all the essential parameters of acceleration (Palmerino, 2010).…”

“…Secondly, from this inverse proportionality, Galileo erroneously infers that times are to each other as the square roots of the spaces (Giusti, 1990, p. xxxv;Blay and Festa, 1998, p. 69;Damerow et al ., 2004, p. 173). I am personally convinced that Galileo abandoned the hypothesis of the speeddistance proportionality, when he understood that it was impossible, on the basis of that hypothesis, to build a diagram representing all the essential parameters of acceleration (Palmerino, 2010).…”

Parti non quante

“…PALMERINO (1999) afirma que "a composição do contínuo como resultado de partes indivisíveis não somente contradiz inteiramente o Livro X dos Elementos de Euclides, como também não possibilitava uma explicação de proporções incomensuráveis entre as magnitudes"(PALMERINO, 1999, p. 314).…”