This paper introduces a little-known episode in the history of physics, in which a mathematical proof by Pierre Fermat vindicated Galileo’s characterization of freefall. The first part of the paper reviews the historical context leading up to Fermat’s proof. The second part illustrates how a physical and a mathematical insight enabled Fermat’s result, and that a simple modification would satisfy any of Fermat’s critics. The result is an illustration of how a purely theoretical argument can settle an apparently empirical debate.

Galileo's refutation of the speed-distance law of fall in his Two New Sciences is routinely dismissed as a moment of confused argumentation. We urge that Galileo's argument correctly identified why the speed-distance law is untenable, failing only in its very last step. Using an ingenious combination of scaling and self-similarity arguments, Galileo found correctly that bodies, falling from rest according to this law, fall all distances in equal times. What he failed to recognize in the last step is that this (...) time is infinite, the result of an exponential dependence of distance on time. Instead, Galileo conflated it with the other motion that satisfies this ‘equal time’ property, instantaneous motion. (shrink)

ABSTRACT. In his Letters on the motion impressed by a moving mover, Gassendi offers a theory of the motion of composite bodies that closely follows Galileo’s. Elsewhere, he describes the motion of individual atoms in very different terms: individual atoms are always in motion, even when the body that contains them is at rest; atomic motion is discontinuous although the motion of composite bodies is at least apparently continuous; and atomic motion is grounded in an intrinsic vis motrix, motive power, (...) while composite bodies simply persist in their state of motion or rest in the absence of outside intereference. Gassendi does not make much effort to explain how his accounts of atomic and composite motion fit together, and it’s difficult to see how they could possible be integrated. My goal is to explain, given this difficulty, why he accepted both the Galilean theory of the motion of composite bodies and the Epicurean theory of atomic motion. (shrink)

Galileo's refutation of the speed-distance law of fall in his Two New Sciences is routinely dismissed as a moment of confused argumentation. We urge that Galileo's argument correctly identified why the speed-distance law is untenable, failing only in its very last step. Using an ingenious combination of scaling and self-similarity arguments, Galileo found correctly that bodies, falling from rest according to this law, fall all distances in equal times. What he failed to recognize in the last step is that this (...) time is infinite, the result of an exponential dependence of distance on time. Instead, Galileo conflated it with the other motion that satisfies this ‘equal time’ property, instantaneous motion. (shrink)