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On chapter 9 of the said book there is an exercise in which Spivak asks the reader to prove that Galileo "got his facts wrong". More specifically, Spivak asks one to to show if a body falls a distance $d(t)$ in $t$ second and $d^{\prime}$ is proportional to $d$ then $d$ cannot be a function of the form $d(t) = ct^{2}$.

Settling it is kind of a no-brainer: yet, did Galileo really claim what Spivak is attributing to him therein? Do you know if this "mistake" by Galileo had been noticed before? If I understand correctly, even Newton took for granted the claim by Galileo according to which "the descent of bodies varied as the square of the time" (cf. p. 21 of vol. I of the University of California Press edition of the Principia)? What's going on here?

Let me thank you for your comments, suggestions, links, answers, etc.

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Yes, Galileo made that error (and so did Descartes). I suggest that you read The new science of motion: A study of Galileo's De motu locali, by Winifred L. Wisan (Archive for History of Exact Sciences, June 1974, 13, Issue 2–3, pp 103–306).

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  • $\begingroup$ Can you state explicitly what the error was? Did Galileo believe that the acceleration of a falling object is proportional to its velocity? $\endgroup$
    – littleO
    Nov 24, 2017 at 0:10
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    $\begingroup$ @littleO He and Descartes believed that the speed of a body moving in free fall was proportional to the distance already covered. Later, Galileo understood that it is proportional to the time ellapsed. $\endgroup$ Nov 24, 2017 at 0:17

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