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.
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).