# Is Spivak right in what he says about Galileo?

On chapter 9 of M. Spivak's book on calculus 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.

• Does Spivak make it sound as though Galileo never realized his mistake?
– user466
Nov 24, 2017 at 1:10

## 2 Answers

Yes, indeed when trying to obtain the law of falling bodies, Galileo's first conjecture was that the speed is proportional to the distance traveled. After some contemplation, Galileo understood that this cannot be the case and eventually came with the correct law.

Good source on Galileo: S. Drake, Galileo at work. (There are many editions).

From the modern point of view, this is a good story to tell to students who begin to study differential equations: equation $$y'=ky$$ and initial condition $$y(0)=0$$ imply $$y=0$$, so the motion never begins. In Galileo's time this was non-trivial.

Yes, Galileo made that error (and so did Descartes). Only later did he realise that the speed is proportional to the time ellapsed, not to the distance already covered. 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).