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Galileo asserts that if a body accelerates uniformly, its velocity increases as the even integers ($1,2,3,4$ etc.) and therefore, the distances passed by the body in equal times increase as the odd integers $(1,3,5,7)$ etc.

This makes no sense to me. If we suppose that velocity is a continuous function of time, with $v(t) = t$, and that $v(0) = 0$ and $v(1) = 1$, for example, it follows that the distance elapsed in the first period of time is $1/2$. Likewise in the second period of time the distance elapsed is $3/2$ etc.

Where does this whole odd-integer business come from?

Edit: you can find the exact statement in Corollary 1 here: https://oll.libertyfund.org/titles/galilei-dialogues-concerning-two-new-sciences

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  • $\begingroup$ It would possibly help if you provided a link and page reference to the source. $\endgroup$ Jul 12, 2020 at 4:43

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“As” means “proportional to”. The sequences $\frac12, \frac32, \frac52, \dots$ and $1, 3, 5, \dots$ are proportional.

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  • $\begingroup$ I just read the actual section in Two New Sciences, it all makes sense now. Thanks m8 $\endgroup$ Jul 12, 2020 at 4:51

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