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