# Galileo's treatment of uniformly accelerated motion

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

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

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