Galileo (and yours truly) on Accelerated Motion

Here is Galileo (in the Two New Sciences) writing on some properties of naturally accelerated motion which are worth knowing:

“That the distances traversed during equal intervals of time by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity”

Amid deepening consultation with Galileo, I am inclined to think that non-calculus-based physics would benefit from a framing in terms of integer sequences and series rather than the changing of d’s to Δ’s.