I opened the same question on Physics Stack Exchange, but it seems more suited for this site.

I've been reading about Galileo's experiment with inclined planes, and he ends up saying something along the lines of "the ratio of distances is equal to the ratio of the times squared"

My initial thought is that, with initial velocity zero. A first distance can be defined as:

$ s_{1} = \frac{1}{2} a t_{1}^2 $

And a second distance as:

$ s_{2} = \frac{1}{2} a t_{2}^2 $

Where I can take the ratio of the distances and end up with:

$ \frac{s_{1}}{s_{2}} = \frac{t_{1}^2}{t_{2}^2} = (\frac{t_{1}}{t_{2}})^2 $

So one doesn't need to know what is the constant of proportionality but can know there's a proportionality if the data matches the previous equation.

However, I'm not sure if this is all there is to it. Is there any other reason for looking at the data of this experiment as ratios? It was customary, back then, to speak of ratios as geometry was the most common way of expressing mathematics?