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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?

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    $\begingroup$ A tantalizing quote, uncited quote concerning this on Galileo's Wikipedia page: "Galileo expressed the time-squared law using geometrical constructions and mathematically precise words, adhering to the standards of the day. (It remained for others to re-express the law in algebraic terms)." $\endgroup$ – Michael Seifert Nov 3 '20 at 21:10
  • $\begingroup$ It may be that geometrical thinking was the standard back then so it was more usual to express things in terms of ratios and proportions? $\endgroup$ – Jon Nov 3 '20 at 22:15
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    $\begingroup$ I’m voting to close this question because this is more suited to the History of Science and Mathematics $\endgroup$ – StephenG Nov 4 '20 at 2:36
  • $\begingroup$ Thanks, @StephenG. I think you may be right. I opened the question in that site as well: hsm.stackexchange.com/questions/12399/… $\endgroup$ – Jon Nov 4 '20 at 11:10
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In Galileo’s time there was no widely agreed system of units for measuring length, weight or small intervals of time. Different regions (and sometimes even different towns) used different units of measure and sometimes referred to them by the same name to add to the confusion. Galileo realised that his readers were more likely to understand and be able to reproduce his experiments if he described them using ratios rather than absolute measurements in local units that they might not be familiar with.

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