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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?

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  • $\begingroup$ Because that was the mathematics available at his time ! In 17th Century symbolic algebra was still in his infancy. $\endgroup$ Nov 4 '20 at 12:54
  • $\begingroup$ Basically, the same question has been already asked here $\endgroup$ Nov 4 '20 at 12:56
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Galileo followed a venerable tradition of distinguishing numbers, magnitudes of different kinds (lengths, times, areas, etc.) and ratios. This is somewhat analogous to the strictures of modern dimensional analysis used in physics, but even stricter, and ancient Greeks did not have dimensional constants to bridge the gaps. They did not even have enough dimensionless numbers, only positive integers were admitted, not even rationals. Geometry was far ahead of arithmetic and algebra in the level of sophistication. And so lengths and areas were not numbers assigned to geometric figures, as we think today, they were literally the figures themselves.

Ratios were defined for both numbers and magnitudes, and were the only "legitimate" way to connect numbers to magnitudes, or magnitudes of different kinds to each other, as their ratios could be equated (being dimensionless), see What did the ratio of two magnitudes mean to ancient Greek mathematicians? And so Euclid does not say that the area of a circle is a constant times diameter squared, but says "circles are to one another as the squares on their diameters". Archimedes does not say that a weight balancing a lever is a constant divided by the length of the leg it is on, but says that the balanced weights are in the opposite ratio to that of the legs, etc.

The distinction was eroding since the late antiquity, as more and more entities were admitted as numbers, but it was still influential in Galileo's time. And on accelerated motion he had a direct predecessor, Oresme (1320-1382), see Nicodemi Galileo and Oresme. Oresme called it "uniformly difform" motion, and developed a theory of it, which included graphing velocities (he used bar graphs, see When do we see for the first time the use of the Cartesian coordinates?). In The Geometry of Qualities and Motions Oresme expresses himself in the same fashion:

"The universal rule is this, that the measure or ratio of any two linear or surface qualities or velocities is as that of the figures by which they are comparatively and mutually imagined... Therefore, in order to have measures and ratios of qualities and velocities one must have recourse to geometry."

In contrast, Galileo in Two New Sciences (1638) is already one step away from geometry. But not from the ratio language:

"If a moveable descends from rest in uniformly accelerated motion,the spaces run through in any times whatever are to each other as the duplicated ratio of their times; that is, are as the squares of those times."

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    $\begingroup$ Thank you very much for your answer. It's very interesting how mathematical thinking changes across generations. $\endgroup$
    – Jon
    Nov 4 '20 at 14:57
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    $\begingroup$ Great answer. I just want to add one thing. At that time people understood that you can multiply length by length and get area. Or you can multiply length three times and get volume. But what did you get if you multiply time by time?! There was no such thing as squared hours! So it is hard to attach meaning to $t^2$. On the other hand, you can take ratio of two times $t_1/t_2$. It is easy to understand, for example, what does it mean that one time interval is twice as much as another. You get dimensionless quantity, and you can square it without problems. $\endgroup$ Nov 5 '20 at 3:48

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