Galileo used balls rolling down ramps to study the relationship between time and distance traveled. However, without any knowledge of physics, it doesn't seem immediately obvious that the time-distance relationship of an object rolling down a ramp is the same as if it were free falling. Why did Galileo assume this? Did he have sufficient physics knowledge to conclude this? Was this just a hypothesis which he confirmed by actual free fall experiments?
From: Discourses and Mathematical Demonstrations Relating to Two New Sciences (Italian: Discorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze), published in 1638.
See: Engl.transaltion by Henry Crew and Alfonso de Salvio (1914):
A motion is said to be equally or uniformly accelerated when, starting from rest, its momentum (celeritatis momenta) receives equal increments in equal times.
This definition established, the Author makes a single assumption, namely,
The speeds acquired by one and the same body moving down planes of different nclinations are equal when the heights of these planes are equal.
By the height of an inclined plane we mean the perpendicular let fall from the upper end of the plane upon the horizontal line drawn through the lower end of the same plane. [...]
THEOREM II, PROPOSITION II
The spaces described by a body falling from rest with a uniformly accelerated motion are to each other as the squares of the time-intervals employed in traversing these distances. [...] considering the total space traversed, that covered in double time will be quadruple that covered during unit time; in triple time, the space is nine times as great as in unit time.
And in general the spaces traversed are in the duplicate ratio of the times, i.e., in the ratio of the squares of the times. [...]
I have attempted in the following manner to assure myself that the acceleration actually experienced by falling bodies is that above described.
A piece of wooden moulding or scantling, about 12 cubits long, half a cubit wide, and three finger-breadths thick, was taken; on its edge was cut a channel a little more than one finger in breadth; having made this groove very straight, smooth, and polished, and having lined it with parchment, also as smooth and polished as possible, we rolled along it a hard, smooth, and very round bronze ball. Having placed this board in a sloping position, by lifting one end some one or two cubits above the other, we rolled the ball, as I was just saying, along the channel, noting, in a manner presently to be described, the time required to make the descent. We repeated this experiment more than once in order to measure the time with an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse-beat. Having performed this operation and having assured ourselves of its reliability, we now rolled the ball only one-quarter the length of the channel; and having measured the time of its descent, we found it precisely one-half of the former. Next we tried other distances, comparing the time for the whole length with that for the half, or with that for two-thirds, or three-fourths, or indeed for any fraction; in such experiments, repeated a full hundred times, we always found that the spaces traversed were to each other as the squares of the times, and this was true for all inclinations of the plane, i. e., of the channel, along which we rolled the ball. We also observed that the times of descent, for various inclinations of the plane, bore to one another precisely that ratio which, as we shall see later, the Author had predicted and demonstrated for them.
For the measurement of time, we employed a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.
The "level of accuracy" of Galileo's measurements is still highly debated, but a lot of hand-written sheets full of drawings and calculations are still extant.
Thus, it is quite clear that Galileo makes assumptions and tested them experimentally (as usual for any good scientist).
Galileo knew that, if a stream of water is ejected from a pipe held horizontally, the stream would follow the shape of a parabola. Mathematically, that means that the motion of the water is linear (uniform) horizontally, and quadratic vertically. And that leads to the realization that the motion horizontally and the motion vertically can be treated independently. He could thus use the inclined plane to stretch the vertical motion out so that the times were long enough to measure easily (probably using his pulse as a timer).
How did he know that the stream of water took the shape of a parabola? Well, he was a man, and he often walked in the woods.
According to Carl Boyer, in his text A History of Mathematics, it is Simon Stevin who is credited with the Law of the Inclined Plane, as justified by his familiar "wreath of spheres" diagram. The law itself had been stated previously by Jordanus Nemorarius.
It is interesting to note that, again according to Boyer, Stevin observed and published Galileo's result some 50 years before the publication of Two New Sciences. Boyer even goes so far to claim that it is doubtful that Galileo ever performed the experiment. Quoting Boyer :
[Stevin] and a friend dropped two spheres of lead, one ten times the weight of the other, from a height of 30 feet onto a board and found the sounds of their striking the board to be almost simultaneous. But Stevin's published report (in Flemish in 1586) of the experiment has received far less notice than the similar and later experiment attributed, on very doubtful evidence, to Galileo.