After many years of effort, Galileo finally came to understand a certain confusing kind of motion -- what we now call uniformly accelerated motion. This is the motion of something in free fall, for example.
The breakthrough experiments seem to have been the ones we will do here. The methods we will use are those reconstructed from Stillman Drake's study of Galileo's laboratory notes from his Padua years. When Galileo finally described these results, many years later, he didn't give the experimental details.
Free fall happens too fast for us to measure easily what happens. So the first clever thing Galileo does is to replace falling by rolling -- on a beam which is nearly horizontal. This slows everything down and makes it much easier to see. We will roll a ball, starting from rest, and see how it gains speed, or perhaps "moment" (as Galileo sometimes said). This idea of moment, which was so successful for him in his theory of floating, was actually a barrier to his understanding accelerated motion. Remember that "moment" for Galileo meant mass in motion -- a thing which moved faster had more moment. So far so good. But his notion of speed was tied to measures of distance -- something which moves farther (in a given time) moves faster. Thus he imagined that something rolling or falling picks up moment proportional to distance. In fact, as he eventually realized, it picks up moment proportional to TIME.
The hard part in using TIME as a variable in the theory is that time is hard to measure: we have already seen what kinds of clocks Galileo had to work with. It is much clumsier to use such clocks than to use, say, a ruler to measure distance. Yet a mathematical theory of motion seems to require clocks. The first clock we will use is our internal sense of rhythm. Can you keep a steady beat? Choose a song and sing it to yourself in strict time. You will thereby have an internal clock. (You may also use a metronome, if you can't do it internally.) Now that you have a clock, try arranging little "frets" on the incline, such that the rolling ball hits the frets at regular intervals in time. You will have to move them around to achieve this. Galileo believed that he had discovered a simple arithmetic progression in the positions of these frets: 1-3-5-7-..., meaning the distance between adjacent frets just increases like the odd numbers! If you measure total distance to the starting point instead, it is 1-4-9-16-..., just the squares!
Our second clock will be a water clock. Taking a cue from the measurement above, see if it is true that the total distance rolled is proportional to the SQUARE of the time. That is, if the time to roll one unit of distance is so much, the time to roll 4 units of distance will be twice that much, the time to roll 9 units of distance will be three times that much, etc. Here time will be represented by quantity of water.