One of Aristotle's claims was the heavy bodies would fall faster than lighter ones. After all, it is clear that a heavier body wants to return to the earth more than a light one does. Galileo asked what would happen if one dropped two identical bodies from a height. They would surely both arrive at the earth at the same time. Now bring them closer together. According to Aristotle, if the two were one body, then they would get to earth faster than either one would on its own. But exactly where did that transition happen. If one held them close together but not touching, surely they would arrive at the same time as one. If they were held so that they just touched, how long would the fall take. If one tied them together with a thin string, would they now suddenly fall faster? If one tied them together tighty? If one welded them together? When did the transition from two lighter to one heavier body, which would fall differently, take place? He suspected it would never do so.
The second idea was to try to look carefully at exactly how bodies fell. Since falling straight down was far to quick for him to be able to do any measurements, he hit on the idea of letting balls roll down an inclined plane. He had absorbed the lessons of the Paris school that air, rather than being a necessity for motion, acted as an impediment. Thus, one must try hard to figure out what would happen in the void-- ie if there was no other matter around to act on the body, no friction to slow it down. He therefore constructed his planes to be as hard and as smooth as possible, so that no interactions with either the wood of the plane would slow down the rolling balls. Since they were rolling more slowly down the plane, the effect of the air should also be less.
Inorder to figure out how they rolled down the plane, he wanted a way of timing them, and he hit on the idea of singing to them. In singing most humans have an extremely good sense of timing, especially if at least partially trained as singers. (Modern experiments have found that people retain the beat of a piece learned in the past for at least 20 years. If asked to sing again they sing at the same beat as they learned it to at least 10%). He therefore would tie very thin threads around the plane, so that when the ball rolled over it, it would produce a very soft bump. He would tie the threads so that each successive thread's bump would be heard to occur in time to the music he was singing. Measureing the distance between the threads he found they were at odd number ratios. If one said that the first thread was a distance of unity from the top where the ball had been released, the second was three times as far from the first, the third 5 times as far from the second as the top was from the first, the fourth was 7 times as far from the third as the first was from the top.
If one looked at the total distance, one found that the second thread, which occured on the second beat, was a total of 4 times the distance from the top to the first, the distance from the top to the third was 1+3+5=9 times the top to the first, the distance from the top to the fourth was 1+3+5+7=16 times the the distance from the top to the first. Expessing the total distance in terms of the beats (the total time) one found that the distance went as the square of the number of beats. The distance moved went as the square of the time. This was exactly the Merton rule for motion where the velocity increased as the time advanced (the acceleration was uniform). The balls were rolling down the planes with uniform acceleration. If he reversed the incline, rolling the ball down one incline and up another, one again found that even going up the distance again obeyed the Merton rule, only backwards. Ie, again the distance to the stopping point went as the square of the time before the stopping time. Whether up or down, the distance obeyed the Merton rule.
This was an astonishing achievement. Whereas before all one could talk about was some vague propesity of objects to fall to earth (and on throwing up one had in some sense for a while cancelled out that propensity) now one had a rule. The change in distance obeyed the motion under uniform acceleration rule.
Ptolomey has managed to parlay (and stretch) the celestial motion rule, uniform motion around circles into a detailed mathematical model of how the heavens moved. Here Galileo had managed to create a mathematical rule of how the earthly, sublunar, sphere into a rule based motion. Noone had had any inkling that this was possible.
At the same time, he also made another tremendous advance. Aristotle's picture of how horizontal motion occured had attracted much criticism throught the ages. In his picture, the air, or some other stuff that things travelled through, was crucial in maintaining that motion. As had been hinted at by earlier workers, for Galileo the stuff that things moved through was instead only an impediment, a friction which changesd the motion. The natural motion horizontally instead was the same as the natural motion of the heavens, namely uniform motion on a circle. The circle in this case was the surface of the earth. If one made the second inclined plane horizontal, instead of slowing down or speeding up, the ball would keep rolling, and, Galileo believed, keep rolling forever along this plane as constant height above the earth.
But in addition he also stated that if one had general motion, motion of way a stone thrown from the hand, or a canon ball blown from a canon, that that motion would be composed of two parts-- its up and down motion, and its sideways motion, and that one could analyse each independently of the other. In the up and down direction, the motion was the constant acceleration motion, obeying the Merton rule, with respect to the distance above the earth, independent of where the object was in a horizontal direction. At the same time, the horizontal motion was independent of the vertical motion. The object would travel with uniform speed horizontally no matter what the vertical motion was doing.
Again, this insight was new. Motion could be decomposed into parts and one could view them independently from each other. Galileo applied this insight into for example the motion of a canon ball "in the void". Ie, under the condition that the vertical motion obeyed the generalisation of the Merton rule with constant acceration, and that the horizontal motion was uniform motion with a constant velocity. The trajectory (at least for distances small compared with the distance over this the surface of the earth curved appreciably) the motion of the canon ball was a parabola, another of the conic sections which had been studied by the Greeks (it is like an ellipse where one of the focii is allowed to become very very large). In order that the canon ball fly the largest distance, the best thing to do would be to aim the canon exactly 45 degrees above the horizon. Aim lower, and the canon ball would go up and down to quickly and the horizontal distance travelled would not be great. Aim it higher and the ball would go higher, but the fraction of the speed that went into horizontal motion would be less, and again the ball would fall before the ball had travelled far. If one divided up the speed to that the horizontal and the vertical speeds were equal, then the distance travelled would be greatest.
This insight was a bit useless for actual military use, since these calculations were all done under the assumption that the travel was in a void. But real canons are used on earth in air, and the air would bleed away speed and alter the trajectory of the canon ball.
As such he had to address the arguments which Aristotle had arrayed against the possibility of the eath moving. Just as Galileo's telescopic observations had destoyed the most naive application of Ptolemy's model (such as having the orbits arranged so as to never have the "crysteline spheres" carrying the planets cross each other) so he now wanted to find arguments for the eath's rotation and its orbit around the sun. Note that there is nothing in Ptolemy's model per se which contradicted his observations. Since the Ptolemeic model could only deliver angles in the sky and said nothing about the distances of the planets and stars, It was easy to rescale the orbits so as to account for Galileo's observations. This process leads in fact to Tycho Brahe's model in which all planets orbit the sun, and the sun orbits the earth.
He developed on of the most influential arguments (It was used crucially again by Einstein 300 years later), that of relativity. Since horizontal motion and vertical motion occured independently of each other, and since natural motion horizontally was uniform motion, he said that one should imagine oneself on a ship, but enclosed in the cabin with no view outside. The ship should be imagined to be moving uniformly across the sea. Now the question was, was there anyway that the resident of the cabin could tell whetehr or not the ship was moving. Vertical motion would carry on as before. Uniform horizontal motion would remain uniform horizontal motion (although perhaps not with the same velocity, but after all the horizontal velocity is not anything inherent, but is what it is because of some other past influence.) There would be, he said, no way whatsoever for the passenger to tell that the ship was moving.
Furthermore, if one had a porthole and saw the masts of some other ship go by, one would have no way of knowing if it was your ship that was moving or if the other ship was moving in the opposite direction. One could say one was moving relative to the other ship. Even if one looked at the water, it could be that one was really at rest with earth, but that the tidal flow was such that your ship appeared to be moving with respect to the water. Natural motion was natural motion even if one looked at it from the viewpoint of someone else who was also engaged in natural motion. This is called the principle of Galilean Relativity
This would mean that anyone on the earth would have no way of knowing if the earth rotated, or the rest of the universe revolved around the earth. Thus, Aristote's argument of the arrow disappeared. On the rotating earth when one shot the arrow into the sky, that arrow would initially not only have the vertical velocity from the bow, but also a horizontal velocity because of the earth's rotation. That horizontal velocity would remain the same (air friction would be unimportant in decreasing it, because the air would also be moving with the same horizontal velocity and whoule thus not alter the arrow;s hrozontal velocity). Thus, yes, in the time the arrow was in the air, the earth's surface and all that was on it would be carried to the east by the rotation. But the arrow also, because of its velocity to the east when it left the bow, would travel the same distance. and come down from straight overhead from the viewpoint of someone on the earth.
Nevermind that the tides do not act like this-- the tides in the Mediteranian are horrendously complicated, not just because of the small size of the water basin, the Mediterainian-- but because of resonances (the time of sloshing of water especially in the Adriatic is almost the same as the time between the tides, which, like a child on a swing that is pushed back and forth at exactly the same timing as the natural swaying of the swing, causes the height of the tides to be much higher than they would be in the open water.). This explanation violated Galilean relativity. Nothing on the earth should be able to tell the difference between an earth tavelling around the sun, or one at rest. But here, he was claiming there was a difference. Futhermore they do not have a regular day long period as his rotating earth would have. It would not be till Newton approached the question that a decent theory of tides would be developed.
He realised that this would be new way to tell time, with unprecedented accuracy. The great cathedrals of Europe by this time all had clocks. But those clocks would be lucky and happy if they could tell time with accuracy better than a half hour per day. They would have to be recalibrated essentially daily, by use of a sundial for example. If it was really true that the pendulum had a tick which was indpendent of the swing, then it could be used to discipline a clock to give much more accurate time. There are engravings from shortly thereafter of choir masters holding little penulums to use as "metronomes" (tell timers) to give the beat to the choir boys.
This isochronicity (equal time) of the pendulum was taken up around Europe almost instantly. Huygens in the Neterlands realised the importance, and began to design and build clocks based on this principle of the pendulum. He also realised that Galileo's observations were not quite good enough. Because of the design of these eary clocks, the pendulum swing was not small-- it was up to 45-50 degrees to each side. While the pendulum is isochronous for small swings, once the swing became larger, the time for one swing cycle became slightly longer. Ie, the clock would run slower with larger swings than with smaller ones. Galileo discovered that the timing of a swing of the penulum was proportional to the square root of the length of the pendulum. Ie, if one multiplied the length by 4 the period of the swing would increase by a factor of 2. Thus if one made the pendulum so that the length of the string decreased for large amplitudes,one could compensate for the increased time it would take for a larger swing. Huygens showed that if the string was arranges so that the bob traced out a cycloid (the curve given by a point on the circumference of a circle, if that circle were rolled along a straight line) the timing of motion would truely be isochronous. By having the string be constrained at the top by a diverging stops, one could have the bob trace out such a curve.
By 1690, clock accuracy was down to seconds per day, rather than a bit under an hour per day of the previous pre-Galileo clocks. Clocks were by then regularly fitted out with minute hands as well as the tradiational hour hands (and soon thereafter even second hands.)
Thus not only had Galileo managed to mathematise motion near the earth, he also managed to mathematise time itself.