It was neither as a designer of calculators nor as a Copernican astronomer that Galileo made his pioneering mark during his years at Padua. His most important work was experiments and investigations in the realms of physics. During those eighteen years he changed the foundations of traditional physics - or, as others see it, established an entirely new science. However, remarkably few people outside Padua realised this. For various reasons he did not make his results public until well into old age, and when he did finally become famous all over Europe, it was for quite different things.
Even with the leaning tower as his laboratory, Galileo had in no way solved the problem of free fall during his time at Pisa. Now he took up the challenge once again.
Galileo's writings - both public and private - are full of attacks on the Aristotelians and their unwillingness to indulge in fresh observation and reasoning. This was clearly partly an expression of his own energetic attempts to find new and more accurate means of describing physical things.
But there was also another side to this. The status of mathematicians in the academic world was low. If he could only demonstrate, with the aid of practical experiments analysed using mathematical methods, that Aristotle's interpreters were wrong, applied mathematics and experimental physics would usurp "natural philosophy's" pre-eminent place in academia, both in terms of prestige - and pay. Galileo had personal experience of how brilliant ideas did not necessarily bring with them money and recognition. He could hardly forget that, at times, his father had to let his wool merchant in-laws keep the family.
Galileo's radical renewal sprang, nevertheless, from the Aristotelian mind set, as it was taught at the Jesuits' Collegio Romano: human reason has a basic ability to recognise and understand the objects registered by the senses. The objects are real. They have properties that can be perceived, and then "further processed" according to logical rules. These logical concepts are also real (if not in exactly the same way as the physical objects).
This is the philosophical foundation of Galileo's subsequent increasing cock-suredness: there is a definite route to knowledge. The world exists independently of us, it is "just" a matter of understanding it correctly.
There was one fundamental problem: if we only perceive individual objects, and these are subject to all kinds of changes, how, on that basis, can we say anything definite about the characteristics common to all such objects - for example, falling bodies? The answer to this question is crucial to all experimentation. Early on Galileo realised that the solution was to sift out the individual and random from the particular to arrive at the general.
His experiments at Pisa had taught him that spheres of the same size but different weights fall at roughly the same speed. The difference between an iron ball and a wooden one was so small that he believed it could be explained by the buoyancy of air. But he had also realised that it was practically impossible to measure distances and times in free fall. The balls simply fell too fast. But it was not actually "free fall" that he was interested in, but rather what Aristotle had called "natural motion", i.e. movement that had no visible outward cause, no hand that pushed or horse that pulled.
At Padua Galileo got the epoch-making idea of using inclined planes instead. A ball on an inclined plane still moves "of its own accord", but not so quickly. Furthermore, the observer can alter the incline and see how the speed alters.
The technique of making many, comparable observations of a phenomenon to enable an underlying connection to be drawn, was not new. This had been the working method of astronomy since antiquity. In Galileo's time, astronomical observations were being made with greater accuracy than ever before, principally by the eccentric and despotic Danish aristocrat Tycho Brahe on the island of Hven in 0resund. The difference - which many Aristotelians would have found insurmountable - was that Brahe observed naturally occurring phenomena. Galileo wanted to arrange the "phenomena" himself, purely for the purposes of observing them.
Another important inspiration for experimentation was Galileo's experience of music. The daily routine of tuning a lute so that its sound was pure, was another sort of experimental trial and error: one had to put more or less tension on the strings, until they fell into an underlying and mathematically describable pattern.
Presumably Galileo's first inclined planes were rigged up with what looked like a tribute to his father: a copy of the finger-board of a stringed instrument, with thin, moveable bands or strings running across it. By altering the distance between these bands and listening for the click as the sphere rolled over them, it was possible for him to gain an insight into the relationship between time and the distance the ball rolled.
The first big problem he encountered, was to measure time accurately. Presumably he first tried to do this by singing. It was not as absurd as it may sound. A trained and skilled musician has a "metronomic" feel for the length of the sub-divided beat.
But neither the finger-board bands nor the rhythmic song were completely satisfactory. The bands disturbed the evenness of the ball's rolling movement, and singing was undeniably somewhat impracticable and imprecise. Galileo worked at getting the groove that the balls ran in as smooth and even as humanly possible. Then he also had the idea of measuring time with a sort of water clock - by simply allowing water to flow from one container, through a thin pipe and into another. If the water flow was constant, he could get a measure of how long had elapsed by weighing the water in the receptacle. The excruciating accuracy that characterised Galileo as a practical man and experimenter was visible in the way he also estimated the weight of water that remained on the walls of the container!
Galileo wanted to find out how the speed of the balls varied over distance and time. But he was operating within a Euclidean, geometrically influenced mathematical framework. In other words, he was not much interested in pure numbers. Instead, he attempted to discover the proportions between various stages. He was a stranger to the new algebra and he did not use decimals, only vulgar fractions. Decimals were on the way in, but it is possible Galileo did not consider the system soundly enough based in logic to be used in work that was to provide one hundred percent logically valid conclusions.
One basic difficulty in analysing the relationship between distance and time for rolling balls, he discovered, was that their speed altered the whole time. He had therefore surmounted the false conclusion from the Pisan period, that any falling (or rolling) body will eventually achieve a constant velocity. (In practical free fall experiments in air, increasing air resistance will eventually slow the object down so much that its speed after a certain time will become roughly constant. Otherwise it would be imprudent, for instance, to do a parachute jump.)
The very concept of "velocity" itself was not easy to grasp. Velocity equals "distance divided by time" - but what happened to the distance when he made the time interval smaller and smaller and finally asked about speed at this instant or at that point and there was no distance to divide nor time to divide it by? What did "velocity at a given point" actually mean?
The mathematical solution is found in the development of differential calculus, a development to which Galileo contributed, but which was outside his sphere of interests. In the absence of this aid, Galileo's concepts of velocity had been linked to completed movements over a certain distance, rather than to points. In the first instance he was content to measure how far down his inclined plane his spheres got if he increased the time they were allowed to roll. He had to keep to average measurements (distance divided by time), but he could study how much the average velocity altered over a given period. He was not, therefore, able to calculate the continual change in velocity, which is the real key to understanding this type of motion.
As he had hoped, his measurements revealed a rule. If the average velocity during the first unit of time was 1, it rose in the second unit to 3, in the third to 5 and so on. Using the arbitrary units "second" and "foot", the arrangement was as follows:
After 1 second 1 foot covered After 2 seconds 3 more feet covered After 3 seconds 5 more feet covered average velocity first sec. average velocity second sec. average velocity third sec.
A New Star in an Unchanging Sky?
Contented, Galileo could conclude that he had found a rule, a proportionality - if somewhat cumbersome - that concerned the increase in average speed, which was clearly proportional with the progression of odd numbers. If, on the other hand, he had added up the distances and looked at the total distance from the start, he would have been within a whisker of a fundamentally important, simple and general law.
But that was to come later. The most important result of the inclined plane experiments on this occasion was that velocity constantly increased as the sphere rolled. There was no "given speed" that a body would naturally reach. This hardly dampened Galileo's belief in his experimental method: he had clearly shown that here, too, Aristotle had made an elementary mistake.
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