Scientific Method, Galileo and Falling Under Gravity From the Leaning Tower of Pisa.
Galileo and his 16 th. Century contemporaries used a theory of scientific method, differing from the one in use today. (Ignoring the disgraceful suggestion that, in practice, contemporary method is: "It's science if it suits powerful interest groups" - in which case the two methods may differ little.) Their method followed from Euclid's Elements - perhaps the most quintessentially methodical and successful exemplar available to them.
The Nature of Proof.
Euclid's Elements is a geometrical text where proofs are produced by the tautological transformation of axioms and specified common notions - given propositions.
To prove that the angle sum of a triangle is 180o, for example, it would not suffice to draw a number of triangles, measure their angles and note that in each case these summed to 180o. This is the experimental, scientific method of the present day. This would only provide an indication that one might be thinking along the right lines. A general proof would necessitate deriving the proposition "angle sum = 180o" by specified logic from specified axioms.
Throwing Things Off Pisa Tower Proves Nothing.
Throwing objects of differing weights off the Leaning Tower of Pisa and observing them hitting the ground at the same moment would not therefore prove that all objects fall at the same rate in a gravitational field, when air resistance is factored out. For this reason - Sporus seems to remember reading in "Scientific Method, An Historical and Philosophical Introduction" - Galileo claimed never to have performed this experiment; though it is thought he may have indeed done so, almost surreptitiously.
Galileo's proof of things falling at equal rates.
Assume lighter objects fall more slowly. Then show that this assumption leads to a contradiction. This proves the assumption must be wrong. (This method of proof was introduced in Euclid's Elements.)
Perform the following thought experiment: Throw a heavier and a lighter object off the tower. The heavier falls faster - by the assumption. Now tie the two objects together at the two ends of a length of rope and throw them off once more. The lighter one lags behind, the rope pulls taught and the laggard, lighter weight slows the fall of the heavier. The time taken for the heavy weight to reach the ground is therefore longer than before - when it was not tied to the other weight.
But imagine shortening the length of rope between the two; clearly this would make no difference. Shorten the rope until the two masses are bound together as one. This single weight is heavier than the heavy weight on its own. Yet we have proved it takes longer to fall.
The assumption that heavy weights fall faster proves that heavier weights fall slower. This is a contradiction. Therefore the assumption is false (by a proof method from the Elements). Hence bodies fall at equal rates irrespectively of their weights, proved.
The scary thing is it seems a very convincing proof.
(See book: "Scientific Method, An Historical and Philosophical Introduction".)