Galileo’s last work, the Two New Sciences,1 was published in 1638. At the time, he was completely blind and had been under house arrest for four years; he would pass away four years later. In this and subsequent posts, I will write about some of the contents of this very influential work. I have previously written the following posts pertaining to the Two New Sciences:
The dialogue of Day One is the longest in the Two New Sciences, and covers numerous topics, each flowing naturally into the next. In this post, I wish to focus on one specific topic, namely the question of falling bodies, and their relationship with respect to the existence of the void. This discussion begins with Simplicio, having been prompted by Salviati, expounding some Aristotelian principles:
Simp. As I recall it, Aristotle does battle against some ancients who introduced the void as necessary for motion, saying that no motion could exist without it. Aristotle, opposing this [view], proves that on the contrary, the occurrence of motion, which we see, destroys the supposition of the void; and these are his steps. He makes two assumptions; one concerning moveables differing in heaviness but moving in the same medium and the other concerning a given moveable moved in different mediums. As to the first, he assumes that moveables differing in heaviness are moved in the same medium with unequal speeds, which maintain to one another the same ratio as their weights [gravità]. Thus, for example, a moveable ten times as heavy as another, is moved ten times as fast. In the other supposition he takes it that the speeds of the same moveable through different mediums are in inverse ratio to the crassitudes or densities of the mediums. Assuming, for example, that the crassitude of water is ten times that of air, he would have it that the speed in air is ten times the speed in water.
From this second supposition he derives his proof [against the void] in this form: Since the tenuity of the void exceeds by an infinite interval the corpulence, though most rare [sottilissima], of any filled medium [mezzo pieno], every moveable that is moved through some space in some time through a filled medium must be moved through the void in a single instant; but for motion to be made instantaneously is impossible; therefore, thanks to motion, the void is impossible. [p.65]
Salviati takes on the first topic, i.e., “moveables differing in heaviness are moved in the same medium,” and splits it into two. He first examines moveables of the same nature, i.e, of the same material, as was the topic of the previous post, Galileo Galilei's and Simon Stevin's Tower Experiments. Salviati asks Simplicio what would happen were a large stone made to drop with a small stone:
Salv. Then if we had two moveables whose natural speeds were unequal, it is evident that were we to connect the slower to the faster, the latter would be partly retarded by the slower, and this would be partly speeded up by the faster. Do you not agree with me in this opinion?
Simp. It seems to me that this would undoubtedly follow. [pp.66-67]
From this exchange, Salviati presents two possibilities. First, if the lighter stone is placed on the larger stone, since the lighter stone should fall more slowly than the heavier stone, then the fall of the heavier stone will not be affected by the lighter stone:
Salv. We feel weight on our shoulders when we try to oppose the motion that the burdening weight would make; but if we descended with the same speed with which such a heavy body would naturally fall, how would you have it press and weigh on us? Do you not see that this would be like trying to lance someone who was running ahead with as much speed as that of his pursuer, or more? Infer, then, that in free and natural fall the smaller stone does not weigh upon the larger, and hence does not increase the weight as it does at rest. [pp.67-68]
Second, were the heavier stone to be placed on the lighter stone, then the heavier would be slowed down by the lighter one, and the lighter one by the heavier one, all of which would be contradicted by the fact that the two together should fall faster, as they are heavier together:
Salv. It would increase the weight if its motion were faster. But it was already concluded that if the smaller were slower, it would partly retard the speed of the larger so that their composite, though larger than before, would be moved less swiftly, which is against your assumption. From this we conclude that both great and small bodies, of the same specific gravity, are moved with like speeds. [p.68]
So now the question of moveables of the same nature have been dealt with, what about moveables of any nature? And what does this have to do with the void, or, as we might say today, the vacuum? The key is to consider how moveables fall, or not, in different mediums. Clearly, Aristotle’s claim that moveables all fall in water at a fixed rate slower than in air was preposterous: a wooden ball, for example, does not fall, but floats. What Galileo explains is that the difference in speeds at which moveables of different densities fall, or not, is higher in denser mediums than less dense ones.
Salv. After assuring myself of this, I began to combine these two phenomena together, noting what happened with moveables of different heaviness placed in mediums of different resistances, and I found that the inequality of speeds is always greater in the more resistant mediums, as compared with those more yielding. This difference is such that of two moveables descending in air and differing little in speed of motion, one of them will be moved in water ten times as fast as the other; or even such that one of them may swiftly descend in air, and not only fail to descend in water, but will remain quite still there, or even move upward. [p.72]
So working backwards from dense mediums to less dense mediums, Galileo posits that in a medium with no density, i.e., the void, not only would moveables move at a finite speed, but all moveables would move at the same speed. Hence Aristotle was wrong on both counts.
Salv. We have seen that the difference of speed in moveables of different heaviness is found to be much greater in more resistant mediums. What now? In mercury as the medium, not only does gold go to the bottom more swiftly than lead, but gold alone sinks, and all other metals and stones are moved upward and float in mercury. Yet balls of gold, lead, copper, porphyry, and other heavy materials differ almost insensibly in their inequality of motion through air. Surely a gold ball at the end of a fall through a hundred braccia will not have outrun one of copper by four inches. This seen, I say, I came to the opinion that if one were to remove entirely the resistance of the medium, all materials would descend with equal speed. [p.75]
So, as we can see, Galileo considered the void to be the limiting case of a succession of mediums, each less dense than the previous. Of course, without the air pump, invented a bit later that same century, he could not test the theory in an experiment.
Salv. We are trying to investigate what would happen to moveables very diverse in weight, in a medium quite devoid of resistance, so that the whole difference of speed existing between these moveables would have to be referred to inequality of weight alone. Hence just one space entirely void of air — and of every other body, however thin and yielding — would be suitable for showing us sensibly that which we seek. Since we lack such a space, let us [instead] observe what happens in the thinnest and least resistant mediums, comparing this with what happens in others less thin and more resistant. If we find in fact that moveables of different weight differ less and less in speed as they are situated in more and more yielding mediums; and that finally, despite extreme difference in weight, their diversity of speed in the most tenuous medium of all (though not void) is found to be very small and almost unobservable, then it seems to me that we may believe, by a highly probable guess, that in the void all speeds would be entirely equal. [p.76]
This result of Galileo’s is vital. Not only do objects of the same material fall at the same rate, in a vacuum, but objects of any material do so as well.
Galileo Galilei. Two New Sciences, Including Centers of Gravity & Force of Percussion. Translated by Stillman Drake. University of Wisconsin Press, 1974.
That's like the hammer and feather drop on the moon by Apollo 15 !