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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| De motu antiquiora | 5/12 | https://en.wikipedia.org/wiki/De_motu_antiquiora | reference | science, encyclopedia | 2026-05-05T08:51:23.985272+00:00 | kb-cron |
=== Chapter 8: Different bodies moving in the same medium maintain a ratio of speeds different from what is said by Aristotle === Galileo states that two bodies may differ in two ways: 1) they are of same material but have different size (i.e., volume), and 2) they are of different material but a) they differ in size and weight, b) differ in weight but not size, or c) differ in size but not weight. He then refutes Aristotle’s claims for each situation. Aristotle claims that, in the case of naturally moving bodies that are of the same material, the larger moves more swiftly, such that a large piece of gold would move more swiftly than a small piece, and that the ratio of their speeds is the same as the ratio of their sizes. Galileo asserts that this is ridiculous because this would imply that, for two lead balls, one ball a hundred times larger than the other, both falling from a great height, then the lighter ball would take a hundred times longer to fall than the heavier ball, but this does not happen. Instead, Galileo argues that objects made of the same material, though different in size, will fall with the same speed, and that anyone surprised by this will also be surprised to realize that a large piece of wood will float no less than a small piece of wood. In another example, Galileo proposes that a piece of wax be mixed with sand so that it becomes slightly heavier than water and begins to sink slowly. When comparing a piece of mixed-wax that is a hundredth part of the first considered wax, Galileo argues that no one would believe that the smaller piece of wax would take a hundred times longer to sink. The same may be said for the analogy of weights on a balance: for two large and equal weights are balanced, and a minuscule weight is added to one side, the heavier side will fall, but it won’t fall any faster than if the two weights were small weights instead. Similarly, for water and wood, where one weight on the balance represents the weight of the wood and the other weight represents the weight of a volume of water that is equal in volume of the wood, if the weight of the volume of water is equal to the weight of the wood, the wood will not sink, but if the wood is made a little heavier so that it sinks, it will not sink faster than a small piece of the same wood, which initially weigh the same as an equally small volume of water, and then is made a little heavier. In another argument, Galileo considers an assumption: if there are two bodies with one body moving with natural motion more swiftly than the other, then a combination of the two bodies will move more slowly than the body that, by itself, moves more swiftly, and also the combination will move faster than the body that, by itself, moved more slowly. For example, a ball of wax and an inflated bladder are both submerged in water and both move upward in the water, but the inflated bladder moves faster than the wax. If the two are connected to each other, the combination will rise more slowly than the bladder alone, but more swiftly than the wax alone. The same may be said for downward-falling bodies: if one is of wood and the other an air bladder, the wood falls faster than the air bladder, but when connected, together they fall with an intermediate speed. With this assumption, Galileo then returns to Aristotle’s claim that heavier bodies of the same material fall faster: if two bodies of the same material but different sizes (and likewise weights) fall with different speeds, then when connected together, the assumption leads us to believe that the combined bodies will have an intermediate speed; however, the combination of the two bodies will have a total weight that is greater than any of the standalone bodies. Therefore, according to Aristotle, the combined weight should fall even faster than either of the standalone bodies, which leads to self-contradiction. The only way to correct the contradiction is to reject Aristotle’s claim and assume that the two bodies of same material but different size (and weight) fall at the same speeds. This same argument will appear again in Galileo’s Two New Sciences. A caveat is then recognized: the weights of the bodies of same material cannot be taken to the extremes, for even a thin plate or even a leaf of the same substance can be made to float on water. Thus, the weight and volume of the smaller must be large enough to not be affected by the viscosity of the medium. However, this caveat does not justify Aristotle’s original claim since it remains that the assumption that great differences in weight correlate to great differences in times is deeply flawed and must be rejected. Galileo then considers the ratios of the speeds of bodies of different material moving in the same medium. Such bodies differ from each other in three ways: either in size but not weight, or in weight but not size, or both in weight and size; however, only the case of those that differ in weight but not in size need be considered since the ratios of the other ways can be reduced to this one. In the case of bodies differing in size but not in weight, we may take from the larger a part that is equal in size to the smaller, thus, the bodies will then differ in weight, but not in size. And the larger body will, with the smaller body, maintain the same ratio as will the part taken from the larger, since it was proved that bodies of the same material, though different in size, move with the same speeds. In the case of bodies differing both in size and weight, if we take from the larger a part equal in size to the smaller, again, we have two bodies differing in weight, but not in size. And the part will, with the smaller, maintain the same ratio in its motion, as will the whole of the larger – again, in the case of bodies of the same material, the part and whole move with the same speed. Aristotle claims that, in the case of the same body moving in different media, the ratio of the speeds is equal to the ratio of the rareness of the media. Galileo proves that this assumption leads to an absurdity and is therefore false. If the speeds have the same ratio as the rareness of the media, then, conversely, the rareness of the media will have the same ratio as the speeds. Since wood falls in air but not in water, and since the speed in air has no ratio to the speed in water, then the rareness of air will have no ratio to the rareness of water, which is absurd. Galileo then investigates the ratio of the speeds of the same body moving in different media during upward motion.