278 manifest to the senses, but have never been demonstrated , and are further not demonstrable, because they are totally false. {2}For he has assumed that the motions of the same mobile in different media observe the same ratio to one another, in swiftness, as that which the subtilties of the media have: that this is assuredly false, has been abundantly demonstrated above. {1}In confirmation of which I shall add this one thing: if the subtlety of air has the same ratio to the subtlety of water as the swiftness of a mobile in air to its swiftness in water, then when a drop or any other part of water goes down fast in air, but in water is not moved downward at all, since the swiftness in air has no ratio to the swiftness in water, then, according to Aristotle himself, the subtlety in air will not observe any ratio to the subtlety of water: which is ridiculous. {1}Thus it is evident that, when one argues this way, Aristotle should be answered in the following manner: for in the first place it is false, as has been shown above {1}, that the difference of slowness and speed of the same mobile comes from the greater or lesser thickness and subtlety of the medium; and even if that were conceded, again it is false that a mobile in its motions observes the ratio that the subtleties of the media have.
And as for what Aristotle has written in the same section {1}, that it is impossible for a number to have the same ratio to a number than a number to nothing, this surely is true for a geometric ratio, and not only for numbers but in every quantity. In the case of geometric ratios, since it is necessary that a lesser quantity could be augmented so many times that it exceeds any quantity whatever, the said quantity must be something and not nothing; for nothing augmented by itself again and again will nevertheless exceed no quantity. However, this is not necessary in the case of arithmetic ratios: for in the case of these, a number can have to a number the ratio that a number has to nothing. For since numbers are in the same arithmetic ratio when the excesses of the larger ones over the smaller ones are equal, it will be perfectly possible for a number to have the same ratio to a number as another number has to nothing: as if we were saying, 20 to 12 is as 8 to 0;