which have been demonstrated above be called back into memory. </s>
</p>
<p>
<s id="id.1.1.13.03.00.fig"/>
<s id="id.1.1.13.03.01">It has been demonstrated {1}, for example, that a certain solid weighs as much less in water than in air, as the heaviness in air of a size of water equal to the size of the solid: so that if there are two solids a, b, and the heaviness of a in air is 8, while the heaviness of b is 6, and their sizes are equal, and the size of water c, whose heaviness in air is 3, is equal to their size, it is evident, from the things said above, that the heaviness of a in water is 5, but the heaviness of b 3. </s>
<s id="id.1.1.13.03.02">Consequently in water the difference [in terms of ratios] between the heavinesses of a and b will be greater, as between 5 and 3 the discrepancy is greater than between 8 and 6. </s>
<s id="id.1.1.13.03.03">But if, on the other hand, there were a certain medium heavier than water, whose heaviness is, for instance, 5, the heaviness of a in it will be 3, but the heaviness of b 1. </s>
<s id="id.1.1.13.03.04">And thus it is evident how in the heavier media the difference between the heavinesses is always greater; for in air the heaviness of a is 4/3 of the heaviness of b; in water, 5/3, and in another heavier medium, 3/1: but who will say that the true heavinesses of the solids are in this medium rather than that one? </s>
<s id="id.1.1.13.03.05"> No one, surely: but it will certainly be truer to say that in none of them are the exact weights found. </s>
<s id="id.1.1.13.03.06">For since in every medium the heavinesses of heavy things are diminished, by the amount that a portion of that medium equal in size to the solid would weigh, it is evident that the entire and undiminished heavinesses of the solids will only be found in that medium whose heaviness would be null: but only the void is such. </s>
<s id="id.1.1.13.03.07">But in other media heavy things weigh and exert weight only to the extent that they are heavier than those media (for if they were equally as heavy as a certain medium, they would exert no weight in such a medium): and since in a void, similarly, solids exert weight only to the extent that their heavinesses surpass the heaviness of the void; they surpass it according to the their own total heaviness, since the heaviness of the void is null; it thus follows, necessarily, that it is only in the void that the true heavinesses of heavy things can be found: that is why the differences of such heavinesses will be present only there.</s>
</p>
<p>
<s id="id.1.1.13.04.00"/>
<s id="id.1.1.13.04.01">Similar considerations also hold concerning the speeds of motions and of their ratios. </s>
<s id="id.1.1.13.04.02">For who will say that they are found in plenum media, if a mobile has one speed in this medium, another in that, still another in another one, and in still another one, null, like wood in water? and similarly, if there is one ratio of speeds in air, another in water, another in a heavier medium, another in a lighter one; as anyone will easily be able to find from these things that have been written above? </s>
<s id="id.1.1.13.04.03">And finally, since the speeds of the mobiles, in the medium in which they are moved, follow the heavinesses; and, consequently, the ratios of the speeds follow the ratios of the heavinesses; and it happens that these are not given except in the void; it must be asserted beyond any doubt that the true and natural differences of speeds also occur in the void only. {1}</s>
</p>
</subchap2>
</subchap1>
<subchap1>
<subchap2>
<p>
<s id="id.1.2.1.00.00"/>
<s id="id.1.2.1.00.01">Chapter 1 of Book II [293.5-302.18][Drabkin ch.14]In which the matter in question concerns the ratios of the motions of the same mobile on different inclined planes. </s>
</p>
<p>
<s id="id.1.2.1.01.00"/>
<s id="id.1.2.1.01.01">The question that we are about to explain has been treated thoroughly {1} by no philosopher, as far as I know: yet, since it concerns motion, it seems that it must necessarily be examined by those who profess to hand on a treatment concerning motion that is not incomplete. </s>
<s id="id.1.2.1.01.02">Now this question is no less necessary, than it is elegant and clever. </s>
<s id="id.1.2.1.01.03">For what is asked is why the same heavy mobile, in descending naturally along planes inclined to the plane of the horizon, is moved more easily and more swiftly on those that will maintain angles nearer a right angle with the horizon; and, furthermore, the ratio of such motions made on diverse inclinations is sought.</s>
<s id="id.1.2.1.01.04">The answer to this question, when first I had tried to investigate it, seemed to have explanations that were not entirely easy: yet, when I examined the thing more carefully and tried to resolve its demonstration into its principles, I finally discovered that its demonstration, like that of others that at first glance seem very difficult, relied on known and manifest principles of nature. </s>
<s id="id.1.2.1.01.05">These ideas, as they are necessary for the explanation of it, we will now expound first. </s>
</p>
<p>
<s id="id.1.2.1.02.00"/>
<s id="id.1.2.1.02.01">And to begin with, in order that all these things may be better understood, let us make the problem clear by means of an example.</s>
</p>
<p>
<s id="id.1.2.1.03.00.fig"/>
<s id="id.1.2.1.03.01">Thus let there be a line ab, directed toward the center of the world, which is perpendicular to a plane parallel to the horizon; and let line bc be in the plane parallel to the horizon; and from point b let there be drawn any number of lines maintaining acute angles with line bc, and let them be lines bd, be. </s>
<s id="id.1.2.1.03.02">It is then asked why a mobile, in going down, goes down most swiftly on line ab; and on line bd more swiftly than on line be, however more slowly than on ba; and on line be, more slowly </s>
