347.24 Which things must be said to be more, which less, and which equally heavy. {1}
Let us first bring forth for everyone to see how it should be said that a thing is more, less, or equally heavy.Indeed, it is necessary to determine this: for it often happens that things that are less heavy are called heavier, while things that are more heavy are called less heavy.Thus, at times we say of a large piece of wood that it is heavier than a small piece of lead, even though, purely and simply, lead is heavier than wood; and of a large piece of lead, we say that it is heavier than a small one, even though lead is not heavier than lead.For this reason, in order that we may escape pitfalls of this kind, those things will have to be said to be equally heavy to one another which, when they are equal in size, will also be equal in heaviness: thus, if we take two pieces, <u>one of silver, the other of steel, </u>which, equal in size, are also congruent in heaviness, they will have to be said to really weigh the same.Thus, wood and lead<u> must not be said to</u> exert weight equally: for a piece of wood which weighs the same thing as a piece of lead will considerably exceed the latter in size. [see note below]Moreover, a thing should be called heavier than another, if when a piece of it is taken, equal to a piece of the other, it is found to be heavier than the piece of the other: as, for example, if we take two pieces, one of lead and one of wood, equal to one another, and the piece of lead is heavier, then we shall surely be justified in asserting that lead is heavier than wood. [see note]That is why, if we <u>assume</u> a piece of wood which weighs the same as a piece of lead, still wood and lead should not be deemed to be equally heavy; for we will find that the size of the lead is considerably exceeded by the size of the wood. [see note below]Finally, the converse must be <u>deemed</u> to hold of things that are less heavy: it must be <u>decreed</u> that a thing is less heavy, if when a piece of it is taken, equal in size to a piece of the other, it turns out to be less in heaviness; as, if we <u>take</u> two solids, one of wood, the other of lead, which are equal in size, but the piece of wood exerts less weight than the piece of lead, then it must be <u>asserted</u> that wood is less heavy than lead. [see note]
<u>These are the things that had to be said concerning the definitions of terms.</u><u>Now, in order that we may more easily come to the things that have to be demonstrated, let the following axiom be assumed: namely, that what is heavier cannot be lifted by what is less heavy, other things being equal.</u>But for the things that have to be said we also need the following lemma.
Lemma for what follows.{1}
Heavinesses of unequal sizes of equally heavy bodies have to one another the ratio which the sizes have.
Thus let a, b be unequal sizes of equally heavy bodies, and let a be greater; a will then be heavier than b.Let c be the heaviness of a and let d be the heaviness of b: I say that heaviness c has the same ratio to heaviness d as size a has to size b.For let the sizes a, b be augmented by any number of multiples whatever: let size efg be a multiple of size a, and let size hk be a multiple of size b, but in such a way that size efg exceeds size hk: and let heaviness nop be as many multiples of heaviness c, as size efg is of size a; and let heaviness lm be taken to be as many multiples of heaviness d, as size hk is of size b.Now, because size efg and weight nop are equal multiples between, respectively, a and c, as much size their is of efg, equal in size to a, as much will their be of weight nop, equal in weight to c: and because weight c is equal to the weight of size a, and that weight c is equal to the weight n, and size a to size g, the weight of n will be equal to the weight of size g.Similarly it will be shown that weight o is equal to the weight of size f, and the weight p with the weight of size e: consequently, the weight of size efg taken as a whole will be nop.Now, it will also be similarly shown, that the weight lm is equal to the weight of size hk.Now, it has been assumed as true that size efg is greater than size hk: hence, the weight of this same efg, which is nop, will be greater than the weight of size hk, which is lm.Similarly we will show, for any multiple whatsoever, that if size efg has been greater than size hk, weight nop also is greater than weight lm; and if efg had been smaller or equal to hk, in the same way would nop have been smaller or equal to hk: and efg are equal multiples of a, also nop, respectively to c; and, for any multiple whatever, hk is an equal multiple of b, also lm for d: accordingly, by definition of an equal ratio, as size a is to size b, weight c is to weight d.Which was what was to be demonstrated.
These things having thus been inspected, let us come near now to explain how the cause of downward motion is the excess of the weight of the mobile over the weight of the medium: because, as we have already made clear earlier {1} it will then be obvious, when that will have been shown, that nothing can naturally be moved downward, less their is such an excess of weight over that of the medium; and that nothing, in a medium which is exceding in weight, can go down in it, unless it is hindered.Now, because the media, through which motions are made, are varied, and because water is a very suitable medium through which can be understood motions, upwards as well as downwards, we will put ourselves in a situation to make observations concerning motions as happening in such a medium: and, in the first place, we will demonstrate, that any solid bodies whatever, equally heavy with water, let down in it, are surely totally submerged; but they are not yet moved downward in water.Now, let it be remembered what was said earlier, that is, those bodies, which between