264 of wood: for the reasoning is the same; thus, if we conceive in our mind that the water, on which a beam and a small piece of the same beam float, becomes imperceptibly and progressively lighter, in such a way that in the end the water gets to be lighter than the wood and the pieces of wood start slowly to go down, who would ever say that the beam would go down first or more swiftly than the small piece of wood? For although a large beam may be heavier than a small piece of wood, the beam must be put into relation with the great quantity of water that must be raised by it, and the small piece of wood with the small quantity of water [that must be raised by it]: and since an amount of water as great in size as the beam itself must be raised by the beam, and similarly for the small piece of wood, these two amounts of water, namely those that are raised by the pieces of wood, will have the same ratio in heaviness to one another as their sizes have (for the parts of homogeneous things are to one another in heaviness as they are in size, something which should be demonstrated{1}), that is, the ratio that the sizes of the beam and the small piece of wood have to one another: hence the heaviness of the beam will have the same ratio to the heaviness of the water that must be raised by it as the heaviness of the small piece has to the heaviness of the water that must be raised by it: and the reluctance of the large quantity of water [to be raised] will be surpassed by the large beam with the same facility as the resistance of a little water will be overcome by the small piece of wood. And if, on the other hand, we conceive in our mind, for example, an amount of wax of considerable size floating on water, and we mix the wax with either sand or some other heavier thing, in such a way that in the end it comes to be heavier than water and it just barely begins to go down very slowly, who would ever believe, if we took a particle of such wax, say one hundredth of it, either that it would not go down or that it would go down a hundred times more slowly than the totality of the wax? Surely no one.
And it will be possible to experience the same thing in the balance: for if very large, equal weights are placed on each side, and then to one of them something heavy, but only modestly so, is added, the heavier will then go down, but not any more swiftly than if the weights had been small.
And the same reasoning holds in water: for the beam corresponds to one of the weights of the balance, while the other weight is represented by an amount of water as great in size as the size of the beam: if this amount of water weighs the same as the beam, then the beam will not go down; if {1} the beam is made slightly heavier in such a way that it goes down, it will not go down more swiftly than a small piece of the same wood, which weighed the same as an [equally] small part of the water, and then was made slightly heavier.