It does not matter whether the weights are commensurable or not.If (a) they are incommensurable the same reasoning holds.For instance, suppose E multiplied by three is rather more than C: the weight of three masses of the full size of BD will be greater than C.We thus arrive at the same impossibility as before.Again (b) we may assume weights which are commensurate; for it makes no difference whether we begin with the weight or with the mass.For example, assume the weight E to be commensurate with C, and take from the infinite mass a part BD of weight E.Then let a mass BF be taken having the same proportion to BD which the two weights have to one another.(For the mass being infinite you may subtract from it as much as you please.)
These assumed bodies will be commensurate in mass and in weight alike.
Nor again does it make any difference to our demonstration whether the total mass has its weight equally or unequally distributed.For it must always be Possible to take from the infinite mass a body of equal weight to BD by diminishing or increasing the size of the section to the necessary extent.
From what we have said, then, it is clear that the weight of the infinite body cannot be finite.It must then be infinite.We have therefore only to show this to be impossible in order to prove an infinite body impossible.But the impossibility of infinite weight can be shown in the following way.A given weight moves a given distance in a given time; a weight which is as great and more moves the same distance in a less time, the times being in inverse proportion to the weights.For instance, if one weight is twice another, it will take half as long over a given movement.Further, a finite weight traverses any finite distance in a finite time.It necessarily follows from this that infinite weight, if there is such a thing, being, on the one hand, as great and more than as great as the finite, will move accordingly, but being, on the other hand, compelled to move in a time inversely proportionate to its greatness, cannot move at all.The time should be less in proportion as the weight is greater.But there is no proportion between the infinite and the finite: proportion can only hold between a less and a greater finite time.And though you may say that the time of the movement can be continually diminished, yet there is no minimum.Nor, if there were, would it help us.For some finite body could have been found greater than the given finite in the same proportion which is supposed to hold between the infinite and the given finite; so that an infinite and a finite weight must have traversed an equal distance in equal time.But that is impossible.
Again, whatever the time, so long as it is finite, in which the infinite performs the motion, a finite weight must necessarily move a certain finite distance in that same time.Infinite weight is therefore impossible, and the same reasoning applies also to infinite lightness.Bodies then of infinite weight and of infinite lightness are equally impossible.
That there is no infinite body may be shown, as we have shown it, by a detailed consideration of the various cases.But it may also be shown universally, not only by such reasoning as we advanced in our discussion of principles (though in that passage we have already determined universally the sense in which the existence of an infinite is to be asserted or denied), but also suitably to our present purpose in the following way.That will lead us to a further question.Even if the total mass is not infinite, it may yet be great enough to admit a plurality of universes.The question might possibly be raised whether there is any obstacle to our believing that there are other universes composed on the pattern of our own, more than one, though stopping short of infinity.First, however, let us treat of the infinite universally.