On The Heavens   

impossibility; but there is a further point. If there is no such thing

as infinite weight, then it follows that none of these bodies can be

infinite. For the supposed infinite body would have to be infinite

in weight. (The same argument applies to lightness: for as the one

supposition involves infinite weight, so the infinity of the body

which rises to the surface involves infinite lightness.) This is

proved as follows. Assume the weight to be finite, and take an

infinite body, AB, of the weight C. Subtract from the infinite body

a finite mass, BD, the weight of which shall be E. E then is less than

C, since it is the weight of a lesser mass. Suppose then that the

smaller goes into the greater a certain number of times, and take BF

bearing the same proportion to BD which the greater weight bears to

the smaller. For you may subtract as much as you please from an

infinite. If now the masses are proportionate to the weights, and

the lesser weight is that of the lesser mass, the greater must be that

of the greater. The weights, therefore, of the finite and of the

infinite body are equal. Again, if the weight of a greater body is

greater than that of a less, the weight of GB will be greater than

that of FB; and thus the weight of the finite body is greater than

that of the infinite. And, further, the weight of unequal masses

will be the same, since the infinite and the finite cannot be equal.

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.)