第7章

Further, neither that which moves towards nor that which moves away from the centre can be infinite.For the upward and downward motions are contraries and are therefore motions towards contrary places.But if one of a pair of contraries is determinate, the other must be determinate also.Now the centre is determined; for, from whatever point the body which sinks to the bottom starts its downward motion, it cannot go farther than the centre.The centre, therefore, being determinate, the upper place must also be determinate.But if these two places are determined and finite, the corresponding bodies must also be finite.Further, if up and down are determinate, the intermediate place is also necessarily determinate.For, if it is indeterminate, the movement within it will be infinite; and that we have already shown to be an impossibility.The middle region then is determinate, and consequently any body which either is in it, or might be in it, is determinate.But the bodies which move up and down may be in it, since the one moves naturally away from the centre and the other towards it.

From this alone it is clear that an infinite body is an 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.