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De caelo A 6 273a21-b27
From these things it is obvious that it is not possible for there to be an infinite body, and, in addition, if its weight is not infinite, nor would any of these bodies be infinite either. For it is necessary that the weight of the infinite body also be infinite. (The same argument will also occur in the case of the light. For if there is an infinite heaviness, so will there be lightness, if what floats above it is infinite.). It is clear from the following arguments.
De caelo A 6 273a27-b10: a) we start with a finite part of the body and its weight measures .
(figure 1) For let the weight be finite, and let the infinite body be AB, and the weight of it be . (figure 2) Subtract from the infinite some finite magnitude, B. (figure 3) Let the weight of it be E. Then E will be less than . For the weight of the smaller is smaller. (figure 4) Let the smaller measure it out many times as you like, (figure 5) and let it come about that as the smaller is to the larger, so B is to BZ. For it is possible to subtract as many times as you like from the infinite. Yet if the magnitudes are proportional to the weights, and if the lesser weight is of the lesser magnitude, then the larger will be of the larger magnitude. Therefore the weight of the finite and the weight of the infinite will be equal.
Moreover, if the weight of the larger body is larger, then the
weight of HB will be greater than the weight of ZB, with the result
that the weight of the finite will be larger than the weight of
the infinite. And unequal magnitudes will have the same weight.
For the infinite is unequal to the finite.
De caelo A 6 273b10-15: b) we start with a finite part of the body and its weight does not measure (is incommensurable
7) It makes no difference whether the weights are commensurable or incommensurable.
(figure 8) For the same argument
will apply when they are incommensurable, e.g., if the third E exceeds in measuring
. (figure 9)For if three B magnitudes are taken together,
their weight will be greater than . Thus the same impossibility
De caelo A 6 273b15-23: c) we start with a commensurable portion of
10) Moreover, it is also possible to take commensurable weights. For it
makes no difference whether we begin with the weight or with the magnitude.
(figure 11) For example, take
E as commensurable with , and subtract that which
has the weight of E from the infinite, e.g. B. (figure
12) Then let it occur that as the weight is to the weight, so B is to some other magnitude, e.g.
to BZ. For it is possible to subtract as much as you like from a magnitude which
is infinite. For when these are taken both their magnitudes and their weights
will be commensurable with one another.
De caelo A 6 273b23-26: d) we treat the distribution of weight as uneven (cf. Swineshead and Oresme for the refutation of this argument)
It doesn't make a difference for the demonstration
whether the magnitude is regularly heavy or irregularly heavy. For it will be
possible to take bodies of equal weight with B, either taking away from
so much of the infinite or adding to it.
De caelo A 6 273b26-29: Conclusion
Thus it is clear from what we've said that the weight of the infinite body will not be finite. Therefore, it is infinite. Therefore, if this is impossible, then it is impossible that there be some infinite body.