ARISTOTLE ON THE INFINITE 
343 
does include one and the same magnitude, whatever it is, you 
will come to the end of the finite magnitude, for every finite 
magnitude is exhausted by continually taking from it any 
definite fraction whatever. In no other sense does the infinite 
exist but only in the sense just mentioned, that is, potentially 
and by way of diminution. 1 And in this sense you may have 
potentially infinite addition, the process being, as we say, in 
a manner the same as with division ad infinitum ; for in the 
case of addition you will always be able to find something- 
outside the total for the time being, but the total will never 
exceed every definite (or assigned) magnitude in the way that, 
in the direction of division, the result will pass every definite 
magnitude, that is, by becoming smaller than it. The infinite 
therefore cannot exist, even potentially, in the sense of exceed 
ing every finite magnitude as the result of successive addition. 
It follows that the correct view of the infinite is the opposite 
of that commonly held; it is not that which has nothing 
outside it, but that which always has something outside it.“ 
Aristotle is aware that it is essentially of physical magnitudes 
that he is speaking: it is, he says, perhaps a more general 
inquiry that would be necessary to determine whether the 
infinite is possible in mathematics and in the domain of 
thought and of things which have no magnitude. 3 
‘ But ’, he says, ‘ my argument does not anyhow rob 
mathematicians of their study, although it denies the existence 
of the infinite in the sense of actual existence as something 
increased to such an extent that it cannot be gone through 
(dSie^LTTjToi')', for, as it is, they do not even need the infinite 
or use it, but only require that the finite (straight line) shall 
be as long as they please. . . . Hence it will make no difference 
to them for the purpose of demonstration/ 4 
The above disquisition about the infinite should, I think, 
be interesting to mathematicians for the distinct expression 
of Aristotle’s view that the existence of an infinite series the 
terms of which are magnitudes is impossible unless it is 
convergent and (with reference to Riemann’s developments) 
that it does not matter to geometry if the straight line is not 
: nfinite in length provided that it is as long as we please. 
1 Phys. iii. 6. 206 a 15~b 13. 
3 lb. iii. 5. 204 a 34. 
2 lb. iii. 6. 206 b 16-207 a 1. 
4 lb. iii. 7. 207 b 27.