Page 64 - Foundations Of Differential Calculus

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On the Infinite and the Infinitely Small

72. Since every quantity, no matter how large, can always be increased,
and there is no obstacle to adding to a given quantity another like quantity,
it follows that every quantity can be increased without limit. Furthermore,
there is no quantity so large that a larger one cannot be conceived, and so
there is no doubt that every quantity can be increased to infinity. If there
is someone who would deny this, he would have to give some quantity that
cannot be increased, and so he needs to give a quantity to which nothing
can be added. This is absurd, and even the idea of quantity rules out this
possibility. He must necessarily concede that every quantity can always be
increased without limit, that is, it can be increased to infinity.

73. For each kind of quantity this becomes even clearer. No one can
easily defend himself if he declares that the series of natural numbers,
1, 2, 3, 4, 5, 6,... has a limit beyond which it cannot be continued. Indeed,
there is no such number to which 1 cannot be added to obtain the following
number, which is greater. Hence, the series of natural numbers continues
without limit, nor is it possible to come to some greatest number beyond
which there is no greater number. In like manner the straight line cannot
be extended to such a point that it cannot be extended further. By this it
is clear that both the integers and the line can be increased to infinity. No
matter what kind of quantity it may be, we should understand that every
quantity, no matter how large, can always be made greater and greater,
and thus increased without limit, that is, increased to infinity.