CHAPTER V.
CONSIDERATIONS ON THE APPLICATION OF THE IDEA OF THE INFINITE TO CONTINUOUS QUANTITIES, AND TO DISCRETE QUANTITIES, IN SO FAR AS THESE LAST ARE EXPRESSED IN SERIES.
31. One of the characteristic properties of the idea of the infinite is application to different orders. This gives occasion to some important considerations which greatly assist to make this idea clear in our mind.
32. From the point where I am situated I draw a line in the direction of the north; it is evident that I may prolong this line infinitely. This line is greater than any finite line can be; for the finite line must have a determinate value, and therefore, if placed on the infinite line, will reach only to a certain point. This line, therefore, seems to be strictly infinite in all the force of the word, because there is no medium between the finite and the infinite, and we have shown that it is not finite, since it is greater than any finite line; therefore it must be infinite.
This demonstration seems to leave nothing to be desired; yet there is a conclusive argument against the infinity of this line. The infinite has no limits, and this line has a limit, because, starting from the point from which it is drawn in the direction of the north, it does not extend in the direction of the south.
33. This line is greater than any finite line; but we may find another line greater still. If we suppose it produced in the direction of the south, it will be greater by how much it is produced towards the south; and if it be infinitely produced in this direction, its length will be twice that of the first line.
34. By the infinite prolongation of a line in two opposite directions we seem to obtain an absolutely infinite line; for we cannot conceive a lineal value greater than that of a right line infinitely prolonged in opposite directions. But it is not so: by the side of this right line another may be drawn, either finite or infinite, and the sum of the two will form a lineal value greater than that of the first line; therefore that line is not infinite, because it is possible to find another still greater. And as, on the other hand, we may draw infinite lines and prolong them infinitely, it follows that none of them can form an infinite lineal value, because it is only a part of the lineal sum resulting from the addition of all the lines.
35. Reflecting on this apparent contradiction in our ideas, we discover that the idea of the infinite is indeterminate, and consequently susceptible of different applications. Thus, in the present instance, it cannot be doubted that the right line, prolonged to infinity, has some infinity, since it is certain that it has no limit in its respective directions.