at its beginning. But if it does not terminate there, we do not concede that it is finite—just as the future and the successive multiplication of a number are not finite, since they do not terminate at their far end, even though a comparison of this kind is possible in their case.

Someone else, however, may offer three replies.

The first concerns his statement:

“If neither falls short of the other at the opposite end, a thing with something added to it would be equal to that thing without the addition.”

We reply: This follows only if we match one of the two series with the other. Yet matching things that do not exist is impossible, as in matching the sequences of numbers from one to infinity, from ten to infinity, and from one hundred to infinity. We know that the numbers obtained by multiplying one are smaller than those obtained by multiplying ten; those obtained by multiplying ten are smaller than those obtained by multiplying one hundred; and those obtained by multiplying one hundred are smaller than those obtained by multiplying one thousand. Yet all these sequences are infinite. This argument is of the same kind as the argument that compares the revolutions of one of two heavenly bodies with those of the other. There, however, the revolutions occurred and ceased to exist; here, past times and motions are conceived as forming shorter and longer series.

Another reply to this argument—their best-known argument—is to say:

We do not concede that such matching is possible. For if neither series has a beginning, and one of them