4. This is lore so elementary to mathematicians that it appears almost puerile to dwell upon it; but this seems to have been overlooked, in the proof that we can have no knowledge concerning infinites. In such proof it is assumed as quite evident, that all infinites are equal. Yet, as we have seen, infinites may differ infinitely among themselves, both in quantity and in kind. A German writer is quoted[301] for an "ingenious" proof of this kind. In his writings, the opponent is supposed to urge that a line _BAC_ may be made infinite by carrying the extremity _C_ infinitely to the right, and again infinite by carrying the extremity _B_ infinitely to the left; and thus the line infinitely extended both ways would be double of the line infinite on one side only. The supposed reply to this is, that it cannot be so, because one infinite is equal to another: and moreover that what is bounded at one end _A_, cannot be infinite: both which assumptions are without the smallest ground. That one infinite quantity may be double of another, is just as clear and certain as that one finite quantity may. For instance, if one leaf of the book which the reader has before him were produced infinitely upwards it would be an infinite space, though bounded at the bottom and at both sides. If the other leaf were in like manner produced infinitely upwards it would in like manner be infinite; and the two together, though each infinite, would be double of either of them. 5. As I have said, infinite quantities are conceived by conceiving finite quantities increased by the transfer of a certain limit, and then by negativing this limit altogether. And thus an infinite number is conceived by assuming the series 1, 2, 3, 4, and so on, up to a limit, and then removing this limit altogether. And this shows the baselessness of another argument quoted from Werenfels. The opponent asks, Are there in the infinite line an infinite number of feet? Then in the double line there must be twice as many; and thus the former infinite number did not contain all the (possible) unities; (numerus infinitus non omnes habet unitates, sed præter eum concipi possunt totidem unitates, quibus ille careat, eique possunt addi). To which I reply, that the definition of an infinite number is not that it contains all possible unities: but this--that the progress of numeration being begun according to a certain law, goes on without limit. And accordingly it is easy to conceive how one infinite number may be larger than another infinite number, in any proportion. If, for instance, we take, instead of the progression of the natural numbers 1, 2, 3, 4, &c. and the progression of the square numbers 1, 4, 9, 16, &c. any term of the latter series will be greater than the corresponding term of the other series in a ratio constantly increasing, and the infinite term of the one, infinitely greater than the corresponding infinite term of the other.