P$.

-   $S$ is infinitely greater than $P$, $S >> P$. (§33).

Bolzano’s reasoning is: “The multitude of terms in both series is certainly the same. By raising every single term of the series $P$ to the square into the series $P$, we alter merely the nature (magnitude) of these terms, not their plurality.”

-   Every term of $S$, except the first one, is greater than the corresponding term of $P$, consequently $$S > P.$$

-   We shall prove that $S$ is infinitely greater than $P$. If we subtract successively the series $P$ from $S$ then we obtain the remainders

    $S - P = 0 + 2 + 6 + 12 + \dots + \text{in inf.} $

    $S - 2 P = -1 + 0 + 3 + 8 + \dots + \text{in inf.} $

    $S - m P = (1-m) + \dots + (n^2-mn) + \dots + \text{in inf.} $

    In these series, only a finite multitude of terms, namely $m-1$, is negative, and the $m$th term is 0, but all successive terms are positive and grow indefinitely. The sum of every series is positive. Thus $$S >> P.$$

Consequently, there is infinitely many infinite quantities. Sometimes we can 