between some infinite multitudes. He is aware that sometimes one multitude is a part of the other one simultaneously. The following quotation begins a little ambiguously.

> Let us now turn to the consideration of a highly remarkable peculiarity which *can occur*, indeed actually *always occurs*, in the relationship between two multitudes *if they are both infinite*, but which previously has been overlooked to the detriment of knowledge …(§20.)

Bolzano describes very carefully the one-to-one correspondence.[^12] He presents two examples of two multitudes such that, *on the one hand*, there is a one-to-one correspondence between them and, *on the other hand*, one of these multitudes contains the other within itself as a mere part (§20).

The first example are the multitudes of quantities[^13] in two open intervals $(0,5)$ and $(0,12)$. “Certainly the latter multitude is greater since the former is indisputably only a part of it.”

The one-to-one correspondence is defined by the equation $$5y = 12x.$$

The second example is similar: the infinite multitudes of points of line segments $ab$ and $ac$, where $b$ is an inner point of line $ac$. “The multitude of points which lie in $ac$ exceeds that of the points in $ab$, because in $ac$ 