as well as all the points of $ab$ there also lie that of $bc$ which do not occur in $ab$.”

Let $x$ be some point in $ab$ then it corresponds to the point $y$ in $ac$ if the following proportion holds $$ab : ac = ax : ay.$$

Bolzano emphatically warns against the assertion that one-to-one correspondence of two multitudes allows for the conclusion of the equal plurality of their parts if they are infinite.

> An equality of these multiplicities can only be concluded if some other reason is added, such as that both multitudes have exactly the same *determining ground \[Bestimmungsrűnde\]*, e.g. they have exactly the same *way of being formed \[Entstehungsweise\]*. (§21).

Bolzano doesn’t explain here what the *determining ground* means exactly. He speaks in other texts about *elements of determination, \[bestimmende Stűcke\]*, though it is not entirely clear there either. To determine an object means to describe all representations that the objects falls under. The determination is complete if the representation of an object is unique. Thus, a point, the area and the diameter completely determine a circle, because all of its properties can be uniquely determined. (Šebestík 1992, p. 460).

Bolzano shows that this is 