not the case in the examples mentioned. (§23). In the first Example, the distance of two quantities in $(0,5)$, say $3$ and $4$, is less than the distance of their corresponding quantities in $(0,12)$ which are $7\frac{1}{5}$ and $9\frac{3}{5}$. It is similar in the second example. The distance $ax$ is different (smaller) than the distance $ay.$ Obviously, Bolzano considers the distance to be a determining element. In this case we can speak about isometry rather than a mere one-to-one correspondence. An isometry is a distance-preserving transformation between two metric spaces.

On the other side, Bolzano presents the positive example of multitudes which have the same plurality of elements. (§29). He denotes the multitude of quantities between two numbers $a$ and $b$ as $Mult(b-a)$. This multitude is infinite and depends only on the distance of the boundary quantities and therefore must be equal whenever this distance is equal. Therefore, we have innumerable equations of the forms $$Mult(8 - 7) = Mult(13-12).$$ $$Mult(b - a) : Mult (d - c) = (b-a) : (d-c)$$

One-to-one correspondence is a necessary condition but not a sufficient one for the equality of two multitudes. The necessary condition in the cited examples is the *isometry*. In 