construct x =ab by using compass alone, if a and b are given segments.

Question

I found the problem in the book "What is mathematics?".

The following is a description of Mohr's constructions.(Macheroni problem)
9) Find $x = ab$, if $a$ and $b$ are given segments.

I found the question didn't give a segment denotes 1.I've no idea how to construct the x.

Can anyone give me a hint?

Answer 1

The question makes no sense unless you have a $1$ given: $a$ and $b$ are lengths, so $ab$ is an area; the required $x$ is the length of a rectangle of equal area to $ab$ and of width $1$.

One construction is to plot axes, and mark $(0,a)$ on the $y$-axis and $(-1,0)$ on the $x$-axis. The line between them has gradient $a$, so prolong that to where it meets the (vertical) line perpendicular to the $x$-axis drawn through $(b-1,0)$. The intersection is at $(b-1, ab)$.