What makes a difference if the ruler is marked?

Question

The classical straightedge and compass construction problems, i.e. squaring the circle, devising a trisecting algorithm for a generic angle, doubling the cube, are they indeed answered in the positive, should the ruler be marked? Why, a marked ruler marks constructible lengths along it, and the compass could draw arcs of constructible radius already, ain't so? I don't see why marked rulers are an advantage. The sources stating so apparently renders this as a most obvious fact.

Answer 1

Traditionally a "marked ruler" has exactly two marks on it, at a distance we can choose to call $1$. The marks on the ruler are supposed to let you do the following

Given a point $A$ and lines $l$, $m$, construct a line through $A$ such that its intersections with $l$ and $m$ are exactly $1$ unit apart.

To do this, imagine sliding one mark on the ruler along line $l$ with one hand while keeping the ruler snug against a pin at point $A$ with the other hand. When you see the other mark pass line $m$, stop and draw a line along the ruler.

In variants of this task, the role of $l$ and/or $m$ may be played by circular arcs.

This is known as a neusis construction, and is what you can't do in general with an unmarked straightedge. It is not the fact that your two intersection points are a known distance apart, but the fact that you can choose both intersection at once such that the resulting line goes through $A$.

With a marked ruler and neusis, you can trisect angles and double cubes (or in general extract cube roots), but the constructible points will still all have algebraic coordinates, so you can't square the circle that way.