As far as I understand, it's not the marked ruler per se that allows to trisect arbitrary angles, but the specific use one makes of it. It's this use that's not allowed in Euclids Elements, not the mere presence of a ruler with two marks on it.

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In fact it turns out, that it's not easy to use a marked ruler: it requires abilities that go beyond the abilities one has to have to make the standard Euclidean constructions.

The problem $\mathcal{P}$ to solve is: Given two lines $l$ and $m$ , a point $P$ not on $l$ and $m$ and a ruler with two marks on it having distance $d$. Draw a line through $P$ which intersects the lines in $P_l$ and $P_m$ respectively, such that the length of $|P_l P_m|$ equals $d$.

Question 1: How can it be proved that $\mathcal{P}$ cannot be solved with compass and straightedge alone (and the things one is allowed to do with them)? Is $\mathcal{P}$ mentioned in Euclids Elements and does he prove its unsolvability? If it was not Euclid: Who gave the first recorded proof?

(It seems unprobable that Euclid could have proved the unsolvability of $\mathcal{P}$ which is equivalent to the problem of angle trisection which to be unsolvable with compass and straightedge alone was only proofed in 1837 by Wantzel.)

So what do we have to do to solve $\mathcal{P}$ with a marked ruler? We have to translate and rotate the ruler around $P$ until one of the marks lies on $l$ and the other one on $m$. To achieve this, translation and rotation must be performed at the same time and highly coordinated: You cannot achieve the goal by finitely many alternating translations and rotations. (This is a my claim.) It requires some practice to do this so-called neusis construction, and it's obviously not a simple basic operation - so we can understand why it was ruled out by Euclid.

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Question 2: How can the statement that the use of the marked ruler requires coordinated translations and rotations at the same time and can not be done by alternating translations and rotations be made mathematically precise? The other way around: How does an algorithm for solving $\mathcal{P}$ with the help of a marked ruler look like?

What's interesting is that the Tomahawk - which gives the same power as the marked ruler, e.g. in angle trisection - is fairly easy to use:

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So the above argument would not suffice to rule it out for reasons of "simplicity" of use: its exclusion looks more arbitrary than the exclusion of the marked ruler resp. the way to use it.

Question 3: Would have Euclid ruled out the Tomahawk (which can be constructed with compass and straightedge and easily used) if he had known it? And for what reasons?

Note that the Archimedian spiral is another tool to trisect an angle. It's hard (or even impossible) to construct with compass and straightedge alone ("Draw a circle with constantly growing radius $r = r_0\varphi$") but easy to use:

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We have (with construction meaning construction with compass and straightedge alone, n-gons meaning constructability of arbitrary regular n-gons ):

tool               |  construction  |  trisection |  n-gons   |  squaring circle
marked ruler       |  easy          |  difficult  |  no       |  no
tomahawk           |  easy          |  easy       |  no       |  no
Archimedian spiral |  difficult     |  easy       |  ??       |  ??
Hippias' quadratix |  ??            |  ??         |  ??       |  yes
cone-and-string    |  easy          |  difficult  |  yes      |  yes

I have added without further discussion a cone-and-string tool which I describe here and the quadratix of Hippias (thanks to user Blue):

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Question 4: Which other tools that allow to trisect angles are there? Which other construction powers to they have (e.g. squaring the circle)? Are there other examples which can both be constructed and used easily? Is there a well-known example that is hard both to construct and to use?

  • $\begingroup$ As for Question 4: The quadratrix of Hippias was devised for angle trisection. $\endgroup$ – Blue Aug 25 '18 at 2:18

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