# Ruler and Compasses without Ruler (Mohr-Mascheroni)

In 1797 Lorenzo Mascheroni published result that

Every geometric construction that can be carried out by compasses and ruler may be done without ruler.

This theorem named after Lorenzo Mascheroni a Mascheroni theorem. By the way it turned out that Georg Mohr proved this theorem back in 1672 already, but this publication remained unknown, since was written in Latin I guess.

I am looking for elementary proof of the theorem.

We have to prove that the following three constructions can be done with only compasses:

1. Points of intersection of two circles given by center and one of the points for each circle
2. Points of intersection of a circle (given by center and one of its points) and a straight line (given by two points).
3. Point of intersection of two straight lines each of them given by two points.

On this route I found the paper "A short elementery proof of the Mohr-Mascheroni Theorem" by Norbert Hungerbuhler.

But it looks unsatisfactory for me since it uses measurement by means of compasses.

So I believe that there must be some other reference on elementary proof of Mohr-Mascheroni theorem which does not use methods like a measurement by means of compasses.

If anyone knows such a text please inform me to.

## 1 Answer

Why do you say that they require measurements by compass?

Is it because of the reflections? Observe that you can construct the reflection of $M$ with respect to the line passing through $P_1$ and $P_2$, by drawing the circle with center $P_1$ and radius $P_1M$ and drawing the circle with center $P_2$ and radius $P_2M$ you get two intersection points $M$ and $M'$.

Besides constructing reflections (which they claim but I didn't see proven) I didn't see any other step that can be seen as using measurements.

The midpoint construction is explicitly there on page $2$. Did I miss some other instance?

Doubling is also OK. Take into account that necessarily one should be allowed to, given two points, place the needles of the compass on each of them, otherwise we wouldn't be able to draw the circle with center on one of them and passing through the other.

This Mir book also has it: Geometrical Constructions Using Compasses Only by A. N. Kostovskii. The link has the translation in Spanish.

The first and second constructions in Kostovskii's book are precisely the reflection of a point, and multiplying a segment by an integer number.

• Well, Euclid's compass could only draw a circle given its center AND a point of the circle. Using a compass to transport a measure was not allowed and Euclid gave a specific construction for that. – Intelligenti pauca Mar 27 '18 at 17:44
• @Aretino There is no transportation used in the construction. At least in the book, let me double check the OP's paper. – saltandpepper Mar 27 '18 at 17:47
• @Aretino It looks like they do assume it possible in construction 4 in the OP's paper, without proving it. But well, maybe they skipped as they skipped the reflection. – saltandpepper Mar 27 '18 at 17:52
• How do they construct circle $K'$ in Construction 1? – Intelligenti pauca Mar 27 '18 at 19:41
• @saltandpepper Construction 1 and 2 are OK I agree, but in Construction 3 we read: (Where $K$ is a circle with center $M$ and $PM$ a segment) Let $K_1$ be a circle through $A$ and $B$ with radius larger than the radius $R$ of $K$ and $M_1$ the center of $K_1$. I do not see how it can be done to construct a circle with radius larger than radius of other circle when I do not know a center of it. The only way I see is to take a legs of compasses in position which is larger than $BM$ and $AM$ but how to do that? – Evgeny Kuznetsov Mar 27 '18 at 20:46