With compass alone…
During the Italian campaign, Napoleon met the Italian geometer Lorenzo Mascheroni (1750–1800). Mascheroni taught at the University of Pavia, the very place where Christopher Columbus had been a student. Mascheroni's best-known work remains his Geometria del compasso ("the geometry of the compass"), published in 1797 and dedicated to Napoleon. Its main result is that any construction carried out in a finite number of steps with straightedge and compass can be carried out with compass alone — that is, without the straightedge, and so without a single straight line!
One certainly cannot construct a straight line with a compass alone, but every point of a line constructible with straightedge and compass is constructible with compass alone.
Mathematics teachers should be careful not to announce this news too loudly in their classrooms, for fear that students might stop bringing their straightedges on the grounds that they are useless. They would be right: it has been proved. To their misfortune, however, most constructions can be carried out with a graduated ruler — but that is another story.
