The Compass and the Mohr-Mascheroni Theorem

Abstract

Napoleon proposed to the French mathematicians the problem of divid-ing a circle into four congruent arcs by using the compass alone. Although not original with Napoleon, the problem has become known as Napoleon’s Problem. During his campaign in northern Italy, Napoleon had encountered the poet and geometer Lorenzo Mascheroni (1750-1800). Mascheroni was a professor at the University of Pavia, where Christopher Columbus had once been a student. Mascheroni’s most famous mathematical work is his Geometria del Compasso, published in 1797. This work, which began with an ode of some literary merit that was dedicated to Napoleon, showed that all the ruler and compass constructions can be accomplished with the euclidean compass alone. Surprisingly, any point that can be constructed with ruler and compass can be constructed without using the ruler at all. In these compass constructions, a line is considered to be constructed as soon as two points on the line are constructed. In practice, we cannot draw a line with only a compass, but we may be able to construct some particular point on the line as the intersection of circles that are drawn with the compass. As usual, we do not expect every point on a constructed line to be constructible.

In December of 1797 there took place in Paris a brilliant gathering of prominent writers and scholars, with the immortal Lagrange and Laplace among them. A most conspicuous member of the company was the young and victorious General Napoleon Bonaparte, who … had occasion to entertain Lagrange and Laplace with a kind of solution of some elementary problems of elementary geometry that was completely unfamiliar to either of the two world-famous mathematecians. Legend has it that after having listened to the young man for a considerable while, Laplace,somewhat peeved, remarked, “General, we expected everything of you, except lessons in geometry.”

N. A. Court