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The Compass and the Mohr-Mascheroni Theorem

1998; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-1-4612-0629-3_3

2197-5604

George E. Martin,

Advanced Operator Algebra Research

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.