Juan Ramón Koenig and the cube doubling problem: contemporary re-reading of Cubus et sphaera geometrice duplicata

In 1696, Juan Ramón Koenig, the Chief Cosmographer of the Viceroyalty of Peru and professor of mathematics at San Marcos University in Lima, published a book titled Cubus et sphaera geometrice duplicata in which he claims to solve the classical problem of doubling the cube by compass and straightedge only. His idea was to start from the neusis solution of this problem given by Nicodemes and somehow transform it into a compass and straightedge construction. We know today that the cube root of 2 is not a constructible number, thus his solution has to be faulty. Since so far nobody has analyzed Koenig’s work in detail, we undertake this task using modern mathematical notation and methods. By scrutinizing his construction step by step, we reveal where he has failed.