Bunch-Hellemans, The origins of doubling the cube, squaring the circle and trisecting the angle

About 430 BCE a great plague struck Athens. The Athenians appealed to the oracle at Delos to provide a remedy. The oracle said that Apollo was angry because his cubical altar was too small. If it were doubled, the plague would end. The Athenians had a new altar built that was twice the original in length, breadth, and height. The plague became worse. Consulting the oracle again, the Athenians learned that Apollo was angrier than ever. The god wanted the volume of the cube doubled, and the Athenians had octupled it. The plague went on until 423. The problem of doubling the cube went on until the 19th century.

Or so the story goes. There are other stories (but not as good) about how the Delian problem, as doubling the cube came to be known, arose. Whatever its origin, the Delian problem together with the problems of squaring the circle and trisecting the angle became the three classic unsolved problems of Greek mathematics.

It is essential to understand that the method of solution for the classic problems was restricted. A solution would count only if it were accomplished by a geometric construction using an unmarked straightedge and a compass that collapsed when lifted from the paper. Tradition has it that Plato was responsible for setting this requirement. Although a great many other problems were easily solved under the restriction, the three classics stubbornly refused to yield.

It is likely that the first classic problem was squaring the circle.

It is thought that Anaxagoras worked on solving it, about 450 BCE, while he was in prison for having claimed that the Sun is a giant red-hot stone and that the Moon shines by reflected light. The problem specifically is to construct a square that has the same area as a given circle.

The trisection problem arose around the same time. In that case, the problem was to find an angle whose angle measure is exactly one-third that of a given arbitrary angle. Trisection appears deceptively simple since among the easiest constructions are the trisection of a line and the bisection of an angle.

Like the other classic problems, it stumped the ancient Greeks, but not completely.

The apparent progress toward squaring the circle by Hippocrates of Chios (not to be confused with Hippocrates of Cos, the father of medicine) around 430 probably encouraged other mathematicians to continue. Hippocrates succeeded in squaring a region bounded by two arcs of circles.

This figure looks like the crescent Moon and is therefore called the lune. Hippocrates made the first successful attempt to convert an area bounded by curves to one bounded by straight lines.

A number of mathematicians decided to stretch the rules and solve the problem ignoring Plato’s restriction. Hippias of Elis is supposed to have found two different ways to square the circle, both explicitly rejected by Plato (according to Plutarch). One of the methods attributed to Hippias, based on the first curve other than a line or circle that was well defined and constructible (the quadratrix), was used to square the circle at a later date. Similarly, while trying to solve the Delian problem, Menaechmus seems to have discovered the conic sections –– the ellipse, the parabola, and the hyperbola. Use of these curves enabled him to duplicate the cube. Archimedes solved two of the classic problems, trisection of the angle and squaring the circle, with his famous Archimedean spiral (a curve invented by Conon of Alexandria). Archimedes also discovered that all that is needed to trisect an angle is a marked straightedge instead of an unmarked one. In a related development, Archimedes managed to square a region bounded by a parabola and a line.

Around 320 CE, Pappus declared that it was impossible to solve any of the classic problems under the Platonic restriction, although he did not offer a proof of this assertion. Nevertheless, many continued to work on the classic problems in the traditional manner. Finally, the 19th century produced the definitive “solutions” to all three problems. Each problem was shown to be incapable of any solution that met the Platonic requirement, just as Pappus had stated. In 1837, Pierre Wantzel supplied a rigorous proof that an angle cannot be trisected with an unmarked straightedge and collapsing compass.

In 1882, Ferdinand Lindemann showed that π, the ratio of the circumference of a circle to its diameter, is a transcendental number, implying that the circle cannot be squared. As for the Delian problem, it had been shown even earlier that it required construction of a line whose length is the cube root of two. With a straightedge and compass, it is only possible to construct square roots.

Although it is clear to mathematicians that the solution of the classic problems under Plato’s restriction is impossible, amateurs continue to offer their “proofs” of success.