Cubes, Conic Sections, and Crockett Johnson - Three Greek Problems (2024)

An ancient Greek problem in geometry was a very specific type of problem. Wilbur Knorr opened his book, The Ancient Tradition of Geometric Problems, with the explanation [Knorr 1986, vii]:

Within ancient geometry, a geometric ‘problem’ seeks the construction of a figure corresponding to a specific description.

Here the constructions are meant to be limited to those that can be carried out using only an unmarked ruler, usually referred to as a straightedge, and a compass. The three classical Greek problems seek

1. the construction of a square with the same area as an arbitrarily given circle,
2. the construction of a cube with twice the volume of an arbitrarily given cube, and
3. the construction of an angle, three congruent versions of which together comprise an arbitrarily given angle.

These problems are commonly called (1) squaring a circle; (2) doubling, or duplication of, a cube; and (3) trisecting an angle. In addition to Knorr’s book, one can find a description of these three famous problems in Sir Thomas Heath’s two-volume history of Greek geometry [Heath 1981, I:218-270] and in general textbooks on the history of mathematics; for example, those by Katz [2009] or by Merzbach and Boyer [2011].

The problems of squaring a circle and doubling a cube were referenced as early as the fifth century BCE. Plutarch wrote that Anaxagoras sought to square the circle while in prison for stating the sun was not a deity. About a year before Anaxagoras’ death in 428 BCE, a severe plague erupted in Greece and it was reported that in response to the plague an oracle was offered by Apollo on the Island of Delos. The oracle advised that there would be relief from the plague if the altar at Delos was doubled, and, according to Theon of Smyrna’s Mathematics Useful for Understanding Plato, “much bewilderment fell upon the builders who sought how one was to make a solid double of a solid” [Knorr 1986, 21]. Because of this oracle, and the lack of success in doubling the Delian altar, the problem of doubling a cube is also called the “Delian problem.”

In response to a request for an explanation of the Delian oracle, Plato is reported as having [Knorr 1986, 2]

said that the god was making sport with the Greeks for their neglect of education, as it were taunting us for our ignorance and demanding that we engage in geometry, and not just as a pastime.

While Plato may have thought the Greeks did not take geometry very seriously, they took it seriously enough to classify geometric problems into solid, planar, and linear problems. Knorr quoted Pappus describing this classification of “the ancients” as [Knorr 1986, 341]

those which are capable of being solved … by means of straight line and circular arc would … be called “planar”; for the lines by means of which such problems are found … have their genesis in a plane. But those problems which are solved when there is assumed toward their discovery … one or several sections of the cone are called “solid”; for their construction … necessarily employs surfaces of solid figures, namely the conic surfaces. Yet a third kind of problem is left, the one called “linear” for there are taken for their construction lines different from those just mentioned, having a more diverse and rather contrived genesis.

Although the third classification is called linear, the term refers to a problem whose solution requires a curve that is not a straight line, a circle, or a conic. Many Greek philosophers, particularly Plato, believed that the ideal geometric construction was planar and therefore was to be drawn without the use of mechanical contrivances; i.e., the drawings were to be done with only a compass and an unmarked straightedge. Once these restrictions were more fully defined, the third problem of trisecting an angle was introduced. This problem possibly arose from attempts to construct regular polygons, which are polygons of equal sides and equal angles. In particular, if one can trisect a \(60^{\circ}\) angle then one can construct a \(20^{\circ}\) angle. Since the measure of an angle of a regular \(n\)-gon is \(\left(\frac{n-2}{n}\right)180^{\circ},\) the angle of a nonagon (a 9-sided polygon) is \(140^{\circ},\) or seven \(20^{\circ}\) angles. Therefore, copying a \(20^{\circ}\) angle makes the construction of a regular nonagon straightforward.

The three ancient problems of squaring a circle, doubling a cube, and trisecting an angle had a profound influence on mathematics. An incredibly wide range of people, from ancient Greece until modern times, have worked on these problems; among them are Archimedes, Leonardo da Vinci, René Descartes, Isaac Newton, and Carl Friedrich Gauss. As Victor Katz wrote in A History of Mathematics [Katz 2009, 40]:

The multitude of attacks on these particular problems … serve to remind us that a central goal of Greek mathematics was geometrical problem solving, and that, to a large extent, the great body of theorems found in the major extant works of Greek mathematics served as logical underpinnings for these solutions.