Cubes, Conic Sections, and Crockett Johnson - Mean Proportionals and Doubling a Cube

As described by Eratosthenes [Knorr 1986, 23],

Hippocrates of Chios was first to come up with the idea that if one could take two mean proportionals in continued proportion between two lines, of which the greater is double the smaller, then the cube will be doubled.  Thus he turned one puzzle into another one, no less of a puzzle.

It may not be clear, however, how considering the equivalent problem of “two mean proportionals” will lead to a solution of doubling a cube; i.e., given a cube with edges of length \(a\) and volume \(a^3,\) how can we construct a cube with edges of length \(\sqrt[{\scriptstyle 3}]{2a^3}=a{\sqrt[{\scriptstyle 3}]{2}}\) and volume \(2a^3\)?  We saw above that a mean proportional, or geometric mean, is the solution to the equation \[{\frac{a}{x}}={\frac{x}{b}},\] i.e. \(x=\sqrt{ab}.\)  Two mean proportionals are the solutions to the three equations \[{\frac{a}{x}}={\frac{x}{y}},\,\,{\frac{x}{y}}={\frac{y}{b}},\,\,{\rm{and}}\,\,{\frac{a}{x}}={\frac{y}{b}},\] which can be written more compactly as \[{\frac{a}{x}}={\frac{x}{y}}={\frac{y}{b}}.\] If \(a\) and \(b\) are taken to be the lines mentioned in Eratosthenes’ account, “of which the greater is double the smaller,” then \(b = 2a\) and the equations can be written \[{\frac{a}{x}}={\frac{x}{y}}={\frac{y}{2a}}.\]

Once again we can discuss the solutions in terms of geometric means.  In general the geometric mean of \(n\) numbers is the \(n\)th root of the product of the \(n\) numbers, so the geometric mean of \(a, b,\) and \(c\) is \(\sqrt[{\scriptstyle 3}]{abc},\) which is the length of the edge of a cube with the same volume as a rectangular prism with edges of length \(a, b,\) and \(c.\)  If we consider a square-based rectangular prism with edges of length \(a, a,\) and \(2a,\) then to find the length of the edge of a cube with volume \(2a^3,\) we need to find the geometric mean of \(x^3=2a^3\) which is \({\sqrt[{\scriptstyle 3}]{2a^3}}\) or \(a{\sqrt[{\scriptstyle 3}]{2}}.\)  This is the smaller of the two mean proportionals of \(a\) and \(2a.\)  (The larger, \({a\sqrt[{\scriptstyle 3}]{4}},\) is the length of an edge of a cube of volume \(4a^3.\))  Thus it is natural to assume that \(b=2a\) when doubling a cube.

In addition to a geometric interpretation, if we look again at \[{\frac{a}{x}}={\frac{x}{y}}={\frac{y}{2a}}\] as three separate equations, we discover that they define two parabolas, \(x^2=ay\) and \(y^2=2ax\) and a hyperbola \(xy=2a^2.\)  Menaechmus, who was mathematically active in the middle of the fourth century BCE, is credited with discovering conic sections while considering mean proportionals.  In a History of Greek Mathematics, Heath wrote [Heath 1981, I:251],

Two solutions by Menaechmus of the problem of finding two mean proportionals are described by Eutocius; both find a certain point as the intersection between two conics, in the one case two parabolas, in the other a parabola and a rectangular hyperbola.  The solutions are referred to in Eratosthenes’ epigram: “do not”, says Eratosthenes, “cut the cone in the triads of Menaechmus.”  From the solutions coupled with this remark it is inferred that Menaechmus was the discoverer of the conic sections.

More recently, Knorr conjectured that Menaechmus did not actually use conic sections but, rather, that he could have constructed curves in a pointwise fashion.  Specifically Knorr proposed “that Menaechmus based his solution on curves defined with respect to second-order relations among the mean proportional lines” [Knorr 1986, 65].  The series of diagrams he showed to support his conjecture [ibid., 64-65] are much more complicated than those used by Crockett Johnson that we describe below.  Gary Stoudt also discussed the question of how Menaechmus arrived at a solution in “Menaechmus’ Constructions” and “Why Do Menaechmus’ Constructions Work?,” the first two sections of his 2004 Convergence paper “Can You Really Derive Conic Formulae from a Cone?”  

Euclid, who lived about fifty years after Menaechmus, discussed in his Elements mean proportionals in the context of cubic numbers such as \(1, 8, 27, 64,\) etc.  The statement of Proposition 12 of Euclid’s Book VIII, begins “Between two cube numbers there are two mean proportional numbers” [Euclid 1956, II:364].  Heath’s commentary following the proof of that proposition notes that “The cube numbers \(a^3,\) \(b^3\) being given, Euclid forms the products \(a^2b,\) \(ab^2\) and then proves” that these are the two mean proportional numbers between \(a^3\) and \(b^3\) [ibid., 365].  Thus Euclid had found solutions to equations equivalent to ours; i.e., \[{\frac{a^3}{x^3}}={\frac{x^3}{y^3}}={\frac{y^3}{b^3}}\] so that \(x^3=a^2b\) and \(y^3=ab^2.\)  In other words, Euclid found that \({\sqrt[{\scriptstyle 3}]{a^2b}}\) and \({\sqrt[{\scriptstyle 3}]{2}}\) are the two mean proportionals between \(a\) and \(b.\)  We, too, could have found these solutions if we first solved the three equations to get \(x^2=ay,\) \(y^2=bx,\) and \(xy=ab.\)