The Duplicators, Part I: Eutocius’ Collection of Cube Duplications - Menaechmus' Notes on Conic Sections

Biographical note: Menaechmus (ca 380 BCE–ca 320 BCE) is responsible for discovering the conic sections during his investigation of how to solve the cube duplication problem. The 10th century Greek encyclopedia, the Suda, also has a reference apparently to him, suggesting he also worked in philosophy. Read more about Menaechmus at MacTutor.

The Greek view of conic sections is considerably different than our own, and so I think it wise to provide a brief introduction. I personally am always surprised when some of my students have no idea why the conic sections are called “conic sections”: in their view, parabolas are synonymous with equations like \begin{equation}y = x^{2}. \end{equation} But we use the term we do precisely because of the Greek mathematical heritage; namely, the conic sections are the results of cutting a cone with a plane. 

Greek mathematics actually had two ways of defining the conic sections, corresponding to two different ways to define a cone. The first, likely due to Menaechmus himself, and still in use by Euclid and Archimedes, was that the three types of cones came from rotating the different kinds of right triangles about one of their legs. The three types of triangles are:

1. A triangle where the adjacent angle, \(\angle ACB,\) is half a right angle (so the triangle is also isosceles):

2. A triangle where the adjacent angle, \(\angle ACB,\) is greater than half a right angle:

3. A triangle where the adjacent angle, \(\angle ACB,\) is less than half a right angle:

Hence the three conic sections came from cutting these three different kinds of cones. In each situation the cutting plane was aligned at right angles to the “side” of the cone, where here “side” refers to the hypotenuse of the generating triangle. The three kinds of sections are:

1. The section of a right-angled cone (our “parabola”). Here the cutting plane is perpendicular to one side of the cone, and parallel to the other. Move point D to change the location of the cut.

2. The section of an obtuse-angled cone (half of our “hyperbola”). The obtuse angle here is the angle at the vertex. Apollonius did consider the "other half" of the cone, but as a different object. For our purposes, it is sufficient to consider only one "half" of the cone, as presented here, giving us only one of the two branches in a “modern” hyperbola. Move point D to change the location of the cut:

3. The section of an acute-angled cone (our “ellipse”). Move point D to change the location of the cut:

Under this view the circle would not have been considered a conic section in the same sense, because the only way to get one from a cone was to have the cutting plane parallel to the base of the cone, not perpendicular to an edge. 

The second view, due to Apollonius and attested to in the Conics, redefines cones in such a way that only one definition is necessary. Apollonius defines a cone by first considering a circle (the “base” and a point not in the plane of that circle (the “vertex”). For each point on the circle, draw the line from that point to the vertex. Hence Apollonius’ cones can be “lopsided,” whereas the Euclidean view does not permit that. Apollonius calls the “lopsided” variety oblique. When the line drawn perpendicular to the plane of the circle, through the circle’s center, passes through the vertex, he calls the cone right (but note: right to Apollonius is not the same thing as right-angled to Euclid). In this way, Apollonius can form all three conic sections (which he dubs parabola, hyperbola, and ellipse) by cutting a single cone in three different ways. 

Since each conic section is formed by the intersection of a cone and a plane, we can restrict our attention to this cutting plane only, and the conic section within it. This will more easily allow us to state the details of the basic properties, or symptoms of each section. This is of course different than proving these basic properties: for this, Apollonius needs to keep the sections and the cone in view. Conics I.11 and I.12 contain the details of Apollonius’ proofs for the parabola and hyperbola, respectively. The case of the ellipse is not necessary for what follows, and so I omit it.

For the parabola, we start with the diameter, which in many cases resembles what the modern reader might call the axis. A segment of the diameter is called an abscissa, from Latin abscindere, to cut or separate. A line drawn from the diameter to the section is called an ordinate, and for our purposes these ordinates are perpendicular to the diameter.^[27] We then consider a segment, called the parameter, so that the square on the ordinate is equal to the rectangle contained by the parameter and the abscissa. 

Above: In this figure, the diameter is AB, the abscissa is AC, and the ordinate is CD.
The parameter is AE, and satisfies sq.(CD) = rect.(AC, AE). Point C is movable.

The case of the hyperbola, as far as Menaechmus’ solutions are concerned, is a bit different than what Apollonius first defines in Conics I.12. Apollonius does not deal with the concept of asymptotes until Book II of the Conics, and hence his symptom of the hyperbola has to be defined in a different manner (see Stoudt’s article [2015] for more details on Apollonius’ approach in I.12). Menaechmus uses the property that if a hyperbola has perpendicular asymptotes, then the rectangle contained by perpendiculars drawn from a point on the hyperbola to each of the asymptotes has a constant area. Apollonius gives the details in Conics II.12; but it should be noted that for Apollonius, this is not the symptom of a hyperbola, but rather a property that follows from it.^[28]

Above: In this figure, the lines ACF and AEB are the asymptotes. The two constant lengths are AF and AE.
For a given point D, the lines CD and BD are such that rect.(BD, CD) = rect.(AF, AE). Point D is movable.

[27] In Apollonius, this is not necessarily the case, but for Menaechmus and what follows, we can assume it is. As usual, see the Conics for full details! 

[28] In his commentary on the Conics, Eutocius uses makes the distinction between the principal properties, τὰ ἀρχικὰ συμπτώματα, and the resulting properties, τὰ παρακολουθήματα συμπτώματα. For more details on the distinction, see my dissertation [2010].