Kepler tells us that Pappus solved the problem of trisecting of the angle using a hyperbola. Before addressing Pappus’s construction as such, ask: from where did this hyperbola come? Who dreamed up these conic sections; who imagined the cone and thought to slice it through? What could be found through intense investigations of the sun, of its settings and risings and the changing shadows it creates? Could we notice the curve of the hyperbola as a momentary shadow cast by the sun as it sinks below the horizon and only subsequently perceive the cone from which it was projected?
This path leads away, into uncharted territory, into optics and deeper studies of the Greeks. Yet, there is something different here than the previous procedures Kepler maps and demonstrates all lead to the same mire. Conic sections seem to provide real paths out of the labyrinth, or at least seem to impart greater substance to the thread that we have grabbed hold of.
Now, leave Kepler’s side for a moment, so we can return to him the better equipped to finish our travels, and quash the creature of our doubts.
Conic Sections
A discovery is attributed to Menaechmus, the student of Eudoxus, about 2400 years ago. This discovery bestows new found power to solve seemingly unsolvable problems, like doubling the cube and trisecting the angle, with the properties of the cone and its sections.
How did the Greeks discover this hidden capability embedded in the cone, a capability which is also reflected in Archytas’s solution for the doubling of the cube? Perhaps it develops naturally out of optical investigations that present themselves necessarily as questions related to astronomical observation. Or perhaps the idea of the cone arises from inquiring into how we see, how the process of vision, as entwined with the phenomena of light, works, not unlike the way in which we can imagine the idea of the sphere arising from conceptualizing the celestial sphere. But that is another story, for another time.
The latent power of cones is first documented in Menaechmus’s solution to the problem of finding the length of the side of a cube double the volume of a given cube. Bruce Director, in pedagogical 33 of his Riemann for Anti-Dummies pedagogical series, covers the essential points of how Menaechmus solved this problem as part of a broader discussion of hyperbolic functions. Follow Menaechmus’s solution in greater detail for a somewhat different purpose, but refer to Director’s pedagogical for a different perspective on this land into which we find we have trodden.
The problem with inquiries into Greek geometry, science, and astronomy is: often the accounts of the discoveries and the discoverers are excerpted from anthological accounts written centuries later then the discovery was made and the discoverer lived. And always ask, “What was the intent of the person writing the account?”
In the short passage on Menaechmus’s solution for the doubling of the cube, written by Eutocius, who was likely one of the successors of Proclus as the head of the Platonic academy in the middle of the sixth century AD, 800 years after Menaechmus was said to have lived, there is a reference to the first known development of the geometry of conic sections. Menaechmus’s conic sections are also mentioned in a letter written by Eratosthenes around 200 BC. In Eutocius’s account, Menaechmus begins with what he knows must be the proportional relations inherent among the magnitude that acts as the side of the doubled cube, the side of the cube of one, and a side of 2, which would produce a cube of 8. He knows that just as there are two “doubled” cubes “between” a cube of one and a cube of 8, namely a cube of 2 and a cube of 4, so there must be two geometric means between the side length of 1 and that of 2, which would produce the cube of 2 and the cube of 4. Many people have written a great deal on this problem, it is covered in a cursory way in Riemann 33 referred to above, and is covered in great depth in other pedagogicals of that series on Archytas, as well as in pedagogicals which follow this one. For the present, remember this famous quote in Plato’s Timaeus, where Timaeus proclaims, “If it were necessary that the universe should be a superficies only, and have no depth, one medium would indeed be sufficient, both for the purpose of binding itself and the natures which it contains. But now it is requisite that the world should be a solid: and solids are never harmonized together by one, but always with two mediums.” (Timaeus) The nature of the problem is plainly the difference between planes and solids.
But what is that nature? Let us not simply substitute one name for another. Plunge ahead into Menaechmus’s discovery.
Parabola
First he sets out the proportion we wish to construct in which two means are located between two extremes. He then declares that were these two lengths to exist, they would have the property of defining a point that lay both on a parabola and a hyperbola of specific characteristics. Eutocius’s description of Menaechmus’s construction, the proof that this intersection of a given parabola and hyperbola does in fact determine two mean proportionals between two extremes, and the following “synthesis,” in which the intersection of the two curves is constructed, is truncated and abrupt. Neither proof nor synthesis gives any insight into how Menaechmus conceived of the conic sections he employs, nor how he knew what he did of their properties. The description itself must be unwound, for the brief instructions related in that passage are not straightforward.
First, the parabola is described as a curve defined as a process of generating all the geometric means between a fixed extreme and a second extreme which increases from 0 indefinitely.
How does this work?
First think of the circle, which uniquely generates all single means between two extremes. If the diameter is held as one extreme, then the other two magnitudes which we seek are: 1. the chords of the arcs swept out as the radius rotates two right angles, tracing the semi-circle that has the doubled radius for diameter, and 2. the segments of the diameter cut off by perpendiculars dropped from the arc swept out to the diameter. The chords of the arcs are then the geometric means between the whole diameter and the segment of the diameter cut off at each point.
In this animation, the chords are drawn in different colors, while the extreme which corresponds to each is drawn on the diameter in the same color. Each cord is then the geometric mean between the corresponding segment of the diameter and the diameter itself.
This animation shows how the parabola could be generated in this manner. Each generated extreme becomes the distance along the horizontal axis to which the height of the cord (the geometric mean) corresponds. As these segments fall out of the circle, see how they form the skeleton of the parabola.
This animation is intended to show how the parabola is then defined by this skeleton. The final part of the animation was created to show more clearly that the curve just drawn is a segment of a parabola.
But this is not how Eutocius describes Menaechmus’s parabola. Eutocius begins with the two mean proportionals between two extremes needed to double the cube. He takes a line that he labels "A" and sets it equal to the larger extreme, in this case 2. He sates that "A" is the "latus rectum" of the parabola. After defining the parabola’s axis, he states: “let the squares of the ordinates drawn at right angles to the (axis) be equal to the areas applied to A having as their sides the straight lines cut off by them towards (the origin, or beginning of the axis.)”
At times, simply deciphering the directions given to get to the next location requires the greatest amount of work, while the trip itself ends up a fairly simple and direct one.
Instead of generating the lengths for the parabola’s construction inside the circle and then subsequently assembling the frame for the parabola, is there a way to generate the parabola itself, as Eutocius describes? And in the previous method employed, in which the parabola’s structure is generated from the circle, what happens after we reach the point where the circle’s diameter and the cord of the angle swept out coincide? Does the parabola stop there?
That moment of coincidence does represent an interestingly singular point, but the process can not possibly stop there.
If we look again at Eutocius’s description, disregarding an unhelpful translator’s insertion of the word “ordinate” in place of what Eutocius actually says, there clearly is a relationship between, on the one hand, the squares drawn on the axis along with the areas applied to A, and, on the other, our first attempt at creating Menaechmus’s parabola. For one of the properties of geometric proportion is that a square built on a geometric mean is equal to the rectangle whose sides are the two extremes between which the mean falls. Thus, as in our circle, the height of the parabola would be the geometric mean between the latus rectum and a quantity that increases from 0. Instead of making the latus rectum the diameter of the circle, now we make it a segment of the diameter of a circle which is growing, so that it is always one extreme, the other segment of the diameter is the other, and the line drawn perpendicularly from the latus rectum to the circle is the geometric mean between the two.
In this animation, we begin with the circle whose diameter is the latus rectum. As this circle grows, this length remains fixed, and so becomes an ever smaller portion of the total diameter. The mean between this length and the remaining diameter is then traced out along the horizontal axis. As demonstrated, this is equivalent to generating a series of right triangles as the circle grows, where the base of each triangle is the diameter of the circle and the vertex always remains on the horizontal axis.
The series of corresponding horizontal and vertical components of the parabola are thus generated as a single process as the initial circle grows.
This animation is intended to reveal this process of generation.
This animation shows that this process maintains the equality of the areas which Eutocius describes. The yellow square constructed on the horizontal axis, and the yellow rectangle constructed from the latus rectum and the other extreme are always equal in area. The animation pauses at the moment when the rectangle and square are identical figures, when the distance traveled along the horizontal and vertical axes is the same. This is the same moment that ended our previous attempt at constructing the parabola.
Hyperbola
After successfully navigating the parabola, Eutocius’s description of the hyperbola’s construction is by far more apparent. He takes the two axes of the parabola and defines a hyperbola with these as asymptotes. At the point he is designating, the rectangle formed from the horizontal and vertical distances delimited by the perpendiculars dropped from this point to the asymptotes has an area equal to the area of the rectangle formed from the initial extremes Eutocius set out in his description. One of these extremes is A, the latus rectum of the parabola we drew. Menaechmus’s construction is a solution to the problem of finding the two geometric means between 1 and 2. If A is equal to 2 then the other extreme is equal to one, and the rectangle formed from them would have an area of two. The rectangular hyperbola is a curve such that, when perpendiculars are dropped from any point on the curve to the two asymptotes, the rectangles formed from these two lengths are always equal. So in the case of Menaechmus’s hyperbola, the horizontal and vertical components of any point will always form a rectangle whose area is 2.
This animation shows the process of generating the rectangular areas of 2 which are defined by Menaechmus’s hyperbola, and which we then use, conversely, to create the hyperbola about which he speaks. We begin with the circle with diameter 2, equal to the latus rectum of the parabola. The radius of this circle is therefore 1. The diagonal of the square built off the radius is the side of a square of two. This in turn is equal to a rectangle whose sides are 1 and 2, the two extremes that Menaechmus laid out. This rectangle is equal to another whose sides are 4 and 1/2. Notice that the square of 2 represents a singular moment in generating the hyperbola and defines this hyperbola’s vertex.
Here we draw out the hyperbola as described so that the axes of the parabola are the asymptotes of the hyperbola. Where they intersect, the shorter, horizontal perpendicular is the cube root of two, while the longer vertical length is the cube root of 4. Given the properties of the hyperbola and parabola with which we have been working, do you see why this is true? (Take note that in some of these animations, the parabola is oriented differently than in others.)
Just as the parabola could be conceived as a single motion instead of as a step-wise activity, so the hyperbola can be seen as a continuous process. The latus rectum of the parabola swings up to become the diagonal of a square of two, which in turn defines the vertex of Menaechmus’s hyperbola. The hyperbola is then described as that square continuously changes into longer and longer rectangles that continue to delimit areas of 2.
Here is the Menaechmus solution generated as part of a continuous development.
