I'd like to turn the clock back to the time before Archytas discovered his solution for doubling the cube, and ask the question, “How could he possibly have come up with this unique construction?”

Certainly, the idea of looking for the intersection of a torus, a cylinder, and a cone, seems to come totally out of the blue! However, if we can get inside the mind of Archytas, and share the kind of thought processes which might lead to this kind of solution, we can learn a valuable lesson about what it means to “think geometrically.” In fact, we should be able to see that Archytas could be said to have “composed” this construction, and that the kind of thinking involved is much the same as the thought process which is required to compose a piece of music.

In reconstructing the work of Archytas and the other Pythagoreans, it's important to understand the true way in which music is integrated into a creative mind. If you read typical academic discussions of the Pythagoreans' work, they will tell you that Archytas put together the idea of an overarching science of mathematics, consisting of a “quadrivium” of arithmetic, geometry, music, and astronomy. Yet, when you look at what these academics say about music, you don't find any music at all! Instead, you find sterile discussions of vibrating strings and the “sensation of tone.” Contrast this with Plato, who states (Republic, 424c) that the musical modes are so integral to a society, that to change them means to change the underlying laws of that society.

On the one hand, Archytas's discovery cannot be reduced to a step-by-step chain of deductive logic, which leads directly from problem to solution. On the other hand, I think we can show how Archytas might have set the stage for a leap of creative genius, by “arranging” in his mind, the following elements:

• the problem;
  
  
• a passionate intention to solve the problem;
  
  
• previously known “facts” and tools which might be used;
  
  
• lessons learned, about how others had solved “intractable” problems in the past;
  
  
• his own prior discoveries about the construction of surfaces through motion;
  
  
• a creative “inkling” about new elements or methods which might be brought to bear in this particular circumstance.

How important is the “passionate intention”? Well, consider that Gauss, throughout his Disquisitiones Arithmeticae, repeatedly uses the metaphor of pursuing and cornering the enemy; while Archytas was, by profession, among other things, a military general. These folks, like Lyndon LaRouche, had a lot of fun with geometry, but they also treated it as a matter of utmost importance.

We will begin by examining a single, constructive principle, which would have been very familiar to Archytas. That principle is the method for inscribing a right triangle in a circle. We will investigate the many things which can be learned from that single, very rich principle. This is exactly analogous to the way in which a musical composer starts with a very simple theme (which, for Bach, Mozart, or Beethoven, could be as little as four notes), and extends it into many directions.

1. Basic Construction (Figure 1)

Construct a semicircle on a given diameter “OA.” From O, take an arbitrary length less than OA, say, 3/4 of OA. Swing an arc of that length, marking where it crosses the circumference, and label that point “M.” Connect OM and MA to form a triangle with a right angle at M. Drop a perpendicular from M to OA, and mark the intersection “B,” noting that it cuts OMA into two additional right triangles. Note that OMB is similar to OMA, so that OA:OM = OM:OB.

OM will therefore be the geometric mean between OA and OB.

Note that this basic construction can be reversed in its order: Given any lengths OA and OB, you can raise a perpendicular from B, to intersect the circumference at M, and thereby construct OM as the geometric mean between the two given extremes.

Now, let's examine two additional constructions which can be carried out easily, which simply involve repeated application of the procedure outlined above, for finding a geometric mean. These constructions would have been second nature to Archytas. During this phase, we're not really trying to solve the underlying problem—we're simply attempting to get some sense of the kinds of operations which it might be natural to carry out to generate a series of geometric relationships.

2. First Extended Construction (Figure 2)

Repeat the basic construction, but label point “A” as “P₀,” “M” as “P₁” and “B” as “P₂.” Next, construct a perpendicular (at an upwards slant) from point P₂ to line OP₁, and label the point of intersection “P₃.” If you examine things carefully, imagining (or, if necessary, drawing) OP₁ as the diameter of a smaller circle, you will find that 0P₃:OP₂ = OP₂:0P₁ = OP₁:OP₀.

If you reflect on what we've done, then you should notice several things:

First, that we now have a construction where OP₁ and OP₂ form two mean proportionals between two extremes (OP₀ and OP₃).

[Does that mean that we've solved the problem? No! Remember: We started this construction with one extreme (OP₀) and one of the means (OP₁). The main problem we're trying to solve, is to start with OP₀ and a given OP₃, and then to find OP₁ and OP₂—i.e., to divide the proportional interval into three identical proportions.]

Second, that each time a perpendicular cuts the prior segment, it forms a similar right triangle, and the same proportions hold.

Third, that we could continue this process indefinitely. The next step would be to drop a perpendicular from P₃ to OP₂ to yield point P₄, and then from P₄ to OP₃ to yield P₅, and so forth. In other words, we've now created a “ruler” or “scale” for finding any term in a specific geometric series.

Fourth, that all of the lines dropped perpendicular to OP₀ form one set of parallels, and all of the perpendiculars to OP₁ form a second set of parallels.

Fifth, that the numbers which designate each point (P₀, P₁, etc.) indicate the number of times that the proportional cut has been made, starting with the original diameter. (We can also refer to a series of lengths r₀ = OP₀, r₁ = OP₁, and so forth.)

Sixth, that two pairs of lengths will be in equal proportions, if the number of steps between the numbers is the same in each pair—for example, r₁:r₃ = r₂:r₄, because each pair is two steps apart on the scale.

Furthermore, you might notice that it is possible to continue the same series outwards. In particular, if we raise a perpendicular at P₀ (i.e., a tangent to the original circle at that point) and extend the line 0P₁, labelling its intersection with the perpendicular as “P_X1,” then 0P_X1:OP₀ = OP₀:OP₁.

Not bad, for such a simple procedure! And that's just the start. We can now use this construction as the basis for others.

3. Second Extended Construction (Figure 3)

Repeat the basic construction again, on a separate piece of paper. This time, label the original diameter “O” and “A₀.” Copy the chord OP₁ (length r₁) from the scale, to a similar point on the perimeter, and label it “B₁.” Drop the perpendicular and label its intersection with the diameter, “B₂.” It should be clear that the length OB₂ is r₂.

Second, take the length r₂ and, swinging upwards around O, mark where it crosses the circumference of the semicircle. Label this point “C₂.” (We will now use the subscript “2” as a mnemonic device to remind us that this point is distance r₂ from O.) Draw OC₂. Drop a perpendicular from C₂ to the diameter, and mentally label the intersection point “X.”

What is the length OX? We know from our basic construction that OC₂ is the geometric mean between OX and OA₀, i.e., that OX:OC₂ = OC₂:OA₀. When it's clear to you that OX = r₄, then you can take the point which you mentally marked “X,” and label it “C₄.”

Third, take the length r₃, from the scale. In the second construction, swinging from point O, mark a point on the semicircle which will yield a chord of that length. Mark the point “D₃.” Draw OD₃, and drop a perpendicular, to the point which will be labelled D₆.

Also, as an aside, note that we can erect a perpendicular to OA₀ (i.e. a tangent to the semicircle) at A ₀, and extend the three chords, to intersect it at points Bx₁, Cx₂, and Dx₃. Also, just as the triangel OB₁A₀, leads to construction of the series of points B₂(B₃,B₄)etc.) the triange OC₂A₀ implies that you can construct, not only C₄, but C₆, C₈ and so forth. Similarly, OD₃A₀ implies D₆, D₉, D₁₂,etc.

Someone who composes music, walks around all the day with different snatches of music playing in his mind's ear. When a composer hears a new melody, a new scale, a new poem, or a new musical idea, he immediately thinks of it as a tool, something which can be used as an element in relationship to other ideas floating around in his head.

Similarly, a geometer is always thinking about how to use new constructions as tools. Archytas would have thought about all the ways that the proportional lengths r₀, r₁, and so forth, could be used, to construct circles, squares, spheres, cubes, cones, and so forth, with the related proportionalities.

For example, if we construct a pair of circles with radii r₀ and r₁ (or pair of squares, with sides r₀ and r₁), then we can assert that the areas of those plane figures in each pair will be in the ratio (r₀:r₁) x (r₀:r₁), which is the same as (r₀:r₂). Similarly, if we create spheres or cubes with those radii or sides, the volumes will be in the ratio (r₀:r₃). (I say “assert that,” because we haven't really discussed what it means to say we will use a number to measure the ratio between two areas, or two volumes—but, we'll leave that topic for another occasion.)

Now, Archytas was also familiar with two additional, complementary principles of construction, both of which are alluded to directly in Riemann's habilitation paper, “On the Hypotheses Which Underlie Geometry.” The opening section of that paper defines manifolds of different degrees, and then describes the kinds of actions (motions, mappings, projections) which can be used to transform a manifold of one degree, into a manifold of a different degree (higher or lower).

The first of these two constructive principles relates to intersecting geometric figures: When two surfaces intersect and interpenetrate, their locus of intersection will be a line (this could be a curved line such as the circumference of a circle, or a polygonal line, such as the border of a triangle). Similarly, when two lines cross, their locus of intersection will be a point. When a line penetrates a surface, again the locus will be a point, and so on.

The second principle is, that you can create a manifold of higher degree through the motion of a manifold of lower degree: If you take a point and move it, you will create a line. If you take a line, then you can form a surface, and so forth.

Archytas is generally given credit for introducing the idea of systematically investigating surfaces from this standpoint, that is: Start with a given shape. Let it undergo a certain motion. What do we now know about the metrical properties of the surface which has been created? Conversely, you may start by knowing certain metrical properties of a line or point in space, and thereby, know how to construct a surface which will contain that line or point. We can assume that, by the time Archytas undertook to solve the problem of doubling the cube, he had investigated many, many different surfaces and solids using this method.

For example, you probably know that you can create a sphere by spinning a circle, or create a cylinder by sliding the center of a circle along an axis.

Of particular importance for us here, is the following method for constructing of a cone: Take a right triangle (Figure 4) and extend one side indefinitely in both directions. This line will become our "axis." Extend the hypotenuse similarly. Spin the entire construction around the axis to create a “double cone.” Now, if you pick any other point on the cone, you can draw a straight line back to the apex, and drop a second line as perpendicular to the axis. Note that these two lines, plus the axis, form a right triangle, and that this triangle is always of the same proportions as the original given triangle. In fact, any point in space which meets those proportions, will be on the surface of that specific cone!

In the first part of this series (New Federalist, May 26), Jonathan Tennenbaum showed how the problem of doubling the cube was equivalent to that of finding two means between two extremes. Let's restate the “two means” problem one more time, using the terminology from our prior constructions: “Can we find a construction, which can be carried out given only r₀ and r₃, in which either r₁ or r₂ can be produced?”

So far, we've discussed the problem, the available tools, and some of Archytas's particular contributions, which he could bring to bear. Now, let's add a few metaphorical elements, which involve some speculation on our part:

• Archytas must have been convinced that his method of construction through motion represented a great increase in man's power over nature in the realm of geometry. What better way to put it to the test, than to take on one of the great “unsolved problems” of his age?
  
  
• There is no simple, deductive method for solving this problem. Therefore, we need to employ the subjunctive: Were there a solution, what would its properties be? This method was known to the Greeks by the name of “analysis.”
  
  
• From the considerations above: Two surfaces will intersect to determine a line, and three to determine a specific point. If we are to use Archytas's method, then we will require the intersection of three surfaces, each of which can be constructed knowing only r₀ and r₃.
  
  
• Finally, an “inkling”: There's some sort of “double projection” we can carry out, in order to “trap” the required proportions, which would otherwise manage to squirm out of our grip like a greased pig.

Now, we've set the stage for Archytas to begin where Tennenbaum did, in Part 1.

Take into account that Archytas would likely have found and explored a series of related constructions which solved the problem at hand, and would have tried out different variants until he was satisfied that he had located the one which exhibited the greatest simplicity and pedagogical value.

Now, if you've read this far, I would strongly suggest that you take the few minutes required to make the following construction (Figure 5) in cardboard, as it will definitely make the whole process come alive to you.

Lay out a copy of our “Second Extended Construction,” flat on the surface of a table. Take a cutout of the semicircle which was our “First Extended Construction.” Stand this second semicircle up vertically, so that its point O coincides with point O on the horizontal semicircle, and P₀ coincides with A₀. Now, hold O fixed, and swing the upright semicircle away from you, until P₂ encounters the circumference of the horizontal semicircle at point C₂.

Mentally, trace a path which represents two projections: from P₁, down to P₂ (C₂) and then back towards you, to C₄. Very interesting! Projecting P₁ down to P₂ gave a line which was smaller, in the ratio r₀:r₁. However, the second projection took us from C₂ to C₄, a ratio of (r₀:r₂), or (r₀:r₁) x (r₀: r₁).

Now, shift your viewpoint, and look at O-P₁-C₄ as a triangle spanning the gap between the vertical and horizontal semicircles. If you think about it, you will recognize that the angle at C₄ is a right angle.

Now, if you were Archytas, you would be filled with a sense of impending discovery, because you would instantly recognize that the ratio of the hypotenuse OP₁, to the side OC₄, is the ratio (r₀:r₁), applied three times—that is, the very ratio, (r₀:r₃), which we've been trying all this time to divide. For us lesser mortals, however, it will probably be necessary to physically construct a right triangle (with side r₄ and hypotenuse r₁), cut it out, and see how it fits in the gap (Figure 6).

Archytas would have also recognized the implication that OP₁ must be a radial line of the cone, constructed from the ratio (r₀:r₃). To see how this works, remove the vertical semicircle from your work area. Hold the new triangle so that it continues to touch line OC₄, and swing it down and back to lie flat (Figure 7). Notice that it aligns exactly with the triangle 0D₃D₆, and that OD₃ is simply a chord of length r₃! If you swing this triangle up and down, holding the side 0C₄ steady, you will see how the motion of its hypotenuse traces a cone.

However, we still have a bit of work left. Remember, that the above construction started with a specific, known length r₁. We still need to re-cast the sequence to show how to find r₁, knowing only r₀ and r₃.

As Tennenbaum pointed out in Part 1, given a length r₀ (OA₀) we can pick any value of r₁, larger than 0 and smaller than OA₀, and generate a structure like Figure 5. For each value r₁, we will get a unique point P₁. True, the specific proportions will change; however, the relationships between the proportions will be invariant. (see Figures 8a, 8b, 8c)

Archytas's method is to look at all possible values of r₁, and see what resulting curved path (the “bold curve”) describes the locus of all possible points P₁. Then, if we have a specific, known value for r₃, we will be able to pick out the one point on the bold curve which we require, because it will be the only point on that curve which intersects the cone.

Specifically, we know that as we vary r₁:

• The family of vertical lines “P₁P₂,” when the lines are extended, traces out a cylinder (also determined only by r₀ alone).
  
  
• The motion of the upright circle (OP₀) traces out a torus (a figure determined by r₀ alone).
  
  
• The bold curve, which is the sequence of points labelled P₁, will therefore consist of the intersection of the torus and the cylinder.
  
  
• And, finally, if we construct the cone as above (from the ratio r₀:r₃), then its intersection with the bold curve will yield the desired point P₁, and we will be able to construct P₂, as well (Figure 8d).

Now that we see what kind of conceptual thinking Archytas employs in geometry, we are in a position to begin investigating the musical ideas of the Pythagoreans, and other Classical Greeks, as well. In particular, we can raise the question: To what extent was their music polyphonic? Witness the way in which Archytas created the crucial breakthrough on his construction: He took a simple principle (the “Basic Construction”) and extended it in at least two different ways—the first and second “Extended Constructions.” Then, by juxtaposing the results of those two constructions in a unique way (by raising one semicircle to a vertical position, and rotating it) he developed a “cross voice” between the two, generating a relationship which otherwise would not be seen to exist. But that cross-voice principle, as a method of composition, is exactly the method which characterizes all polyphonic music. Therefore, since Archytas binds music and geometry so closely together, as areas of science, we cannot help but suspect that when he and his contemporaries talk of music, they mean polyphony.

Nevertheless, so little exists, in written form, of the music of Classical Greece, that there is no reliable, direct representation of exactly what that music sounded like. What representations do exist, have been subject to the prejudices of those who chose to translate and interpret the few writings that survive. I will therefore leave it as a challenge, to rediscover the true nature of the music which Archytas and Plato might have sung.


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