Since the individual grasp of Architas’ solution to the Delian problem can provide evidence for understanding the nature of insight, and increasing the ability of other minds to acquire it, this is a short account of mine, along with a consideration for the thinking that may lead toward such a solution.
We would consider, of course, the restatement of the problem of doubling a cube as the problem of finding two geometric means between two extremes.
Having experienced the relative ease of finding one geometric mean by means of a circle with respect to the challange of finding two, an epistemological question arises: Whence this difference between (dividing a ratio into) two and three (parts)? Or, also: How is the three to be known from the two? Considering that this question may be viewed easier not in terms of duplicate and triplicate ratios (sometimes called square and cube ratios), but rather by arithmetic means, we consider how this difference is expressed in the difference between bisecting and trisecting a given length.
We can arrive, more or less directly at a half and at a quarter of the length, but not at a third. However, the desired ratio (1 : 3) is expressed when a length is divided into four equal parts; namely, between one and three of them. The typical solution to finding a third part of the initial length is to, then, replicate the ratio between the 1 part and the 3 parts, but now both proportionally enlarged in concert until the 3 parts coincide with the inital length (the one to be trisected). Then the 1 part, enlarged in the same proportion as the 3, had become the sought third part of the initial line.
Such is the practical solution to this problem, without giving us a solution, but only an example, to the wonder of any of the causes of the ontological difference between the half and the third, such as the difference between coincidence with itself and coincidence with another (identity and duplication), as found in literature and in geometry.
If we shall apply this practical lesson to the matter of trisecting a proportion (such as 1 : 2), we shall resort to the literary method of analogy&emdash;with the trisection of a length. Were we to start with a ratio between two extremes, such as AQ : AO, placing them on top of each other, so that they coincide at one end point, we have the known construction of the geometric mean AP as a chord in the semicircle APO by erecting QP perpendicular to AO in circle with diameter AO.
Having split the proportion AQ : AO in half, we construct the mark dividing the 3 parts (first three quarters) of this proportion from its last quarter by splitting the proportion AP : AO by means of the same construction: taking AS = AO along the line AP and PR as perpendicular to AS, R being on the semicircle ARS. Thus, AR : AS is half of the proportion AP : AS, and a quarter of the proportion AQ : AO, so that of the four parts into which we have split AQ : AO we lack the boundary separating the first two, but only have them combined as AQ : AP, whereas the third part is AP : AR, and the fourth one AR : AS. Thus, AQ : AR is the triplicate of the proportion AP : AR = AR : AS.
Our analogy now leads us to reduce AQ until the lacking boundary would coincide with the first of our extremes, while AP would be proportionally reduced to the first of our means, AR to the second one, AS = AO remaining the second extreme all the while (AQ becoming smaller than the first extreme the way in the analogous problem the first endpoint of the line divided in 4 goes outside the line to be trisected).
If concerns are now arising over how we are to carry this construction out, since this process of enlarging the proportional distance between AQ and AO is to stop when our first extreme coincides with a phantom boundary, let the purpose of concerns be not to suppress us, but to provide our free will with a struggle that can lead us to ingenuity, remembering that the insight to overcome challenges is what we can leave to others even after our time has passed.
We can then see that, since we are required to proportionally enlarge proportions, and that until a definite proportion is reached between each of these proportions and its triplicate, the true boundary of the process of enlargement is not a line, an extreme, but rather only a proportion between lines, regardless of their specific size. Thus, if it be required for the proportion between the two extremes to be 1 : 2 (to double the cube), it is simply required to stop the process of enlargement at the point any of the proportions has a triplicate which is 1 : 2. Therefore, we can stop the process of shrinking AQ when AQ : AR (the triplicate of AP : AR = AR : AS) becomes 1 : 2.
Our construction has now become the following. We would like to rotate circle ARS around point A, generating different points P, and having a perpendicular to AS from P always generate the corresponding location of point R (on circle ARS) for the different positions of P. We would like to stop that process when AQ : AR becomes 1 : 2. Then, if AS is twice the edge of the inital cube, AP will be the edge of its double, AR of its quadruple, etc.
How might we physically construct this? Circles, movable points on them, as well as lines between given points are easy enough to construct. What remains is to construct the means by which to stop the motion of P on the circle APO at the moment that AQ : AR becomes 1 : 2. How might we determine when two lines of changing length have reached a given proportion?
What is common between all possible pairs of lines, such that the lines in each pair have the same given proportion between themselves? Geometricallly, this is, of course, expressed as a triangle with fixed angles. A trangle with the same set of angles, and proportions between the sides can be formed by moving one of its sides so it stays parallel to itself.
We could use such a construction, if, for example, we could keep either angle ARQ or angle AQR constant while P was moving, stopping the process when RAQ becomes the angle which corresponds to the given proportion 1 : 2.
Could that be done?
To execute it, should we, as Gauss did, take our mind off the ground, and think not of the traces we have left on the plane, but rather of the physical (and mental) processes that had cast them there, we may, as Archytas did, erect circle ARS to make its plane perpendicular to that of circle APO, the rotation of ARS around point A now tracing out half a torus, and the line RP lying on a half cylinder, perpendicular to its base APO. We need to stop the process of rotating ARS around A when AR becomes twice AQ. In other words, when the perpendicular from point R (lying on both the torus and the cylinder) to a point Q on AO forms a right triangle RQA in which the hypotenuse AR is twice the leg AQ. All such possible triangles share the same angle RAQ, so, even if we do not know the specific sides of the one we are looking for, if we take a line forming the correct angle with AO (60° for AR = 2AQ) and, keeping that angle constant, spin the line around AO, it would trace all the possible positions for point R to lie on, its trace being what we call a cone.
* * *
Thus, we acquire the fruit to nourish our mind. And, like on the path form seed to grape, the turmoil has turned into sweetness under the light of the sun.