Doubling the Cube or Archytas’s sublime Discovery

1. Doubling the Cubeor Archytas’s sublime Discovery

2. Delian Paradox “Eratosthenes, in his work entitled “Platonicus’ relates that, then the God proclaimed to the Delians through the oracle that, in order to get rid of a plague,they should construct an altar double that of the existing one, their craftsmen Fell into great perplexity in their efforts to discover how a solid could be made The double of a similar solid; they therefore went to ask Plato about it, and he Replied that the oracle meant, not that the God wanted an altar of double size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contemt of geometry.”

3. The Narrow Path • Greeks (Pythagoreans and Plato) • Johannes Kepler • Pierre de Fermat • G. W. Leibniz • Karl Gauss • Bernhard Riemann

4. The Narrow Path • This narrow path, was designed to provide an insight into human creativityper se, rather than the mathematical reformulations found in textbooks and on internet encyclopedias.

5. Archytas' solution for the problem of doubling the cube a method as advocated by Archytas himself: • "To become knowledgeable about things one does not know, one must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find."

6. A citizen in 2010 A.D., wishing to muster the conceptual power necessary to comprehend today's historical, political and economic crisis, and to act to change it, will find it of great benefit to bind into one thought, Archytas' construction for finding two mean proportionals between two extremes, (circa 400 B.C.), with Bernhard Riemann's 1854 lecture, "On the Hypothesis which Underlie the Foundations of Geometry". Two thoughts, separated temporally by 2400 years, recreated simultaneously by one mind yours.

7. "As for me, I cherish mathematics only because I find there the traces of the Art of Invention in general, and it seems to me I have discovered, in the end, that Descartes himself did not yet penetrate into the mystery of this great science. I remember he once stated, that the excellence of his method, which appears only probable in terms of his physics, is proven in his geometry. But I must say, that it is precisely in Descartes' geometry that I recognized the principle imperfection of his method... I claim that there is an entirely different method of geometrical analysis, than that of Vieta and Descartes (i.e. algebra), who did not go far enough, because the most important problems do not depend at all upon the equations, to which Descartes reduces his geometry." (Leibniz, Letter to Princess Elisabeth, late 1678)

8. It must be kept in mind, that Archytas, like Riemann and Gauss, was anti-Euclidean. In fact, even Euclid was more anti-Euclidean than today's neo-Aristotelean followers of Galileo, Newton and Kant, who assert that physical space-time conforms, a-priori, to the axioms, postulates and definitions of Euclidean geometry. No where in Euclid's Elements is such a preposterous assertion stated. Rather, the Elements are a compilation of earlier discoveries that could never have been produced by the logical deductive methods used in the Elements. The actual method of discovery is best indicated when the Elements are read backwards; from the 13th book on the spherical derivation of the five regular solids, to the 10th book on incommensurables, to the middle books on proportions, to the opening sections on the circular derivation of constructions in a plane.

9. "Let the two given lines be OA [= a] and b; it is required to construct two mean proportionals between a and b. Draw the circle OBA having OA as diameter where OA is the greater [of the two]; and inscribe OB [as a chord on the circle] of length b, and prolong it to meet at C the tangent to the circle at A. ... Imagine a half-cylinder which rises perpendicularly on the semicircle OBA, and that on OA is raised a perpendicular semicircle standing on the [base] of the half-cylinder. When this semicircle is moved from A to B, the extremity O of the diameter remaining fixed, it will cut the cylindrical surface in making its movement and will trace on it a certain bold curve. [The latter motion generates a section of a torus--JT.] Then, if OA remains fixed, and if the triangle OCA pivots about OA with a movement opposite to that of the semicircle, it will produce a conical surface by means of the line OC, which, in the course of its movement, will meet the curve drawn on the cylinder at a particular point P....«  • …Eudemus:

10. LaRouche: • « There is a certain tendency which is, unfortunately, typical of modern classroom and related mathematics. This involves a trait which is likely to be noticed for its frequent occurrence among otherwise presumptively qualified secondary and university students and graduates. It is an acquired mental disorder known as “positivism,” a type of disorder expressed as a viciously systemic form of reductionism, notoriously common to certain types of university professors and their students.

11. It is occasionally appropriate to attack the symptoms of that mental disorder simply because it is a mental disorder with relevant practical implications for an individual person, or society generally. It becomes necessary to do so when the effect of the subject’s mental disorder is a policy which is a specific threat of some kind to the welfare of mankind, as in this present report, on the subject of what is fairly identified as “positivism.”

12. Positivism as such, is a mental problem whose common habitat includes “science departments” of secondary and university classrooms. In globally extended European cultures, it is usually associated with the virtually hereditary influence, directly or indirectly, of Aristotle’s role in promoting the diseased mental condition often known as Euclidean geometry, or, under a broader label, the “positivist” dogma asserted by Paolo Sarpi and the modern representatives of his following. »