Fidelio, Volume 11, Number 3-4, Summer-Fall 2002
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Journal of Poetry, Science, and Statecraft Summer/Fall 2002 $10.00 Archytas’s Construction for Doubling the Cube Archytas developed a construction to find two geometric means between two magnitudes, AC and AB. Magnitude AC is drawn as the diameter of circle ABC; AB is a chord of the circle. Using this circle as the base, generate a cylinder. The circle is then rotated 90° D about AC, so it is perpendicular to the plane of circle ABC; it is then rotated about point A, to form a torus with nil diameter. (The intersection of the torus and the cylinder produces a curve of double curvature.) Chord AB is extended until it intersects the perpendicular to AC at point D; this forms triangle ACD, which lies in plane of circle ABC, AB, and AC. Triangle ACD is then rotated around AC, producing a cone. The cone, torus, and cylinder all intersect at point P. Perpendicular PM is then dropped from P along the surface of the cylinder, until it intersects circle ABC at point M; this forms right triangle AMP. Through this construction, a series of similar right triangles (only partially shown) is generated, which produces the continued proportion, AB:AM::AM:AP::AP:AC. Thus, AM and AP are shown to be the two geometric means between magnitudes AC and AB. Bringing the Invisible to the Surface: The Legacy of Plato physical concept of magnitude the German word Kraft. higher powers, was generated by a Awas already fully developed by That is, a linear magnitude has succession of different types of the circle associated with Plato, and the power to double a line, whereas action. Specifically, a simply- expressed most explicitly in the Meno, only a magnitude of a different extended magnitude was produced Theatetus, and Timaeus dialogues. species has the power to double a from linear action, doubly-extended Plato and his circle demonstrated square, and a still different species magnitudes from circular action, and this concept, pedagogically, through has the power to double a cube. In triply-extended magnitudes from the paradoxes that arise when Bernhard Riemann’s terminology, extended circular action, such as the considering the uniqueness of the these magnitudes are called, rotational actions that produce a five regular solids, and the related respectively, simply-extended, cone, cylinder, or torus. Plato’s problems of doubling a line, square, doubly-extended, and triply- collaborator, Archytas, demonstrated and cube. As Plato emphasized, each extended. Plato’s circle emphasized that the magnitude with which a species of action generated a that magnitudes of lesser extension cube is doubled, is not generated by different species of magnitude. He lacked the potential to generate circular action, but by extended denoted such species by the Greek magnitudes of higher extension, circular action, i.e., conic sections. word dunamis, the root of the creating, conceptually, a succession of English ‘dynamo,’ translated as higher powers. [SEE ‘The Fundamental ‘power.’ The meaning of the term Plato’s circle also emphasized, that Theorem of Algebra: Bringing the dunamis is akin to Leibniz’s use of this succession of magnitudes of Invisible to the Surface’] “It is through beauty that one proceeds to freedom.” —Friedrich Schiller Vol. XI, No. 3-4 Summer/Fall 2002 SPECIAL DOUBLE ISSUE EDITOR-IN-CHIEF William F. Wertz, Jr. 4 EDITOR How To Turn a Dark Age Kenneth Kronberg Into a New Renaissance ART DIRECTOR Gabriele Liebig Alan Yue EDITORIAL ASSISTANT 17 Denise Henderson The Fundamental Theorem of Algebra: BOOKS Bringing the Invisible to the Surface Katherine Notley Bruce Director Fidelio (ISSN 1059-9126) is Afterword by Lyndon H. LaRouche, Jr. published by the Schiller Institute, Inc., P.O. Box 20244, Washington, D.C. 20041-0244. 32 www.schillerinstitute.org Homer, The Blind Seer Editorial services are provided Rosa Tennenbaum by KMW Publishing Company, Inc. © Schiller Institute, Inc. Fidelio is dedicated to the S PECIAL F EATURE promotion of a new Golden Renaissance based upon the 62 concept of agap¯e or charity, as LaRouche Launches that is reflected in the creation of artistic beauty, the scientific Revolutionary Youth Movement— mastery of the laws of the physical For a New Renaissance! universe, and the practice of republican statecraft for the benefit of our fellow man. Subscriptions by mail are $20.00 Editorial 2 Lyndon LaRouche’s International Youth Movement for 4 issues in the U.S. and Canada. Airmail subscriptions Translation 49 Wilhelm von Humboldt: On the Greek Ideal to other countries are $40.00 for News 91 D.C. Meet: ‘The Courage To Lead in a Time of Crisis’ 4 issues. Payment must be made 93 Italian Parliament Spurs New Bretton Woods in U.S. currency. Make check or 97 LaRouche Keynotes Conference on China money order payable to Schiller Institute, Inc. 98 ‘Boynton Weekend’ Honors Heroes of Selma Obituary 100 William Warfield Commentary 102 Beware of the Athenians, Mr. Bush! On the Cover Nike Unbinding Her Sandal Art 105 Leonardo da Vinci: Painter of Movement (detail) (Temple of Athena Nike, Acropolis, Athens, c. 410-407 B.C.). Books 110 LaRouche’s Emergency Infrastructure Program Mid-motion-change: what is 111 Nothing Like It in the World going to happen. (Nimatallah/ 112 Supreme Command Art Resource, NY) 114 Festschrift for Lyndon LaRouche FIDPD-11-03 What kind of a life, what kind of a nation, what kind of a world are you going to give them? Now, you can’t give that to them all at once. Because, it’s going to take a quarter-century to build this country out of the mess we’re in, economically, right now. But you can, potentially, as leaders of a new generation, create the circumstances which will enable the children of your generation to succeed. And people Lyndon LaRouche’s will honor your generation for centuries to come because of that! International Now, you need the help of your parents’ generation. But they’re off sleeping like the dormouse in the teacup, drowning themselves in their delusions. You Youth Movement have to do what has been done before: The reason that youth movements create revolutions, is because the This Double Issue of Fidelio is devoted to the creation of youth, when faced with a prospect of no future, or a Lyndon LaRouche’s International Youth Movement, as very bad future, say, “Well, we can’t change the lever for transforming a world in the throes of everything. But we can get out there and begin to economic depression and on the brink of perpetual war. organize our own generation, and our parents’ Lyndon LaRouche addressed the following edited remarks generation, to waken them, to act to get us all out of to approximately 85 young people from the East Coast this mess.” of the U.S.A., who attended a two-day conference on In other words, you have to change your parents. November 2-3, 2002. You have to educate your parents. You do it largely by example. You do it by doing the right thing. And that our parents’ generation, generally, has shows them that something is possible in this society, withdrawn from reality, and are living a kind because you’re doing it. Because, no matter how stupid Yof Baby Boomer fantasy life, a state of denial, they are, they know that you are the future. When trying to imagine that they’re push comes to shove, when happy; and very rarely are they’re facing the grave, they happy. EDITORIAL they’re going to say that what We’re a dying society. We they have, going into the are a consumer society. And if grave, is what they’ve left you compare the ideas, the behavior, of people in your behind in the form of your generation. parents’ generation, and what’s going on in your Now the problem is, that most individuals fail generation; and you look at the moral degeneration of because they fail morally. They don’t have that sense of your parents, and what they were 20 years ago,— if identity. We’re all going to die. It’s inevitable. you look back on them 20 years ago, you would say Mortality is mortality; it means ultimate death. So they were vibrant, active, and more like you. In the therefore, what is your interest in life, if you know you past 10-20 years, they’ve gotten prematurely old. are going to die, sooner or later? The pleasures you get You, in your generation, can only defend yourself by out of mortal life? No, of course not. The money you becoming leaders. You have a bunch of dormice—your get? No. All of these things disappear the minute you parents’ generation. Maybe not your particular parents; go into the box. but the generation is hopeless. You know, it’s like Alice What’s important is, what is the meaning of your in Wonderland, where the dormouse keeps drowning mortal life? What are you doing for humanity? It’s himself in the teacup; the Mad Hatter has to intervene what you would demand of a President. You want a to pull him out so that he doesn’t drown himself. Your President who is not concerned about his personal parents are like that. They’re dormice—not all of them, self-interest. You want a President who says, “I am but some of them. What do you do? You are the future. the guy you can rely upon, to make a decision based You’re not the future itself; the future is your children. on the interest of coming generations, as well as the 2 present generation, the present nation.” around things of relevance to humanity and saying, Now, that is the basis for leadership.