By Bruce Director This afternoon, I will introduce you to the mind of Carl Friedrich Gauss, the great 19th century German mathematical physicist, who, by all rights, would be revered by all Americans, if they knew him. Of course, in the short time allotted, we can only glimpse a corner of Gauss's, great and productive … Continue reading Mind Over Mathematics: How Gauss Determined the Date of His Birth →

By Bruce Director In 1818, Karl F. Gauss accepted the assignment to conduct a geodesic survey of a large part of the Kingdom of Hannover, or, in other words, to measure a section of the surface of the Earth. The project involved many difficulties, and requires, first, that one reflect on the general concept of … Continue reading Measurement and Divisibility →

by Bruce Director As the pedagogical series on spherical geometry has indicated, a profound discovery arises, when you attempt to map spherical action on to a flat plane. Any such effort, immediately presents to the mind, the existence of two distinct types of action. Basic investigations of the physical universe, astronomy and geodesy, immediately confront … Continue reading The Importance of Good Maps →

by Jonathan Tennenbaum Last week my esteemed colleague Bruce Director poked into a real hornets' nest, when he asked: What makes people so susceptible to the kinds of frauds now perpetrated routinely by the mass media? Is there something {sinister} involved, a vulnerability inside the minds of our fellow-citizens, that leads them to desire a … Continue reading Transfinite Principle of Light, Part I: Prologue →

by Bruce Director The foolishness of relying on pure mathematical models for the design and production of automobiles, nuclear weapons, or any other physical device, would be obvious to anyone with a minimal level of knowledge of the discoveries of Cusa, Kepler, Leibniz, Gauss, Riemann, et al. Unfortunately, such knowledge is virtually non-existent among the … Continue reading Don’t Vote for Anyone Who Doesn’t Know Kepler →

by Jonathan Tennenbaum The issue of analysis situs becomes unavoidable, when we are confronted with a relationship of two or more entities A and B (for example, two historical events or principles of experimental physics), which do not admit of any simple consistency or comparability, i.e., such that the concepts and assumptions, underlying our notion of ``A,'' … Continue reading Incommensurability and {Analysis Situs}, Part I →

by Bruce Director Let us turn our investigations to the domain of manifolds of a Gauss-Riemann hypergeometrical form. There is no need, as too often happens, for your mind to glaze over as you read the above mentioned words. Lyn has given us ample guidance for this effort, most recently in his memo on non-linear … Continue reading Hypergeometric Curvature →

by Bruce Director Last week's pedagogical discussion ended with the provocative question: "If there exists no grand mathematical system which can combine and account for the various cycles, then how can we conceptualize the `One' which subsumes the successive emergence of new astronomical cycles as apparent new degrees of freedom of action in our Universe? … Continue reading Higher Arithmetic as a Machine Tool →

By Pierre Beaudry "The heavens are filled with gods." -Thales of Miletus Let me start with a provocative question: what do the height of the Egyptian pyramids and the distance to the Moon have in common? How did Thales of Miletus (600 B.C.) discover a principle by means of which one is able to determine … Continue reading How The Greeks Measured The Invisible, Part 1 The Discovery Of Principle Of The Thales Theorem →

by Bruce Director After spending the last three months determining where the asteroid Ceres is, it is more than appropriate to ask where you are? The question of determining one's location, while elementary, is not exactly simple. The same principles underlying Gauss' method for the determination of the orbit of Ceres, were applied by Gauss, … Continue reading Where Are You? →

by Bruce Director The following is provided to provoke some thinking, with respect to matters raised in previous pedagogical discussions, and to lay a conceptual basis for subjects to be taken up in the near future. Plato's dialogue of the Laws, continues in the short appendix known as the Epinomis: "Let us then first consider … Continue reading The Epinomis and the Complex Domain: A Fragmentary Dialogue in the Simultaneity of Eternity →

by Jonathan Tennenbaum Last week's inquiry concerning Leonardo da Vinci's principles of machine-tool design, brought us face-to-face with an old friend: the significance of the so-called Golden Section. Although this topic has been discussed many times in our organization, I think there still exists a residue of mystification, remaining to be cleared away. Often enough … Continue reading Demystify The Golden Section! →

By Larry Hecht Just as Cusa's principle of {weighing}, the balance, was the basis for progress in chemistry, leading to the 1869 discovery of the Periodic Table, Gauss's 1832 development of the magnetometer is the basis of all later discoveries in physics. This is so in a twofold sense. First, because the determination of the … Continue reading Demonstrate the Principle that Measurement is Hypothesis →

by Bruce Director Before the excitement from his stunning determination of the orbit of Ceres had begun to diminish, Gauss found, in the tiny variations of the orbit of the newly discovered asteroids, a further means for extending Kepler's discovery of the non-linear harmonic ordering of the solar system. As those who have worked through … Continue reading Dance with the Planets →

by Jonathan Tennenbaum In the course of our discussion of " the first measurement of the Universe" the concept of curvature arose at first in a {negative way}: the {impossibility} of representing the visible arrangement of stars in the heavens on a flat surface. Any attempt to create such a star map inevitably distorts the constellations … Continue reading Curvature: True Versus Apparent →

By Bruce Director Most people, because of an apparent childhood recollection, associate whole numbers with the counting of objects, and, as such, the numbers 1, 2, 3, ... to whatever large number imaginable seems to be the natural order of things, hence the name "natural numbers." But as St. Paul says in the oft-cited 1 … Continue reading What Counts or How Your Days Are Numbered →

Can You Solve This Paradox? by Jonathan Tennenbaum In some of his letters concerning the ``Characteristica Universalis,'' Leibniz notably refers to the virtues of rational methods of entrepreneurial bookkeeping and budget-allocation, as such were originally introduced (according to some credible accounts) by Leonardo da Vinci's collaborator Luca Pacioli. Leibniz remarks, that rational deliberation and discourse … Continue reading Circular Action And the Fallacy of “Linearity in the Small”–Part I →

by Bruce Director Among the most interesting and provocative investigations of the thinkers of the ancient Greek speaking world, were problems concerning the construction, with straight edge and compass, of certain geometrical figures; specifically, the doubling of the cube, the trisection of the angle, the construction of the regular heptagon, and the quadrature of the … Continue reading The Circle Is Not Simply Round →

by Jonathan Tennenbaum The following discussion begins a long journey, along a pathway of <astronomical paradoxes> leading from our discussion of "the simplest discovery," via the revolutionary work of Johannes Kepler, to the birth of a physics characterized by non-algebraic, elliptic and hypergeometric functions. In his "Commentaries on Mars" (also known as "Astronomia Nova"), Kepler … Continue reading How Johannes Kepler Changed the Laws of the Universe, Part I →

by Jonathan Tennenbaum Contrary to British-authored mythologies, the intense interest on the part of Greek geometers from Pythagoras to Eratosthenes, in so-called "unsolvable problems" of geometry, had nothing to do with an idle fascination in mathematical puzzles. At issue, in the investigation of such problems as doubling a cube, trisecting an arbitrary angle, constructing a … Continue reading From Cardan’s Paradox To The Complex Domain, Part I →

by Bruce Director In previous pedagogical discussions on Higher Arithmetic, we investigated the ordering of numbers with respect to arithmetic (rectilinear) progressions, (as in the case of linear and polygonal numbers) and geometric (rotational) progressions, as in the case of geometric numbers, and prime numbers. (See Doc.#'s 97267bmd01; 97316bmd001; 97326bmd001;) The deeper implications of these … Continue reading Beyond Counting — A Preparatory Experiment →

by Jonathan Tennenbaum This week's pedagogical discussion illuminates the issue of "non-linearity in the small" from a somewhat different angle than the preceeding ones. This week we shall take a look at the celebrated case of Mercedes' famous "A-Class" automobile. The following documentation was researched by Rudiger Rumpf and Ralf Schauerhammer, and will be featured … Continue reading The Fraud of Benchmarking →

By Robert Trout "But Aristarchus of Samos brought out a book consisting of certain hypotheses, in which the premises lead to the conclusion that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain motionless, that the earth revolves about the sun in … Continue reading How Aristarchus Measured the Universe →

Bruce Director To Archimedes came a youth desirous of knowledge. "Tutor me," spake he to him, "in the most godly of arts,  Which such glorious fruit to the land of our father hath yielded  And the walls of the town from the Sambuca preserv'd!"  "Godly nam'st thou the art?" She is't, "responded the wise one;  … Continue reading Archimedes and The Student →

by Jonathan Tennenbaum Lyn has emphasized, how Carl Friedrich Gauss' 1799 dissertation on the so-called "Fundamental Theorem of Algebra", constituted a devastating refutation of the leading scientific authorities of his day, including Jean-Louis Lagrange and Leonard Euler. Gauss first points out fundamental flaws in purported proofs of the "Fundamental Theorem", put forward by D'Alembert, Euler … Continue reading Gauss vs. Empiricism →

By Jonathan Tennenbaum Things are not what they appear, nor does the world function as the naive varieties of "common sense" (horse sense) would have it do. Those who subscribe to Rene Descartes' doctrine of "clear and distinct" truths, and pride themselves on not listening to anything that smells of "theory," or cannot be explained … Continue reading Motion Is Not Simple! →

by Jonathan Tennenbaum Once our prehistoric predecessors created the concept of a day, year, and other astronomical cycles, a new fundamental paradox arose: By its very nature, a cycle is a "One" which subsumes and orders a "Many" of astronomical or other events into a single whole. But what about the multitude of astronomical cycles? … Continue reading The astronomical origins of number theory, Part 1 →

By Bruce Director ``Can we deny that a warrior should have a knowledge of arithmetic?... ``.... It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the … Continue reading Prime Numbers →

Can You Solve This Paradox? by Jonathan Tennenbaum The following two-part discussion is intended to prompt a richer reflection on what was presented earlier, concerning Analysis Situs, the paradox of ``incommensurability'' in Euclidean geometry, and Nicolaus of Cusa's discovery of a higher geometry based on ``circular action.'' At the same time, I will set the … Continue reading From Nicolaus Of Cusa To Leonardo Da Vinci: The “Divine Proportion” As A Principle Of Machine-Tool Design, Part I →

by Jonathan Tennenbaum One of the reasons why you don't really understand the significance of Plato's five regular polyhedra, is because you have never questioned your own, completely unfounded assumption, that the sphere is a figure in 3-dimensional space. We all remember the type of horror movie, where the Earth has been invaded by alien … Continue reading How To Purge Your Mind of “Artificial Intelligence”: Introduction to a new pedagogical series →

by Pierre Beaudry "The history of astronomy is an essential part of the history of the human mind." Jean Sylvain Bailly THE SHADOW OF A DOUBT Over 50 centuries ago, it was the Egyptians, not the Greeks, who invented and built the Five Platonic Solids. This can now be proven with no more than a … Continue reading The Twelve Star Egyptian Sphere That Generated The Great Pyramid And The Platonic Solids →

by Jonathan Tennenbaum Our pedagogical discussions concerning the problem "incommensurability" in Euclidean geometry demonstrated, among other things, that the shift from linear to plane, or from plane to solid geometry cannot be made without introducing new principles of measure, not reducible to those of the lower domain. Thus, the relationship of the diagonal to the … Continue reading The curvature of “rectangular numbers” Part I →

By Larry Hecht The law for the reflection of a ray of light, was known since ancient times. Imagine a plane mirror, resting on a table-top. A beam of light, directed at the mirror, forms an angle with the mirror's surface called the ``angle of incidence.'' About 2,000 years ago, scientists knew that the beam, … Continue reading The Refraction Of Light And The Circle →

By Jonathan Tennenbaum The following three-part series is ostensibly devoted to some crucial paradoxes raised by the discovery of the so-called "mitogenetic" or "biophoton" radiation of living organisms, by the great Russian biologist Alexander Gurwitsch. At the same time, I hope to provoke reflection on one of the unique and so far irreplaceable functions, which … Continue reading Science and Life: The Importance of Keeping People in a Healthy, Unbalanced State →

by Jonathan Tennenbaum The fundamental crisis of civilization is forcing the question: Where do the ideas come from, which we find in our own heads and those of our fellow human beings, and which determine Mankind's ability or inability to survive the onrushing crisis? The "short answer" is, that apart from the products of oligarchical … Continue reading The Simplest Discovery, Part I →

by Jonathan Tennenbaum  "You have to prove your case..." "Demonstrate to me on my own terms, that what you say is right..." "Where are your facts? Give me the hard facts!" "I agree with you, but my wife..." "That sounds exaggerated. How can you be so sure?..." "Why should I believe you? I heard something … Continue reading Spring Cleaning For The Mind: On `Proof,’ Part I →

by Jonathan Tennenbaum Since at least the time of Plato and Aristotle, and most likely even long before Pythagoras, the struggle between oligarchical and republican conceptions of physics has turned on the relationship between what the Greeks called {dynamis} and {energeia}. To a rough first approximation, the Greek {dynamis} might be rendered, in its broad … Continue reading {Dynamis} vs. {Energeia} — A Sketch →

by Jeremy Batterson Archimedes' discovery of the method of determination of the volume of a sphere was a discovery of such beauty and with such astonishing implications, that Archimedes, before his death, instructed that it be engraved upon his tombstone. And, yet, almost none, in our day, have ever worked through its proof, although it … Continue reading An Incredible Discovery Of Archimedes →

Lyndon LaRouche recently described classical Greece as the "child of Egypt." The great figures of the sixth century B.C., Solon, Thales and Pythagoras, were, in fact, the children of Egypt, each having travelled to Egypt and studied under the Egyptian astronomer- and geometer-priests. Through them, and others, Egypt transmitted a science -- a method of … Continue reading Greece: Child Of Egypt, Pt. I →

By Jonathan Tennenbaum It is a fair guess, that the Crab Nebula will play a comparable role, in a coming series of revolutions in astrophysics, to that of Mars' anomalous motion in Johannes Kepler's launching of modern astronomy four centuries ago. The anomalies of the Crab Nebula confront us directly with the issue of the … Continue reading The Crab Nebula And The Complex Domain →

by Pierre Beaudry, Leesburg, October 30, 2003 Some people said that the design for the city of Washington D.C. came from the heavens; that the French architect, Pierre L'Enfant, determined the location of the House of Congress, and the House of the President, in accordance with a divine plan written in the stars, and that … Continue reading How Benjamin Banneker Discovered The Principle Of Proportionality In A Mathematical Puzzle: A Peace Of Westphalia Pedagogical →

by Pierre Beaudry In ancient Egypt, an astronomer once asked an architect: "If you were an astronomer, how would you start building an astronomical observatory, which would be perfectly in line with a meridian circle, from which one could observe and teach young people how to determine the transit of all of the stars in … Continue reading The Angular Determination Of The Great Pyramid →