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On the very eve of his unjust incarceration, Lyndon LaRouche issued a short, but substantial, memo on the catenary function, that was vigorously maligned by a few, and, unfortunately, largely ignored or not understood by many. The principles identified there, are critical at this stage of this pedagogical review of the GaussRiemann theory of functions, and also more generally. The crucial issue is the distinction between defining a principle from the standpoint of abstract geometry, versus real physics. From the standpoint of abstract geometry, least time and equaltime are represented by a cycloid, but from the standpoint of real physics, the catenary function reflects these principles. The investigation of the gap between what abstract geometry leads us to believe, and what we come to know by real physics, is at the center of the method of Cusa, Kepler, Leibniz, Gauss, Riemann, et al. "But, wait a minute," some might protest, "If I make a pendulum wrap around a cycloid, or, if I make a ball roll along a cycloidal path, it's motion conforms to equaltime and leasttime. Doesn't that show that the cycloidal path corresponds to a physical principle?" The difficulty, or even downright hostility, with which some people might react to this paradox, is paradigmatic of the mediocrity associated with relying on secondary sources, and popularly accepted gossip, instead of becoming to know, by reliving an original discovery. While LaRouche draws new, revolutionary, implications from this paradox, the distinction he makes between abstract geometry versus real physics, contrary to academically accepted chatter, is identical to the standpoint of the original discoverers; Huygens, Johann Bernoulli and Leibniz. During the 1680's and 1690's these thinkers engaged in a dialogue concerning the development of the new mathematics, demanded by Kepler's confirmation of Cusa's hypothesis, that action in the physical universe is nonuniform. Because physical action of this type is always changing nonuniformly, it is impossible to determine the position of, for example, a planet based simply on its past positions, or as LaRouche has put it, "by connectthe dots, statistical methods". Rather the position of the planet is determined by an underlying characteristic of change that governs the whole orbit. This problem is exemplified by the question, "How does the planet know how to move?". To answer that question, we must first ask, and answer, "What intention is this action of the planet fulfilling?", and, "How is that intention manifest at each moment?" Cusa expresses this in his dialogue De Ludo Globi (The Bowling Game). The dialogue concerns a game played with a nonuniform ball that is rolled on a surface on which 9 concentric circles are drawn. The object of the game is roll the ball as close to the center as possible. But, since the ball is nonuniform, it follows a spiral, rather than a straight path. The player intends to roll the ball on a path that ultimately winds up in the center of the circle, but to do that, he must start the ball with a speed and trajectory, that after changing nonuniformly, ends up at the center. Nicholas of Cusa draws an analogy from this game to the relationship between God, Man and Nature:
Leibniz and his collaborators, Johann and Jakob Bernoulli, developed the calculus to increase the mind's capacity to grasp the nature of the intentions governing nonuniform action in the universe. The effectiveness of the calculus is illustrated by Bernoulli's determination of the brachistrone, discussed in the last installment. In that example, Bernoulli derived the cycloid as the leasttime path that results, if at each moment, the speed of the body is proportional to the square root of the distance dropped. This week, we look at another example of the application of the calculus, in the determination of the geometry of the hanging chain, by Bernoulli and Leibniz. In 1691 Bernoulli published his "Lectures on the Integral Calculus", which remains the best elementary textbook on the integral calculus to this day. (Anyone comparing this work to the postCauchy calculus textbooks widely used today, will be immediately struck by how fraudulent all such treatments of the calculus are. It is a testament to the bankruptcy of modern science education, that Bernoulli's book, rather than being the standard for all introductory courses in calculus, can be found only in obscure places in some university libraries.) Contrary to the Cauchy fraud, Bernoulli defines the calculus from the standpoint of the integral as the solution of a differential equation. In other words, the integral, for Leibniz and Bernoulli, expresses the underlying nature (intention) of a physical process which at each moment has a certain characteristic of action. The characteristic which expresses the change at each moment is what Leibniz called the differential. A whole physical processes (integral) can, thus, be expressed as a function of its characteristic change at each moment, by what Leibniz called a differential equation. The example of the problem of the hanging chain will illustrate this relationship. Bernoulli justly claims that Leibniz' method, "Which to a certain extent, stretch into the deepest regions of geometry," is capable of solutions, "that until now the power of ordinary geometry had ridiculed and were unable to produce." Bernoulli shows how the calculus was developed to solve certain physicalmechanical problems, such as determining the path of leasttime and equal time, or the shape of the hanging chain. However, Bernoulli also issued the same caveat, that had previously been sounded by Kepler with respect to the methods of Ptolemy, Brahe and Copernicus. For example, he says that investigations of the cycloid as the path of leasttime and equaltime, start with certain physical assumptions, and then, as Bernoulli said, "dress[es] them up so as to transform the mechanical principle into a purely geometrical one." The physical mechanical principles that result, such as the leasttime and equaltime properties of the cycloid, are, thus, products of abstract geometry, and not true physical principles. Now, look at the problem associated with the catenary, to which Bernoulli and Leibniz also applied the methods of the calculus. A chain or rope hanging under its own weight assumes a unique geometrical shape. That shape, however does not conform to any curve found in any textbook on geometry. Here the calculus is employed to determine, "What is the geometry that characterizes this physical process." It is important to restate this inversion. The cycloid is the path that geometry produces, on the assumption that the universe acts in a certain way. The catenary is the path the universe produces to enable the hanging chain to assume a stable, "orbit". In the former, geometry produces the principles, in the latter, the principles produce the geometry. The Catenary To grasp this distinction, look at the catenary, as Leibniz and Bernoulli did.^{1} Bernoulli's treatment is found in a German translation of his 1691 "Lectures on the Integral calculus".^{2} The physical properties of the hanging chain are described in Chapter 4 of "How Gauss Determined the Orbit of Ceres" Fidelio, which the reader should review.) As emphasized there, the catenary shape, formed by the hanging chain, is akin to a planetary orbit, in that every position along the curve, is a function of the physical principles that produce the curve. If any part of the curve is changed, the entire curve reorients itself, so as to maintain the nonuniform curvature of the catenary. (See Figure 1.) Galileo attempted to apply his "ivory tower" methods to investigations of this phenomenon by trying to fit the hanging chain into his preexisting assumptions of geometry. The closest shape he could find, was that of a parabola. However, reality didn't want to be girdled, no matter how hard Gallileo tried. Joachim Jungius, by experiment, definitively proved that Gallileo was wrong, but he could not determine what the curve of the hanging chain was. So the question remained, what was the geometry of the hanging chain? Or, more generally, what does the curvature of the hanging chain show us about the geometry of the physical universe? Since none of the curves of pure geometry fit this physical process, the physical process required the development of a new geometry. Begin then with the physical properties of the hanging chain. Hang a chain and it assumes a characteristic shape. (See Figure 2.) Contrary to naive intuition, that shape is the same, regardless of the material of which the chain is made, or the position of the suspension points, or other factors. The shape, thus, reflects a universal physical principle. But the chain is not just sitting there doing nothing. It is always in motion, so to speak. Each point along the chain is feeling a tension. The link on one side of the point is pulling it in one direction along the curve, and the link on the other side of the point is pulling it equally in the opposite direction along the curve. These countervailing tensions are the same for every point along the chain, regardless of how much chain is hanging between them. This is also contrary to naive intuition, which would assume that the points closer to the suspension points, for example, would have more tension on them, and thus have to be made of stronger material, since they have more of the weight of the chain to support. If the length of the chain is increased or decreased between any two points, the amount of weight supported by those points changes, but the equality of tension at each point doesn't change. The hanging chain assumes a shape, such that as the length (weight) of chain changes, the principle of equal tension remains. Thus, each position of this nonuniform curve, is a function of a physical principle. It is to this physical property, that Bernoulli applied Leibniz' calculus. Taking the above described property of equality of tension as the "differential", Bernoulli sought to determine what is the nature (integral) of the curve that would produce this characteristic at each point. He began with an experimental corollary. He demonstrated that the force (Kraft) the chain exerted between any two points on opposite sides of the catenary would be the same as if the entire weight of the chain between those points, was concentrated in a body, that hung from strings that were tangent to the catenary at those points. (See Figure 3.) The relationship of the forces at these points is dependent on the sine of the angles the tangents make with a vertical line drawn through the weight. The reader can conduct a simple experiment to discover this for himself. (See Figure 4.) The lowest point on the chain is a singularity, as it is the one place where the force doesn't change, regardless of whether the length of the chain is increased or decreased on either side of it. Paradoxically, this point supports no chain, while supporting all the chain. (See Figure 5.) Bernoulli shows that the shape of the hanging chain, which Huygens called the catenary curve, is that path that must be followed, so as to maintain an equal force on this lowest point. In order to satisfy this intention, the chain must manifest a unique geometrical configuration. (See Figure 6.) This contradicts any assumption that space conforms to a uniform geometry, that is infinitely extended in three dimensions, such as is suggested by the axioms, definitions and postulates of Euclidean geometry. Rather, the physical properties of the chain interacting with the Earth produces a unique type of curvature to which the chain must conform in order to be stable. It is not the geometry that determines the shape of the chain, but the physics that determine the geometry. Coincident with Bernoulli's discovery, Leibniz discovered another principle underlying the geometry of the hangting chain. In the next installment, we will present Leibniz's side of the story. to PART II Notes


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