Justice for the Catenary
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 Gauss-Riemann 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 equal-time 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 equal-time and least-time. 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 re-living 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 non-uniform. Because physical action of this type is always changing non-uniformly, 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 connect-the 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 non-uniform 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 non-uniform, 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 non-uniformly, ends up at the center. Nicholas of Cusa draws an analogy from this game to the relationship between God, Man and Nature:
“Analogously, the rational soul intends to produce its own operation; with its steadfast intention persisting, the soul moves the hands and instruments when a sculptor chisels on a stone. Intention is seen to persist immutably in the soul and is seen to move the body and the instruments. In a similar way, nature (to which certain men give the name “world-soul”), moves all things while there persists ts unchanging and permanent intention to execute the command of the Creator. And the Creator, with His eternal, unchanging, and immutable intention persisting, creates all things. “Now, what is an intention except a conception, or a rational word, in which all the respective exemplars of things are present?…”
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 non-uniform 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 least-time 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 post-Cauchy 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 physical-mechanical problems, such as determining the path of least-time 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 least-time and equal-time, 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 least-time and equal-time 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 re-state 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 re-orients itself, so as to maintain the non-uniform 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 pre-existing 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?
Figure 1 |
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, Figure 3 , and Figure 4.)
Figure 2 |
Figure 3 |
Figure 4 |
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 non-uniform 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 5.)
Figure 5 |
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 6 and Figure 7.)
Figure 6 |
Figure 7 |
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 8 and, Figure 9). 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 10.)
Figure 8 |
Figure 9 |
Figure 10 |
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.
Riemann for Anti-Dummies Part 10a
JUSTICE FOR THE CATENARY (CONTINUED)
The last installment presented Bernoulli’s discovery of the unique geometry exhibited by a hanging chain. While Bernoulli discovered the characteristics of the catenary, it was Leibniz who asked, and answered, “Why does the chain assume this shape and not some other?”.
To summarize Bernoulli’s discovery: a chain hanging under its own weight, in order to form a stable “orbit”, assumes a unique shape, that does not correspond to any geometrical configuration that was known to mathematicians in Bernoulli’s time. Huygens called this shape the catenary curve. Bernoulli derived the geometrical properties of the catenary from the physical properties of the chain; specifically that in order for the chain to be stable, it must distribute the tension equally throughout its length. The catenary is: that which produces this physically determined characteristic of change, or, what Leibniz called, the integral. That physically determined characteristic is manifest at all positions along the chain. The nature by which that characteristic changes from position to position, Leibniz called the differential. Thus, the geometrical shape the chain assumes, is, that shape which expresses this unique physical property.
Such thinking enrages mathematicians of the Newton, Euler and Cauchy variety. “Mathematics first, physical reality second”, might as well be their motto, which is just another version of the same psychosis exhibited today by those who attribute some magical economic value to money, particularly, “my money”. But it is quite natural for thinkers in the tradition of Plato, Kepler and Cusa, who comprehend that mathematics is only a metaphor to express a certain level of knowledge about the intention that a physical process is carrying out.
Cusa expresses this way of thinking in his dialogue De Ludo Globi (The Bowling Game). The dialogue concerns a game played with a non-uniform 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 non-uniform, 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 non-uniformly, ends up at the center.
Cusa draws an analogy from this game to the relationship between God, Man and Nature:
“Analogously, the rational soul intends to produce its own operation; with its steadfast intention persisting, the soul moves the hands and instruments when a sculptor chisels on a stone. Intention is seen to persist immutably in the soul and is seen to move the body and the instruments. In a similar way, nature (to which certain men give the name “world-soul”), moves all things while there persists its unchanging and permanent intention to execute the command of the Creator. And the Creator, with His eternal, unchanging, and immutable intention persisting, creates all things.
“Now, what is an intention except a conception, or a rational word, in which all the respective exemplars of things are present?…”
In the case of the hanging chain, the universe was presenting a paradox not unlike the one Kepler confronted in his determination of the geometry of the non-uniform motion of a planet, or like the case of Pierre de Fermat’s discovery that light travels according to the path of least-time. In these examples, the physical action measured did not conform to a geometry that could be deduced from the axioms, postulates and definitions of Euclidean geometry. In fact, in each case, the physical action contradicted the conclusion, deduced from those axioms, postulates and definitions, that space was a uniform continuum, infinitely extended in three orthogonal directions.
Like Kepler, Bernoulli rejected the “curve fitting” methods typified by Ptolemy, Copernicus, Brahe, Gallileo and Newton. These Aristoteleans assumed that space was a sort of infinite empty box, in which physical objects interacted with one another along straight lines or perfect circles. For them, man’s knowledge of such physical action was limited to mapping whatever observations were made onto perfect circles, straight lines, or some combination of same. On the other hand, Kepler, Bernoulli, and Leibniz made no such a priori assumption about the nature of space. Rather they sought to determine what is governing the physical process “in between”, so to speak, what is seen. The irony is that what is actually governing the physical process is not directly observable, but it must be discovered from paradoxes that are produced, when ,what is seen, contradicts our assumptions.
A further comparison with Kepler’s astronomical discoveries and Fermat’s work on light, is useful. Kepler showed that the unique path of a planet in the solar system is governed, not by a pair-wise interaction between the planet and the sun, but, by what Gauss and Riemann would later call a “hypergeometry”. The characteristics of that hypergeometry were expressed by Kepler’s principles of planetary motion, which have been discussed at length in earlier installments of this series. Similarly, Fermat showed that the path the light took was governed, not by a Euclidean notion that the shortest path is the shortest distance, but by a “hypergeometry” in which the shortest path is the path of least-time.
A further review of Fermat’s discovery will prove relevant. When reflected in a mirror, light assumes the geometry such that its angle of incidence and angle of reflection are equal. But, the question remains, “Why does the light assume this geometry, and not some other?” While Aristoteleans bristled at the mere posing of this question, Plato’s followers were compelled to ask and then answer it, leading to a discovery of a characteristic of the “hypergeometry” governing the phenomenon. When confronted with this observation, Plato’s followers demonstrated that the equal angles were a consequence of the hypergeometric requirement that light must follow the path of shortest distance.
However, under refraction, the light does not travel the shortest distance, nor are the angles of incidence and refraction equal. Kepler and others, particularly Willlibroad Snell, determined that the geometry of light under refraction, was such, that the sine of the angles of incidence and refraction were proportional. But again, this is the geometry of the observed phenomenon, not the characteristic of the hypergeometry governing it. In other words, the question, “Why are the sines of the angles of incidence and refraction proportional?” remained unasked, and unanswered. It was Fermat’s great discovery to show, that this geometrical relationship was itself a consequence of the universal principle, that light travels in the path of least-time. Upon reflection, the shortest distance exhibited by reflecting light, is simply a special case of the principle of least-time, expressed by refraction.
So, why does the chain assume the shape that it does? Or, in other words, “what are the characteristics of the hypergeometry governing the chain’s action?”
Leibniz’ discovery was based on his and Bernoulli’s re-working of the discoveries of Pythagoras, Theodorus, Theatetus, and Plato, as recounted in part in Plato’s dialogue, “The Theatetus”. These investigations concern the first level of paradoxes that arise, when considering the difference between linear action and rotational action.
To grasp these paradoxes, conduct the following exercise:
First, draw a line segment, then double it, then double it again, and so forth. Then, begin with the a similar segment, and triple, once, twice, three times, etc.
Now, do the same thing again, except instead of beginning with a line segment, begin with a square.
Notice that when doubling, or tripling the line, the result is always a line. However, when doubling or tripling a square, the result is an alternating series of squares and rectangles. In the Theatetus dialogue, Plato presents the paradox that the rectangles are incommensurable with the squares. (Re-draw the alternating series of squares and rectangles as all squares. Begin with a square; draw its diagonal. Using that diagonal as a side draw another square. Now draw the diagonal of that square, and so on. This should produce a spiral of squares.)
Thus, doing the same thing in two different geometries, produces two different results. To use the terminology of Gauss and Riemann, the dimensionality of the manifold determines the nature of the action in it.
The Greeks expressed these two different manifolds, in numbers, as arithmetic (linear) and geometric (rotational), and measured the relationship of action in each manifold, by the characteristic intervals, or “means” that each process defined. The “arithmetic mean” is the characteristic interval between two linear magnitudes, specifically the midpoint of a line. The “geometric mean”, is the characteristic interval of rotation, specifically, half a rotation.
Bernoulli, Huygens, and Leibniz investigated this paradox in a new light. Bernoulli discovered that both the arithmetic and geometric could be combined in one representation, by an equiangular spiral.
Leibniz represented both the arithmetic and geometric by what he called the “logarithmic” curve. (Leibniz’ logarithmic curve is constructed such that the horizontal change is arithmetic, while the vertical is geometric. See http://www.schillerinstitute.com/fid_97-01/011_catenary.html)
It was Leibniz’ surprising discovery that the catenary curve can be constructed from the logarithmic curve. Thus, the catenary is the arithmetic mean between two logarithmic curves, and inversely, the logarithmic curve is the geometric mean of the catenary!
Now compare Bernoulli’s discovery with Leibniz’s. Bernoulli discovered the geometry of the catenary as a consequence of the physical characteristics of the hanging chain. Leibniz showed, that, that geometry, is itself a consequence, of a characteristic of the hypergeometry governing the chain’s physical action. In other words, the chain is being “guided”, so to speak, by an unseen (logarithmic) curvature. It is a demonstration of Leibniz’ method of “analysis situs” that he discovered the nature of that unseen curvature, from the seen. The guide, from what is being guided.
(Think of the little experiment described above. [See Figure 6, and Figure 7] The curve one must follow in order to maintain the equal force is being “guided” by the curvature of the logarithmic curve. This is a physical demonstration that space is not Euclidean.)
Figure 6 |
Figure 7 |
Just as the planet’s action is an expression of the principles underlying the solar system, and light’s path an expression of the principle of least-time, so to the hanging chain’s path is an expression of the principle expressed by the logarithmic curve. But, is there a multiplicity of “hypergeoemtries”, or is there some unifying principle that unites these three seemingly disparate phenomena?
That is the discovery of Gauss and Riemann.