Author: Bruce Director

Bernhard Riemann’s “Dirichlet’s Priniciple”

by Bruce Director

In his revolutionary essay of 1857, {Theory of Abelian Functions}, Bernhard Riemann brought to light the deeper epistemological significance of the complex domain, through a new and bold application of a principle of physical action which he called, “Dirichlet’s Principle”. Riemann’s approach, combined with what he enunciated in his habilitation dissertation of 1854, not only ushered in a revolution in scientific thinking: it ignited a counter-reaction as fierce as the one launched, for the same reasons, against Cusa, Kepler, Fermat and Leibniz by the Venetian-British controlled empiricist school of Gallileo, Newton, Euler and Lagrange; a counter-reaction that continues to rage to this day, and with implications that reach far beyond the specific setting of Riemann’s 1857 paper. Despite the volumes that have been written on this subject, from Riemann’s time to ours, an honest examination of the history of the matter reveals that, just as Gauss demonstrated the fraud of Euler, Lagrange and D’Alembert in his 1799 proof of the fundamental theorem of algebra, Riemann was right, and his critics, like today’s Straussian controllers of Bush and Cheney, were malevolent frauds.
We cannot know for sure whether, when Riemann chose to call this method an application of “Dirichlet’s Principle”, he expected to provoke the reaction he received, or if he was merely stating what would have been obvious for anyone within the extended network of Abraham Kaestner’s students. Nevertheless, it is fortunate for us that he used that name, as it enables us to fairly accurately reconstruct, not only the scientific origins of Riemann’s thought, but the historical-political process from which it arose.

Lejeune-Dirichlet

Johann Peter Gustav Lejeune-Dirichlet was a pivotal figure in early 19th Century science. Born in 1805 to a family of Belgian origin living near Aachen, his early education took place in Bonn. At the age of 16, with a copy of Gauss’s {Disquisitiones Arithmeticae} under his arm, he went to Paris to audit the lectures at the College de France and the Faculte des Sciences. After a year, Dirichlet became employed as a tutor by General Maximilien Sebastien Foy, a republican member of the Chamber of Deputies, who introduced him to Alexander von Humboldt. After Foy’s death in 1825, von Humboldt recruited Dirichlet back to Germany, arranged for him to get a degree (even though Dirichlet refused to speak Latin), and eventually succeeded in obtaining for him a professorship at the University of Berlin. There, in addition to meeting, and marrying, Moses Mendelssohn’s granddaughter, Rebecca, (the sister of the composer Felix Mendelssohn), Dirichlet developed a fruitful collaboration with Jacobi and Jakob Steiner , including a tour with both to Italy in 1843 under Alexander von Humboldt’s sponsorship.
In 1847 Riemann arrived in Berlin to study with Dirichlet, Jacobi and Steiner, having spent the previous two years studying with Gauss. In 1849 he returned to Goettingen to complete his studies and in 1851, under Gauss’s direction, published his doctoral dissertation, {The Foundations for a general Theory of Functions of a Complex Variable Magnitude}, in which he first applied his principle, without mention of Dirichlet. When Gauss died in 1855, Dirichlet was appointed his successor, bringing himself back into contact with Riemann, who, just seven months earlier, had received permission to teach, after delivering his habilitation lecture, {On the Hypotheses which Lie at the Foundations of Geometry}. In 1857, Riemann published the {Theory of Abelian Functions} in which, for the first time, he identified the principle on which his new theories were based, as “Dirichlet’s Principle”. Two years later, Dirichlet died, and Riemann, now 33 years old, was appointed to his chair, a position he held until his own, unfortunately too early death, only seven years later.

The Potential

What Riemann called “Dirichlet’s Principle”, arose out of Gauss’s application of the complex domain to his investigations in geodesy and terrestrial magnetism; the former organized in collaboration with Schumacher beginning in 1818, and the latter initiated by Alexander von Humboldt in 1832. Both projects had enormous practical benefits. Each produced detailed maps of their respective physical effects which were vital for infrastructure development, and Humboldt’s project organized, for the first time, an international collaborative network of scientists who would impact the development of the physical economy from the Americas to Eurasia for generations. But Gauss recognized that both projects posed deeper epistemological questions for science. Writing in his {General Theory of Earthmagnetism} in 1839, Gauss said a complete and accurate map of the observations is not, in itself, a proper goal for science. “One has only the cornerstone, not the building, as long as one has not subjugated the appearances to an underlying principle.” Citing the case of astronomy as an example, Gauss said that mapping the observations of the apparent motion of the heavenly bodies onto the celestial sphere is just a beginning. Only once the underlying principle of gravitation is discovered, can the actual orbits of the planets be determined.
Gauss recognized that the first step in geodesy and geomagnetism is the measurement of changes in the effect of both phenomena on the measuring instruments. In the case of geodesy, that meant changes in the direction of a plumb bob, or plane level, as those changes are mapped onto the celestial sphere. The case of geomagnetism is more complicated. Here he was measuring changes in the direction of a compass needle, with respect to three directions and time. The general question was: what is the characteristic nature of the principle of gravitation or geomagnetism that would produce these apparent effects? The specific task was: how, from these infinitesimally small measured changes in the apparent effects, can that general characteristic be determined?
It is the second question that brings us more directly into contact with what Riemann called “Dirichlet’s Principle”. However, the task of understanding Dirichlet’s Principle will be made much easier if we first look at the elementary, but congruent case of the catenary.
The relevant focus for this discussion is the devastating rebuke which Leibniz and Bernoulli delivered to Gallileo and Newton over the case of the catenary. Gallileo had insisted that all that need, or could, be known about the catenary was a description of its visible shape. On the other hand, Leibniz and Bernoulli insisted that the shape of the catenary was merely the visible effect of an underlying physical principle, and the correct shape could not be determined until the underlying principle was known. As was developed in previous installments of this series, Leibniz and Bernoulli determined the characteristic nature of that principle, by determining first, the changing physical effect of that principle in the infinitesimally small, and then, by inversion, the overall characteristic of the principle. The result was Leibniz’s discovery that the shape of the hanging chain reflected the least-action effect of the principle of universal gravitation, and that this effect could be expressed geometrically as the arithmetic mean between two contrariwise exponential functions.
It is of extreme importance to emphasize that we are speaking here of the physical hanging chain, not a formal mathematical expression. In a formal mathematical expression the exponential curves have no boundary. The physical hanging chain does–the positions of the hanging points. Consequently, the specific shape of the chain is determined by the position of the hanging points relative to the weight and length of the chain. If the positions of the hanging points change, the position of every link in the chain also changes, but always in accordance with the relationship cited above. In other words, as the boundary conditions of the physical chain change, so does the chain’s specific path, but that path’s general form, required by the principle of least-action, is always a catenary. It will never become a parabola or any other curve. (See Figure 1.)

This example illustrates an aspect of the method that Leibniz originally called “analysis situs”, or what Gauss and Carnot later called “geometry of position”, that is relevant to an understanding of Riemann’s “Dirichlet’s Principle”. The positions of the individual links in the chain are a function of the relationship of the boundary conditions (position of the hanging points relative to the length of the chain) to the characteristic curvature of the principle of gravitation, and not by a pair-wise relationship among the links themselves. In other words, the position of any individual link is not determined by a distance to the right or left and a distance up and down from its neighbors, as the Cartesians and Newtonians would insist. Rather, the position of each link is a function of the characteristic of change of the physical action as a whole. Any change in the boundary conditions changes the position of every link, { as a whole} in conformity with the least-action principle of the catenary. Thus, the unseen physical principle’s effect in the visible domain is expressed by the characteristic of change the principle of least-action demands. This is what determines the specific positions of the links. That is, position is a function of change.
Gauss recognized that the principles underlying geodesy and geomagnetism could be understood by an extension of Leibniz’s method. He rejected the popularly accepted, but provably false method of Newton, that attempted to explain these phenomena as the pair-wise interaction of material bodies, according to the algebraic formula of the inverse square. (See {Riemann for Anti-Dummies–Part 53 “Look to the Potential”}. Instead, Gauss insisted that these phenomena, like the catenary, must be understood as one process, and that the local variations in the position of the plumb bob or the compass needle are a function of the characteristic of the principle governing the phenomenon as a whole. That whole, Gauss called “the potential”, which is the Latin equivalent of the Greek, “dynamis” or Leibniz’s “kraft” or “vis viva”. Gauss invented the idea of a “potential function” to express the least-action effect of the physical principle over an area or volume, in a similar, but extended, manner to that used by Leibniz to express the effect of gravity in producing the curvature of the hanging chain. To accomplish this, Gauss extended Leibniz’s idea of a function into the complex domain.
This transformed Leibniz’s functions, which characterized a single minimal pathway, into Gauss’s “potential function,” which characterized a whole class of minimal pathways: in effect, a function of functions. In other words, if Leibniz’s catenary is understood to be a minimal pathway determined by one set of two functions, Gauss’s potential function takes the next step: to a function that unifies two (or more) {sets} of functions. Riemann would later show that these sets of minimal pathways implicitly defined minimal surfaces, as, for example, the catenoid formed by a soap film suspended between two circular rings.
These sets of functions are not arbitrary. They are related by a special type of relationship, called by the descriptive names, “spherical functions”, or “harmonic functions.” An harmonic or spherical function is a set of orthogonal functions all of whose curvatures are changing at the same rate.
This can be most easily illustrated pedagogically with some geometric examples. A set of concentric circles and radial lines comprises an harmonic function because both the circles and the radial lines intersect orthogonally and both have constant curvature. (See Figure 2)

A more illustrative example is a set of orthogonal ellipses and hyperbolas. (See Figure 3.)

To get an intuitive grasp of their harmonic relationship, carry out the following thought. Each ellipse is associated with a confocal orthogonal hyperbola. Beginning at the point where both curves meet the axis, create in your mind a connected action that moves simultaneously on both curves. (See Figure 4.)

Note that as the curvature on the hyperbola becomes less curved, so does the curvature on the corresponding ellipse, and at the same rate.
Thus, harmonic functions relate two sets of different curves such that the rate of change of their respective curvatures is always equal. (Using Leibniz’s calculus, we could calculate this relationship precisely, but an intuitive understanding is sufficient for present purposes.)
Furthermore, a set of harmonic functions need not be familiar curves such as circles, lines, ellipses, or hyperbolas. In fact, very complicated sets of functions can be harmonic. (See Figure 5.)

By contrast, a set of circles and hyperbolas is not harmonic, because the curvature of the circle is constant, while the curvature of the hyperbola is changing. Consequently, the two sets of curves are not orthogonal. (See Figure 6.)

Gauss recognized that Leibniz’s principle of least-action with respect to the surfaces and volumes encountered in phenomena like terrestrial gravitation and magnetism, could be expressed by harmonic functions. One set of curves of the harmonic function expressed the pathways of minimal change in the potential for action, while the other, orthogonal curves expressed the pathways of maximum change in the potential for action. For example, if the Earth were perfectly spherical, its minimum and maximum of potential action could be expressed by a series of concentric spherical shells and orthogonal planes. A cross-section of such a configuration would be harmonically related circles and radial lines. If the Earth were perfectly ellipsoidal, its potential would be expressed by a set of triply orthogonal ellipsoids and hyperboloids whose cross sections would be the harmonically related set of ellipses and hyperbolas illustrated above.
But, as Gauss emphasized, the shape of the Earth is much more complicated than a sphere or an ellipsoid, with respect to both gravity and magnetism and the pathways of minimal and maximal potential for action were not such simple and well known curves as circles, lines, ellipses or hyperbolas. Thus, a more complex harmonic function must be found to express these principles. Such a function could not be determined a priori, but only from the measured changes in the effect of the Earth’s gravity or magnetism.
The question for Gauss was: how to determine the true physical shape of the Earth, or the characteristic of the Earth’s magnetism, from the measured infinitesimally small changes in its potential obtained by his geodetic and magnetic measurements?
This begins to get us closer to a first approximation of what Riemann called “Dirichlet’s Principle”.
To make a precise determination of the Earth’s surface, or magnetic effect, as Gauss did, is quite complicated, but the principle on which his method was based is within the scope of this pedagogy. If one recognizes, as Gauss did, that the changes in the direction of the plumb bob are measuring the changes in direction of the potential function, then the physical shape of the Earth has the same relationship to this potential as the hanging points have to the catenary. In other words, the surface of the Earth must be understood as merely the boundary of the potential, or, as Gauss put it: “The physical surface of the Earth is, in a geometric sense, the surface that is everywhere perpendicular to the pull of gravity.”
A reference to the ancient Pythagorean problem of doubling the line, square and cube can shed some light on this idea. The line is bounded by points, the square by lines and the cube by squares. The size and position of these boundaries is determined by the length, area or volume they enclose. For example, it is the square that determines the size and position of its sides, even though it is the latter that you see and the former that you don’t. The sides of the square are lines, but they are produced by a different power, (potential), than the lines produced from other lines. Similarly, the size and position of the squares that form the boundaries of a cube are produced by a different power (potential), than the squares formed by the diagonal of another square. Thus, even though the power can not be seen, it can be measured by its unique, characteristic effect on the boundaries of its action.
Now apply this same method of thought to the physical principles discussed above. The catenary is a curve whose boundaries are points. A catenoid is a surface whose boundaries are curves. The surface of the Earth is the boundary of a gravitational volume. The magnetic effect of the Earth is still more complicated, and will be taken up in more detail in a future pedagogical.
This connected relationship between the boundary conditions of a physical process and the expression of the principle of least-action with respect to that physical process, is the relationship to which Riemann is referring when he speaks of “Dirichlet’s Principle.”

From Gauss to Dirichlet to Riemann

After succeeding Gauss in 1855, Dirichlet began lecturing on Gauss’s potential theory at Goettingen while Riemann was preparing his {Theory of Abelian Functions}. What Gauss, Dirichlet and Riemann all recognized, was that complex functions, as the extension of Leibniz’s concept of the catenary and natural logarithms, were uniquely suited to express the least-action pathways of potential functions.
Gauss had already demonstrated this in his 1799 proof or the fundamental theorem of algebra, where he showed that a complex algebraic expression produces two surfaces whose curvatures are harmonically related. What Riemann attributes to Dirichlet, is the principle that given a certain boundary condition, the function that minimizes the action within it is a complex harmonic function.
Warm up to this idea on the familiar territory of the catenary. The boundary conditions here are the positions of the hanging points. The “interior” of this boundary is the curve itself. Within the curve is a singular point–the lowest point. If the boundary conditions change, by changing the position of the hanging points, so does the position of the lowest point. To state Dirichlet’s principle in this simplified context, the catenary is the least-action pathway of a hanging chain with these specified boundary conditions and singularity. If the boundary conditions change, the shape of the curve changes correspondingly, in accordance with the preservation of the principle of least-action.
Riemann inverted Dirichlet’s principle: {since the physical principle of least-action is primary, the position of the hanging points and the lowest point completely determine the shape of the chain!}
Now, make this same investigation with respect to a catenoid formed by a soap film between two circular rings. This catenoid is a physical least-action, or minimal surface. Embedded in this surface is an orthogonal set of curves of minimal and maximal action. (Riemann later showed that these curves are harmonically related. This will be illustrated in a future installment of this series.) Experiment by changing the shape of these boundaries from circles, to ellipses, to irregular smooth shapes, to polygons. When you change the position or shape of the boundaries of this surface, the shape of the surface and the embedded curves change accordingly, but the least-action principle is preserved.
Now, generalize this idea with some other pedagogical examples, illustrated in the following animations. In Figure 7 we see a set of harmonically related circles and radial lines that intersect at the center of the circles, being transformed while maintaining their harmonic relationship.

If the position of that intersection point changes, the radial lines must be transformed into circular arcs, and their endpoints move along the boundary in order to maintain their harmonic relationship. In the animation, we see this effect as the point of intersection moves, first away from the center, and then in a circular path around the center. This motion causes all positions inside the boundary to change {as a whole}. What doesn’t change is the harmonic, i.e. least-action, relationship.
This could also be thought of inversely: that the changes in position of the intersection of the radial lines at the boundary, cause their point of intersection to move in a circular arc, and their form to change from lines to circular arcs.
Or, infinitesimally small changes in the curvature of the pathways are determined by the conditions at the boundary with respect to the position of the singularity.
Compare this action with the change in the position of the lowest point of the catenary as the positions of the hanging points change, as illustrated in the animation Figure 1. There, a change in the boundary points produced a change along a single curve. Here, a change in the boundary curve produces a change in a set of harmonically related curves within a surface.
Compare this with the problem Gauss confronted in, for example, determining the location of the Earth’s magnetic poles from infinitesimally small changes in the Earth’s magnetic effect. Gauss understood that those small changes were connected to the position of the singularities, i.e. magnetic poles, of the Earth’s magnetic effect. However, the exact location, or even the number, of those poles was still unknown in Gauss’s time. On the basis of the measurements obtained by von Humboldt’s network, Gauss determined where those poles must be located. The famous American Wilkes expedition of 1837 was launched, in part, to confirm Gauss’s findings, which it did.
In Figure 8, this same effect is illustrated by moving the point of intersection of the radial lines along the path of a lemniscate.

Notice again how this change in the position of the singularity, changes the condition at the boundary, so that all the resulting relationships remain harmonic.
Figure 9 animates the same process in which the shape of the boundary has been changed to an ellipse, which correspondingly changes the shape of the orthogonal curves into hyperbolas, and the intersection point into two foci.

Of course, it could also be said that the radial lines are changed into hyperbolas, which changes the circles into ellipses, and the intersection point in to two foci. Or, that the intersection point is changed into two foci, which changes the the boundary into an ellipse, and the radial lines into hyperbolas.
In short: {a physical process of least action is a connected action. Changing any aspect of the process, changes everything else in the process correspondingly, so as to preserve the least-action characteristic of the process. That is, it is the physical principle of least-action that is primary.}
It was Riemann’s genius to recognize, through this application of Dirichlet’s Principle, that the principle of least-action of a physical process could be understood completely by the relationship between the boundary conditions and the singularities, and that this relationship could be expressed uniquely by Riemann’s geometric concept of complex functions. Moreover, Riemann showed that the characteristic of least-action of a physical process could be changed, in a fundamental way, only by the addition of a new principle. That change in principle is expressed in a complex function, as a corresponding increase in the number of singularities. In his {Theory of Abelian Functions} Riemann demonstrated this by applying Dirichlet’s Principle to the higher transcendental functions of Abel.
The deeper significance of this discovery can only be hinted at in this installment, and will be taken up in more depth later, but it can be illustrated by the animation of Figure 10, which expresses the principle of least-action with respect to an elliptical function.

Riemann demonstrated that all elliptical functions, being functions formed by the interaction of two connected principles, are expressed in the complex domain as surfaces with two boundaries. (These boundaries are marked in green in the animation.) In this animation you can see each boundary changing differently, but connectedly, with the other, causing corresponding changes in the minimal pathways, while at all times maintaining the overall harmonic relationship of the function. In other words, the characteristic curvature of these least-action pathways is determined, in this case, by the connected interaction of two distinct principles.
A comparison between this and the previous examples indicates what Riemann emphasized, that the only way to fundamentally change the characteristic of action of a physical process is by the addition of the action of a new principle. This more advanced question will be investigated more thoroughly in future pedagogicals.
A suggestive example from economics can also help illustrate this principle. What is the relationship between all physical economic relationships and the economic boundary conditions of physical infrastructure and cultural development? What is the relationship between these boundary conditions and the singularities represented by the introduction of new technologies? What is the effect on all economic relationships, of a change, positive or negative, in these physical economic boundary conditions?
Four years after Riemann’s death Karl Weierstrass criticized Riemann’s application of Dirichlet’s Principle on formal mathematical grounds. Weierstrass contended that it was inappropriate to speak mathematically of least-action, unless a formal mathematical proof could be presented proving that a mathematical minimum, or maximum, existed. While it is possible to produce a formal mathematical example which has no minimum, all {physical} process are characterized by bounded least-action. For example, as Cusa showed, there is no absolute maximum or absolute minimum polygon because the polygon is bounded maximally by a circle (which is not a polygon) and minimally by a line (which is also not a polygon). Or, while a mathematical catenary can be extended into infinity, the physical one is always bounded by the hanging points. For Riemann, as for Gauss and Dirichlet, Weierstrass’s demand for a formal mathematical proof of a minimum was less than unnecessary, it was a sophistry. The universal physical principle of least-action was sufficient to supply the proof.
Weierstrass’s critique was seized upon by the formalists who were desperate to roll back the achievements of Kaestner, Gauss, Dirichlet, Jacobi, Abel, Riemann et al. and return science to the slavish days of Euler, Lagrange and D’Alembert. Consequently, while the form of Riemann’s discoveries has been widely discussed, the substance of his thinking has, by and large, been suppressed, until, it found new life in the more advanced discoveries LaRouche.

Pythagoras as Riemann Knew him

There is a widely circulated report that when Pythagoras discovered the incommensurability of the side of a square to its diagonal, he sought to conceal its discovery on pain of death to whomever would disclose it. But such an account is of dubious veracity, as it attributes to Pythagoras an attitude more appropriate to his enemies than to his collaborators. For it was the Eleatics, Sophists and Aristotle, who insisted that what was inexpressible could not be known; and it was Aristotle’s Satanic disciples, as Bertrand Russell would come to exemplify, who demanded physical death for those who posed the potential for discovering new ideas; and it was Aristotle’s method itself, when practiced as directed, that caused so much mental disease from his day to ours. For Aristotle: control what can be expressed, and you control what can be known.

On the other hand, those who considered themselves Pythagoreans realized that the inexpressible was the frontier, not the barrier, of human thought. As Plato expressed it in the {Laws}, those who don’t know the significance of the incommensurability of the line with the square, and the square with the cube, were closer to “guzzling swine” than human beings. The issue for the Pythagoreans was not that the inexpressible could not be known, but simply that it could not be expressed, in terms consistent with an {a priori} set of axioms, postulates and definitions, as Aristotle insisted. Thus, for the Pythagoreans, the discovery of something inexpressible was not a cause for alarm, but a joyful occasion to demonstrate, that man was not constrained by mere Aristotelean logic, but was, unlike a swine, free and unbounded.

Therefore, as Plato insisted, it is of great benefit, and to be highly recommended, that political leaders discover for themselves the significance of incommensurability, in the terms that that discovery was known to Pythagoras and Plato. However, the true profundity of that discovery becomes much more fully illuminated when viewed from the standpoint of its more advanced development–the complex domain of Gauss and Riemann as that concept is expressed by Gauss’s 1799 {New Proof of the Fundamental Theorem of Algebra}, and Riemann’s crucial 1854 Habilitation lecture, and his 1855-57 lectures and writings on elliptical, Abelian and hypergeometric functions. These breakthroughs show that the principles discovered by the Pythagoreans were simply the first of an extended, and virtually unbounded, succession of transcendental functions, that express the increasing power of the human mind to discover, and communicate, ideas concerning universal physical principles.

Knowing Is Not Calculating

Much to the disdain of the Leibniz-hating followers of Euler, Kant, Lagrange and Cauchy, Riemann insisted that physical principles could be known, and given a mathematical expression, “virtually without calculation.” In taking this approach, Riemann was directly in the Pythagorean tradition of Plato, Cusa, Kepler and Leibniz, who all recognized, that to know a physical principle, meant to have an {idea} concerning that principle’s generative power, the which could never be discovered, nor expressed, by merely calculating that power’s visible effects. As Gauss noted in comparing Euler’s attempt to determine the orbit of a comet by calculation (an effort that left poor E. blind in one eye), with his own uniquely successful determination of the orbit of Ceres, “I too would have gone blind had I calculated like Euler!”

Gauss’s comment was consistent with, and inspired by, Kepler’s earlier attack on the Aristotelean Petrus Ramus’s diabolical demand that the tenth book of Euclid, (which concerns the incommensurables) be banned. Ramus insisted, as did Aristotle, that since only ratios of whole numbers were susceptible to finite calculation, no physical action was knowable, that could not be calculated thus. (Ironically, Gauss’s, {Disquisitiones Arithmeticae}, {Treatises on Biquadratic Residues I & II} and the subsequent work of Lejeune Dirichlet and Riemann on the subject of prime numbers show, that even the principles governing whole numbers cannot be expressed by the linear arithmetic advocated by Aristotle and Ramus.)

In his {Hamonices Mundi}, Kepler demonstrated that the physical principles that govern planetary motion cannot be expressed by the ratios of whole numbers, but only by those magnitudes which the Aristoteleans considered “inexpressible”, specifically the magnitudes associated with the regular divisions of the circle, the five regular spherical solids, and the harmonic relations of the musical tones.

This posed an ontological paradox for the Aristotelean. The principles governing physical action were inexpressible in terms acceptable to the Aristotelean. Therefore, as Aristotle’s syllogism went, the physical universe was unknowable.

But for Kepler, the principle governing physical action could be {discovered}, by physical hypothesis, and {known} as a simple, i.e. unified, idea ({Geistesmasse}) . The effect of that principle could be expressed mathematically only by the appropriate, “inexpressible”, magnitudes. An inexpressible magnitude was thus known, not in itself, but as that which was produced by the effect of a discovered physical principle.

{In other words, the principle is not known by a magnitude. Magnitude is known by the principle whose effect it expresses.}

Here, Kepler took his approach directly from Nicholas of Cusa, who, citing the Pythagoreans in {The Laymen on Mind}, insisted that such inexpressible magnitudes, such as the proportion of the side of a square to its diagonal, or the relationships among the musical tones, lead to an understanding of ” a number that is simpler than our mind’s reason can grasp”:

“By comparison then, see how it is that the infinite oneness of the Exemplar can shine forth only in a suitable proportion a proportion that is present in terms of number. For the Eternal Mind acts as does a musician, who desires to make his conception, visible to the senses. The musician takes a plurality of tones and brings them into a congruent proportion of harmony, so that in that proportion the harmony shines forth pleasingly and perfectly. For there the harmony is present as in its own place, and the shining forth of the harmony is made to vary as a result of the varying of the harmony’s congruent proportion. And the harmony ceases when the aptitude-for-proportion ceases.”

John Keats makes clear in his great poem, {Ode on a Grecian Urn}, that all human knowledge is gained in this way. Looking at the urn, Keats sees the images of an ancient Greek society– images of real people who lived and died, with passions much like ours. Yet all the questions he poses, which attempt to determine what the formalist would consider precise knowledge of those people and their culture, go unanswered. However, what is completely known, with absolute precision, is that {principle} of whose effect this urn is an image–the eternal power of human thought:

When old age shall this generation waste,
Though shalt remain, in midst of other woe
Than ours, a friend to man, to whom thou say’st,
“Beauty is truth, truth Beauty” that is all
Ye know on earth, and all ye need to know.

Toward an Extended Class of Higher Transcendentals

To understand Riemann’s essential discovery, we must take a quick look back, at the early development of the knowledge of inexpressibles, from the higher standpoint of Riemann’s work.

Begin with the magnitude which doubles the line. It can double the line but not a square. Yet, the magnitude that doubles the square is inexpressible, in terms of the magnitude that doubles the line. Inexpressible, but known–as that magnitude, that expresses the effect, of the physical principle, that has the {power}, (i.e., {dynamis}), to double a square. Thus, this simple, yet inexpressible magnitude, is known.

The magnitude that doubles the square, however, cannot triple, nor quadruple, nor quintuple, etc., a square. These magnitudes are associated with different physical actions. Though each is distinct, they are nevertheless mutually related, and expressed by the general relationship, which the Pythagoreans called one geometric mean between two extremes. Thus, each particular square power is generated by a still higher species of power–the power that generates all individual square powers.

This higher power can be given a clear mathematical expression as the geometrical relationships among the sides of the connected right triangles formed by a certain motion in a semi-circle. (See Figure 1.) While this construction expresses the effect of this power, as one unified action, it is not the power itself. The power is in the {idea} of that which has the power to generate all individual square powers. By giving the effect of this idea such an expression, our {mind’s} power to control, and act on this {physical} power, is increased.

Figure 1

But to know more of this idea, we must know not only what it can do, but what it cannot. This square power, while unlimited with respect to squares, is impotent to double a cube. The doubling, tripling, etc. of the volume of a cube, is the effect of a different species of power, which the Pythagoreans understood could be expressed as two geometric means between two extremes.

As Archytas’s construction demonstrates, the generation of this cubic power, can be given a mathematical expression by the proportions generated by a series of connected right triangles formed by the relative motion of two orthogonal semi-circles. (See Figure 2.) The relationships among the right triangles so produced, though changing, always express two geometric means between two extremes.

Figure 2

This construction expresses not only the effect of the cubic power, but also the connection between the cubic and the square power, because here, the effect that generates the square powers, is itself generated as an effect, of the motion that generates the cubic.

Even more importantly, the Archytas construction provides an insight, if seen from the standpoint of Cusa, into that still higher power, from which the square and cubic powers are themselves generated. While the specific magnitudes that correspond to the edges of squares and cubes are generated in the above construction as specific relationships among the lines forming the sides of right triangles, those relationships are determined not solely by lines, but by the connected effect of circular and rectilinear action.

This can be seen clearly in the above cited figures. In figure 1, the relationships among the sides of the triangle are formed as an effect of the connection between the uniform motion of “P” along the circular arc which generates the non-uniform motion of “Q” along a line. But in figure 2, “Q” now moves both along a straight-line, {and} around a circular arc, while the motion of “P” is along both a circular arc {and} along the curve formed by the intersection of a torus and cylinder.

Thus, it is a type of doubly-connected circular action that generates the rectilinear relationships that determine the effective changes in squares and cubes. Cusa, in {On the Quadrature of the Circle}, became the first to identify, and prove, that this circular action was an effect of an entirely different species of power, than the cubic and square powers. Leibniz identified this species of power as {transcendental}, as distinct from the lower species of powers (such as the cubic and square), which he called {algebraic}.

Power From the Standpoint of the Complex Domain

The above review is pedagogically helpful as a starting point for approaching the work of Gauss and Riemann. As these simple examples illustrate, physical processes are the effects of a connected action of physical powers (principles). Each power is expressed by a distinct species of magnitude. But, when a physical action is generated as the effect of a connected action among a group of powers, it generates a manifold, the which expresses a new, and completely different, characteristic species of magnitude. Riemann called such manifolds, “multiply-connected”.

A strong word of caution is in order. As will become more clear as we work through Riemann’s ideas, by “multiply-connected”, Riemann did not mean the Aristotelean idea of a set of theorems connected to one another through a lattice of logical formalism. Rather, Riemann’s multiply-connected manifold is a unity of demonstrable physical principles, which, like Leibniz’s {monads}, are distinct, but connected, not directly to each other, as if point-wise, but only through the higher organizing principle of the manifold itself.

A few physical examples, with which readers of this series will be familiar, will help illustrate this point:

–As Kepler’s principles of planetary motion illustrate, the planet’s motion, at every infinitesimal moment, is being determined by the connected action of all those principles that govern action in the solar system. This action is expressed mathematically by the combined effect of Kepler’s treatment of the five regular solids, the principles of elliptical motion, and the harmonic relationship among the musical pitches. As Gauss later showed through his determination of the orbit of Ceres, and his later work on the secular perturbations of the planets and asteroids, there are an even larger number of physical principles affecting the motion of the planet at each moment, than those expressed by Kepler. Gauss showed that the manifold of these connected principles can only be expressed in the complex domain. (See pedagogical discussion {Dance With the Planets.})

–The case of the intersection of a beam of light with a boundary between two different media, such as air and water, in which some of the beam is reflected and some of the beam is refracted. On the macroscopic level, we can see that this action must be thought of as occurring in a manifold that connects the two principles, reflection and refraction. But as we take this investigation into the microscopic domain, many more principles, those governing action in the atomic and sub-atomic domain, come into play, requiring a re-conceptualization of the manifold, into one with the power to connect a greater number of principles.

–The catenary’s expression of the universal principle of least-action as the arithmetic mean between two, oppositely directed exponentials. Each exponential itself denotes a manifold that transcends all algebraic powers. The catenary, therefore, must exist in a manifold that connects two such transcendental manifolds. In this higher manifold, both exponentials are acting, not only arithmetically, as indicated by their visible relationship, but also geometrically, the latter acting in the direction perpendicular to the visible plane of the hanging chain. (See Figure 3.) As Gauss showed, a manifold with the power to act on both exponentials arithmetically and geometrically, must be expressed as a surface in the complex domain.

Figure 3

In all of the above examples, the powers determining the physical action, are acting, from outside the visible domain, but their effects are present everywhere. Therefore, as Riemann made clear in his 1854 Habilitation lecture, to understand physical action, we must ban from science all considerations of geometry formed from a set of {a priori} axioms, postulates and definitions, and consider only {ideas} concerning physical manifolds, whose “modes of determination” are physical principles. With axiomatic assumptions now eliminated from geometry, the characteristic of action associated with Euclidean geometry, i.e., infinitely extended linearity, in three directions, disappears as the phantasm it always was. Instead, the characteristics of such a physical manifold are determined only by the physical principles which form the “modes of determination” of the physical action under consideration.

In his work, Riemann established the elementary principles to construct an image that faithfully reflects the means by which such physical “modes of determination” determine the characteristic of action in such a multiply-connected manifold, by showing how the effect of these principles determines the topology and characteristic curvature of the image. Most importantly, what is gained by Riemann’s method, is a means to determine and express the type of change that occurs, by the discovery of a new physical principle.

Riemann based his discovery on the previous work of Gauss, most notably, Gauss’s 1799 treatise on the fundamental theorem of algebra, and Gauss’s work on the general characteristics of curvature. Thus, it is most efficient pedagogically, to begin with a quick review of these features of Gauss’s work.

In rejecting the methods of Euler, Lagrange, and D’Alembert, Gauss showed that any formalist treatment of algebraic expressions, according to the logical rules of algebra, lead to a contradiction, (i.e. the square root of -1), within the domain of the formal system of algebra itself. This was not the result, Gauss insisted, of some hidden flaw within the logical system. It was a flaw of the system itself, arising from the fact that the algebra of Euler, Lagrange and D’Alembert was merely a logical system. As Gauss emphasized, the system could not be reformed, it had to be abandoned all together. In other words, Gauss did not come to save the system of algebra. He came to free science from its mind-killing constraints.

As Gauss showed, the inherent flaw in the formalist’s algebra, was the treatment of an algebraic power by simple rules of arithmetic. Gauss, in referring back to the Pythagorean principles of the doubling of the line, square and cube, insisted that the “power” in an algebraic expression must be understood to reflect a physical principle. For example, an algebraic expression of the second degree, must concern what Riemann would later call a “doubly- extended” relationship such as areas; an expression of the third degree, must concern a “triply- extended” relationship such as among volumes. A change from one power to another, therefore, denoted a change in the number of principles under investigation, not the number of times one number is multiplied by another. By constructing his surfaces as images that reflect this physical idea of power, the addition of a new power is reflected in the image, as a change in what he called the geometry of position, or topology, of the surface. (See Figure 4.) Thus, what is counted in algebra is not numbers, but powers. For Gauss, it was mind deadening brainwashing to consider an algebraic expression as a set of formal rules. Instead, he insisted, such expressions are, at best, only a short-hand description of a physical action, whose real characteristics could only be truthfully expressed through his geometric constructions.

Figure 4

Riemann insisted that only a method similar to Gauss’s could be applied when investigating the transcendental, elliptical and Abelian functions. As Leibniz had already indicated, such functions, by their very nature, could never be expressed by any formal algebraic- type means. For example, assigning a set of rules for calculating the expression “sine of x” does not give us any knowledge of the transcendental relationship between circular and rectilinear motion, let alone the profound connection that Leibniz discovered between circular, hyperbolic and exponential functions. Yet, as Leibniz emphasized, following Kepler and Cusa, universal {physical }action could only be expressed by such non-algebraic, “inexpressible” magnitudes.

Thus, for Riemann, to “know” a transcendental function, meant to know its geometrical characteristics, because all attempts at formal expression, as typified by the work of Euler, Lagrange, and the bigoted Cauchy, were always impotent. (See The Dramatic Power of Abelian Functions, Riemann for Anti-Dummies Part 54.)

In Riemann’s geometrical expressions, as in Gauss’s, the change from one transcendental power to another, is reflected as a change in the topology of the Riemann surface. For example, the circular/hyperbolic transcendental, which is associated with the catenary, is simply periodic, has two branch-points, and thus can be characterized by the topology of the sphere. (See Figure 5.) Whereas the elliptical transcendental associated with the elliptical orbit of a planet, or the motion of a pendulum, is doubly periodic, with four branch-points, and is characterized by the topology of the torus. (See Figure 6. See Riemann for Anti-Dummies 49, 52, 54, and 56 ).

Figure 5

Figure 6

Just as in the case of Gauss’s treatment of algebraic powers, each transcendental power is distinct. Consequently, the transition from one transcendental to another, because it involves the addition of a new principle, is not continuous. Like a discovery of a revolutionary new idea, the shift to a new transcendental, suddenly and completely, transforms all pre-existing relationships, that had been considered, until then, fundamental.

For example, think of how Riemann expressed the effect of a simply periodic transcendental function, through the image of a stereographic projection of a sphere onto a plane. In this image, the circles of latitude on the sphere are images of concentric circles in the plane, and, as such, are orbicular. But, the circles of longitude are images of radial lines which converge at the image of the “infinite”, i.e., north pole. Consequently, motion along these longitudinal circles can never be periodic, as a complete rotation must always “cross over the infinite”.

In this way, Riemann’s image fixes in our mind the idea of a physical process in which simple periodicity is a physical characteristic, not simply a mathematical formalism.

On the other hand, a doubly periodic action is a connected action with two distinct periods. Such an action could never be represented on a sphere with an infinite boundary. As Riemann showed in his treatment of the elliptical transcendentals, the type of surface on which these elliptical transcendentals “live”, must correspond topologically to a torus, whose “hole” allows for these two distinct, but connected, periods. However, as Riemann emphasized, the transformation from a sphere to a torus is discontinuous, because an entirely new possibility of action is added. In this way Riemann showed, that the essential characteristics of a transcendental function, {and} the characteristic of a change in transcendental power, could be made intelligible, even though such characteristics were utterly “inexpressible” in formal algebraic terms.

Riemann called the type of transformation just illustrated, a change in the “connectivity” of the manifold. For Riemann, the sphere is “simply-connected”, because it has no hole and requires only one closed curve to cut it into two distinct parts. The torus, on the other hand, is a surface that Riemann called “doubly-connected”, because it has one hole and requires two closed curves to cut the surface into two distinct parts. A “triply-connected” surface is one that has two holes, etc. (See Figure 7.).

Figure 7

Riemann emphasized that connectivity is a characteristic, like the number of “humps” in Gauss’s surfaces, that is independent of all measure relations of that surface, or calculations within the formal expression. For example, in the case of an algebraic expression, it doesn’t matter how wildly the coefficients of the expression vary, the physical characteristics of the action that expression describes are determined solely by the number of principles involved, as denoted by the expression’s highest “power”. This is what is reflected by the topology (number of “humps) of the corresponding Gaussian surface. In the case of Riemann’s investigation of the higher transcendentals, the “power” of the transcendental is expressed by a similar type of invariant characteristic, the surface’s connectivity.

It is important to note here, but reserve for the future its more complete development, that Riemann showed that this characteristic change in the topology of the image, is a function {solely} of the “power” of the transcendental function, which, in turn, is determined by the number of characteristic singularities generated by that transcendental function. Thus, the “holes” in a Riemann surface do not signify “nothingness”, or that something is missing or left out. Rather the number of holes signifies the density of singularities associated with the power of the transcendental function.

In this way, Riemann showed, in his lectures on Abelian and hypergeometric functions, that Abel’s “extended class” of transcendentals could be expressed by surfaces of increasing degrees of connectivity, or what Riemann called “multiply-connected” surfaces. A change in the number of singularities associated with a transcendental function, is expressed as a change in the connectivity of the surface that expresses that function.

Connectivity and Curvature

But, there is another significant characteristic of these higher transcendental functions which Riemann emphasized, but which only comes to light when Gauss’s general principles of curvature are taken into account. This can be introduced pedagogically by taking note of the change in the characteristic curvature of the surface associated with different transcendental functions. For example, a sphere, which is simply-connected, is everywhere positively curved, but a torus, which is doubly-connected, is positively curved only on the “outside”, but negatively curved on the “inside”. (Ironically, and interestingly, this combination of positive and negative curvature gives the torus a total curvature of zero!) Thus, a higher transcendental power is associated not only with a change in connectivity, corresponding to a change in the density of singularities, but also with a change in the characteristic curvature. Thus, a change in the power of a transcendental function , which occurs through the revolutionary discovery of an existing, but previously undiscovered universal principle, changes the characteristic curvature of the manifold of physical action.

To illustrate this, we must again turn back to the work of Gauss. In his {General Investigations of Curved Surfaces}, Gauss showed that on a positively curved surface the sum of the angles of a triangle is always greater than two right angles (180 degrees), whereas on a surface that is negatively curved, the sum of the angles of a triangle is always less than two right angles. Inversely, the characteristic curvature of a surface can be determined by the characteristics of the triangles that exist on it.

Furthermore, this characteristic curvature of a surface determines what Kepler called the types of congruences (harmonics) possible on that surface. For example, on a surface of zero curvature, six equilateral triangles can form a perfect congruence, because these triangles will all have angles of 60 degrees, and six such angles form one complete rotation. On the other hand, on a sphere, since any equilateral triangle will always have angles that are greater than 60 degrees, three, four or five triangles, but never six, will form a perfect congruence. Thus, from Gauss’s standpoint, the uniqueness of the five regular solida can be demonstrated to be a consequence of the characteristic curvature of spherical action.

But something very different happens on surfaces of negative curvature. Since here the angles of an equilateral triangle are always less than 60 degrees, perfect congruences can be formed by any number of triangles greater than six.

The problem Gauss understood, was that while surfaces of positive curvature could be represented as objects in visible space, such as a sphere, negative curvature acted on the visible domain from outside. Consequently, no negatively curved surface could be faithfully represented directly as a visible object! Gauss discovered, however, that the relationships of negatively curved surfaces could be represented visibly, but only as projections in the complex domain. Although Gauss never published his results, his notebooks document the direction of his thinking. Figure 8 shows one of Gauss’s drawings depicting the projection of a congruence formed by eight triangles, each with three 45 degree angles. Such triangles could only exist outside the visible domain, on a negatively curved surface.

Figure 8

To understand this projection, think of it as an analogy to the stereographic projection of the sphere onto the plane. In that case, the circles of longitude are projected onto radial lines, and the circles of latitude are projected onto concentric circles. (See Figure 9.)

Figure 9

The circles of longitude are orthogonal to all circles of latitude, as are the radial lines to the concentric circles in the plane. But, whereas the circles of longitude all converge on the north pole, the radial lines spread out, approaching Cusa’s infinite circle. Note, that these radial lines will, therefore, be orthogonal to the “infinite”. Spherical triangles on the sphere are projected onto the plane as triangles whose sides are circular arcs, and whose angles are the same as on the sphere. (See Figure 10.)

Figure 10

But, though the angles are preserved by the stereographic projection, distance is not. Consequently, as the distances measured approach the north pole of the sphere, the distances in the image on the plane increase exponentially.

Now look at Gauss’s projection of a negatively curved surface. Instead of an infinitely extended plane, the negatively curved surface projects onto a bounded disc. Here the sides of the triangles are formed by circular arcs, which, like the radial lines of the stereographic projection, are orthogonal to the boundary of the surface. Also, as in the stereographic projection, angles are preserved, but distances are not. But unlike in the projection of a sphere, where the distances become exponentially large as the boundary (“infinite”) is approached, the distances in the projected image of a negatively curved surface, become exponentially shorter. (See Figure 11.)

Figure 11

With this work of Gauss in mind, we can now begin to illustrate the relationship Riemann showed, between the increasing density of singularities associated with higher transcendental functions, and a change in the characteristic curvature of the manifold.

This can be illustrated pedagogically by comparing the difference between the elliptical transcendental and the hyper-elliptical. As developed earlier, the elliptical transcendental, which generates four singularities, is expressed as a Riemann function on a torus, on which there are two distinctly different types of curves that go around the torus and the curves that go “through the hole”. (See Figure 12.)

Figure 12

This doubly-connected action maps into a network of rectangles. (See Figure 13 & Riemann for Anti-Dummies 56). As we just discovered through Gauss, such a congruence of rectangles can only be formed on a surface of zero or positive curvature.

Figure 13

But the next highest transcendental, the hyper-elliptical, generates six singularities, and as Riemann showed, must be expressed on a triply-connected surface, such as a torus with two holes. On such a surface there are four distinct closed curves, instead of the two for the torus. (See Figure 14.) A mapping of these four pathways yields an octagonal congruence. (See Figure 15.)

Figure 14

Figure 15

As Gauss showed, such a congruence can only exist on a surface of negative curvature, and so its appearance in the case of the hyper-elliptical transcendental is the image of a physical action, characterized by negative curvature, acting from outside the visible domain.

Thus, as we now think of the hierarchy of the so-called “inexpressibles”, from the algebraic, to the circular transcendentals, to the elliptical transcendentals, to the hyper-elliptic and higher, we can understand a successive transformation in curvature from zero (rectilinear/algebraic), positive (spherical/exponential), to positive/negative (elliptical/toroidal), to negative (hyper-elliptical/Abelian).

Riemann emphasized that it is the relationship among these three characteristic curvatures, positive, zero and negative, that characterizes physical action. We cannot think of physical action as being characterized by any one type of curvature, but must consider the change in curvature that corresponds to the “power” governing the action. In the Habilitation lecture, Riemann posed a pedagogical construction of three such surfaces, represented by a sphere, cylinder, and the inside of a torus, all intersecting at one circle. (See Figure 16.) The circle is the unique pathway that at all times exists on all three types of curvature at once. Think of this circle as a new type of “infinitesimal”, a moment of change from one manifold to another of greater transcendental “power”.

Figure 16

This relationship between curvature and the higher transcendentals is of extreme importance for the future development of modern physical science. As Riemann stated in his Habilitation lecture, the characteristics of physical action change when extended from the observable range, into the astronomically large, such as the Crab Nebula and the microscopically small, such as the sub-atomic domain. Such changes correspond to an increasing density of universal principles, i.e., singularities, which in turn is reflected as changes in the characteristic curvature, and connectivity, of the manifold of physical action.

As science extends its investigations into these domains, an ever increasing number of universal physical principles will be discovered and incorporated into our knowledge of the universe. Such increases are associated with transcendental functions of increasingly higher power, of the type suggested by Riemann a type whose power is akin to that which connects us, through the mind of Keats, to those ancient people depicted on that Grecian urn.

RIEMANNIAN SPHERICS

When Carl Friedrich Gauss repeatedly stated his conviction that Euclidean geometry was not true, his thoughts were connected to the pre-Euclidean science of the Pythagoreans and Plato. However Gauss’s “{anti}-Euclideanism” was not a mere restatement of its antecedent. Rather, Gauss, and later Riemann, sublimated the ancient Egyptian-Pythagorean science of “spherics” with a new spherics, that had been demanded by Cusa’s discovery of transcendental physical action, the development of Kepler’s discovery of the harmonics of elliptical planetary orbits, and Leibniz’s generalization of both, as the universal principle of least-action.

This is the vantage point from which to gain a firmer grasp of Riemann’s treatment of Abelian and hypergeometric functions.

The relevant characteristics of the Pythagorean concept of spherics are summarized in the Plato dialogue, named for the astronomer Timaeus, who, attributing the origin of his discourse to the testimony of wise men from more ancient times, recounts the nature and creation of the universe. Timaeus begins by noting that investigations of the created world must focus on the eternal universal principles from which it is patterned. Though “apprehensible by reason and thought,”, those principles are, of necessity, reflected in the world’s physical form. Accordingly, when speaking about the physical universe, Timaeus emphasizes, it is most important to distinguish between the original principles and their created copy, and one must be careful to recognize that a simple description of the latter cannot suffice for an explanation of the former.

“Accordingly, in dealing with a copy and its model, we must affirm that the accounts given will themselves be akin to the diverse objects which they serve to explain; those which deal with what is abiding and firm and discernible by the aid of thought will be abiding and unshakable; and in so far as it is possible and fitting for statements to be irrefutable and invincible, they must in no wise fall short thereof; whereas the accounts of that which is copied after the likeness of that model, and is itself a likeness, will be analogous thereof and possess likelihood; for as Being is to Becoming, so is Truth to Belief. Wherefore, Socrates, if in our treatment of a great host of matters regarding the Gods and the generation of the Universe we prove unable to give accounts that are always in all respects self-consistent and perfectly exact, be not thou surprised; rather we should be content if we can furnish accounts that are inferior to none in likelihood, remembering that both I who speak and you who judge are but human creatures, so that it becomes us to accept the likely account of these matters and forbear to search beyond it.” (29c-d)

Timaeus goes on to say that the Creator, being good, intended for that goodness to be reflected in the creation and so he endowed it with intelligence and soul, and brought it into existence as a self moving, self-subsisting living creature.

“He fashioned it to be One single Whole, compounded of all wholes, perfect and ageless and unailing. And he bestowed on it the shape which was befitting and akin. Now for that living creature which is designed to embrace within itself all living creatures, the fitting shape will be that which comprises within itself all the shapes there are; wherefore he wrought it into a round, in the shape of a sphere, equidistant in all directions from the center to extremities, which of all shapes is the most perfect and most self-similar, since he deemed that the similar is infinitely fairer than the dissimilar….

“Such then was the reasoning of the ever-existing God concerning the god which was one day to become existent, whereby He made it smooth and even and equal on all sides from the center, a whole and perfect body compounded of perfect bodies, and in the midst thereof He set soul, which He stretched throughout the whole of it; therewith He enveloped also the exterior of its body; and as a circle revolving in a circle He established one sole and solitary Heaven, able of itself because of its excellence to company with itself and needing none other beside, sufficing unto itself an acquaintance and friend. And because of all this He generated it to be a blessed God..” (34-b.)

This spherical form, Timaeus explains, expresses not only the goodness and perfection of a universe patterned from reason and intelligence, but it also expresses the motions of the visible objects in the heavens as well. These visible objects, whose individual motions trace circular arcs onto the celestial sphere, form the “moving image of eternity” we call time. Timaeus proceeds to recount, in detail, the diverse motions of these celestial objects, the totality of which are visible expressions of that which is eternal and universal.

But Plato emphasized that these characteristic spherical motions, while a means to express the characteristics of an idea, cannot, by themselves, be taken as knowledge of the idea. But, being patterned after the eternal, their study can aid us in grasping what can only be apprehended by reason and thought:

“Thus, said I. These sparks that paint the sky, since they are decorations on a visible surface, we must regard, to be sure, as the fairest and most exact of material things, but we must recognize that they fall far short of the truth, of the movements, namely of real speed and real slowness in true number and in all true figures both in relation to one another and as vehicles of the things they carry and contain. These can be apprehended only by reason and thought, but not by sight, or do you think otherwise?

By no means, he said.

Then, said I, we must use the blazonry of the heavens as patterns to aid in the study of those realities, just as one would do who chanced upon diagrams drawn with special care and elaboration by Daedalus or some other craftsman or painter. For anyone acquainted with geometry who saw such designs would admit the beauty of the workmanship, but, would think it absurd to examine them seriously in the expectation of finding in them the absolute truth with regard to equals or doubles or any other ratio.” (Rep. 529e-530b)

The Boundaries of the Sphere

Because these visible expressions could also mislead reason, Plato cautioned against the methods of the Sophists and Eleatics, (forerunners of today’s mathematical formalists and information theorists), whose practice, like their modern counterparts, was to induce insanity in their victims by restricting their attention to the visible form of mathematical objects that have been disconnected from the universal principles from which they arose, and, from the ideas whose shadows they were intended to represent. As he warned in the {Republic}:

“This at least, said I, will not be disputed by those who have even a slight acquaintance with geometry (earth-measure), that this science is in direct contradiction with the language employed in it by its adepts.

How so? he said.

Their language is most ludicrous, though they cannot help it, for they speak as if they were doing something and as if all their words were directed toward action. For all their talk is of squaring and applying the adding and the like, whereas in fact the real object of the entire study is pure knowledge.

This is absolutely true, he said.

And must we not agree on a further point?

What?

That it is the knowledge of that which always is, and not of a something which at some time comes into being and passes away.

That is readily admitted, he said, for geometry is the knowledge of the eternally existent.” (Rep. 527a.)

The Sophists insisted that geometry could state nothing truthful concerning “knowledge about that which always is” because geometrical objects, being objects of sense, lie in the domain of that which is always changing. Consequently for the sophist, geometrical objects, like all other objects of sense, could be connected to each other by a set of formal rules, described and manipulated, but such actions were {by definition} divorced from the eternal physical principles governing the universe itself. By limiting scientific inquiry to such objects of sense perception, as Aristotle also maintained, the Sophists could awe their audiences with dazzling demonstrations, such as the simultaneous existence and non-existence of the one and the many, while convincing them that knowledge of universal principles was, in the end, beyond their grasp.

By contrast, the Pythagorean-Platonic scientists admitted no objects into geometry that did not arise in connection with an effort to grasp an unseen, eternal, universal principle. Such objects, such as the Pythagorean concept of the sphere, were brought into existence by an action of a mind intending to gain greater mastery over the principles that govern the universe itself. Consequently for these Socratics, all such geometrical objects are representations of ideas that are apprehensible only by reason and thought. “Thought-objects” so generated contain within them, the essential characteristics of the universal principles they are intended to grasp. Being thus connected to universal principles, paradoxes generated by the investigation of such objects, point to the existence of unknown principles, that, once discovered, lead to the generation of new ideas, that demand new types of geometrical expression.

In the science of Classical Greece, this is exemplified by the uniqueness of the five regular solids. Nothing in the smooth, everywhere same, form of a sphere gives any indication of the existence of a bounding principle that determines that there be five, and only five, unique solids that equally divide that everywhere constant surface. But, the fact that only five such solids can be constructed, reveals the existence, and reflects the characteristics, of that bounding principle. Thus, even though the sphere is the shape “that comprises all shapes”, the five regular solids indicate the existence of a still higher principle from which the sphere’s power is derived. This higher power, though outside the visible boundary of the sphere, is, nevertheless, expressed on the sphere’s surface, by the uniqueness of the five regular solids.

That that higher principle can be found only by surpassing the boundaries of the sphere, was already evident to the Pythagoreans, as indicated by Archytas’s construction for the doubling of the cube by the intersection of a torus, cylinder and cone. This stands in contrast to the doubling of the square, whose generating principle can be expressed by circular action existing within the same manifold (plane) in which the square itself is generated. On the other hand, the cube, which is formed in the sphere, cannot be doubled by means of spherical action. As Archytas demonstrated, a more complex unfolding of the sphere is required. Thus, the characteristic of the cube, as well as the other four regular solids, indicated to the Pythagoreans and Plato, the existence of a still higher principle that bounds the apparently self-bounded spherical form.

Because of paradoxes like these, Plato insisted that the study of these solids should be emphasized, in addition to the Pythagorean quadrivium of arithmetic, geometry, astronomy and music, in the education of political leaders, so that those leaders could develop the cognitive powers to recognize universal principles and act only on the basis of truth, not on the basis of arbitrary opinion.

For opposing reasons, Aristotle introduced the arbitrary fiction that the form of the universe is an unbounded, empty, linearly extended void, as codified in the Elements of Euclid. Under this scheme, spherical action is not the reflection of the universal principles from which it is generated. It is but only one, of infinitely many undifferentiated possible types of action. For Euclid, the sphere’s existence, like the rest of the objects of Euclidean geometry, rests only on the authority of an {a priori} set of definitions, axioms and postulates. For Euclid, and those indoctrinated by sophism, empiricism, etc. this false, fantasy world of linear action becomes the only true one from which all physical processes deviate.

Ironically, the Pythagorean-Platonic method expressed as self-bounded spherical action reflects the unlimited potential for the discovery of new physical principles, because the existence of these new principles is indicated through the emergence of paradoxes, such as the uniqueness of the five regular solids. On the other hand, the Aristotelean-Euclidean fantasy of an apparently unbounded, linearly extended void, that, {by definition}, permits no physical principles within its midst, acts as a barrier to the introduction of new ideas, restricting human knowledge to a definite and finite domain: a barrier that held back human progress for nearly 1500 years, from the murder of Archimedes in 231 B.C. until the dawn of the European Renaissance in the 14th century.

In sum: the self-bounded is unlimited potential; the unbounded is forever constrained.

Surpassing the Boundaries of the Self-Bounded Sphere

The Aristotelean barrier was cracked when Nicholas of Cusa re-introduced the Pythagorean-Platonic concept of spherics, in the new, more advanced form of his method of “On Learned Ignorance”. Like his forerunners, Cusa understood that mathematical objects, generated as metaphors, were a means for expressing new discoveries:

“Therefore, in mathematicals the wise wisely sought illustrations of things that were to be searched out by the intellect. And none of the ancients who are esteemed as great approached difficult matters by any other likeness than mathematics….

“Proceeding on this pathway of the ancients, I concur with them and say that since the pathway for approaching divine matters is opened to us only through metaphors, we can make quite suitable use of mathematical signs because of their incorruptible certainty.”

Cusa’s approach was emphatically anti-Aristotelean. Instead of the unlimited, linearly extended infinite void, in which geometrical objects rattled around, devoid of intention, Cusa saw the apparent infinite merely as the domain of yet to be discovered universal principles, that bounded the finite and determined the characteristic of action within it. The characteristics of that set of principles can be discovered by ascending upward from their finite manifestation towards the higher expression:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined; if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a symbolism.”

Cusa’s approach redefined the self-bounding characteristic of the Pythagorean sphere. For Cusa, the visible sphere, though apparently self-bounded, was actually bounded by a higher domain of universal principles that lay outside the sphere itself. Because the domain of these principles exists outside the domain of finite physical objects, it appears, from the finite perspective of the objects, to be infinitely far away.

Thus, the characteristics of the visible sphere, most importantly for Cusa, the incommensurability between the spherically curved and the linearly straight, reflected the higher principle from which the curved and the straight were both generated. Thus, for Cusa, in opposition to Aristotle, the curved was not a deviation from straightness. Rather, the incommensurability between them, reflected one single principle, in unfolded form. Cusa expressed this relationship by the ironical juxtaposition of the idea of a finite sphere and line with an hypothesized infinite one:

“I maintain, therefore, that {if} there were an infinite line, it would be a straight line, a triangle, a circle and a sphere. And likewise {if} there were an infinite sphere, it would be a circle, a triangle, and a line.” (emphasis added).

From this standpoint, Cusa redefined the physical expression of Pythagorean spherics. There are no perfectly circular or spherical motions in the created world, Cusa correctly insisted. Nor is there perfect equality in harmonics, statics, or other physical phenomena. There is no absolute center to the universe, nor is there a single set of poles. However, this imprecise nature of the physical world does not indicate its deviation from the false-perfection of the {a priori} linearly-extended fantasy world of Aristotle’s Euclidean-type geometry. Rather, this imprecision indicated the characteristics of the true physical-geometry of the universe, which cannot be described by simple abstract geometrical shapes, but must be discovered by physical investigations:

“Thereupon you will see through the intellect, to which only learned ignorance is of help that the world and its motion and shape cannot be apprehended. For the world will appear as a wheel in a wheel and a sphere in a sphere having its center and circumference nowhere, as was stated.”

But, the fact that the physical universe cannot be described by simple, static shapes, does not make it unknowable. It is through this very imprecision, as projected into the visible domain, (as for example, onto the Pythagorean sphere) that the true higher ordering principle can be discovered, as knowledge of the Creator’s intention:

“Who would not admire this Artisan, who with regard to the spheres, the stars, and the regions of the stars used such skill that there is though without complete precision both a harmony of all things and a diversity of all things? This Artisan considered in advance the sizes, the placing, and the motion of the stars in the one world; and He ordained the distances of the stars in such a way that unless each region were as it is, it could neither exist nor exist in such a place and with such an order nor could the universe exist….”

“But all things reply to him, who in learned ignorance asks them what they are, or in what manner they exist, or for what purpose they exist…”

On this basis, Kepler, who called Nicholas of Cusa “divine”, broke the centuries-old barrier that had been established by Aristotelean dogma, and established a new, modern physics not tied to any {a priori} considerations, but only to physical ones. As Kepler noted in the opening of his {New Astronomy}, it may be the first presumption of reason that action in the physical universe is circular, but physical experiment determined this to be untrue. Consequently, all {a priori} geometrical descriptions of the motions of the planets, such as those adopted by Ptolemy, Copernicus, and Brahe must be abandoned in favor of a physical-geometry based only on physical causes. For Kepler, the paradoxes among the tracings of the motions of the planetary orbits onto the celestial sphere provided the clues that led him to discover the principles that were determining them. Thus, Kepler investigated not the objects of the sky, nor simply their motions as projected onto the celestial sphere. Rather, he investigated the relationships between these projected motions, from which he apprehended the characteristics of the unseen principles from which those motions were determined.

It is from this standpoint that Kepler was able to elaborate the harmonics of the non-uniform, essentially elliptical motion of the planetary orbits, giving an entirely new meaning to the physical significance of the ancient Pythagorean discoveries concerning the boundaries of spherical action, such as the five regular solids, conic sections, and musical harmonics. The question posed by Kepler to all future generations, was to develop a new type of mathematics that could express these higher principles. Standing on Leibniz’s shoulders, this is what Gauss and Riemann developed as the complex domain.

The Physical “Shape” of the Universe

As Riemann emphasized in his 1854 habilitation lecture, his philosophical fragments, and his lectures on Abelian and hypergeometric functions, and as Gauss had spoken earlier, the common, intuitive notions concerning geometrical objects presupposes a set of assumptions concerning the fundamental nature of the “space” in which those objects arise. Like Cusa, Kepler, and Leibniz before them, both Gauss and Riemann insisted it was the action of physical principles that determined the characteristics of what is called “space”, not a set of arbitrary {a priori} assumptions. While these physical principles are unseen, the effect of their action is measurable.

To express the connection between these measurable effects and the unseen principles themselves, Gauss and Riemann developed a new form of spherics, by extending Leibniz’s infinitesimal calculus into the complex domain.

For Gauss this meant rejecting all arbitrary notions of the sphere as a three-dimensional object bouncing around in an empty, linearly-extended void. Instead, Gauss considered the sphere to be what Riemann would later call, a doubly-extended manifold. In this way, Gauss eliminated from the sphere (and all other doubly-extended manifolds), all arbitrary ideas of linear extension. Linear distance between any set of points on the surface is determined solely as a function of angular displacement. The relationship between angular displacement and linear extension reflected the effect of an invariant characteristic of the surface. That characteristic Gauss called “curvature”.

By “curvature” Gauss did not mean “not-straight”. Gauss’s idea of “curvature” is an expression of the characteristic principles that determine action on that surface. But, since the characteristic curvature is not visible, it must be discovered by its effect on all action on that surface.

For example, the elliptical arc of a planetary orbit is determined by the harmonic characteristics of the principle of universal gravitation. To the planet, such an arc is a “straight-line”. To our senses the planet’s orbit is a variable arc of a great circle on a sphere. But to our minds, as Kepler showed, we can discover that those arcs are merely the projection onto a sphere of an action that is occurring in a different surface a surface we cannot see. However we can imagine this other surface, not as a visible image, but as that set of physical principles which has the power to produce the physical effects whose visible images are projected onto the celestial sphere. Furthermore, since we are moving in that surface ourselves, we must apprehend its characteristic curvature, from within it, by measuring the effect of that curvature in the relatively infinitesimally small–a method that Leibniz called “analysis situs”.

Gauss’s idea of a surface, therefore, is not a simple visible shape but a manifold of physical principles. To better determine the characteristics of these manifolds, Gauss developed the means to map these manifolds onto a sphere. Under these mappings the characteristic curvature of the unseen surface is expressed as anomalies within the domain of spherical functions. Working backwards, we can then “unfold” these projections, to determine the characteristics of the manifold from which they are projected. (See Riemann for Anti-Dummies Parts 44-46.)

The Riemann Sphere

Armed with Gauss’s work, Riemann generalized the method. As indicated in his fragment on “n-dimensional” manifolds, Riemann considered the geometrical ideas of line, surface and solid as only special intuitive examples of a singly, doubly and triply extended manifold, whose modes of determination are physical principles, not linear extensions. However, Riemann emphasized as did Gauss, Leibniz, Kepler, Cusa, and Plato before him, that the construction of the appropriate metaphorical representation was essential to achieve a greater understanding of these principles, and to communicate these discoveries to others, so as to bring them under the willful control of mankind.

Riemann developed these ideas in his lectures on Abelian and hypergeometric functions, whose deeper implications were not realized until LaRouche’s more advanced, revolutionary breakthroughs in the science of physical economy.

Riemann’s starting point was Gauss’s original idea that the physical characteristics of doubly-extended manifolds, (i.e. surfaces) could be represented in the complex domain as functions of complex numbers. This is based on Gauss’s idea of representing complex numbers as a function of position on a surface.

This is {not} the idea of representing complex numbers as points on an abstract Cartesian grid as is universally taught in reductionist mathematics education today. We emphasize, as Gauss did, that any function of position must be in reference to some physically determined starting point and a physically determined direction. For example, the determination of position on the celestial sphere, must be in reference to some starting point (the pole), and some direction (the direction of the position of sunrise on the vernal equinox). Once these two parameters (origin and direction) are determined, all other positions on the surface can be expressed with respect to them. This is distinct from the Aristotelean, Euclidean, Cartesian, Kantian idea of an undifferentiated, linearly-extended space. As Gauss pointed out with reference to Kant, it is impossible to determine anything about the nature of space except by reference to physical objects. In other words, Gauss did not begin with a Cartesian-Kantian idea of an empty surface and then assign some arbitrary origin and direction to it. Rather, physics establishes the origin and reference direction, which in turn established the surface.

Riemann showed that once such a physically determined doubly-extended surface has been established, the characteristics of action within that surface can be expressed as a complex function. Here again, Riemann adopted Gauss’s approach of the “Copenhagen Prize Essay” on conformal mappings. For example, the stereographic projection of the sphere onto the plane is an expression of the complex exponential. (See Figure 1.)

Figure 1

Thus, complex functions considered as conformal mappings, express the transformation of one surface, with a certain characteristic of action, into another, with a different characteristic of action.

For pedagogical purposes let us investigate this matter in more detail with respect to the ancient Pythagorean problem of doubling the square, as seen from the higher perspective of the complex domain. As Gauss demonstrated in his new proof of the fundamental theorem of algebra, the most general expression of an algebraic power is represented geometrically as a specified change in angle and length. That is, squaring is expressed by doubling the angle and squaring the length with respect to that physically determined origin and direction. Cubing is expressed by tripling the angle and cubing the length, etc. (See Figure 2.)

Figure 2

So, for example, in the simplest case of squaring as a complex function, a semi-circular pathway will be mapped onto a full circle. (See Figure 3.)

Figure 3

Thus, a complete circle will be mapped onto a complete circle twice. (See Figure 4.)

Figure 4

Transforming an entire surface in this way, thus maps half the original surface on top of the other. (See Figure 5.)

Figure 5

Since it is impossible, in a Cartesian empty space, to distinguish the first mapping from the second, Riemann invented the idea of a surface that has two layers with the second mapping laying on top of the first. Thus, by squaring, every point on the original surface is transformed into two different points, one laying directly over the other, on the “squared” surface. Those points that have only one value (in this example, the origin), Riemann called a “branch point”. (See Figure 6.) These branch points correspond to physical singularities on the surface as the characteristic of action with respect to them is different than the rest of the surface. A surface with no branch points, Riemann called, “simply-connected”; a surface with one branch point he called “doubly-connected”; a surface with two branch points he called “triply-connected”, etc.

Figure 6

Riemann emphasized that this type of mapping can not occur in a linear, infinitely-extended surface, but only in a bounded one. The reader is encouraged to discover this for yourself. You will discover that it is impossible to think of transforming one doubly-extended manifold (i.e. surface) into another without a boundary. Even if the nature and characteristics of that boundary are not yet determined, its necessary existence is.

This is another indication of why physical action can only be expressed in the complex domain, as there is no physical process that is not bounded. For this reason, Riemann insisted that one could not express complex functions on a Cartesian plane, but only on a bounded surface, as represented by what today is called, “The Riemann Sphere”. (See Figure 7.)

Figure 7

Now, back to Riemann’s surfaces. Riemann confronted the problem that when a mapping produces a multi-sheeted surface, a paradox results, because a continuous curve on one surface becomes two separate curves on two (or more) different layers of the second surface. For this reason he specified the need for a cut, which he called a “branch-cut” that extended from the branch points to the boundary. The various layers of the surface are then connected to one another along these branch cuts, so that all pathways on one of the layers can be continued onto all the other layers continuously. (See Figures 8a, 8b, 8c, 8d, and 8e. These figures include some computer generated representations as well as a hand drawn representation by Riemann himself.)

Figure 8a

Figure 8b

Figure 8c

Figure 8d

Figure 8e

On the Riemann sphere, the entire function can be represented because unlike the Cartesian plane which has no boundary, the Riemann sphere is bounded. The accompanying animation, (See Figure 9.) illustrates the squaring function projected onto Riemann’s sphere. As the boundary on the plane extends outward from the origin, its projection approaches the “north pole” of the sphere. In this example, the squaring function forms a two-sheeted surface on the sphere, whose branch cut runs from one pole to the other.

Figure 9

These animations illustrate the action of the singularity on the Riemann sphere. Here two closed pathways, one encircling the branch point at the origin (10a.) and the other not, (10b.) are illustrated.

Figure 10a

Figure 10b

Under squaring, the pathway encircling the branch point at the origin folds over itself and is mapped onto two sheets. The pathway that does not contain the branch point remains entirely on one layer. (See Figure 10c.)

Figure 10c

However, something is revealed on the Riemann sphere that would otherwise remain hidden. The sphere reveals the existence of another branch-point at the north pole, which in the Cartesian plane can only be considered “the infinite”. This can be seen by performing the same squaring action, but under the inversion that turns the function “inside out”. What was the origin is now the “infinite” and vice versa. (See Figure 11.) This has the same effect on closed pathways. Inversion turns the pathways inside out, which now encircle the branch point at the “infinite”, or the north pole on the sphere. (See Figure 12.) Just as the branch point at the origin acted as a singularity, so now does the other branch point, that had been hidden in the “infinite” in the Cartesian plane, but on the Riemann sphere it is plainly brought into view.

Figure 11

Figure 12

Beyond the Sphere

On this basis, Riemann was able to demonstrate, using Leibniz’s principle of “analysis situs”, that all the essential characteristics of a complex function, are determined by the relationship of the boundary to the branching points, or the other singularities.

This discovery was crucial to Riemann’s treatment of elliptical and Abelian functions. As we discussed in previous installments (See Riemann for Anti-Dummies Parts 49, 52 and 54) these functions, that initially arose from the investigations of Kepler’s elliptical planetary orbits, express a succession of higher transcendental powers. Riemann demonstrated, using a method similar to the one used by Gauss in his proof of the fundamental theorem of algebra, that those complex functions associated with these higher transcendentals generate surfaces with two sheets, but with more than the two branch points expressed by the algebraic, circular, hyperbolic or exponential functions. (See Figure 13.) As Riemann showed, each successively higher power, generates a new set of branch points.

Figure 13

For example, as illustrated in the previous installments, Gauss showed that the elliptical transcendentals were distinguished from the circular, hyperbolic, and elliptical, by their double periodicity. (See Figure 14.)

Figure 14

Riemann showed that this characteristic, by necessity, produces four branch points, which require two branch cuts. (See Figure 15.)

Figure 15

A surface with two branch cuts must be doubly connected, as for example a torus (See Figure 16.) Thus, each new species of transcendental function is characterized by a change in the characteristic topology of the surface, from an “n” connected manifold to an “n+1” connected manifold.

Figure 16

This is the type of transformation associated with the discovery and application of a new physical principle.

Now looking back from the standpoint of this new type of Riemannian spherics, we can recognize that the Pythagorean spherics, is only the simplest type of a succession of self-bounded surfaces of increasingly higher “connectivity”. The provocative question now posed is: What bounds this succession of self-boundedness?

This same question was posed by Gauss in a different way concerning the determination of what is the true geometry. In a letter to Olbers on April 28, 1817, Gauss asserted, like Cusa, that the true geometry could never be seen directly but must be discovered:

“Perhaps in another life we will come to another insight into the essence of space, which is now unreachable for us. Until then, one must not put geometry with arithmetic, purely a priori, but closer to in rank with mechanics…”

The Dramatic Power of Abelian Functions

To understand Riemann’s treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche’s unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at once, as a dramatic history, leap over time, project the past into the future, the future into the past, and both into the present, so that centuries of accomplishment are telescoped into a single, instantaneous thought.

Dynamis

In the opening scene of Plato’s {Theatetus} dialogue, Euclides of Megara informs Terpsion that he has just seen Theatetus being carried to Athens, near death, after being wounded in the battle at Corinth. This experience prompts Euclides to recall that Socrates, when near his own death many years before, had told him that “Theatetus will be a remarkable man if he lives.” Now, contemplating Theatetus’s life at its end, Euclides reports that Socrates had also recounted his first conversation with Theatetus when the latter was merely a boy. Having written down Socrates’s report, Euclides now reads the history to Terpsion, the which forms the bulk of Plato’s dialogue. As Euclides tells it, the conversation began with the young Theatetus being praised by his teacher Theodorus, because the former had surpassed the insights of the latter. As Theatetus explained, Theodorus had taught him about the incommensurablity of lines that double, triple, quintuple, etc. a square, by demonstrating each as a separate and distinct power, beginning with tripling and ending with seventeen, where, “for some reason he stopped.” But, Theatetus continued, “Since the number of powers are innumerable, the notion occurred to us of including them all under one name or species.”

Then by reference to a simple geometrical construction, Theatetus indicates the existence of three distinct {species} of powers: those that generate lines, those that generate squares and those that generate solids. Each species comprises an entire manifold of separate and distinct individual powers. But, each manifold, Theatetus explains, can be thought of under a single principle: linear, quadratic, or cubic.

Not mentioned directly in the dialogue, but prominent in the background, is the related discovery of Archytas, who showed that each species is associated with a different type of physical action: linear, circular and toroidal, respectively.

These discoveries of Theatetus and Archytas demonstrate a power of the mind that the Eleatics, Sophists and later, Aristotle, denied existed, and one that is essential to Riemann’s treatment of Abelian functions: {the capacity of the mind to rise above the finite determinations of sensible objects, and to recognize the higher universal principles that determine them}.

As Theatetus demonstrated, it is possible to {know} an entire species of powers, without having to construct each one individually, by knowing the principle that determines what each individual power can do e.g., double a square– and what they can’t– double a cube. To know that all these individual powers are of one species, {and} that the principle that generates square powers can never generate cubic ones, requires the mind to rise above the characteristics unique to each individual magnitude and recognize the nature of the entire species, {and} the nature of its boundaries.

Aristotle insisted that the human mind did not have this power because, being limited and mortal, it could not rise from finite determinations to universal ones:

“Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self- subsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also….”

“It is plain, too that the infinite cannot be an actual thing and a substance and a principle….

“Thus the view of those who speak after the manner of the Pythagoreans is absurd….

“This discussion, however, involves the more general question, whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject matter, the objects of sense, and we have to ask whether there is, or is not, among them a body which is infinite in the direction of increase.” (Physics, Book 5)

Aristotle’s argument is pure sophistry. By {defining} physics to concern only sensible objects, he excludes all consideration of the universal principles that generate those objects. Once excluded, he asserts that such principles play no role in the physical world, because being infinite, they cannot actually exist.

Contrary to Aristotle, as Theatetus demonstrates, in and through Plato’s dialogue, or as Socrates himself makes clear in numerous other locations, most notably the {Phaedo}, immortality, not temporal boundaries, characterizes the human spirit. It can freely transcend the finite limits of the sensual domain through its power to comprehend, not only things, such as lines, squares and cubes, but the powers that generate them. It can transform itself, as Theatetus had shown, from comprehending different powers individually, as Theodorus had taught, to comprehending the concept of an entire species of innumerable individual powers, as he had discovered.

And, as the above account of Plato’s account, of Euclides’s account, of Socrates’ account, of Theatetus’s account of his own discovery illustrates, the human spirit is not constrained by the mortal life in which it originates.

Powers that Generate Powers

The method described in the {Theatetus }dialogue is an elementary, but historically and pedagogically important example of the type of thinking that underlies Riemann’s method, elaborated in his {Theory of Abelian Functions}. There Riemann is dealing with the principles that generate higher and higher species of functions, beyond those contemplated by the Pythagoreans, Theodorus, Theatetus, Archytas and Plato. While Riemann is looking back on what was anticipated by these great ancient minds, he does so from the vantage point of the discoveries of Cusa, Kepler, Leibniz, Gauss and Abel. What these later discoverers demonstrated, was that the species of powers indicated by Plato et al., were superceded by a succession of higher species of successively greater power. Riemann, extending Gauss’s concept of the complex domain, created the means to comprehend the higher power that generates this extended class of higher powers. His treatment of Abelian functions gives expression to the quality of mind that can reach beyond the domain of sense perception, transcend what {appears} to be infinite, recognize the existence of new species of transcendental powers, and {generate} them as higher forms of cognition.

To understand Riemann’s elaboration of Abelian functions it is crucial to emphasize the epistemology underlying Cusa’s earlier discovery of a new species of power, by the method that Cusa called {“Learned Ignorance “}.

Cusa discovered that the incommensurability of the curved to the straight, as exemplified by the incommensurability between a circle and its inscribing and circumscribing polygons, indicated the existence of a new species of power, that transcended those named by Plato, et al. Cusa proved, and Leibniz later elaborated, that this new species, which Leibniz called {transcendental}, could generate all lower, {algebraic} powers, whereas the inverse–to generate the transcendental from the algebraic–is impossible.

Like Plato, Cusa recognized that to comprehend these higher species of powers, the mind had to look beyond the boundaries of the domain of sense-perception, to discover the higher powers that govern it from the outside:

“I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….” (Actualized-Potential)

Cusa gave his method a mathematical representation, but {not by the formal, logical deductive procedures of Aristotle, or his later followers, Euler, Lagrange, and the bigoted Cauchy}. In Cusa’s mathematical representation, what appears to be mathematically infinite, that is, what is over, or beyond, the finite, (what Cantor would later call the transfinite) indicates the existence of a new power or principle acting on the domain of sense-perception from outside. Thus, pushing finite mathematical expressions to the boundary, and beyond, reveals the characteristics of those sought for, but yet undiscovered, higher powers:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined: if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a metaphor.” (On Learned Ignorance, Book I.)

Cusa elaborated this metaphor extensively. He examined carefully the difference between the finite relations of the point, line, polygon and circle, and their transfinite relations. In the finite, these objects are each distinct, but transformed into their infinite forms, they become congruent, indicating the existence of the higher principle from which their finite relations are unfolded. Thus, the lower, finite forms are bounded (and governed) by the higher, which, from the standpoint of the lower, {appears} to be infinitely far away. Therefore, to know anything about the relationships in the finite, one must view them from the standpoint of the conditions at that seemingly infinite boundary. As we will soon illustrate, this relationship reaches a more developed form in Riemann’s treatment of Abelian functions.

Like Plato, Cusa associated this capacity of the human mind, to reach beyond the boundary of the apparent infinite, and identify new species of numbers as powers, with the immortality of the soul:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible? Moreover, no number can deplete the mind’s power of numbering. Hence, since the motion of the heavens is numbered by the mind, and since time is the measure of motion, time will not exhaust the mind’s power. Rather, the mind’s power will continue on as the limit, measure, and determination of all things measurable. The instruments for the motions of the heavens instruments produced by the human mind attest to its not being the case that motion measures mind rather than mind’s measuring motion. Hence, mind seems to enfold by its intellective operation all movement of succession, and mind brings forth from itself rational operations, or rational movement. Thus, mind is the form of moving. Hence, if when anything is dissolved, the dissolution occurs by means of motion, then how could the form of moving be dissolved through motion? Since mind is an intellectual life that moves itself i.e., is a life that gives rise to the life which is its understanding how would it fail to be always alive? How could self-moving motion cease? For mind has life conjoined to it; through this life it is always alive. (By way of illustration: a sphere is always round, by virtue of the circle that is conjoined to it.) If mind’s composition is as the composition of number, which is composed of itself, how would mind be dissolvable into not-mind?…

…thus, he who numbers unfolds and enfolds. Mind is an {image} of Eternity, but time is an {unfolding} of Eternity…”(The Laymen on Mind)

Unlike the heathen Aristotle, for Cusa, the infinite is not outside the universe. What appears to be infinitely far away from the standpoint of sense-perception, is only the boundary marking the transition to the unseen, but efficient, domain of universal physical principles. Those principles reach through that boundary, into the visible domain, determining the characteristics of action within it. The mind, in turn, reaches back, transcending the apparently infinite, and discovering the characteristics of those principles. {Thus, the appearance of the infinite in mathematical expressions, simply indicates the necessity for a transformation of the mind’s idea a transformation that makes a universal principle that had appeared to be inaccessible, known.}

The Physical Generation of Powers

From Cusa, the road to the discovery of those higher species of transcendental powers that are the subject of Riemann’s work, goes through the investigations of Gauss and Abel who were provoked by:

–Kepler’s discovery that the principle of universal gravitation generated planetary orbits that were essentially elliptical, and;

–Leibniz’s determination that these orbits reflected a universal principle of physical least- action expressed by the catenary curve.

In rejecting the Aristotelean formalism of Ptolemy, Copernicus, and Brahe, and basing his investigation only on physical considerations, Kepler was compelled to reject the mathematically simpler circular orbits for the physically determined elliptical ones. This produced a crisis, because, whereas in a circular orbit, there is an incommensurability between the angle and the sine, in an elliptical orbit, there is an incommensurability between the arc and the angle as well. Kepler’s attempts to measure elliptical motion from the standpoint of the circular functions (i.e., as the connected action of a rectilinear triangle and a circular arc), produced the paradox known as “The Kepler Problem”. Kepler developed a method to approximate this measurement for practical purposes but, for him, this was insufficient. Kepler was not a pragmatist. Though practical solutions were vital to his work, Kepler was insistent on knowing the principle as well. When knowledge of that principle eluded him, Kepler demanded that future geometers solve the problem, insuring that even his own death would not end his quest.

While Kepler did not discover the higher principle of elliptical motion, he indicated where to look. In his related work on optics, Kepler proposed to look at the ellipse as merely one phase in a single function that generated all the conic sections. He expressed this geometrically through his famous projection that carried the circle into a hyperbola. (See Figure 1.) The emergence of an apparently infinite boundary between the ellipse and the hyperbola, within an otherwise continuous function, indicated the direction in which this undiscovered principle could be found.

Figure 1

The appearance of the infinite boundary between the ellipse and the hyperbola reflects, of course, the double cone construction of the conic section. On one side of the “infinite”, the circle, ellipse and parabola, only the lower cone is cut by an intersecting plane. It is when that plane touches the upper cone, that the hyperbola is formed. (See Figure 2.)

Figure 2

Leibniz, through an application of his infinitesimal calculus, showed that the curvature of the hanging chain can be expressed as the arithmetic mean between two exponential curves. (See Figure 3.) These exponential curves do not exist in the perceptible domain of the hanging chain. Nevertheless, they express a characteristic of the {physical manifold} that is acting, from outside the domain of sense-perception, on the chain, in every infinitesimal interval.

Figure 3

As Leibniz emphasized, these exponential curves express that transcendental species of power that generates all the lower algebraic ones. (See Figure 4.)

Figure 4

This same relationship is also expressed, in different forms, by the equiangular spiral and the hyperbola. (See Figure 5a, and Figure 5b.)

Figure 5a

Figure 5b

In connection with this investigation of the catenary, Leibniz coined the term {“function”} to denote a specified transformation. So, for example, the exponential function transforms arithmetic relations into the geometric ones. Its inverse, the logarithmic function, transforms geometric relations into arithmetic ones. As the arithmetic mean between two exponential functions, the catenary, therefore, is a function of two functions.

When Leibniz pushed this investigation to its boundary, by attempting to determine the function that generates the logarithms of negative numbers, he met the appearance of the square root of -1, which he identified as existing in a real, but “imaginary” domain.

That Undiscovered Country

Thus, the young Gauss, confronted with the appearance of what Leibniz called, “a fine and wonderful recourse to the divine spirit, almost an amphibian somewhere between being and non-being”, granted the square roots of negative numbers their “full civil rights”. From this standpoint he reexamined the “Kepler Problem”, and discovered that elliptical motion was governed by a higher species of transcendental, that had been anticipated by, but not known to Kepler or Leibniz.

The physical significance of complex numbers emerged quite naturally from Leibniz’s catenary principle and the paradox of negative logarithms. (See Figure 6.)

Figure 6

Since the catenary is formed as the arithmetic mean between {two} exponential curves, Leibniz sought the function that generated both, which could be expressed as that which generated their geometric mean. But, there is no continuous transformation, within the plane of the catenary, and the exponentials hanging behind it, that can transform one exponential into the other. The only physical action that has the power to produce both curves, is a rotation orthogonal to the plane of the catenary. (See Figure 7.)

Figure 7

If the direction of one exponential is designated as positive and the other direction negative, the geometric mean between them is expressed as the square root of a negative number. (See Figure 8.)

Figure 8

This gives rise to the following paradox: In the plane of the two exponentials, the geometric mean between them is a straight-line. But, when looked at from the higher standpoint of physical action, that straight-line is actually the axis of rotation from which is generated an entire surface orthogonal to the plane of the chain. Thus, to represent the physical action of the chain, the unseen exponentials that hang behind it, and the unseen principle that generates the exponentials themselves, requires both the plane of the chain, and the plane of the action orthogonal to it.

Gauss conceived the idea of representing this complex domain on one surface. That surface acts as a stage for the imagination on which {both} the physical action and the universal principles that govern it can be represented.

The two exponentials in the plane of the catenary are generated by an action that is completely outside their visible domain. The point at which the axis of rotation touches the surface generated by that rotation, i.e. the center of the circle formed by that rotation, is the only intersection between the visible domain and the surface on which the principle of generation is represented.

Just as on the Classical stage, a single line, (“Its Greek to me.”) in the context of the drama, can indicate an historical transformation of an entire culture, on the surface representing the complex domain, an entire principle, in the context of a complex function, can be represented by the action at a single point.

Hidden Harmonies Become Heard

By adopting Gauss’s approach, Riemann was able to create a stage, on which an innumerable class of transcendentals of successively higher power could be brought within the scope of the imagination. Each new species of transcendental, beginning with the simplest species of transcendentals discovered by Cusa, to the elliptical functions of Gauss, to those higher species discovered by Abel, is distinguished from its predecessor by the addition of a new principle. Under Riemann’s idea, each new principle is represented by a change in the geometrical characteristics, {the topology} of the surface.

Here Riemann adopted Leibniz’s method of {analysis situs} as applied by Gauss in his 1799 dissertation on the fundamental theorem of algebra. Gauss showed that formal algebra could not distinguish the implications of a change from one power to another. This is reflected, as Gauss ironically emphasized, in the inability, from the standpoint of formal algebra, to answer algebra’s most important question: “How many solutions are there for a given algebraic expression?” It was only when these expressions were thought of geometrically, as surfaces, that their essential characteristics could be made known. Then, it was clear, that the highest power of the function determined a topological characteristic (the number of “humps”) that was independent of any variations in the lower powers or the coefficients of that function. (See Figure 9a, and Figure 9b.)

Figure 9a

Figure 9b

Thus, Gauss’s employment of Leibniz’s geometry of position, {analysis situs}, expressed the epistemological relationship that the principle associated with the highest power dominated all lower ones. A change in the degree of power is, therefore, a change in the governing principle, and is expressed by a transformation of the entire geometry of the surface. Thus, the essential characteristics of an entire species of algebraic functions could be known completely, without the useless formal calculations of Euler, Lagrange and D’Alembert.

To illustrate Riemann’s approach, we begin with a look at the simplest species of transcendental, those associated with the circular, hyperbolic and exponential functions. As Leibniz and Bernoulli demonstrated the catenary, expressing the universal physical principle of least-action, embodied all three of these functions in one single physical action.

However, from the standpoint of the “simple” domain, these three functions seemed to have entirely different characteristics. For example, the circle and the hyperbola are conic sections, the exponential is not. In the visible domain, the circle is periodic, the hyperbola and exponential appear infinitely extended.

Leibniz imagined that a higher principle united all three transcendentals, but he never elaborated his idea. The sophisticated, but Leibniz-hating Euler, tried to obscure Leibniz’s insight with a series of algebraic formalisms, (such as his infamous equations: e^i*Pi-1=0; and e^i*x=cos[x]+i*sin[x]) which have bedeviled generations of students and scientists to this day.

Yet through Gauss’s and Riemann’s physical conception of the complex domain it can be easily demonstrated that these three functions express one unified species of transcendentals, and are all derived from the complex exponential. Gauss recognized this as early as August 14, 1796, writing in his notebook: “By the way, (a+b*?-1)^m+n*?-1 has been explained”.

Here Gauss indicates he has extended the idea of the exponential function to the complex domain. As noted above, the exponential function is the transformation of arithmetic relationships into geometric ones. In the “simple” domain, this appears geometrically as the characteristic curvature of a single curve, as, for example, the equiangular spiral, hyperbola or exponential curve. In Gauss’s complex domain, the complex exponential function transforms the arithmetic relations of an entire surface into geometric ones, transforming all curves into new ones.

This can be seen most easily from the geometrical example in Gauss’s Copenhagen Prize Essay on conformal mapping (on which Riemann relied.). There Gauss shows that the complex exponential function corresponds to the stereographic projection of a sphere onto a plane a projection first developed by the ancient Greeks in connection with the mapping of the celestial sphere. (See Figure 10.)

Figure 10

Under this projection, the circles of “latitude” on the sphere are projected onto concentric circles on the plane, whose radii are expanding exponentially. Circles of “longitude” on the sphere are transformed into radial lines on the plane.

Don’t think of this transformation as a static image. Think of it from the standpoint of physical action. Think of the motion along a circle of latitude on the sphere. What is the corresponding motion on the plane? Think of the motion along a circle of longitude on the sphere. What is the corresponding motion in the plane? Think of motion along a path of equal heading on the sphere (loxodrome). What is the corresponding motion along in the plane? When you think of it in this way, you can begin to see how all relationships on the sphere are transformed, lawfully, into different relationships on the plane.

Here a crucial new idea emerges that cannot be seen in the “simple” domain. What appeared to be “infinitely” long radial lines of the plane, become “finite” circles of longitude on the sphere. Action on the plane that appears to go off into the “infinite” approaches a single point on the sphere. From the standpoint of Cusa, if the terminus of all the radial lines in the plane is thought of as the “infinite” circle, its image is a single point on the sphere. In other words, the principle bounding the action on the plane, that appears to be infinite and unknowable, is brought into the imagination, by the action associated with a single point on the sphere. Thus, in the complex domain, we can faithfully represent Cusa’s notion that in the infinite, the center and circumference, minimum and maximum, coincide.

It is important to emphasize, however, that physical action, such as the hanging chain, is never “infinite”. The catenary is the curve of a chain hanging between two positions. Thus, “this side of the infinite”, the physical exponential is always bounded. It is the universal principle that it reflects that is {transfinite}.

Another characteristic now comes into view that otherwise remained hidden with the “simple” view of the exponential. In the simple domain, the exponential curve appears aperiodic But, this is only an illusion caused by viewing the exponential too simply. As can be seen in the image of the stereographic projection, the radial lines are unbounded, but the circles of latitude are periodic. (As you can imagine, this period is real. It is Euler’s fraud to insist that the period of the complex exponential is “imaginary”.)

The hidden periodicity of the complex exponential can be seen more dramatically when viewed as a transformation of an orthogonal grid of curves on a plane, or, as Riemann suggested, on the surface of a very large sphere. (See Figure 11.)

Figure 11

Here all “straight” lines are transformed into a network of equiangular spirals, bounded by spirals with maximum rotation and minimum extension (i.e., circles), and spirals with maximum extension and minimum rotation (i.e., radial lines).

Think of this image from the standpoint of action. Under the transformation of the complex exponential, motion along a “straight”line is transformed into spiral motion, including the extreme cases of the circles and the radial lines. The non-periodic motion along the vertical lines is transformed into periodic circular action. (See Figure 12.)

Figure 12

The aperiodic motion along horizontal lines is transformed into motion along the exponentially extending radii, which remains aperiodic. Motion along a diagonal line is transformed into ever expanding spiral action. (See Figure 13.)

Figure 13

Now compare this result with what was noted above concerning the two exponential curves associated with the hanging chain. In the visible domain, this action is aperiodic. However, the rotation orthogonal to the plane of the hanging chain has a periodicity. Here we can recognize the physical reality of the exponential function’s “imaginary” period.

As Leibniz and Bernoulli demonstrated in their work on the catenary, the hyperbolic functions are functions of the exponential: the hyperbolic cosine is one-half the sum of two exponentials and the hyperbolic sine is one-half the difference of two exponentials. (See Figure 14.)

Figure 14

As Leibniz also insisted, the circular functions are also functions of the exponential, but only in a different, “imaginary”, domain. (See Figure 15.) The circular cosine is one-half the sum of two complex exponentials, while the circular sine is one-half the difference of two complex exponentials.

Figure 15

When viewed from the standpoint of Gauss’s and Riemann’s complex domain, all these functions can be expressed by the same type of action: the arithmetic mean between two complex exponentials moving in opposite directions, as illustrated by the circles in the accompanying animation. (See Figure 16a, Figure 16b, Figure16c, Figure 16d, and Figure 16e.) Together, the hyperbolic and circular functions form the four different variations of the same type of action. Thus, generated by the same type of action, they are all of the same species of transcendental.

Figure 16a

Figure 16b

Figure 16c

Figure 16d

Figure 16e

The Elliptical Transcendentals

The young Gauss, contemplating the origin of the “Kepler Problem”, was puzzled by the inability to express elliptical motion by the circular, hyperbolic or exponential functions. But, when he looked at a similar problem, that of the lemniscate, from the standpoint of the complex domain, he discovered a different species of transcendental that was expressed by elliptical motion. When the characteristics of that species are viewed on Riemann’s stage, the difference between the elliptical and the lower transcendentals becomes obvious.

While Gauss did not state it this way directly, his January 1797 decision to begin serious investigation of the lemnsicate alludes to the clue suggested earlier by Kepler’s concept of the conic sections.

As noted above, Kepler thought of all the conic sections as being generated from one continuous function. The emergence of an infinite boundary, between the ellipse and the hyperbola, reflects the transition to the inclusion of the upper cone in the double conical function. Thus, the entire function involves action in two different directions (parallel and perpendicular to the base of the cones). When thought of from the standpoint of Cusa, the appearance of the infinite signifies that a higher, undiscovered principle exists that encompasses both the perpendicular and parallel action as one. It is from this higher type of function that Kepler’s conic section function is unfolded.

This function must have the same relationship to the circular and hyperbolic functions as those functions have to the algebraic. That is, the higher species must have the capacity to generate the lower, but not the inverse. This is akin to the relationship between the quadratic and cubic species of magnitudes as viewed from the standpoint of Archytas. The principle of toroidal motion that generates cubic magnitudes can generate the quadratics as well, but not the inverse.

Now what is the nature of these elliptical transcendentals and how can we discover it? Let’s reconstruct Gauss’s investigation of the lemniscate from the standpoint of the clue supplied by Kepler, and the later solution supplied by Riemann.

That the lemniscate would become the focus of Gauss’s attention, flows quite naturally from the extension of Kepler’s concept of the conic sections into the complex domain. As presented in previous installments of this series, the lemniscate can be generated as the inversion of the hyperbola in a circle, or the stereographic projection of a hyperbola onto a sphere. (See Figure 17 and Figure 18.) From the standpoint of these two projections, it can be seen that the circle is the geometric mean between the hyperbola and the lemniscate. Thus, we seek a higher function that transforms the hyperbola, through the circle, into the lemnsicate.

Figure 17

Figure 18

It is important to take note of the method of inversion employed here. In the case of the two exponential curves and the catenary, the geometric mean between the two exponential curves could not be found within the visible domain of the curves and the hanging chain. The search for the principle that would generate both exponentials led us into the complex domain. In the case of the conic sections, Kepler’s function generates one from the other. But the appearance of the infinite in the middle of the function coaxes us to consider a higher function. That function generates, not another conic section, but a lemnsicate.

Now look at Kepler’s function, as he envisioned it projected onto a plane, and, from Riemann’s standpoint, projected onto a sphere. (See Figure 19.) In the former case, the circle is transformed into an ellipse, which is transformed into an hyperbola, and then into a line. In the latter, the circle is transformed into a projected ellipse, which is transformed into a lemnsicate, and then into two coincidental hemispheric circles.

Figure 19

Thus, to grasp the higher principle we have to think of both functions happening simultaneously: the one on the stage of the plane that is generating the conic sections, and the one on the stage of the sphere, that is generating a lemniscate. But from our vantage point we see both on {one} stage the stage of the complex domain.

Note that the infinite boundary that appears in the plane, appears in the spherical projection as the crossing point of the lemniscate. This presence of the {transfinite}, represented as a single point in the lemniscate, signifies the existence of an additional principle, that is outside the domain of the conic sections, but, {inside} the domain in which the lemnsicate resides. As noted in a previous installment, Gauss showed that the crossing point on the lemniscate also reflects the boundary between the regions of positive and negative curvature on the torus. (See Figure 20.)

Figure 20

Does the presence of this new principle indicate that the lemniscate is associated with a different species of transcendental than the circular or hyperbolic functions? Gauss said yes, and Riemann provided the basis to recognize this geometrically.

Gauss recognized the existence of this added principle in the lemniscate by the double periodicity of the lemnsicatic functions. Where, for example, the circular functions are periodic with respect to the interval 0 to 2Pi, the lemnsicatic functions are periodic with respect to the interval 1 to -1 {and} the interval \/-1 and -\/-1. (See Riemann for Anti-Dummies 52.)

Gauss expressed the geometrical characteristic of double periodicity in his work on bi-quadratic residues, which he published in 1832. However, its discovery was much earlier, as indicated by his notebook entry, “I have discovered a remarkable connection between the lemniscate and bi-quadratic residues”.

That connection is illustrated by the geometrical representation Gauss gave, with respect to simple and complex moduli, in his {Second Treatise on Bi-Quadratic Residues}. There Gauss shows that for real numbers, the modulus can be expressed geometrically by a simple line segment or curve. (See Figure 21a and, Figure 21b.)

Figure 21a

Figure 21b

However, in the complex domain, a complex modulus is expressed by a parallelogram on a surface. (See Figure 22.)

Figure 22

This same geometrical distinction, as Gauss noted, also appears in comparing the lemniscate with the circular functions in the complex domain. In the case of the circular, hyperbolic or exponential functions, motion in one direction was periodic, while motion perpendicular to it was not. As illustrated above, this type of motion could be mapped onto a surface such as a sphere.

However, for the lemnsicate, the motion is periodic in two directions simultaneously. (See Figure23a, Figure 23b, and Figure23c.)

Figure 23a

Figure 23b

Figure 23c

Such motion can not be represented on Riemann’s sphere, where one direction, such as the circles of latitude, are periodic, but, in the orthogonal direction, the circles of longitude, are not periodic without crossing the infinite.

Riemann showed that the elliptical functions could only be expressed on a surface that allowed for two distinct periodic curves. Such a surface can be constructed, Riemann showed, by taking the parallelogram of Gauss’s complex modulus and connecting the opposite sides to each other, forming a surface with the topology of a torus. (See Figure 24.)

Figure 24

On a torus, there are two distinct curves, one that goes around the outside, and one that goes through the hole. (See Figure 25.)

Figure 25

Riemann emphasized that the torus is an entirely different type of surface than a sphere. {And, there is no continuous function that can transform a sphere into a torus, without the introduction of a new principle. Once introduced, that new principle effects a complete transformation of all relationships on that surface.}

Through this investigation into the distinction between the higher species of elliptical transcendentals and the lower species, we are able to recognize that the difference between these two species of transcendentals reflects a fundamental change in the number of principles acting through each species. As Abel showed, an entire class of transcendentals can be constructed of successively higher power . In Riemann’s elaboration, the change from one species to the higher, conforms to a transformation from a domain of “n” principles to a domain of “n+1” principles. As in Archytas’s construction for the doubling of the cube, or Gauss’s method of his 1799 dissertation on the fundamental theorem of algebra, such transformations can be made intelligible on the stage of the complex domain, as a discontinuous transformation in the {topology} of that domain.

More importantly, however, armed with this new power to make intelligible the generation of higher species of transcendentals, we can now envision the transformation that must occur, in any domain when a new principle is added. Thus, through the willful employment of the mind’s power of discovery, we can reach past the boundary of what seems to be infinite, and bring new principles into our conscious possession.

And this brings us back to Theatetus, who, as a young boy, shows us today, that an entire species can be known, completely without calculation.

The Dramatic Power of Abelian Functions

To understand Riemann’s treatment of Abelian functions, situate that discovery within the context of the history in which it arose, reaching back to the pre-Euclidean Pythagoreans of ancient Greece, and forward to LaRouche’s unique and revolutionary discoveries in the science of physical economy. Imagine that entire sequence, all at once, as a dramatic history, leap over time, project the past into the future, the future into the past, and both into the present, so that centuries of accomplishment are telescoped into a single, instantaneous thought.

Dynamis

In the opening scene of Plato’s {Theatetus} dialogue, Euclides of Megara informs Terpsion that he has just seen Theatetus being carried to Athens, near death, after being wounded in the battle at Corinth. This experience prompts Euclides to recall that Socrates, when near his own death many years before, had told him that “Theatetus will be a remarkable man if he lives.” Now, contemplating Theatetus’s life at its end, Euclides reports that Socrates had also recounted his first conversation with Theatetus when the latter was merely a boy. Having written down Socrates’s report, Euclides now reads the history to Terpsion, the which forms the bulk of Plato’s dialogue. As Euclides tells it, the conversation began with the young Theatetus being praised by his teacher Theodorus, because the former had surpassed the insights of the latter. As Theatetus explained, Theodorus had taught him about the incommensurablity of lines that double, triple, quintuple, etc. a square, by demonstrating each as a separate and distinct power, beginning with tripling and ending with seventeen, where, “for some reason he stopped.” But, Theatetus continued, “Since the number of powers are innumerable, the notion occurred to us of including them all under one name or species.”

Then by reference to a simple geometrical construction, Theatetus indicates the existence of three distinct {species} of powers: those that generate lines, those that generate squares and those that generate solids. Each species comprises an entire manifold of separate and distinct individual powers. But, each manifold, Theatetus explains, can be thought of under a single principle: linear, quadratic, or cubic.

Not mentioned directly in the dialogue, but prominent in the background, is the related discovery of Archytas, who showed that each species is associated with a different type of physical action: linear, circular and toroidal, respectively.

These discoveries of Theatetus and Archytas demonstrate a power of the mind that the Eleatics, Sophists and later, Aristotle, denied existed, and one that is essential to Riemann’s treatment of Abelian functions: {the capacity of the mind to rise above the finite determinations of sensible objects, and to recognize the higher universal principles that determine them}.

As Theatetus demonstrated, it is possible to {know} an entire species of powers, without having to construct each one individually, by knowing the principle that determines what each individual power can do e.g., double a square– and what they can’t– double a cube. To know that all these individual powers are of one species, {and} that the principle that generates square powers can never generate cubic ones, requires the mind to rise above the characteristics unique to each individual magnitude and recognize the nature of the entire species, {and} the nature of its boundaries.

Aristotle insisted that the human mind did not have this power because, being limited and mortal, it could not rise from finite determinations to universal ones:

“Some, as the Pythagoreans and Plato, make the infinite a principle in the sense of a self- subsistent substance, and not as a mere attribute of some other thing. Only the Pythagoreans place the infinite among the objects of sense (they do not regard number as separable from these), and assert that what is outside the heaven is infinite. Plato, on the other hand, holds that there is no body outside (the Forms are not outside because they are nowhere), yet that the infinite is present not only in the objects of sense but in the Forms also….”

“It is plain, too that the infinite cannot be an actual thing and a substance and a principle….

“Thus the view of those who speak after the manner of the Pythagoreans is absurd….

“This discussion, however, involves the more general question, whether the infinite can be present in mathematical objects and things which are intelligible and do not have extension, as well as among sensible objects. Our inquiry (as physicists) is limited to its special subject matter, the objects of sense, and we have to ask whether there is, or is not, among them a body which is infinite in the direction of increase.” (Physics, Book 5)

Aristotle’s argument is pure sophistry. By {defining} physics to concern only sensible objects, he excludes all consideration of the universal principles that generate those objects. Once excluded, he asserts that such principles play no role in the physical world, because being infinite, they cannot actually exist.

Contrary to Aristotle, as Theatetus demonstrates, in and through Plato’s dialogue, or as Socrates himself makes clear in numerous other locations, most notably the {Phaedo}, immortality, not temporal boundaries, characterizes the human spirit. It can freely transcend the finite limits of the sensual domain through its power to comprehend, not only things, such as lines, squares and cubes, but the powers that generate them. It can transform itself, as Theatetus had shown, from comprehending different powers individually, as Theodorus had taught, to comprehending the concept of an entire species of innumerable individual powers, as he had discovered.

And, as the above account of Plato’s account, of Euclides’s account, of Socrates’ account, of Theatetus’s account of his own discovery illustrates, the human spirit is not constrained by the mortal life in which it originates.

Powers that Generate Powers

The method described in the {Theatetus }dialogue is an elementary, but historically and pedagogically important example of the type of thinking that underlies Riemann’s method, elaborated in his {Theory of Abelian Functions}. There Riemann is dealing with the principles that generate higher and higher species of functions, beyond those contemplated by the Pythagoreans, Theodorus, Theatetus, Archytas and Plato. While Riemann is looking back on what was anticipated by these great ancient minds, he does so from the vantage point of the discoveries of Cusa, Kepler, Leibniz, Gauss and Abel. What these later discoverers demonstrated, was that the species of powers indicated by Plato et al., were superceded by a succession of higher species of successively greater power. Riemann, extending Gauss’s concept of the complex domain, created the means to comprehend the higher power that generates this extended class of higher powers. His treatment of Abelian functions gives expression to the quality of mind that can reach beyond the domain of sense perception, transcend what {appears} to be infinite, recognize the existence of new species of transcendental powers, and {generate} them as higher forms of cognition.

To understand Riemann’s elaboration of Abelian functions it is crucial to emphasize the epistemology underlying Cusa’s earlier discovery of a new species of power, by the method that Cusa called {“Learned Ignorance “}.

Cusa discovered that the incommensurability of the curved to the straight, as exemplified by the incommensurability between a circle and its inscribing and circumscribing polygons, indicated the existence of a new species of power, that transcended those named by Plato, et al. Cusa proved, and Leibniz later elaborated, that this new species, which Leibniz called {transcendental}, could generate all lower, {algebraic} powers, whereas the inverse–to generate the transcendental from the algebraic–is impossible.

Like Plato, Cusa recognized that to comprehend these higher species of powers, the mind had to look beyond the boundaries of the domain of sense-perception, to discover the higher powers that govern it from the outside:

“I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….” (Actualized-Potential)

Cusa gave his method a mathematical representation, but {not by the formal, logical deductive procedures of Aristotle, or his later followers, Euler, Lagrange, and the bigoted Cauchy}. In Cusa’s mathematical representation, what appears to be mathematically infinite, that is, what is over, or beyond, the finite, (what Cantor would later call the transfinite) indicates the existence of a new power or principle acting on the domain of sense-perception from outside. Thus, pushing finite mathematical expressions to the boundary, and beyond, reveals the characteristics of those sought for, but yet undiscovered, higher powers:

“…when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined: if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figure. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of a metaphor.” (On Learned Ignorance, Book I.)

Cusa elaborated this metaphor extensively. He examined carefully the difference between the finite relations of the point, line, polygon and circle, and their transfinite relations. In the finite, these objects are each distinct, but transformed into their infinite forms, they become congruent, indicating the existence of the higher principle from which their finite relations are unfolded. Thus, the lower, finite forms are bounded (and governed) by the higher, which, from the standpoint of the lower, {appears} to be infinitely far away. Therefore, to know anything about the relationships in the finite, one must view them from the standpoint of the conditions at that seemingly infinite boundary. As we will soon illustrate, this relationship reaches a more developed form in Riemann’s treatment of Abelian functions.

Like Plato, Cusa associated this capacity of the human mind, to reach beyond the boundary of the apparent infinite, and identify new species of numbers as powers, with the immortality of the soul:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible? Moreover, no number can deplete the mind’s power of numbering. Hence, since the motion of the heavens is numbered by the mind, and since time is the measure of motion, time will not exhaust the mind’s power. Rather, the mind’s power will continue on as the limit, measure, and determination of all things measurable. The instruments for the motions of the heavens instruments produced by the human mind attest to its not being the case that motion measures mind rather than mind’s measuring motion. Hence, mind seems to enfold by its intellective operation all movement of succession, and mind brings forth from itself rational operations, or rational movement. Thus, mind is the form of moving. Hence, if when anything is dissolved, the dissolution occurs by means of motion, then how could the form of moving be dissolved through motion? Since mind is an intellectual life that moves itself i.e., is a life that gives rise to the life which is its understanding how would it fail to be always alive? How could self-moving motion cease? For mind has life conjoined to it; through this life it is always alive. (By way of illustration: a sphere is always round, by virtue of the circle that is conjoined to it.) If mind’s composition is as the composition of number, which is composed of itself, how would mind be dissolvable into not-mind?…

…thus, he who numbers unfolds and enfolds. Mind is an {image} of Eternity, but time is an {unfolding} of Eternity…”(The Laymen on Mind)

Unlike the heathen Aristotle, for Cusa, the infinite is not outside the universe. What appears to be infinitely far away from the standpoint of sense-perception, is only the boundary marking the transition to the unseen, but efficient, domain of universal physical principles. Those principles reach through that boundary, into the visible domain, determining the characteristics of action within it. The mind, in turn, reaches back, transcending the apparently infinite, and discovering the characteristics of those principles. {Thus, the appearance of the infinite in mathematical expressions, simply indicates the necessity for a transformation of the mind’s idea a transformation that makes a universal principle that had appeared to be inaccessible, known.}

The Physical Generation of Powers

From Cusa, the road to the discovery of those higher species of transcendental powers that are the subject of Riemann’s work, goes through the investigations of Gauss and Abel who were provoked by:

–Kepler’s discovery that the principle of universal gravitation generated planetary orbits that were essentially elliptical, and;

–Leibniz’s determination that these orbits reflected a universal principle of physical least- action expressed by the catenary curve.

In rejecting the Aristotelean formalism of Ptolemy, Copernicus, and Brahe, and basing his investigation only on physical considerations, Kepler was compelled to reject the mathematically simpler circular orbits for the physically determined elliptical ones. This produced a crisis, because, whereas in a circular orbit, there is an incommensurability between the angle and the sine, in an elliptical orbit, there is an incommensurability between the arc and the angle as well. Kepler’s attempts to measure elliptical motion from the standpoint of the circular functions (i.e., as the connected action of a rectilinear triangle and a circular arc), produced the paradox known as “The Kepler Problem”. Kepler developed a method to approximate this measurement for practical purposes but, for him, this was insufficient. Kepler was not a pragmatist. Though practical solutions were vital to his work, Kepler was insistent on knowing the principle as well. When knowledge of that principle eluded him, Kepler demanded that future geometers solve the problem, insuring that even his own death would not end his quest.

While Kepler did not discover the higher principle of elliptical motion, he indicated where to look. In his related work on optics, Kepler proposed to look at the ellipse as merely one phase in a single function that generated all the conic sections. He expressed this geometrically through his famous projection that carried the circle into a hyperbola. (See Figure 1.) The emergence of an apparently infinite boundary between the ellipse and the hyperbola, within an otherwise continuous function, indicated the direction in which this undiscovered principle could be found.

Figure 1

The appearance of the infinite boundary between the ellipse and the hyperbola reflects, of course, the double cone construction of the conic section. On one side of the “infinite”, the circle, ellipse and parabola, only the lower cone is cut by an intersecting plane. It is when that plane touches the upper cone, that the hyperbola is formed. (See Figure 2.)

Figure 2

Leibniz, through an application of his infinitesimal calculus, showed that the curvature of the hanging chain can be expressed as the arithmetic mean between two exponential curves. (See Figure 3.) These exponential curves do not exist in the perceptible domain of the hanging chain. Nevertheless, they express a characteristic of the {physical manifold} that is acting, from outside the domain of sense-perception, on the chain, in every infinitesimal interval.

Figure 3

As Leibniz emphasized, these exponential curves express that transcendental species of power that generates all the lower algebraic ones. (See Figure 4.)

Figure 4

This same relationship is also expressed, in different forms, by the equiangular spiral and the hyperbola. (See Figure 5a, and Figure 5b.)

Figure 5a

Figure 5b

In connection with this investigation of the catenary, Leibniz coined the term {“function”} to denote a specified transformation. So, for example, the exponential function transforms arithmetic relations into the geometric ones. Its inverse, the logarithmic function, transforms geometric relations into arithmetic ones. As the arithmetic mean between two exponential functions, the catenary, therefore, is a function of two functions.

When Leibniz pushed this investigation to its boundary, by attempting to determine the function that generates the logarithms of negative numbers, he met the appearance of the square root of -1, which he identified as existing in a real, but “imaginary” domain.

That Undiscovered Country

Thus, the young Gauss, confronted with the appearance of what Leibniz called, “a fine and wonderful recourse to the divine spirit, almost an amphibian somewhere between being and non-being”, granted the square roots of negative numbers their “full civil rights”. From this standpoint he reexamined the “Kepler Problem”, and discovered that elliptical motion was governed by a higher species of transcendental, that had been anticipated by, but not known to Kepler or Leibniz.

The physical significance of complex numbers emerged quite naturally from Leibniz’s catenary principle and the paradox of negative logarithms. (See Figure 6.)

Figure 6

Since the catenary is formed as the arithmetic mean between {two} exponential curves, Leibniz sought the function that generated both, which could be expressed as that which generated their geometric mean. But, there is no continuous transformation, within the plane of the catenary, and the exponentials hanging behind it, that can transform one exponential into the other. The only physical action that has the power to produce both curves, is a rotation orthogonal to the plane of the catenary. (See Figure 7.)

Figure 7

If the direction of one exponential is designated as positive and the other direction negative, the geometric mean between them is expressed as the square root of a negative number. (See Figure 8.)

Figure 8

This gives rise to the following paradox: In the plane of the two exponentials, the geometric mean between them is a straight-line. But, when looked at from the higher standpoint of physical action, that straight-line is actually the axis of rotation from which is generated an entire surface orthogonal to the plane of the chain. Thus, to represent the physical action of the chain, the unseen exponentials that hang behind it, and the unseen principle that generates the exponentials themselves, requires both the plane of the chain, and the plane of the action orthogonal to it.

Gauss conceived the idea of representing this complex domain on one surface. That surface acts as a stage for the imagination on which {both} the physical action and the universal principles that govern it can be represented.

The two exponentials in the plane of the catenary are generated by an action that is completely outside their visible domain. The point at which the axis of rotation touches the surface generated by that rotation, i.e. the center of the circle formed by that rotation, is the only intersection between the visible domain and the surface on which the principle of generation is represented.

Just as on the Classical stage, a single line, (“Its Greek to me.”) in the context of the drama, can indicate an historical transformation of an entire culture, on the surface representing the complex domain, an entire principle, in the context of a complex function, can be represented by the action at a single point.

Hidden Harmonies Become Heard

By adopting Gauss’s approach, Riemann was able to create a stage, on which an innumerable class of transcendentals of successively higher power could be brought within the scope of the imagination. Each new species of transcendental, beginning with the simplest species of transcendentals discovered by Cusa, to the elliptical functions of Gauss, to those higher species discovered by Abel, is distinguished from its predecessor by the addition of a new principle. Under Riemann’s idea, each new principle is represented by a change in the geometrical characteristics, {the topology} of the surface.

Here Riemann adopted Leibniz’s method of {analysis situs} as applied by Gauss in his 1799 dissertation on the fundamental theorem of algebra. Gauss showed that formal algebra could not distinguish the implications of a change from one power to another. This is reflected, as Gauss ironically emphasized, in the inability, from the standpoint of formal algebra, to answer algebra’s most important question: “How many solutions are there for a given algebraic expression?” It was only when these expressions were thought of geometrically, as surfaces, that their essential characteristics could be made known. Then, it was clear, that the highest power of the function determined a topological characteristic (the number of “humps”) that was independent of any variations in the lower powers or the coefficients of that function. (See Figure 9a, and Figure 9b.)

Figure 9a

Figure 9b

Thus, Gauss’s employment of Leibniz’s geometry of position, {analysis situs}, expressed the epistemological relationship that the principle associated with the highest power dominated all lower ones. A change in the degree of power is, therefore, a change in the governing principle, and is expressed by a transformation of the entire geometry of the surface. Thus, the essential characteristics of an entire species of algebraic functions could be known completely, without the useless formal calculations of Euler, Lagrange and D’Alembert.

To illustrate Riemann’s approach, we begin with a look at the simplest species of transcendental, those associated with the circular, hyperbolic and exponential functions. As Leibniz and Bernoulli demonstrated the catenary, expressing the universal physical principle of least-action, embodied all three of these functions in one single physical action.

However, from the standpoint of the “simple” domain, these three functions seemed to have entirely different characteristics. For example, the circle and the hyperbola are conic sections, the exponential is not. In the visible domain, the circle is periodic, the hyperbola and exponential appear infinitely extended.

Leibniz imagined that a higher principle united all three transcendentals, but he never elaborated his idea. The sophisticated, but Leibniz-hating Euler, tried to obscure Leibniz’s insight with a series of algebraic formalisms, (such as his infamous equations: e^i*Pi-1=0; and e^i*x=cos[x]+i*sin[x]) which have bedeviled generations of students and scientists to this day.

Yet through Gauss’s and Riemann’s physical conception of the complex domain it can be easily demonstrated that these three functions express one unified species of transcendentals, and are all derived from the complex exponential. Gauss recognized this as early as August 14, 1796, writing in his notebook: “By the way, (a+b*?-1)^m+n*?-1 has been explained”.

Here Gauss indicates he has extended the idea of the exponential function to the complex domain. As noted above, the exponential function is the transformation of arithmetic relationships into geometric ones. In the “simple” domain, this appears geometrically as the characteristic curvature of a single curve, as, for example, the equiangular spiral, hyperbola or exponential curve. In Gauss’s complex domain, the complex exponential function transforms the arithmetic relations of an entire surface into geometric ones, transforming all curves into new ones.

This can be seen most easily from the geometrical example in Gauss’s Copenhagen Prize Essay on conformal mapping (on which Riemann relied.). There Gauss shows that the complex exponential function corresponds to the stereographic projection of a sphere onto a plane a projection first developed by the ancient Greeks in connection with the mapping of the celestial sphere. (See Figure 10.)

Figure 10

Under this projection, the circles of “latitude” on the sphere are projected onto concentric circles on the plane, whose radii are expanding exponentially. Circles of “longitude” on the sphere are transformed into radial lines on the plane.

Don’t think of this transformation as a static image. Think of it from the standpoint of physical action. Think of the motion along a circle of latitude on the sphere. What is the corresponding motion on the plane? Think of the motion along a circle of longitude on the sphere. What is the corresponding motion in the plane? Think of motion along a path of equal heading on the sphere (loxodrome). What is the corresponding motion along in the plane? When you think of it in this way, you can begin to see how all relationships on the sphere are transformed, lawfully, into different relationships on the plane.

Here a crucial new idea emerges that cannot be seen in the “simple” domain. What appeared to be “infinitely” long radial lines of the plane, become “finite” circles of longitude on the sphere. Action on the plane that appears to go off into the “infinite” approaches a single point on the sphere. From the standpoint of Cusa, if the terminus of all the radial lines in the plane is thought of as the “infinite” circle, its image is a single point on the sphere. In other words, the principle bounding the action on the plane, that appears to be infinite and unknowable, is brought into the imagination, by the action associated with a single point on the sphere. Thus, in the complex domain, we can faithfully represent Cusa’s notion that in the infinite, the center and circumference, minimum and maximum, coincide.

It is important to emphasize, however, that physical action, such as the hanging chain, is never “infinite”. The catenary is the curve of a chain hanging between two positions. Thus, “this side of the infinite”, the physical exponential is always bounded. It is the universal principle that it reflects that is {transfinite}.

Another characteristic now comes into view that otherwise remained hidden with the “simple” view of the exponential. In the simple domain, the exponential curve appears aperiodic But, this is only an illusion caused by viewing the exponential too simply. As can be seen in the image of the stereographic projection, the radial lines are unbounded, but the circles of latitude are periodic. (As you can imagine, this period is real. It is Euler’s fraud to insist that the period of the complex exponential is “imaginary”.)

The hidden periodicity of the complex exponential can be seen more dramatically when viewed as a transformation of an orthogonal grid of curves on a plane, or, as Riemann suggested, on the surface of a very large sphere. (See Figure 11.)

Figure 11

Here all “straight” lines are transformed into a network of equiangular spirals, bounded by spirals with maximum rotation and minimum extension (i.e., circles), and spirals with maximum extension and minimum rotation (i.e., radial lines).

Think of this image from the standpoint of action. Under the transformation of the complex exponential, motion along a “straight”line is transformed into spiral motion, including the extreme cases of the circles and the radial lines. The non-periodic motion along the vertical lines is transformed into periodic circular action. (See Figure 12.)

Figure 12

The aperiodic motion along horizontal lines is transformed into motion along the exponentially extending radii, which remains aperiodic. Motion along a diagonal line is transformed into ever expanding spiral action. (See Figure 13.)

Figure 13

Now compare this result with what was noted above concerning the two exponential curves associated with the hanging chain. In the visible domain, this action is aperiodic. However, the rotation orthogonal to the plane of the hanging chain has a periodicity. Here we can recognize the physical reality of the exponential function’s “imaginary” period.

As Leibniz and Bernoulli demonstrated in their work on the catenary, the hyperbolic functions are functions of the exponential: the hyperbolic cosine is one-half the sum of two exponentials and the hyperbolic sine is one-half the difference of two exponentials. (See Figure 14.)

Figure 14

As Leibniz also insisted, the circular functions are also functions of the exponential, but only in a different, “imaginary”, domain. (See Figure 15.) The circular cosine is one-half the sum of two complex exponentials, while the circular sine is one-half the difference of two complex exponentials.

Figure 15

When viewed from the standpoint of Gauss’s and Riemann’s complex domain, all these functions can be expressed by the same type of action: the arithmetic mean between two complex exponentials moving in opposite directions, as illustrated by the circles in the accompanying animation. (See Figure 16a, Figure 16b, Figure16c, Figure 16d, and Figure 16e.) Together, the hyperbolic and circular functions form the four different variations of the same type of action. Thus, generated by the same type of action, they are all of the same species of transcendental.

Figure 16a

Figure 16b

Figure 16c

Figure 16d

Figure 16e

The Elliptical Transcendentals

The young Gauss, contemplating the origin of the “Kepler Problem”, was puzzled by the inability to express elliptical motion by the circular, hyperbolic or exponential functions. But, when he looked at a similar problem, that of the lemniscate, from the standpoint of the complex domain, he discovered a different species of transcendental that was expressed by elliptical motion. When the characteristics of that species are viewed on Riemann’s stage, the difference between the elliptical and the lower transcendentals becomes obvious.

While Gauss did not state it this way directly, his January 1797 decision to begin serious investigation of the lemnsicate alludes to the clue suggested earlier by Kepler’s concept of the conic sections.

As noted above, Kepler thought of all the conic sections as being generated from one continuous function. The emergence of an infinite boundary, between the ellipse and the hyperbola, reflects the transition to the inclusion of the upper cone in the double conical function. Thus, the entire function involves action in two different directions (parallel and perpendicular to the base of the cones). When thought of from the standpoint of Cusa, the appearance of the infinite signifies that a higher, undiscovered principle exists that encompasses both the perpendicular and parallel action as one. It is from this higher type of function that Kepler’s conic section function is unfolded.

This function must have the same relationship to the circular and hyperbolic functions as those functions have to the algebraic. That is, the higher species must have the capacity to generate the lower, but not the inverse. This is akin to the relationship between the quadratic and cubic species of magnitudes as viewed from the standpoint of Archytas. The principle of toroidal motion that generates cubic magnitudes can generate the quadratics as well, but not the inverse.

Now what is the nature of these elliptical transcendentals and how can we discover it? Let’s reconstruct Gauss’s investigation of the lemniscate from the standpoint of the clue supplied by Kepler, and the later solution supplied by Riemann.

That the lemniscate would become the focus of Gauss’s attention, flows quite naturally from the extension of Kepler’s concept of the conic sections into the complex domain. As presented in previous installments of this series, the lemniscate can be generated as the inversion of the hyperbola in a circle, or the stereographic projection of a hyperbola onto a sphere. (See Figure 17 and Figure 18.) From the standpoint of these two projections, it can be seen that the circle is the geometric mean between the hyperbola and the lemniscate. Thus, we seek a higher function that transforms the hyperbola, through the circle, into the lemnsicate.

Figure 17

Figure 18

It is important to take note of the method of inversion employed here. In the case of the two exponential curves and the catenary, the geometric mean between the two exponential curves could not be found within the visible domain of the curves and the hanging chain. The search for the principle that would generate both exponentials led us into the complex domain. In the case of the conic sections, Kepler’s function generates one from the other. But the appearance of the infinite in the middle of the function coaxes us to consider a higher function. That function generates, not another conic section, but a lemnsicate.

Now look at Kepler’s function, as he envisioned it projected onto a plane, and, from Riemann’s standpoint, projected onto a sphere. (See Figure 19.) In the former case, the circle is transformed into an ellipse, which is transformed into an hyperbola, and then into a line. In the latter, the circle is transformed into a projected ellipse, which is transformed into a lemnsicate, and then into two coincidental hemispheric circles.

Figure 19

Thus, to grasp the higher principle we have to think of both functions happening simultaneously: the one on the stage of the plane that is generating the conic sections, and the one on the stage of the sphere, that is generating a lemniscate. But from our vantage point we see both on {one} stage the stage of the complex domain.

Note that the infinite boundary that appears in the plane, appears in the spherical projection as the crossing point of the lemniscate. This presence of the {transfinite}, represented as a single point in the lemniscate, signifies the existence of an additional principle, that is outside the domain of the conic sections, but, {inside} the domain in which the lemnsicate resides. As noted in a previous installment, Gauss showed that the crossing point on the lemniscate also reflects the boundary between the regions of positive and negative curvature on the torus. (See Figure 20.)

Figure 20

Does the presence of this new principle indicate that the lemniscate is associated with a different species of transcendental than the circular or hyperbolic functions? Gauss said yes, and Riemann provided the basis to recognize this geometrically.

Gauss recognized the existence of this added principle in the lemniscate by the double periodicity of the lemnsicatic functions. Where, for example, the circular functions are periodic with respect to the interval 0 to 2Pi, the lemnsicatic functions are periodic with respect to the interval 1 to -1 {and} the interval \/-1 and -\/-1. (See Riemann for Anti-Dummies 52.)

Gauss expressed the geometrical characteristic of double periodicity in his work on bi-quadratic residues, which he published in 1832. However, its discovery was much earlier, as indicated by his notebook entry, “I have discovered a remarkable connection between the lemniscate and bi-quadratic residues”.

That connection is illustrated by the geometrical representation Gauss gave, with respect to simple and complex moduli, in his {Second Treatise on Bi-Quadratic Residues}. There Gauss shows that for real numbers, the modulus can be expressed geometrically by a simple line segment or curve. (See Figure 21a and, Figure 21b.)

Figure 21a

Figure 21b

However, in the complex domain, a complex modulus is expressed by a parallelogram on a surface. (See Figure 22.)

Figure 22

This same geometrical distinction, as Gauss noted, also appears in comparing the lemniscate with the circular functions in the complex domain. In the case of the circular, hyperbolic or exponential functions, motion in one direction was periodic, while motion perpendicular to it was not. As illustrated above, this type of motion could be mapped onto a surface such as a sphere.

However, for the lemnsicate, the motion is periodic in two directions simultaneously. (See Figure23a, Figure 23b, and Figure23c.)

Figure 23a

Figure 23b

Figure 23c

Such motion can not be represented on Riemann’s sphere, where one direction, such as the circles of latitude, are periodic, but, in the orthogonal direction, the circles of longitude, are not periodic without crossing the infinite.

Riemann showed that the elliptical functions could only be expressed on a surface that allowed for two distinct periodic curves. Such a surface can be constructed, Riemann showed, by taking the parallelogram of Gauss’s complex modulus and connecting the opposite sides to each other, forming a surface with the topology of a torus. (See Figure 24.)

Figure 24

On a torus, there are two distinct curves, one that goes around the outside, and one that goes through the hole. (See Figure 25.)

Figure 25

Riemann emphasized that the torus is an entirely different type of surface than a sphere. {And, there is no continuous function that can transform a sphere into a torus, without the introduction of a new principle. Once introduced, that new principle effects a complete transformation of all relationships on that surface.}

Through this investigation into the distinction between the higher species of elliptical transcendentals and the lower species, we are able to recognize that the difference between these two species of transcendentals reflects a fundamental change in the number of principles acting through each species. As Abel showed, an entire class of transcendentals can be constructed of successively higher power . In Riemann’s elaboration, the change from one species to the higher, conforms to a transformation from a domain of “n” principles to a domain of “n+1” principles. As in Archytas’s construction for the doubling of the cube, or Gauss’s method of his 1799 dissertation on the fundamental theorem of algebra, such transformations can be made intelligible on the stage of the complex domain, as a discontinuous transformation in the {topology} of that domain.

More importantly, however, armed with this new power to make intelligible the generation of higher species of transcendentals, we can now envision the transformation that must occur, in any domain when a new principle is added. Thus, through the willful employment of the mind’s power of discovery, we can reach past the boundary of what seems to be infinite, and bring new principles into our conscious possession.

And this brings us back to Theatetus, who, as a young boy, shows us today, that an entire species can be known, completely without calculation.

Look to the Potential

In his 1857 {Theory of Abelian Functions}, Bernhard Riemann stated that the foundation of his theory of higher transcendental functions depended on what he called “Dirichlet’s Principle” and the method of thinking discussed by Gauss in his lectures on forces that act in proportion to the inverse square. These references help situate the historical specificity of Riemann’s discovery, the understanding of which, is essential to understanding the discovery itself.

The Gauss lectures to which Riemann referred were summarized by Gauss in an 1840 memoir titled, “General Propositions relating to Attractive and Repulsive Forces acting in the inverse ratio of the square of the distance”. There Gauss implicitly cast the fight between the physics of Kepler and Leibniz vs. the mystical pseudo-physics of Newton, Euler, Lagrange, and LaPlace, within the 2500 year ongoing drama between the two diametrically opposed views of man, as typified by the conflict between Plato’s concept of {power}, and Aristotle’s concept of {energy}. However, Gauss’s memoir is only one scene in the history. It was written under the political pressures of the post-1789 attack against the American Revolution that asserted, by force, the Aristotelean concept of man as animal associated with such frauds as Newton, Euler, and Lagrange. As such, it does not explicitly reveal all the elements of the play. But, as in any Classical drama (as well as physical processes), the organizing principles of the entire composition are always present and active in every part of every scene. Consequently, from a knowledge of the history of ideas, and the historical context in which Gauss was educated, lived and worked, it is possible to recreate, in the imagination, that great play to whose scene our attention is now drawn.

The Fraud of Newton’s Inverse Square “Law”

Given that Isaac Newton, by his own admission, was a fraud and a man convinced he had no soul (“Hypothesis non fingo”), it is a cause of some amazement that he was ever held in high esteem. This, of course, can be more easily understood, once one realizes that the high regard with which Newton has been held by some, was never for his physics, which, on its face, doesn’t work, but rather, for his degraded view of man from which his physics was derived.

A great mythology has been built up around Newton’s so-called theory of gravity, that asserts as a “law”, that every material body in the universe possesses an innate force that attracts every other material body in the universe according to the product of the masses divided by the square of the distances between them. Most modern textbooks and professors of physics demand blind obedience to this “imperial” law, from all who wish to be indoctrinated into their freemasonry. Such obeisance is demanded, not because this law is held to be true, but because it is so obviously false. In the tradition of Mephistopheles, these high priests seek to bind their supplicants to their cause, by forcing them to willingly embrace a doctrine that is contrary to human reason. And in the tradition of Marlowe’s Faust, too often these supplicants pay the price.

In fact, as Riemann enjoyed pointing out, Newton himself confessed his own immorality to Reverend Richard Bentley in 1693:

“Newton says: `That gravity should be innate, inherent, and essential to matter, so that one body can act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking can ever fall into it.’ See third letter to Bentley.” (Riemann’s Gesammelte Werke).

To accept Newton’s fraud is to accept the Aristotlean concept on which it is based: that man, like animals, is limited to interacting with the physical universe (and by implication other human beings) only through the objects of sense perception, i.e. material bodies, the which exist in an empty, infinitely-extended Euclidean type space. Any change, or motion, that occurs among these bodies (or humans), originates in something innate to the bodies themselves, and not in any higher principle.

As Aristotle stated it:

“This then is one account of `nature’, namely, that it is the immediate material substratum of things which have in themselves a principle of motion or change.” (Aristotle’s Physics, Book II.)

The person adopting such a view is eliminating from the universe the active principles, or “powers”, as Plato called them, that exist outside the domain of sense-perception, but which make possible that which the senses perceive. For Aristotle, all that can be known about the interaction of physical bodies, is contained in the visible attributes of the bodies themselves (extension), and not in any higher principle. Similarly, all human interaction, for Aristotle (or Newton), is limited to the pair-wise interaction of human bodies, and not on such overarching governing influences as the individual human being’s willful interaction with history, language- culture and the simultaneity of eternity.

By contrast, Plato, and later Cusa, insisted that it is those higher powers, not the objects of sense, on which the mind of the scientist must focus. As Cusa expressed this Platonic principle in his dialogue “On Actualized-Potential”:

“CARDINAL: Temporal things are images of eternal things. Thus, if created things are understood, the invisible things of God are seen clearly for example, His eternity, power, and divinity. Hence, the manifestation of God occurs from the creation of the world.

BERNARD: The Abbot and I find it strange that invisible things are seen.

CARDINAL: They are seen invisibly just as when the intellect understands what it reads, it invisibly sees the invisible truth which is hidden behind the writing. I say “invisibly” (i.e. mentally) because the invisible truth, which is the object of the intellect, cannot be seen in any other way.

BERNARD: But how is this seeing elicited from the visible mundane creation?

CARDINAL: I know that what I see perceptibly does not exist from itself. For just as the sense of sight does not discriminate anything by itself but has its discriminating from a higher power, so too what is perceptible does not exist from itself but exists from a higher power. The Apostle said, “from the creation of the world” because from the visible world as creature we are elevated to the Creator. Therefore, when in seeing what is perceptible I understand that it exists from a higher power (since it is finite, and a finite thing cannot exist from itself; for how could what is finite have set its own limit?), then I can only regard as invisible and eternal this Power from which it exists….”

As a follower of Cusa, Kepler looked to this Platonic concept of higher powers to develop his notion of universal gravitation. Kepler understood the principle of universal gravitation to be that immaterial “species” (which is the Latin equivalent for the Greek word “idea”) that had the “power” to move the material planets. This species could be “seen” mentally, as Cusa indicates, as an object of the intellect (Geistesmassen in the language of Riemann and Herbart). This thought-object itself is recognized in the visible domain, again as Cusa indicates, by its harmonic characteristics, expressed, as Kepler showed, by such relationships as the five regular solids, the equal area principle of planetary motion, the musical relationships among the planets, etc. A characteristic of this species was, (as Leonardo da Vinci had shown for light), that its effect diminished with distance according to the inverse of the square of the distance.

{Thus, for Kepler, as for Plato and Cusa, the action of material bodies was not the result of some nature innate to the material itself, but was the visible, measurable effect of an invisible, but efficient and cognizable, physical “power”.}

This power to cognize such physical powers, and to act directly on those powers to control the physical processes themselves, expresses the efficient power of human cognition. This itself is a demonstration, contrary to Aristotle, Descartes and Newton, that the universe is not indifferent to human thought.

As Leibniz later wrote in his {Discourse on Metaphysics}:

“I do not accuse our new philosophers (Descartes), who claim to banish final causes from physics. But I am nevertheless obliged to confess that the consequences of this opinion appear dangerous to me, especially if I combine it with the one I refuted at the beginning of this discourse, which seems to go so far as to eliminate final causes altogether, as if God proposed no end or good in acting or as if the good were not the object of his will. As for myself, I hold, on the contrary, that it is here we must seek the principle of all existences and laws of nature, because God always intends the best and most perfect.”

Led by the Venetian monk Paolo Sarpi, the enemies of Cusa and Kepler sought to extricate from human practice the cognitive powers of the mind by eliminating consideration of physical powers from science. The lackeys Galileo, Descartes, and Newton, elaborated Sarpi’s intention, by developing a new pagan religion, called empiricism, based on an imperial decree that science should concern itself solely with the description, by precise mathematical laws, of the objects of sense-perception.

Exemplary, is Newton’s “law” of gravity. Where Kepler’s mind recognized universal gravitation as an idea that governed the motion of the celestial bodies, Newton saw only the bodies. Where Kepler recognized that the motions of the bodies were the visible effect of an invisible physical power, Newton saw bodies copulating, instantaneously at a distance, through an, ultimately mysterious, force, whose measure is proportional to the inverse of the square of the distance between them.

Leibniz’s Notion of Vis Viva

The exemplary fraud of empiricism immediately relevant to this discussion, was emphasized by Leibniz in his refutation of Descartes’ theories of motion. Adhering to Sarpi’s empiricism, Descartes insisted on Aristotle’s idea that material bodies were totally separate from immaterial principles, ideas or powers. Consequently, what could be known about the universe, according to Descartes, was limited to the perceptible characteristics of material bodies, such as size, shape, mass, and speed. Nothing, at least nothing knowable, existed outside these visible characteristics, and so, the motion of material bodies was assumed to occur in an empty Euclidean-type space.

Thus, disregarding the existence of physical principles, Descartes insisted that motion of a material body could only be measured by its visible effects. From this he deduced that a moving body contained a certain “quantity of motion” which, as Aristotle had indicated, was innate to the body itself. Descartes measured this “quantity of motion” as the mass of the body times its speed. When two bodies collided, the quantity of motion of one body was transferred to the other, which Descartes elevated to a universal principle that he called “conservation of quantity of motion.” When a falling body hit the ground, its effect was measured by the mass times its speed on impact.

Leibniz rejected Descartes’ empiricism on epistemological grounds. Writing, for example, in his 1690, {On The Nature of Body and the Laws of Motion}:

“There was a time when I believed that all the phenomena of motion could be explained on purely geometrical principles, assuming no metaphysical propositions, and that the laws of impact depend only on the composition of motions. But, through more profound meditation, I discovered that this is impossible, and I learned a truth higher than all mechanics, namely, that everything in nature can indeed be explained mechanically, but that the principles of mechanics themselves depend on metaphysical and, in a sense, moral principles, that is, on the contemplation of the most perfectly effectual, efficient and final cause, namely God, and cannot in any way be deduced from the blind composition of motions. And thus, I learned that it is impossible for there to be nothing in the world except matter and its variations, as the Epicureans held.”

For Leibniz, the motion of a material body was the result, not of an innate nature of the body, but of a higher power, a capacity for motion that he called vis viva or “living force”. Thus, Leibniz did not seek to measure the visible effect of the motion. He sought to measure the “metaphysical” principle on which this physical effect depended.

In numerous locations, most notably, {Specimen Dynamicum}, Leibniz demonstrated that the higher principle, “living force”, was measured by the mass times the {square} of the speed. (fn 1.) As Leibniz demonstrated, the same force was necessary to raise a 1 pound body 4 feet as to raise a 4 pound body 1 foot, but the “quantities of motion” would be different. For the 1 pound body will hit the ground with a “quantity of motion” of 2 while the 4 pound body will hit the ground with a “quantity of motion” of 4. Thus, to produce the same quantity of motion requires a force proportional, not to the mass times the speed, but to the mass times the {square} of the speed.

Leibniz wrote, “There is thus a big difference between motive force and quantity of motion, and the one cannot be calculated by the other, as we undertook to show. It seems from that that {force} is rather to be estimated from the quantity of the {effect} which it can produce; for example, from the height to which it can elevate a heavy body of a given magnitude and kind, but not from the velocity which it can impress upon the body.”

The reader is encouraged to work through Leibniz’s demonstration for themselves. What is crucial to this discussion is that Leibniz’s physics is based on measuring the physical powers, not the visible effects. Descartes’ “quantity of motion”, measured by mass times speed, is an effect. Leibniz’s ” living force”, mass times the {square} of the speed, is the measurement of the principle that determines the effect.

Gauss’s Concept of Potential

While Leibniz’s discrediting of Descartes established the Leibnizian concept of quantity of force as the necessary measure of physical action, the fraud that Newton’s inverse square law was an adequate measure of universal gravitation persisted–enforced by the enemies of the American Revolution associated with Voltaire, Euler, Lagrange and the oligarchical controllers of Napoleon.

Kaestner, most notably, insisted, repeatedly and polemically, that Kepler and Leibniz were correct: that Descartes and Newton were only measuring the observable effects of physical action, while Kepler and Leibniz measured the physical principles themselves. Gauss’s 1799 new proof of the fundamental theorem of algebra demonstrated that even deeper physical principles could be measured, when expressed by the physical relationships of the complex domain.

After Kaestner’s death and Napoleon’s rise to power, Gauss was forced to be more cautious in his public discussions of these deeper issues, but he always insisted on focusing on physical principles. For example, in his great treatise on celestial mechanics, {The Theory of the Heavenly Bodies Moving About the Sun in Conic Sections}, Gauss treats Kepler’s principles as primary, and the inverse square law as a derived effect:

“The laws above stated differ from those discovered by {our own} Kepler in no other respect than this, that they are given in a form applicable to all kinds of conic sections…If we regard these laws as phenomena derived from innumerable and indubitable observations, geometry shows what action ought in consequence to be exerted upon bodies moving about the sun, in order that these phenomena may be continually produced. In this way it is found that the action of the sun upon the bodies moving about it is exerted just as if an attractive force, the intensity of which is reciprocally proportional to the square of the distance, should urge the bodies towards the center of the sun…” (fn. 2)

That there had even been any debate over the fraud of Newton’s inverse square law indicates only the viciousness of the political power that was used to enforce the adherence to such a foolish doctrine. Even on its own terms, the efforts to explain physical phenomena by taking the inverse square law as primary, leads to an even greater absurdity than that pointed out by Leibniz about Descartes “quantity of motion”.

For example, to apply the inverse square law, Newton treated all physical bodies as if their entire mass were concentrated in a simple Euclidean point. Furthermore, even with this contrived assumption, it is only possible to calculate the pair-wise relationship of two bodies. Once a third body is introduced, the calculations become, in principle, unsolvable.

But obviously, the universe is not limited to point-masses interacting pair-wise. As with his fundamental theorem of algebra, Gauss used an obvious fallacy–the one everyone danced around with sophistry, making fools of themselves by introducing into science, wild-eyed pagan rituals, in propitiation of an arbitrary authority– to establish, rigorously, the higher principles of science.

This is the standpoint of Gauss’s above mentioned lectures on forces acting according to the inverse square, where he treats the general problem, as it manifests itself in gravity, magnetism and electricity.

“Nature presents to us many phenomena which we explain by the assumption of forces exerted by the ultimate particles of substances upon each other, acting in the inverse proportion to the squares of their distance apart,” Gauss begins his treatise, indicating that the inverse square law is only an assumption, not a physical principle.

Gauss then demonstrates that the effort to explain these natural phenomena by the assumption of the inverse square leads to an inherent contradiction. First in the case of the difficulty of establishing the physical relationships among the interactions of many material bodies, and then as that problem applied to the obvious an extended body. That is, since no point-masses exist, how is it possible to calculate the infinite number of pair-wise interactions between all the parts of one body on all the parts of another?

Gauss’s solution is to throw out the inverse square law completely, and return to the method of Leibniz’s idea of measuring the underlying principle that produces the visible effect. In this case, however, Gauss goes beyond Leibniz, establishing a higher function that determines the physical geometry in which Leibniz’s living force exists.

Gauss called this function {potential}. His choice of the term{ potential }was deliberate, as potential is one of the Latin translations of the Greek idea of “power”. From Gauss’s standpoint, potential denotes a physical geometry, a phase-space, whose characteristic curvature, determines the nature of what action is possible in that phase-space. Thus, the motion of a material body is not occurring in an assumed Euclidean-type empty space, it occurs in a physical phase space whose characteristic curvature expresses the {potential} for the action.

The potential is not seen physically. It is seen only as an object of the mind. Yet, it is this mental object that governs the relationships among the objects of sense. Not being visible, it must be created, in the imagination, using the method of Plato and Cusa, from the paradoxes arising experimentally from physical action.

For purposes of pedagogical efficiency, this idea is best illustrated by examples.

First, take the example of Gauss’s measurements of the surface of the Earth. As Gauss emphasized, these are not measurements of an abstract mathematical geometric shape. These are physical measurements. A physical plumb bob is held on a string, a plane leveller is used to determine its perpendicular, and an angle is turned relative to these two directions. What is being measured, the material shape of the Earth, or that which is causing the plumb bob and bubble to move? If the latter, then what are the characteristics of that cause? Is it the combined interaction of every particle of the plumb-bob acting pair-wise with every particle of the Earth, as Newton’s method would demand? Or, does the motions of the plumb bob indicate the characteristic curvature of the {potential} for action of universal gravitation with respect to the Earth, as Gauss indicated?

Or similarly, when Gauss measured the Earth’s magnetism by the motion of a magnetic needle. Is that motion the result of pair-wise interactions of every magnetic particle of the needle with every magnetic particle of the Earth? Or, do the motions of the magnetic needle indicate the characteristic curvature of the {potential} for action with respect to the Earth’s magnetic properties?

Or, the case of two statically charged electrical objects? Is the repulsive or attractive force between these objects the result of the pair-wise interaction of every electrical particle of one object on every electrical particle on the other?

Gauss rejected such obviously foolish attempts as attempting to add up all the pair-wise interactions. Instead, he investigated these physical effects only as a consequence of the characteristics of the potential function: “[t]he investigation of the properties of this function will be itself the key to the theory of the attracting or repelling forces,” Gauss wrote.

As Gauss showed in his memoir, to investigate the properties of the potential function means to discover the characteristics of its curvature from the physical action.

Riemann Functions

To get an intuitive idea of Gauss’s concept of the potential, it will be pedagogically easier to first illustrate a simple example of Riemann’s more general concept of complex functions. With Riemann’s geometrical concepts in mind, we can then look back on the physical geometry of Gauss’s potential function.

Take the case of the simple “squaring” of a complex number. As Gauss showed in his fundamental theorem of algebra, complex numbers can be represented as a quantity of spiral action. “Squaring” that action is applying that spiral action twice. (See Figure 1.)

Figure 1

Riemann developed a geometrical concept for the general form of a function of a complex variable. Representing complex numbers on a surface, as Gauss did, Riemann considered a function of a complex variable as a Gauss mapping of that surface onto another surface. For example, the complex “squaring” function defines for every point on surface “a”, an image on surface “b”, which is the square of the original point on “a”. (See Figure 2.)

Figure 2

Riemann, following Gauss’s investigation of curved surfaces, showed that such a function transforms all relationships of one surface into a different set of relationships on the other surface. So, for example, an orthogonal grid of straight-lines on surface “a” is transformed into an orthogonal grid of parabolas on surface “b”. (See Figure 3.)

Figure 3

An orthogonal grid of circles and straight-lines on “a” is transformed into a similar grid on “b”. (See Figure 4.)

Figure 4

An orthogonal set of ellipses and hyperbolas on “a” transforms into a more complex set of relationships on “b”. (See Figure 5.)

Figure 5

Riemann then generalized two principles from Gauss. The first from Gauss’s study of conformal mappings and curved surfaces (See Riemann for Anti-Dummies Parts 44 through 48.), the second from Gauss’s investigation of the potential function.

In the first principle, Riemann recognized that if the geometrical conditions Gauss identified in his paper on conformal mapping were met, than images on surface “b” would all be conformal to their pre-images on surface “a”. (fn 3.) In other words, all angles would be preserved. (See Figure 6.)

Figure 6

In the second case, Riemann generalized a crucial concept from Gauss’s potential function. In a complex function there arise unique singularities that determine the general characteristic curvature, but around which that characteristic curvature changes. In this example of the squaring function, that singularity is the origin. Any action that includes this singularity will be different than a similar action that does not include the singularity. In animation 1 you see the effect of a square moving around a phase-space free of singularities. While it changes its shape, the angles between the sides remain orthogonal. But also notice the changes in the shape of the figure that includes the singularity within its boundaries. The presence of the singularity inside that shape, changes the whole shape’s relationship to the curvature and its shape changes dramatically differently than the shape of the singularity-free square.

Animation 1

In animation 2 you can see that even if the singularity free square is inside the boundary of the shape that contains the singularity, it still is unaffected, as long as the singularity remains outside of it. Its as if the singularity and the boundary are acting on each other, not at a distance, but as an effect of the organizing principle of the function.

Animation 2

Riemann also identified another geometrical characteristic that arises in this example of the squaring function. A point on surface “b” will correspond to two different points on surface “a”. From this, Riemann invented a new type of geometrical thought-object, now known as a “Riemann surface.” In this example, surface “b” will consist of two sheets connected both at the origin and the boundary. (See Figure 7.) In this way, the two different points of “a” will now correspond to two different points of “b”, each being on a different sheet.

Figure 7

It must be emphasized that the Riemann surface is a mental object, a metaphor, arising from the interaction of the mind with the physical universe. It is not an object of sense. However, it is more real than an object of sense, because it expresses the principles that govern an object of sense. But, it is not merely an abstract mental object divorced from the physical world, as its existence is completely connected to the physical processes whose investigation give rise to its formation. (A modern mathematician, degraded by his or her acceptance of the method of Descartes and Cauchy, will snarl at this paragraph.)

Gauss’s Potential from the Standpoint of Riemann Functions

Much more will be developed in future pedagogicals concerning these Riemann functions, but with these geometric ideas in mind, look again at Gauss’s concept of a potential function.

From this standpoint, Gauss’s potential function can be understood to express the curvature of a physical phase-space with respect to those universal principles acting in that phase-space, that are being investigated. The investigation of a different set of universal principles will define a different physical phase-space with a different curvature.

To keep it simple, consider the phase-space (potential function), of a material body with respect to universal gravitation. As expressed by a Riemann surface function, this phase-space has a characteristic curvature defined by the relationship between the boundary of the phase- space and the relationship of that boundary to the singularity.

For example, think of a physical action, from the standpoint of Leibniz’s concept of “living force”, that traverses a course in this phase space outside the surface of the body. Such as, the motion of a pendulum on the Earth or some other planet. Gauss’s potential function expresses the changes in the relationship of the pendulum’s “living force” to the phase space, as that pendulum swings within that potential phase-space. Note that the path of the pendulum does not encircle the Earth or the planet.

Now, think about the path of a satellite orbiting the Earth or the planet. A certain amount of force would have to be applied to move the satellite into its orbit, but the potential function indicates the existence of certain physical pathways, which, once achieved, require no changes in the relationship between Leibniz’s “living force” and the continued motion of the satellite. Note that such orbits complete encircle the planet.

(We omit, for now, the very curious question of the potential function inside the planet, or the characteristic of the potential function at the surface. These will be investigated in a future pedagogical.)

Note the relationship between this physical geometry expressed by Gauss’s potential function and the more general case of Riemann functions. Note the relationship of the material body in Gauss’s potential function to the singularity in Riemann’s surface functions.

It is in this more general form that the significance of Riemann’s theory of Abelian functions, for the frontiers of science, politics and art, is found.

FOOTNOTES

1. The fact that modern physicists refer to this quantity “mv²” as “kinetic energy” is itself a type of fraud whose purpose is to obscure the epistemological significance of Leibniz’s idea by clothing it with the name from Aristotle.

2. Emphases added by BMD. The use of the designation, “our own Kepler” is a direct reference to Kaestner who repeatedly criticized German science for turning its back on “our Kepler” and embracing the inferior British science of Newton.

3. It is one of the great frauds of the history of science, that these Gaussian conditions are referred to as the Cauchy-Riemann equations. The significance of these conditions was first identified by Gauss and then generalized by Riemann. Cauchy’s fraudulent formal treatment of these equations was an attack on the geometrical ideas of Gauss. To attach Cauchy’s name to Riemann’s in this regard is equivalent to the fraud of claiming the philosophy of John Locke instead of Leibniz’s as underlying the American Declaration of Independence and the American Revolution.

Riemann For Anti-Dummies Part 52

ABELIAN FUNCTIONS AND THE DIFFERENCE BETWEEN MAN AND BEAST

All Aristoteleans are liars. In fact they must lie. For Aristoteleans believe that their minds are empty vessels, indifferent to what is put in them. They project this view of themselves onto the Universe, which, they insist, must conform to their degraded view of man: an empty box devoid of principles, and subject to no cognizable lawfulness. Devoid of principles both within and without, all statements, in the view of such Aristoteleans, can not be true, but only consistent within a logical deductive framework that rests not on universal principles, but on some arbitrary authority that determines a set of axioms, postulates and definitions.

Leibniz posed this issue in the preface to his {New Essays On Human Understanding}, which was written in response to the pro-beast-man John Locke’s {Essays On Human Understanding}:

“Our differences are about subjects of some importance. There is the question about whether the soul in itself is completely empty like tablets upon which nothing has been written (tabula rasa), as Aristotle and the author of the {Essay} maintain, and whether everything inscribed on it comes solely from the senses and from experience, or whether the soul contains from the beginning the source [principe] of several notions and doctrines, which external objects awaken only on certain occasions, as I believe with Plato and even with the Schoolmen, and with all those who find this meaning in the passage of St. Paul (Romans 2:15) where he states that the law of God is written in our hearts.”

Under the doctrine of Aristotle and Locke, man is no different than a beast– a point also posited by Liebniz, in the {New Essays on Human Understanding}:

“Also, it is in this respect that human knowledge differs from that of beasts. Beasts are purely empirical and are guided solely by instances, for, as far as we are able to judge, they never manage to form necessary propositions, whereas man is capable of demonstrative knowledge [sciences demonstratives]. In this, the faculty beasts have for drawing consequences is inferior to the reason humans have. The consequences beasts draw are just like those of simple empirics, who claim that what has happened will happen again in a case where what strikes them is similar, without being able to determine whether the same reasons are at work. This is what makes it so easy for men to capture beasts, and so easy for simple empirics to make mistakes. Not even people made skillful by age and experience are exempt from this when they rely too much on their past experiences. This has happened to several people in civil and military affairs, since they do not take sufficiently into consideration the fact that the world changes and that men have become more skillful in finding thousands of new tricks, unlike the stags and hares of today, who have not become any more clever than those of yesterday.”

Recently this author, along with several members of the LaRouche Youth Movement in the United States and Mexico, witnessed, on several occasions, a pedagogical demonstration of the above described conflict. In response to the distribution of LaRouche’s {Visualizing the Complex Domain} and {The Pagan Worship of Isaac Newton} in the context of a presentation on the epistemological and historical significance of Gauss’s attack on Euler, Lagrange and D’Alembert in his 1799 proof of the Fundamental Theorem of Algebra, various professors and students of mathematics were observed reacting with an hysterical defense of Newton and Euler and insisting that all knowledge must be stated in the form of logical, deductive mathematics. These individuals were observed insisting that obvious historical falsehoods were in fact true, solely because they could utter them with great energy. Such objections, coming from Aristoteleans, were, of course, disingenuous lies. It was not their beloved idols, Newton and Euler, alone that they were defending. More fundamentally, they were defending their right to lie through the Aristotelean methods which Newton and Euler exemplify. The insistence on knowable truth “was against the rules” and warranted the observed outbursts, much as an enraged animal defends his perceived boundaries of his territory.

With this in mind, now take a look at the history of the development of what have become known as “Abelian Functions”.

Niels Henrik Abel

In 1826, the Norwegian Niels Abel, then 24 years old, arrived in Paris as part of a tour of continental Europe. Abel had been sent to the continent by his teachers, Bernt Holmboe and Christopher Hansteen. Holmboe was Abel’s first teacher and he introduced the young Abel to Gauss’s {Disquisitiones Arithmeiticae} which Abel mastered quickly. Abel was particularly intrigued by Gauss’s remark in Section VII of that work, where Gauss states that his theory of the divisions of the circular functions could be applied to the lemniscate and elliptical functions as well.

Hansteen was a direct collaborator with Gauss, Alexander von Humboldt, and Dallas Bache, in the Magnetic Union and he was responsible for taking magnetic measurements throughout northern Europe. Recognizing Abel’s potential, Holmboe and Hansteen arranged to finance a tour of the continent so that Abel could interact with the leading thinkers of his day.

After a stay in Berlin, Abel proceeded to Paris, where he met the Jesuit-controlled Augustin-Louis Cauchy, whom he called, “a bigoted Catholic a strange thing for a man of science.” Cauchy had made his reputation writing excessively long dissertations on the formal manipulations of algebraic equations and complex functions (which were directed mainly as an attack on Leibniz’s idea of the infinitesimal) and he had established himself as a type of inquisitor within the French scientific community.

Abel, who had already published numerous ground-breaking discoveries, submitted a treatise titled, “Memoire on a General Property of a Very Extensive Class of Transcendental Functions” to the French Academy of Sciences. As the leading mathematical figure in the Academy, Cauchy was entrusted with the manuscript.

Disgusted with the reactionary climate of Paris, Abel left for Vienna, and then went back to Norway where, devoid of any income, he lived in poverty, contracted tuberculosis and died at the age of 26.

Meanwhile, the bigoted Cauchy took Abel’s manuscript home and kept it from being published. Only in 1829, after Abel’s death, when C. G. J. Jacobi heard of Abel’s memoire from Legendre, did its existence come to light. On reading a copy Jacobi wrote:

“What a discovery is this of Mr. Abel’s… Did anyone ever see the like? But how comes it that this discovery, perhaps the most important mathematical discovery that has been made in our century, having been communicated to your Academy two years ago, has escaped the attention of your colleagues?”

Despite Jacobi’s insistence, Cauchy still sat on the manuscript, allowing it to be published only after great pressure in 1841, 15 years after it was submitted, and 12 years after Abel’s death.

The subject of Abel’s discovery is indicated in the opening of the memoir:

“The transcendental functions hitherto considered by mathematicians are very few in number. Practically the entire theory of transcendental functions is reduced to that of logarithmic functions, circular and exponential functions, functions which, at bottom, form but a single species. It is only recently that some other functions have begun to be considered. Among the latter, the elliptic transcendentals, several of whose remarkable and elegant properties have been developed by Mr. Legendre, hold the first place. The author has considered, in the memoir which he has the honor to present to the Academy, a very extended class of functions…..”

The History of Abelian Functions

It was still not until Riemann’s 1857 “Theory of Abelian Functions” that the full significance of Abel’s discovery was brought to light, and it was not until LaRouche’s discoveries in the science of physical economy that the true significance of Riemann’s insights are made clear. In future pedagogicals we will go into more detail concerning the actual constructions of Abel and Riemann, from the standpoint of the higher development of these ideas by LaRouche. However, before embarking on those investigations, it is necessary to set the stage from the historical standpoint.

While the development of the higher transcendentals of Abel and Riemann properly begins with Kepler, it is essential to recognize Kepler’s discoveries from the standpoint of the Pythagorean, Platonic concept of power, as distinct from the Aristotelean concept of energy.

As Plato demonstrates in the Meno, and Theatetus, objects in the visible domain, such as lines, squares and cubes, are generated by powers that are not knowable through the senses. Nevertheless, such powers are perfectly cognizable, through the power of number. The Pythagorean/Platonic idea of number as a proportion that signifies a power, is distinct from the Aristotelean idea of a number that counts objects of the visible domain.

Speaking on this same subject in {The Laymen On Mind}, Nicholas of Cusa distinguishes these two concepts of number:

“I deem the Pythagoreans who, as you state, philosophize about all things by means of number to be serious and keen philosophers. It is not the case that I think they meant to be speaking of number qua mathematical number and qua number proceeding from our mind. (For it is self-evident that that sort of number is not the beginning of anything.) Rather, they were speaking symbolically and plausibly about the number that proceeds from the Divine Mind of which number, a mathematical number is an image. For just as our mind is to the Infinite, Eternal Mind, so number that proceeds from our mind is to number that proceeds from the Divine Mind. And we give our name “number” to number from the Divine Mind, even as to the Divine Mind itself we give the name for our mind…”

From this standpoint, we recognize the existence and characteristics of the powers that generate the actions we observe, from the characteristics of the numbers associated with the proportions those actions produce. For example, the number associated with the doubling of a square, is a special case of one geometric mean between two extremes. In the particular case of the square, the generating power is expressed by the incommensurability between the side of the square and its diagonal. The Aristotelean sees this type of number as “irrational” because it is more complicated than the simple whole number ratios that express simple linear proportions. But for Cusa, this incommensurability is actually simpler, because it indicates the existence of a higher power:

“Moreover, from the relation of the half-tone to a full tone and from the relation of a half a double proportion, this relation being that of the side of a square to its diagonal I behold a number that is simpler than our mind’s reason can grasp…”

In more general terms, the number associated with the diagonal of a square to its side is a special case of a whole class of magnitudes one geometric mean between two extremes which is a function of a type of curvature, i.e. circular rotation. As Archytas demonstrated for the case of the doubling of the cube, there exists a still higher class of magnitudes two geometric means between two extremes which are generated by a different curvature, i.e. conical action, as illustrated by his construction of the torus, cylinder and cone.

Cusa later demonstrated that these classes of numbers, which Leibniz would later call “algebraic”, are all subsumed by a higher class of numbers, that Leibniz called transcendental. Further, Cusa indicated that the generation of these classes of numbers is governed by the succession of discoveries of new physical principles:

“Likewise, the exhibiting of the mind’s immortality can suitably be pursued from a consideration of number. For since mind is a living number, i.e., a number that numbers, and since every number is, in itself, incorruptible (even though number seems variable when it is considered in matter, which is variable), our mind’s number cannot be conceived to be corruptible. How, then, could the author of number [viz., mind] seem to be corruptible?”

Starting from Cusa’s epistemological standpoint, Kepler demonstrated that the motion of the individual elliptical orbits of the planets are governed by a universal principle that cannot not be expressed by the numbers associated with simple circular action. To resolve this problem, Kepler demanded the development of a new mathematics.

That mathematics was supplied by Leibniz’s infinitesimal calculus, which, when applied to the problem of the catenary, demonstrated that the circular functions and the logarithmic/exponential functions, were united by the principle of least-action expressed by the catenary. That unified relationship pointed to the discovery of what Gauss would later call the complex domain. (See Riemann for Anti-Dummies Part 50. ).

But, when Leibniz’s calculus was applied to the elliptical orbits directly, a paradox resulted. This paradox was not a mathematical one, rather it indicated the existence of a new physical principle that could only be characterized by, a hitherto undiscovered, new type of number. (See Riemann for Anti-Dummies Parts 49 & Part 51.)

Kepler had already anticipated the existence of this higher type of transcendental in his investigation of the implications of conic sections for optics. Here Kepler recognized that all the conic sections could be generated by one continuous function. He expressed that function by the motion of the focus of the conic section. Thinking of a circle as an ellipse in which both foci are coincident, the other conic sections are generated by the motion of one focus. (See Animation 1.) Kepler noted the existence of a discontinuity, between the ellipse and the hyperbola, a discontinuity straddled by the parabola, which Kepler said had “one side toward the curved and the other side toward the straight.”

Animation 1

The discovery of this higher type of elliptical function drew the attention of the young Gauss through his investigation of the lemniscate. As discussed in previous installments, the lemniscate expresses the higher unifying principle among all the conic sections, as exemplified by its relationship as the inversion of a hyperbola in a circle (fn.1). (See Figure 1.)

Figure 1

And, from this relationship, Gauss understood the lemniscate to be the expression of a new type of transcendental that was higher than the circular and logarithmic transcendentals. This type of transcendental, like the algebraic magnitudes, or the circular and logarithmic functions, could not be expressed directly, but only by inversion. In other words, it could only be known as “that which expresses the power that generates the characteristic of this species of action.”

To get an intuitive grasp of Gauss’s insight, think about the generation of the circle from the circular functions. This requires the mind to get out of the domain of sense perception and into the domain of principles. For the circle is characterized by uniform motion. Yet, the circular functions, i.e., the sine and cosine, are non-uniform. From the domain of sense perception, it is more “comfortable” to generate the non-uniform from the uniform. But, from the domain of principles, it is the other way around. The non-uniform motion of the sine and cosine express the higher generating principle that produces what appears to be uniform motion. As Cusa insisted, and Kepler demonstrated, uniform motion does not exist in the physical universe. It is only an artifact of non-uniform, transcendental action.

Gauss’s method of the division of the circle proved, from the standpoint of the complex domain, this dependence of uniform on non-uniform motion. It was this investigation of the circle, which Gauss saw as a special case of ellipse, that led him to investigate the lemnsicate, the which later inspired the young Abel.

One way to illustrate this relationship was presented in the last installment of this series. (See Animation 2.) Another way is the following.

Animation 2

A circle can be generated by the uniform motion of one end of a line of fixed length (radius) which rotates while the other end is stationary. From the Pythagorean theorem, the relationship of the cosine and sine to the radius is proportional to the square root of 1minus a square. (See Figure 2.)

Figure 2

A circle can also be generated by varying the length of the moving line according to the cosine (or sine) of the angle it makes with a fixed line. (See Animation 3.) Expressing the length of the moving line (cosine) in terms of the sine, makes the arc of the circle vary according to the square root of 1-sine².

Animation 3

However, when we allow the length of the moving line to vary by the cosine of double the angle, we produce two perpendicular lemniscates. (See Animation 4.) If we vary the length by the square root of double the angle, we generate one lemnsicate.

Animation 4

It can be remembered from the pedagogicals on the fundamental theorem of algebra, that doubling the angle squares the sine. (See Bringing the Invisible to the Surface: Gauss’s Declaration of Independence, Summer/Fall 2002 Fidelio.) Thus, if we express the length of the moving line in terms of the sine, the arc of the lemnsicate varies according to the square root of 1minus the square of a square, or the square root of 1-sine⁴.

From this relationship, Gauss recognized that in the complex domain the principle that generated the lemniscate expressed a fundamentally different type of relationship than the principle that generated the circle. First of all, the circular function, albeit non-uniform, generates uniform motion. But, the lemniscate function generates non-uniform motion. In the case of the circle, the sine (or cosine) is periodic. For example, the sine varies from 0,1,0,-1,0 for each rotation around the circle. (See Figure 3.)

Figure 3

But, since the functions that generated the lemniscate vary according to the fourth power, these functions have two periods. 0,1,0,-1,0 and 0,i,0,-i,0. (where i =square root of -1.) (See Figure 4.)

Figure 4

Thus, the power that generates all conic sections, as expressed by the lemniscatic functions is a higher type of transcendental, which generates the non-uniform action of the ellipse by two distinct, but connected, relationships.

These early investigations of Gauss were never published and they didn’t become known until Gauss’s notebooks were discovered in the 1890’s. But, from the intriguing remark in the {Disquisitiones Arithmeticae} the young Abel reconstructed Gauss’s discovery for himself and then went still further. Abel recognized that the lemniscate, and the related elliptical functions, were only the first step of an “extensive class of higher transcendental functions.” Thus, the circular and logarithmic functions were but a special case of the elliptical, which in turn were a special case of what have since become known as “Abelian” functions.

But, such functions were not supposed to exist in the bigoted animal world of the Aristotelean Cauchy, so he tried to cover them up with a lie.

The truth won out. And there, in part, begins Riemann’s theory of Abelian Functions. Inversion expresses the proportion that the distance from the center of the circle is to a point on the hyperbola is to the radius of the circle, as the radius of the circle is to the distance from the center of the circle to the corresponding point on the lemniscate.

Riemann for Anti-Dummies Part 51

THE POWER OF NUMBER

Nicholas of Cusa begins “On Learned Ignorance”, by reaching back to the method of Pythagoras:

“Therefore, every inquiry proceeds through proportion, whether an easy or difficult one. Hence, the infinite qua infinite, is unknown; for it escapes all proportion. But since proportion indicates an agreement in some one respect and, at the same time, indicates an otherness, it cannot be understood independently of number. Accordingly, number encompasses all things related proportionally. Therefore, number, which is a necessary condition of proportion, is present not only in quantity but also in all things which in any manner whatsoever can agree or differ either substantially or accidentally. Perhaps for this reason Pythagoras deemed all things to be constituted and understood through the power of numbers.”

Here, Cusa adopts the view of Plato, that numbers arise from the inseparable interaction between the human mind and the physical universe. The mind expresses the concepts it creates, about the principles it discovers, through the power of numbers. These concepts themselves become objects of the mind’s investigation, and the relationships among these “thought-objects”, also give rise to concepts, which are themselves expressible through the power of number.

Numbers in and of themselves have no power. They are like ironies that point, by inversion, to the principles that govern the physical universe. Those principles don’t exist in the numbers. They exist “behind” the numbers. Thus, the significance of Cusa’s reference that the Pythagoreans deemed all things to be {constituted and understood} through the power of numbers.

Against Cusa’s understanding of Pythagoras is the far different fraud perpetrated by the Sophists who, then and now, insist on separating the conjoined idea {constituted and understood}. For them, as for Aristotle, how things are {constituted}, and how things are {understood}, are two separate and mutually exclusive actions. In his “Metaphysics”, Aristotle attacks both Plato and the Pythagoreans for their insistence that ideas are an active principle in the Universe:

“But the lauded characteristics of numbers, and the contraries of these, and generally the mathematical relations, as some describe them, making them causes of nature, seem, when we inspect them in this way, to vanish; for none of them is a cause in any of the senses that have been distinguished in reference to the first principles. In a sense, however, they make it plain that goodness belongs to numbers, and that the odd, the straight, the square, the potencies of certain numbers, are in the column of the beautiful. For the seasons and a particular kind of number go together; and the other agreements that they collect from the theorems of mathematics all have this meaning. Hence they are like coincidences. For they are accidents…”

Aristotle’s method is pure sophistry. As Cusa and Plato both indicate, number arises in the {human} mind through the effort to discover the unseen universal principles that govern the world behind the senses. Thus, the power of numbers is {deliberate}, not accidental. However, Aristotle plays the trick of separating the world of mind and the world of matter, and so, for him, any connection between number and the physical world is purely accidental.

But Aristotle does not have clean hands. He is the hired-gun who provides the oligarchy with the method it needs, to create the cultural basis it requires, to assume its arbitrary authority over humanity. By introducing this false separation between the universe of sensible things, the unseen principles that govern them, and the thoughts by which we understand those principles, Aristotle excises human cognition as an active principle from the Universe. Once cleansed of cognition, he creates a false universe, indifferent to the power of human thought, unknowable, and governed by mysterious forces accessible only to those with access to special incantations, such as the moans of the oracles of ancient Delphi, or the formulas of formal mathematics.

That is the form of sophistry from which Cusa rescued civilization.

This is not an arcane technical argument, but one that goes to the heart of the difference between man and beast.

As Cusa stated this explicitly in “On Conjectures”:

“The natural sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality…

“…we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

Cusa’s view was later adopted by Leibniz, who wrote in “Reflections on the Souls of Beasts”:

“However, lest we seem to equate man and beast too closely, it should be known that there is an enormous difference between the perception of humans and beasts. For besides the lowest degree of perception that is found even in insensible creatures, and (as been explained) a middle degree which we call sensation and acknowledge in beasts, there is a certain higher degree which we call thought. But thought is perception joined with reason, which beasts so far as we can observe do not have.

“…However, a human being, insofar as he does not act empirically but rationally, does not rely solely on experience, or a posteriori inductions from particular cases, but proceeds a priori on the basis of reasons. And this is the difference between a geometer, or one trained in analysis, and an ordinary user of arithmetic, teaching children, who learn arithmetical rules by rote, but do not know the reason for them, and consequently cannot decide questions that depart from what they are used to: such is the difference between the empirical and the rational, between the inferences of beasts and the reasoning of human beings….Thus, brutes (as far as we can observe) do not acquire knowledge of the universality of propositions, because they do not understand the ground of necessity. And even if empirics are sometimes led by inductions to universally true propositions, this nonetheless happens only accidentally, not by force of entailment.”

Creating Number

This concept of number is what Gauss had in mind when, provoked by the paradoxes associated with the “Kepler Problem”, he sought the discovery of those hitherto unknown higher transcendentals indicated by Kepler’s discovery. Gauss recognized, in the tradition of the Pythagoreans, Plato, Cusa and Leibniz, that these new numbers, like all numbers, cannot be defined by any set of formal, deductive rules, such as in the methods of Aristotle, or his philosophical protege, the Leibniz-hating Euler. Rather, Gauss understood that these new transcendentals, like all numbers, could only by defined by {inversion} with respect to a physical process.

To illustrate this point, take a case you think you are familiar with, and look at it, as Gauss did, from an entirely new standpoint: the number associated with the relationship of two squares whose areas are in the proportion 1:2, which is sometimes called, “the square root of two”.

Construct a square from a circle. Now construct a square whose area is double. How do you know the areas of these two squares are in the proportion 1:2? Not by looking at them. There is nothing in the visible appearance of the two squares from which you can {know} the proportion of their areas. You can know this only from the method of construction, as Plato discusses that method in the Meno dialogue.

Now, look at the diagonal and the side of the square. These look the same. Both are lines. There is nothing in the visible appearance of these lines from which you can {know} their proportion, other than to say one is a little longer than the other. Yet, the Pythagoreans {proved} that the two lines are {incommensurable}, or in other words, had no common measure. This incommensurability cannot be demonstrated from the visible appearance of the lines, but, only, as the Pythagoreans did, by investigating the relationship itself between magnitudes that are commensurable in length and those commensurable as squares. The Pythagoreans showed that no linear proportion could possibly exist, whose square is in the proportion 1:2. (See Jason Ross article Fall 2003 21st Century.).

Most importantly, this characteristic of incommensurability is independent of the actual size of the squares. It depends only on the proportion between their areas, which is determined only by construction. As such, this incommensurability expresses a characteristic of the physical process of the construction of the squares. Consequently, the number called “the square root of 2”, cannot be defined by any set of rules, definitions or procedures such as a logical deductive method, but only by an inversion, as {that which expresses the principle with the power to generate two squares whose areas are in the proportion 1:2}.

In the visible domain, we proceeded in the opposite manner. We constructed the squares and produced the magnitude called, “the square root of two”. But from the standpoint of physical principles, it is the incommensurability of the magnitude called the square root of two which expresses the {power} (possibility) to produce two squares whose areas are in the proportion 1:2.

As Thales, Pythagoras, Theodorus, and Theatetus, further demonstrated, the square root of two is merely a special case of a more general species of relationships generated from circular action, which they called one geometric mean between two extremes. (See Figure 1.) As point P moves from A to O, the length of the line QP will always be the geometric mean between the lengths of line OQ and QA. When OQ is one half of QA, then QP is the magnitude called “the square root of 2”.

Figure 1

The particular proportion is independent of the size of the circle or the actual lengths of the lines. It depends solely on the position of P with respect to A and O, the which is determined by the circular “orbit” on which P travels. Thus, the circular “orbit” on which P travels produces, as a whole, a complete “type” or “species” of proportions, with respect to the magnitudes OQ, QP and QA.

This is one of the simplest examples of the Greek method of the geometry of position (“topos”) known today by the Latin word, “loci”, in which proportions are understood as generated from some physical action.

Another famous example of this method of loci is Archytas’s construction of cubic magnitudes. Here, an entirely new species of magnitudes is generated by combining two degrees of circular rotation orthogonal to each other, producing the torus and cylinder whose intersection defines a different “orbit” on which P travels. Again, these relationships are dependant only on the characteristic of the “orbit”, not on the size of the circles, or the surfaces generated from them.

The solid loci, or conics, of Menaechmus and Apollonius, follows the same principle, of generating both square and cubic magnitudes from {action acting on action}.

But here we seem to have run into a boundary. The change in the motion that generates one mean between two extremes (squares) to the motion that generates two means (cubes), was effected by introducing a second degree of circular action acting orthogonally to the first. Visible space, however, does not permit the addition of a third degree of circular action acting orthogonally to these two. Does that mean that no higher powers are possible? And, if they are possible, how are they manifest physically, as distinct from some formal mathematical algebraic definition?

Ultimately, it was Leibniz’s discovery of the catenary principle that demonstrated the physical existence of these higher powers. But that discovery rested on a previous one by Cusa, which is necessary to review for the sake of the subject matter to follow.

Look again at the generation of geometric means from circular action, but this time from the standpoint of inversion. As was demonstrated above, the circular “orbit” of P generates the entire species of geometric means. But what about the inverse? Can the entire species of geometric means produce a circle?

Cusa demonstrated that the answer was no, that, in fact, there exists a higher principle that generates the circle, which Leibniz later called, “transcendental”. These “circular transcendentals” are identified with those interrelated magnitudes known as trigonometric functions, and the relationships among them express the incommensurability between the curved and the straight. (See Figure 2.)

Figure 2

As Cusa showed, this incommensurability between the curved and the straight, is a different type than the incommensurability expressed by squares or cubes, and, the higher “algebraic” powers. Later, Leibniz, through his discovery of the catenary principle and its relationship to natural logarithms, demonstrated that all species of algebraic powers are generated by these transcendental functions, and it is these transcendental magnitudes, not the algebraic, that express the relationships that arise in the physical universe.

It is crucial to restate this point in this form:{ The transcendental functions, not the algebraic, are those functions that are inverse to circular rotation}. The significance of this statement will become more clear from the standpoint of the discovery of the higher, elliptical functions, by Gauss and Riemann.

Elliptical Functions From Kepler to Leibniz

Through his education by E.A.W. Zimmerman and A.G. Kaestner, both leading defenders of Leibniz and Kepler, Gauss was focused, from his early adolescence, to investigate the implications of the paradoxes arising from the “Kepler Problem” that were left unresolved by Leibniz’s invention of the infinitesimal calculus. (See Riemann for Anti-Dummies, Part 49, Aug. 16, 2003.)

As Kepler demonstrated, the ellipticity of a planetary orbit demanded a new type of geometry of position, one that expressed the position of the planet as a function of the characteristic of change of the orbit as a whole. Kepler’s effort to develop this concept resulted in his famous principle of equal areas. He recognized that in every interval of an elliptical orbit, no matter how small, the motion of the planet at the beginning of that interval is different than at the end. The only interval excepted is one entire orbital period. In that case, the planet is doing the same thing at the beginning and end of the interval. Thus, Kepler made the entire orbit the primary interval of action, and measured the planet’s intermediate motion as portions of the whole orbit.

In The New Astronomy, Kepler, citing Archimedes, measures this relationship by the proportion of the total area of the planet’s orbit, to the area swept out in a given interval. He defines the area swept out by the planet as the “sum” of the infinite number of radial distances:

{“For the whole sum of the radial distances is, to the whole periodic time, as any partial sum of the distances is to its corresponding time.”}

For Kepler, the planet’s motion was thus measured by the changing proportion between the part of the orbit and the whole. This proportion, Kepler understood to be the “time-elapsed” within any given interval.

It is extremely important to recognize the difference between Kepler’s actual principle and the Newtonian-algebraic formulation, misidentified as “Kepler’s second-law”, and stated in text-book gossip circles as: “the planet sweeps out equal areas in equal times”. This historically and epistemologically false characterization is a clinical example of an Aristotlean-type sophistry in two important ways. First, it falsely characterizes Kepler’s discovered principle as a mathematical “law”, thereby excising out the cognitive action. Second, by stating this law in the form of a proportion between area and time, it transforms time and space, by fiat definition, into independent absolute magnitudes. In this way, the Universe is turned on its head. Instead of recognizing the characteristics of space and time as functions of the physical motion of the planet, the fantasy pseudo-world of absolute time and space is held to define the planet’s motion.

Responding to Kepler’s demand for a generalization of his principle, Leibniz developed the infinitesimal calculus, by extending Kepler’s proportionality between the part and the whole, into infinitesimal intervals of action.

For Leibniz, the infinitesimal is physically determined as a proportionality, as Cusa understood proportionality, between the inseparable part and whole of a physical process. His critics reacted by reaching back to the ancient sophistries of the Eleatics, and, like Zeno, posed the paradoxes of physical motion from the standpoint of an arbitrary formal mathematical definition of a curve. Leibniz defended himself from these attacks during his lifetime, but after his death, the oligarchy recruited Euler to give a more “academic” imprimatur to these attacks, as expressed most blatantly in his Letters to a German Princess.

In response to Euler, Leibniz was posthumously defended, on precisely this point, by those who were fighting to establish the American Republic against the oligarchy that employed Euler, most notably by A.G. Kaestner. (See Appended essay by Kaestner, Moral from the History of the Infinitesimal Calculus.)

Elliptical Functions of Gauss

Ironically, while Leibniz’s discovery of the catenary principle demonstrated the power of his infinitesimal calculus, its application to the elliptical orbit, the problem that had provoked its invention, led to the paradox known as the “Kepler Problem”. The essential characteristics of this paradox can be understood by investigating the difference between a circular orbit and an elliptical one from the standpoint of Kepler and Leibniz.

In a circular orbit the area swept out can be measured directly by the angle. In the elliptical orbit, as Kepler showed, the area swept out is measured by a circular sector and a rectilinear triangle. (See Figures 3).

Figure 3

The incommensurability between the triangle and the circular sector leads to the “Kepler Problem”. (See Figure 4.)

Figure 4

Gauss recognized that this paradox arose from the effort to measure the elliptical action of the planet by circular functions. The problem here was that while the circular functions reflect the incommensurability between the arc and the line, in elliptical action there exists an incommensurability between the arc and the angle as well. (See Figure 5.) Gauss realized that since these two types of incommensurability were connected, both arising from a unified elliptical action, there must exist a new, more general type of elliptical transcendental, of which the circular functions were only a special case. The complex domain was required to make intelligible the relationships associated with these new transcendentals.

Figure 5

This investigation forms the core of Gauss’s youthful work, as a review of his early notebooks and correspondence reveals. Gauss presented some aspects of these discoveries more formally in his 1799 new proof of the Fundamental Theorem of Algebra and his Disquisitiones Arithmeticae, as well as his later works in astronomy, geodesy, and curvature. But because of the tyranny imposed on Europe in the post-1789 reaction to the American Revolution, (especially after the 1799 consolidation of Napoleon’s rise to power and its aftermath) some of these discoveries were never published, and only found their expression in the work of the next generation of scientists, most notably, Dirichlet, Riemann, Abel, and Jacobi, who were themselves influenced by Gauss, and the Leibnizian networks with which Kaestner was associated, such as the Humboldts, Schiller, Herbart et al.

Because of this fragmentary nature of Gauss’s early writings, much of Gauss’s thinking is expressed in abbreviated form. It becomes pedagogically much easier, therefore, to present the nature of his discovery, from the standpoint of his later work on curvature and Riemann’s elaboration of those ideas, most notably, in his famous treatise on Abelian functions. As in Gauss’s 1799 proof of the fundamental theorem of algebra, both Gauss and Riemann indicated the superiority of geometrical construction to algebraic formulas for conveying ideas.

The crux of Gauss’s method was that of the ancient Greeks, Cusa, Kepler and Leibniz: that nothing could be known from the visible manifestation of the circle or the ellipse. Rather, these visible characteristics, such as the uniformity or non-uniformity of the arcs, were a function of some underlying principle. That principle, however, was not visible, and could only be discovered by inversion. In other words, those principles could not be seen, but could be known, as, that which produces the visible characteristics of curvature.

To illustrate Gauss’ inverse method geometrically, take a new look at the trigonometric relationships. In the visible domain, the trigonometric relationships are generated as an effect of circular motion. But, as Cusa indicated (for which Kepler called him “divine”), the harmonic characteristics are found not in circular action alone, but in the incommensurability between the curved and the straight. Thus, Gauss thought of uniform circular motion as being merely the visible artifact of the more complex motion associated with circular functions.

This is illustrated in the accompanying animations. In animation 1, the circular rotation generates the trigonometric relationships. Animation 2 illustrates the inverse, where the visible effect of the circle is created as an artifact of the movement of the cosine and sine. In the visible domain, this seems impossible. How can you know the relationship of the cosine to the sine without first drawing the circle? But in the domain of reason, the essential nature of the sine and cosine can be {known} by inversion, as those functional relationships that produce circular areas.

Animation 1

Animation 2

In the case of the circle, this method may seem a bit arcane and clumsy, and so it was resisted by virtually every scientific thinker of Gauss’s time. But Gauss recognized that this method of inversion, rooted in the method of the ancient Greeks, was required to discover the nature of the elliptical transcendentals. For because of the incommensurability of the arc to both the line and angle, there was no way to generate, from the visible characteristics of an ellipse, a characteristic elliptical function, and all efforts to measure elliptical motion from circular transcendentals failed.

Gauss posed the elliptical problem in exactly these terms. He {knew} the higher elliptical functions must exist, and that they could not be defined directly. “What characteristics must such functions have to produce the ever changing elliptical motion?” Gauss can be imagined to have asked. “How can such characteristics be made intelligible?” This is the only way these functions can become known. Not directly, but only as that which is inverse to elliptical motion.

This thought will undoubtedly provoke psychological resistance in the modern reader, steeped in the culture of Aristotle and empiricism, who so strongly desires logical proofs presented in the visible domain. But it is the method of all Classical science and Classical art. It requires the mind to move; to willfully create new concepts. Hence, the benefit of reliving Gauss’s true discovery.

When the algebraists, such as Euler, Lagrange, et al. had tried to express the elliptical motion using their formalized, non-physical and, therefore, false version of the calculus they produced a formal representation of the elliptical motion that defied all their efforts at calculation. Gauss flanked them all by focusing on the simplest such case, the lemniscate of Bernoulli, which had been shown to be a special case of an elliptical type non-uniform curve.

Gauss’s choice of flank was rooted in Kepler’s insight into conic sections. Kepler had generalized Apollonius’s conics by recognizing the projective relationship among the conic sections as a whole. (See Hyperbolic Functions: A Fugue Across 25 Centuries.). Kepler had shown, by inversion, that all conic sections were generated by a single principle. However, he made particular notice of the significance of the discontinuity in the visible manifestation of this principle, expressed as an infinite boundary between the circle/ellipse and the hyperbola.

It was Gauss’s insight that the lemniscate expressed the higher, unified, principle that generated the conic sections. In its projected, visible form, the lemniscate is the locus of positions in which the product of the distances from a point on the curve to two foci, is equal to the square of ? the distance between those foci. (See Figure 6.). Or, in other words, the distance from one focus to the center of the lemniscate is the geometric mean between the two distances from the curve to each of the foci respectively.

Figure 6

However, the lemniscate has a more general relationship to the generating principle of the conic sections, and to the elliptical functions in particular, that can be grasped intuitively from the higher standpoint of Gauss and Riemann. On the one hand, the lemniscate can be generated as the inversion of the hyperbola in a circle. (See Figure 7.)

Figure 7

From this more advanced standpoint, the hyperbola can be seen as the stereographic projection from a sphere onto a plane of a lemniscate. (See Figure 8)

Figure 8

Here we can begin to see emerge the essential characteristics of the elliptical transcendentals, from the standpoint of Gauss’s principles of curvature. (See On Principles and Powers, Fidelio, Summer 2003.) The lemniscate on a sphere is generated as a mapping of the transition between the sections of positive and negative curvature of a torus. (See Figure 9.) And as Riemann would later demonstrate, the torus expresses a different topology (geometry of position) than the sphere or the ellipsoid. On the sphere or ellipsoid any closed curve separates the surface into two parts. But this is not the case on the torus where there are two distinct types of pathways, one around the torus, and the other through the “hole”. This characteristic, Riemann showed, expressed the double periodicity of the elliptical functions. (See Figure 10.)

Figure 9

Figure 10

Thus, enfolded in the lemniscate, and also in the ellipse, is the characteristic of double periodicity which is unfolded in the form of the torus. Additionally, hidden in the torus, if cut in the right way, one finds, the lemnsicate.

Here is the geometry of position that establishes the unseen, but nevertheless real, “orbit” which exists behind the elliptical functions.

We will come back to this discussion in future installments of this series. But for now, from this high perch, look back to Archytas with a justified sense of happiness.

Appendix

Moral from the History of the Infinitesimal Calculus

by A.G. Kaestner

When the question of calculating the infinite first arose, the most famous mathematical wise men had an aversion to it. Their habitual methods of discovering mathematical truths appeared to them to be clear and secure; whereas with the new one, they found dark secrets, much that was uncertain, and in the main, a degree of subtlety which they would rather forgo.

To convert these scorners, a cure was supplied by the camp of Leibniz and his friends, roughly as follows:

It was demonstrated that the calculation of the infinite was in agreement with all prevailing customary theories, in that it easily and comfortably led to truths which previously could only be attained by tiresome cogitation, and, finally, because it enlarged hitherto existing knowledge, such that the summit of Archimedes’ discovery was its lowest boundary; with it, one could answer, in total completeness, questions which could only be answered incompletely, or not at all, by the previously known feats of mathematical skill. And thus, the calculation of the infinite won the respect of an eye which, even without it, had made so many, so great discoveries.

How much would the Christian faith not gain, if its followers were to show, through their own acts, that with regard to the exercise of virtue, it has the same superiority over every other religion, as makes a Christian deserving of being admired by Socrates?

But many of these followers, and even their teachers, strike me today like someone who would go around constantly spouting higher mathematics and dropping Euler’s name, and who would declare anyone who could not integrate to be a dunce, but who would personally make errors as frequently as he was asked to calculate a Rule of Three!

Riemann For Anti-Dummies Part 50

THE GEOMETRY OF CHANGE

In his famous letter to Hugyens concerning his discovery of the significance of the square roots of negative numbers, G.W. Leibniz stated clearly his recognition that this investigation originated with the scientists of ancient Greece: “There is almost nothing more to be desired for the use which algebra can or will be able to have in mechanics and in practice. It is believable that this was the aim of the geometry of the ancients (at least that of Apollonius) and the purpose of loci that he had introduced….”

Understanding the implication of Leibniz’ statement is crucial to grasping the deeper significance of Gauss’ 1799 treatment of the fundamental theorem of algebra.

Leibniz’ statement will either baffle, or enrage, a modern academic, but such reactions only typify a broader social disease the inability, as LaRouche has repeatedly emphasized, to recognize the essential difference between human and beast. Like any disease, this one spreads through infectious agents that attack the defenses of the victim, causing the victim’s own system to act as an agent for the aggressor. The cure for such conditions is to strengthen the targeted population’s natural immunities, enabling them, not only to fight the disease, but to become permanently resistant to its effects. In this case, those natural immunities are the cognitive powers of the human mind. Hence, the therapeutic effects of pedagogical exercises and classical art.

What Leibniz, Gauss, and their ancient predecessors understood, is that the essential distinction between man and animal is the capacity of the human mind to reach behind the domain of the senses and discover those unseen principles that govern the changes perceived in the physical universe. However, being unseen, those principles can only be discovered through changes (motions) within the domain of the senses, which in turn give rise to paradoxes concerning the relationship of the seen to the unseen. Consequently, it is the coupled interaction between the seen and the unseen that must be comprehended. Physical motion gives rise to the willful motion (passion) of the mind from one state to a higher one. As Leibniz indicates, no formal system, such as algebra or Euclidean geometry, is capable of representing this characteristic of change that emerges from the interaction between the seen and the unseen. Only a geometry of change, such as the pre-Euclidean “spherics” of Thales and the Pythagorean school, the geometry of motion associated with Archimedes, Eratosthenes, and Apollonius, Leibniz’ infinitesimal calculus, or Gauss’ concept of the complex domain, has such power.

Just as the origins of the discovery of the complex domain begin in the ancient Mediterranean cultures of Egypt and Greece, so do the roots of its adversary. The mode of attack has been to induce the false belief that the physical world which is seen, and the immaterial world which is unseen, do not interact, but are hermetically separated. This belief is typified by the mystery cults of ancient Babylonian and Persian cultures. The Eleatics, (such as Parmenides and Zeno) sought to introduce this corruption into Greek culture, against Heraclites and the Pythagoreans, by insisting that change is merely an illusion and does not exist. (fn. 1)

Socrates made mincemeat of Parmenides’ Eleatic argument, so those who would today be called satanic, switched tactics, expressing the same evil intent through forms of Sophistry, such as admitting that change exists, but then arbitrarily defining change as the opposite of the Good and defining the Good as that which does not change and is not corrupted by change.

After Plato discredited the trickery of Sophistry, Aristotle, while distancing himself formally from the Sophists, nevertheless propounded the same evil in a new guise. For example, writing in his “Nichomachean Ethics”, Aristotle said :

“This is why God always enjoys a single and simple pleasure; for there is not only an activity of movement but an activity of immobility, and pleasure is found more in rest than in movement. But change in all things is sweet, as the poet says, because of some vice; for as it is the vicious man that is changeable, so the nature that needs change is vicious; for it is not simple nor good.”

Aristotle adopted this same view towards physical motion, stating in his “Physics” that motion originates only from within a body, and that irregular motion, because it contains more change, is of a lesser degree than regular motion, which is of a lesser degree than rest.

Like the Sophists and the Eleatics, Aristotle was not developing an original argument, but reacting against Plato’s repeated demonstration that the material and the immaterial are coupled:

” for this creation is mixed being made up of necessity and mind. Mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection, and thus and after this manner in the beginning, when the influence of reason got the better of necessity, the universe was created.” (Timaeus).

And it is the power to gain knowledge of the universe through the interaction of the seen with the unseen, the temporal with the eternal, that is human nature. Change is a characteristic, not of viciousness and vice, but of perfection:

“But, now the sight of day and night, and the months and revolutions of the years, have created number, and have given us a conception of time and the power of enquiring about the nature of the universe; and from this source we have derived philosophy, than which no greater good ever was or will be given by the gods to mortal man…God invented and gave us sight to the end that we might behold the courses of intelligence in the heaven, and apply them to the courses of our own intelligence which are akin to them, the unperturbed to the perturbed; and that we, learning them and partaking of the natural truth of reason, might imitate the absolutely unerring courses of God and regulate our own vagaries. The same may be affirmed of speech and hearing;…Moreover, so much of music as is adapted to the sound of the voice and to the sense of hearing is granted to us for the sake of harmony; and harmony, which has motions akin to the revolutions of our souls, is not regarded by the intelligent votary of the Muses, as given by them with a view to irrational pleasure, which is deemed to be the purpose of it in our day, but as meant to correct any discord which may have arisen in the courses of the soul, and to be our ally in bringing her into harmony and agreement with herself; and rhythm too was given by them for the same reason, on account of the irregular and graceless ways which prevail among mankind generally, and to help us against them.” (Timaeus.)

The tension of this Socratic irony, of the unchanging principles of change, is the means by which man, and the universe as a whole, perfects itself. As Kepler notes in the “New Astronomy”, it is the tension from the discovery that the planetary orbits are not circular, “that gives rise to a powerful sense of wonder which at length drives men to look into causes.”

Remove that tension, as Aristotle, Euler, Lagrange, et al., do, and you excise from Man his human nature, rendering him defenseless against those oligarchical forces who seek to enslave him.

The Square Root of -1 and Motion

Riemann for Anti-Dummies Part 49

THE HIDDEN HISTORY OF THE COMPLEX DOMAIN

When Kepler discovered the elliptical nature of the planetary orbits, he uncovered a paradox whose solution would require the development of an entirely new way of thinking, and he called on future generations to develop it. This “Kepler Problem”, as it has since become known, was not merely a mathematical lacuna, but reflected the ontological paradox indicated by Nicholas of Cusa in “On Learned Ignorance” and other locations. Kepler’s demand provoked Leibniz to develop the infinitesimal calculus, which revealed a new manifestation of that same paradox. This led Leibniz to indicate that the solution existed in a higher, yet to be discovered, domain of the imagination. Reflecting on these developments, the young Carl F. Gauss discovered that what both Kepler and Leibniz had sought. He called it the complex domain.

The above sketch is the true history of the origin of the discovery of the complex domain. It was known to Gauss’s immediate collaborators and followers, but today it lies hidden, even to the relatively best scientific thinkers. What has been substituted is the myth that complex numbers arise as “impossible” solutions to formal algebraic equations a myth whose malignancy has infected today’s popular thinking far beyond the domain of pure mathematics. The source of the myth is not new, but begins with Venice’s Paolo Sarpi’s launching of modern empiricism, and continues as a progression of degeneration through Hobbes, Descartes, Newton, Euler, Kant, Lagrange, Hegel, Cauchy, Klein, and down into the modern forms of existentialism and information theory associated with Synarchism. Today, as always, its target is the intellectual and emotional powers of mind associated with the development of the nation state, as embodied in the hard won Declaration of Independence and Constitution of the United States.

Consequently, anyone wishing to know the principles of science, must understand the connection between the history of the development of the complex domain, and the fight for the founding and preservation of the United States. Conversely, anyone wishing to know the latter, must understand its connection to the discovery of the complex domain.

For this reason, it is crucial that the record be set straight.

What follows is a summary overview. An opening statement, so to speak, designed to lay out what the investigation will show. It is best worked through slowly in small discussion groups. Over the coming installments of this series, we will explore these matters more deeply. This will take us, albeit not without a certain amount of hard work, directly to Riemann’s investigation of the higher transcendentals which he called Abelian functions.

The “Kepler Problem”

The ultimate source of Kepler’s discovery of the planetary orbits lies not in some particular astronomical theorem, but, as Kepler repeatedly emphasized, in his conception of Man. Unlike Ptolemy, Copernicus, and Tycho Brahe, Kepler understood that Man was distinguished from lower forms of life, by the capacity of mind that enabled him to rise above the limitations of sense-perception and discover those unsensed principles that govern the universe. As such, Kepler rejected the Aristotelean methods of Ptolemy, Copernicus and Brahe, all of which sought only to model the appearances of the motion of the heavenly bodies. Instead, as Kepler emphasized, his approach was to derive the apparent motion of the planets as a function of their physical causes. Consequently, where Ptolemy, Copernicus, and Brahe looked only to the domain of sense-perception, Kepler looked to the interaction between what is perceived and the principles that efficiently control what is perceived.

This latter point is crucial. Kepler did not ignore the world of sense-perception. Rather, he understood that what appeared to the senses was caused by principles that were unseen. But since those principles could not be sensed directly, they could only be known through contradictions and paradoxes that emerge in the domain of the senses. From these paradoxes, the mind has the power to form a synthetic visualization, so to speak, through which it can grasp for itself and communicate to other minds, those unseen principles indicated through the paradox. This synthetic visualization is emphatically not a symbolism. Rather it is a metaphor that preserves the paradox so as to guide the mind to the indicated principle that lies behind it.

It is in this light that the “Kepler Problem” must be seen. This problem arises when trying to determine the non-uniform motion of a planet. Cusa had already indicated that motion in the physical universe did not conform, as Aristotle had insisted, to fixed perfect circles. Instead, Cusa indicated that the perfection of the universe were better expressed by the principle of change, and, as such, action in the physical universe must be non-uniform, i.e., changing.

By deriving motion of the planets from physical causes, Kepler was led to the discovery that Cusa was right. Kepler demonstrated that the non-uniform motion of the planet through its orbit, as measured by the speeding up or slowing down of the observed movement of the planet against the background of stars on the inside of the celestial sphere, was not just an appearance, as Ptolemy, Copernicus and Brahe all held, but actually reflected the true physical motion of the planet. Thus, the planet didn’t just appear to be always changing, it {is} always changing. Consequently, the planet’s motion was being governed, not by a fixed principle such as would be characterized by perfect circles, but by a principle of change.

Kepler showed that that principle of change was embodied in the Sun, which had the power to move the planets by an immaterial “species” (idea), just as an idea in the mind moves the body. The measured speeding up and slowing down of the planet indicated the existence of an elliptical orbit in which the planet’s distance to the Sun is always increasing or decreasing. The always changing speed and direction of the planet, thus reflects the principle of change inherent in the principle of universal gravitation, as that principle was understood by Kepler, not the dumbed down bastardization associated with Newton.

However, the elliptical orbit presented a new type of paradox. As Kepler stated it, the relationship between the planet’s position and the time elapsed could be measured by his famous principle of equal areas. But these areas were measured by the combination of a circular sector and a rectilinear triangle. (See Figure 1.) Because of the “heterogeneity” of these two areas, this relationship could only be determined retrospectively: that is, if one knew where the planet had been, it was possible to measure the portion of the planet’s total period that had elapsed during that interval. However, Kepler said, there was no elegant way to determine, prospectively, where the planet would be, in a specified interval of time.

Figure 1

“But given the mean anomaly (time elapsed-bmd), there is no geometrical method of proceeding to the equated, that is, eccentric anomaly (position bmd). For the mean anomaly is composed of two areas, a sector and a triangle. And while the former is numbered by the arc of the eccentric, the latter is numbered by the sine of that arc multiplied by the value of the maximum triangle, omitting the last digits And the ratios between the arcs and their sines are infinite in number. So, when we begin with the sum of the two, we cannot say how great the arc is, and how great its sine, corresponding to this sum, unless we are previously to investigate the area resulting from a given arc; that is, unless you were to have constructed tables to have worked from them subsequently.

“This is my opinion. Insofar as it is seen to lack geometrical beauty, I exhort the geometers to solve me this problem:

Given the area of a part of a semicircle and a point on the diameter, to find the arc and the angle at that point, the sides of which angle and which arc, encloses the given area. Or, to cut the area of a semicircle in a given ratio from any given point on the diameter.

It is enough for me to believe that I could not solve this a priori, owing to the heterogeneity of the arc to the sine. Anyone who shows me my error and points the way will be for me the great Apollonius.”

Thus, having rejected Aristotle’s perfect circles, Kepler had to measure the planet’s motion not merely by the arc of a circle, but by the relationship of that arc to its sine. However, as Cusa had indicated, the ratio of the sine to the arc is “infinite”. Thus, the physical motion of the planet depended on a quantity whose characteristics could be precisely known, but could not be precisely calculated. The ellipse was not the orbit, it was that which is lawfully produced by the principle of universal gravitation.

Kepler did not directly observe this elliptical orbit, and to this day neither has anyone else. Nevertheless, the elliptical orbit was known to Kepler (and is known to anyone who relives his thoughts), more surely than if it had been directly seen. However, there is something more to the planet’s orbit. The ellipse is only the visible manifestation of the principle of universal gravitation, produced by the always changing action of gravitation on the planet at every moment of its motion.

The question posed by the paradox of the “Kepler Problem” is: what is the hidden characteristic of universal gravitation that produces the elliptical shape of the orbit?

Since that characteristic cannot be seen, it must be discovered through the paradox it presents in its visible expression. That paradox is the indeterminacy expressed by the “Kepler Problem”. The paradox does not exist for the planet. The planet “knows” at all times what it is doing. It moves seamlessly; always changing. The paradox exists in the expression of that continuously changing orbit within the visible domain of the elliptical orbit, through, as Kepler expresses it, the infinite ratio of the sine to the arc.

Leibniz’ Infinitesimal Calculus

What was required to solve this paradox was a new type of geometry that could express the relationship between the seen and the unseen. That is what Kepler demanded, and that is what Leibniz supplied.

Working from Fermat’s method of inverse tangents and maximum and minimum, Pascal’s investigations of conic sections, and Huyghen’s development of involutes and evolutes, Leibniz invented the infinitesimal calculus as a new type of geometry that expressed not only what was seen, but the relationship of what is seen to the principles that lay behind it. For reasons that will become clear below, Leibniz’ calculus, though inspired to solve the “Kepler Problem”, didn’t provide its solution. Such a solution would require Gauss’ discovery of the complex domain.

The most direct means of grasping the principles of Leibniz’ calculus is through his, and Johann Bernoulli’s application of the calculus to the solution to the catenary problem. As has been developed more thoroughly in other locations, the catenary expresses a principle of universal least-action. That principle is what produces the unique shape of the catenary, which, like the planetary orbit, acts on the hanging chain differently at each point. But unlike the planetary orbit, the catenary does not conform to a conic section.

To determine the shape of the catenary, Leibniz and Bernoulli first investigated the physical manifestation of this continuously changing least-action. This is most easily demonstrated pedagogically by the often cited experiment showing the physical determination of the catenary curve using a string and a hanging weight. (See Riemann for Anti-Dummies 46.) As Bernoulli showed in his treatise on the integral calculus, the total effect of the principle of least action is expressed by the general shape of the catenary, while its continuously changing manifestation at each (infinitesimal) point along the chain, is expressed by the relationship between the sines of the angles formed by the tangents. (See Figure 2.) In other words, the principle of least action exists outside the catenary, but “touches” it at every point as if it were acting tangent to the curve. Consequently, the continuously changing relationship of these tangents produces a visible expression of the everywhere present, but non-visible, principle of least action.

Figure 2

As this crucial example of the catenary illustrates, the infinitesimal calculus is not the geometry of the domain of sense-perception. It is the geometry of the interaction between the domain of sense-perception and the principles that lie behind it. It is a generalization of Kepler’s method of measuring the planet’s motion in any given interval, by its relationship to the whole orbit. It is in that relationship, of the whole (integral) to the part (differential) that expresses the interaction of the unseen to the seen. That which is seen is produced by the unseen acting according to a discoverable principle at each infinitesimal interval.

As mentioned above, Bernoulli’s application of the calculus to the catenary problem demonstrated that, like the planetary orbit, the physical principle governing the catenary was dependent on the sine of the angle. But, Leibniz further demonstrated that the catenary curve was also determined by another type of relationship that he called logarithmic. (See Figure 3.)

Figure 3

As has been developed more fully in earlier installments, this logarithmic relationship is the generalization of the principle of higher powers that was developed by the Pythagorean- Socratic current in Classical Greece, as typified by the investigations into the doubling of the line, square and cube. As Plato emphasizes, each type of action is associated with a magnitude of a different power, which are each mutually incommensurable. The more general form of these types of powers is expressed by the logarithmic spiral and the exponential function. Provoked by the type of discovery expressed by the catenary, Leibniz investigated this more general form of these “Platonic” powers, and discovered that they were related to a still higher power, expressed through the form of the so-called “natural logarithm”.

This type of discovery led Leibniz to distinguish between two types of magnitudes transcendental and algebraic. The algebraic are those types of magnitudes exemplified by the powers that double the line, square and cube. The transcendental are those types of magnitudes exemplified by the trigonometric and the exponential functions. As Cusa had insisted, the transcendental powers are a higher type. All algebraic magnitudes can be generated from transcendentals, but not the inverse.

As mentioned above, the planetary orbit, while dependent on a transcendental function, i.e., the sine, is expressed by a conic section, i.e., an ellipse. Apollonius had already demonstrated, as was later expanded by Fermat and Pascal, that the conic sections expressed the relationship of the second (square) algebraic power.

As such, Leibniz distinguished between those types of curves, such as conic sections, that could be expressed algebraically, and those, like the catenary, that could only be expressed by transcendental functions.

The catenary in particular, however, expressed a relationship between the two types of transcendentals. On the one side, as Bernoulli showed, it expressed the trigonometric, whereas on the other side, Leibniz showed it expressed the exponential. Thus, the physical principle of universal least-action indicated a connection between these two types of transcendentals. Yet, mathematically these two types of transcendentals appeared to be generated differently.

It is in the investigation of the connection between these two types of transcendentals that Leibniz encountered the square root of minus 1. He expressed this as the paradox of logarithms of negative numbers. (See Riemann for Anti-Dummies Part 38.) He insisted that these logarithms existed, but not in the visible domain, but in a domain still to be imagined. As he discussed the matter in a letter to Huygens, “… there is almost nothing more to be desired for the use which algebra can or will be able to have in mechanics and in practice. It is believable that this was the aim of the geometry of the ancients (at least that of Apollonius) and the purpose of loci that he had introduced…. “

He was convinced that the square roots of negative numbers provided the paradox that would open the door to this new domain where the interaction between the seen and the unseen could be expressed. He referred to them, as “amphibians somewhere between being and non- being”.

Gauss, The Kepler Problem, and The Complex Domain

It was left to the young Gauss, when he was between the ages of seventeen and twenty, to discover that the higher transcendentals to which Leibniz had pointed, demand the development of the complex domain as that thought-object through which the interaction between the seen and the unseen could be expressed. Herein would emerge the source of the “Kepler Problem” that the motion of the planets in elliptical orbits are governed not by the circular transcendentals, but by a higher form of “elliptical” transcendental, the characteristics of which could only be discovered and expressed in the complex domain.

Gauss never published his findings, except to indicate in a famous remark in the “Disquisitiones Arithmeticae”, that his method for the division of the circle could also be applied to other transcendentals, particularly the lemniscate. Gauss promised to publish these results in a future time. But, due to the oppressive conditions brought on by the rise of Napoleon and its aftermath, Gauss never brought forth the promised text. His remark, however, prompted two brilliant youth, the Norwegian, Niels Henrik Abel, and the German, C. G. Jacobi, (both of whom were students of Gauss’s collaborators, Hansteen and Dirichlet , respectively) to independently develop these results. Upon seeing their work, Gauss remarked that he had already developed this theory when he was a youth. At the time, these comments were dismissed as arrogant boasting by those in the scientific community who were submitting to fascist terror, as exemplified by the supreme bigot, Augustin Cauchy.

Nevertheless, Jacobi believed Gauss was right, writing in Crelle’s Journal in May 1839, that it was the investigation of elliptical transcendentals which led Gauss to introduce complex numbers . Yet, it wasn’t until 1898, when Gauss’ youthful diaries became available, that the truthfulness of Gauss’s statements was fully confirmed.

From a review of these diaries, it is clear that from the beginning, Gauss was engaged in one unified investigation: the development of the complex domain as that idea necessary to express those higher transcendental relationships governing the physical universe.

While popular academia grudgingly admitts today that Gauss was the actual discoverer of elliptical functions, they still falsely present this discovery as an extension of the formal algebraic approach of Euler and Lagrange. While Gauss’ 1799 dissertation already exposes this as a lie, the diaries give an even more complete picture of Gauss’ train of thought.

The diaries begin on March 30, 1796, with the announcement of the discovery that the 17-gon is constructible by straight-edge and compass. As has been developed in other locations (Riemann for Anti Dummies Parts 30 and 31 and JBT’s lecture on the heptadecagon in L.A.) Gauss’ discovery depends on recognizing that the visible circle is an artifact of a process that can only be expressed in the complex domain. From the standpoint of the visible domain, it appears that the circle can only be divided into 2, 3, or 5 parts by straight-edge and compass, as was believed for more than two thousand years. Yet Gauss showed that the principle on which the division of the circle depends, can only be expressed in the complex domain. From this standpoint, the division of the circle is seen as the more general form of the ancient Greek problems of doubling the square and the cube.

The doubling of the square and cube are determined by those principles that generate one or two means, respectively, between two extremes. Gauss showed, that from the standpoint of the complex domain, the division of the circle into “n” parts is the problem of finding the principle that generates “n-1” means between two extremes. If “n-1” is 2 to a power of 2, then the problem can be transformed into a succession of square roots, and is susceptible of being constructed by straight edge and compass.

Work through Gauss’ method for dividing the circle for yourself, using the above cited pedagogicals as guide. For this discussion, bear in mind this crucial point. An action that is carried out in the visible domain, dividing a circle into “n” parts, is determined by a set of relationships that cannot be expressed in the visible domain. Rather, those relationships can only be expressed in the complex domain, where the interaction between the visible and the principles that lie behind it, can be made manifest.

Is the complex domain real? You can actually, with straight-edge and compass, divide a visible circle into 17 equal parts. But you cannot know how, or that you even can, accomplish such a task, unless you investigate, as Gauss did, the principle on which that act is accomplished, in the complex domain.

During this time, Gauss was also investigating the “Kepler Problem”, and began to recognize what had blocked its solution. The general characteristics of the problem can be illustrated pedagogically in the following way:

The circular trigonometric transcendentals vary periodically according to the angle a moving radius makes with a fixed diameter. (See Figure 4, Animation 1.) The circular sine has a period of 0 to 1 to 0 to -1 and the cosine from 1 to 0 to -1 to 0, as the radius rotates around the circle. For every one rotation, this cycle repeats. Thus, for example, the sine of 30 degrees, 390 degrees, 750 degrees are all the same.

Figure 4

Animation 1

However when this same relationship is extended to an ellipse, something dramatically changes. In an ellipse the length of the radius changes as it moves around the circle. (See Figure 5, Animation 2.)

Figure 5

Animation 2

But how much it changes relative to the angle depends on the eccentricity of the ellipse. (See Figure 6, Animation 3.)

Figure 6

Animation 3

If the idea of the circular sine and cosine are extended to the ellipse, then both the sine, cosine, and length of the radius each vary periodically with the angle, but, how they vary depends on the eccentricity of the ellipse. This, obviously, is a much more complicated relationship than the circular functions. Whereas the circular functions exhibit a very simple periodicity, the elliptical ones are much more complicated. For this reason, the elliptical functions were generally disregarded and all efforts to solve the “Kepler Problem” focused on measuring elliptical action by circular functions, which proved very elusive.

At Kaestner’s prompting in the Spring of 1796, Gauss sought the more general principle on which his method for dividing the circle depended, by extending that investigation into the division of non-uniform curves. While aiming for the elliptical functions, he set his first attack on the simplest case of such a function: the lemniscate.

The lemniscate is a special case of an elliptical curve. It looks like a figure 8 (See Figure 7)

Figure 7

It was originally investigated by Jacob Bernoulli, who was investigating the physics of elastic rods. As such, it was originally known as the “curva elastica” until Bernoulli gave it the Latin name, “lemniscate”, which means ribbon. The lemniscate is the locus of positions the product of whose distances from two foci is always equal to the square of the distance between the foci. (See Figure 8).

Figure 8

This property of the lemniscate has a similarity to the property of the ellipse. In the ellipse the sum of the distance from the foci to the curve is always a constant. (See Figure 9.) For the lemniscate it is the product of the distances. However, unlike the ellipse which is an algebraic curve of the square power, the lemniscate expresses the quartic power. (There is a simple geometric demonstration of the above statement which would be a digression to develop here. The reader is encouraged to discover it as a pedagogical exercise.)

Figure 9

The lemniscate has another geometrical property important for this discussion. It can be formed by the inversion of a hyperbola in a circle. (See Figure 10.)

Figure 10

Gauss’ investigation into the lemniscate began no later than August 1796, when he wrote out a series of eleven “Mathematical Exercises” on various subjects. Of significance for this discussion is the fifth exercise, in which Gauss begins to develop the characteristics of what he would later call sinus lemniscatus, (lemniscate sine).

On January 8, 1797, Gauss wrote, ” I have begun to investigate carefully curves dependent upon the leminiscate integral .”

This followed on March 19, 1797, by an entry announcing that the division of the lemniscate into “n” parts depends on “nn^(t?-1) power, accompanied by the additional notation: “Imaginary quantities: The general criteria are sought according to which it is possible to distinguish complex functions of many variables from the non-complex ones.” (the cited symbolic notation reads: n times n raised to the t times square root of -1 power.)

Gauss’ method of the division of the circle and his generalization to the lemniscate and the elliptic functions, makes use of his clear geometrical understanding of the complex exponential. Protests from formalist sycophants not withstanding, Gauss’ idea is completely different than Euler’s formal algebraic mish mash peddled in virtually every text book today.

Gauss had this concept early on. On August 14, 1796, he wrote, “By the way, (a + b?-1)^{(m + n?-1)}, has been explained.” And two days later, he wrote, “The highest things are already of the mind. Let it stand firm in order that they be protected.” (Note: the cited notation reads: a plus b times the square root of -1 raised to the m plus n to the square root of -1 power.)

The Lemniscate and Elliptical Functions

We can get a preliminary taste for Gauss’ discovery of the elliptical transcendentals by investigating them using the geometrical methods established by Gauss and developed more fully by Riemann. In this way we will obviate, at least for now, the unnecessary use of formalism, which otherwise might be required for a more complete investigation. This is a method that both Gauss and Riemann emphasized, stating in numerous locations, that while certain relationships can only be expressed initially as formulae, the results can be given much better clarity through geometrical representation. It must be born in mind, as Gauss emphasized:

“The demonstration is presented using expressions borrowed from the geometry of position, for in this way, the greatest acuity and simplicity is obtained. Fundamentally, the essential content of the entire argument belongs to a higher domain, independent from space, in which abstract general concepts of magnitudes, are investigated as combinations of magnitudes connected by continuity, a domain, which, at present, is poorly developed, and in which one cannot move without the use of language borrowed from spatial images.”

Look at figure 11 showing the lemniscate as the inversion of the hyperbola in the circle. Now, let’s investigate the principles that generate this relationship among these three functions, through the method of complex mappings as developed by Gauss in his work on curved surfaces and later extended by Riemann.

Figure 11

The mappings we will investigate all depend on the complex exponential as Gauss indicated in his August, 14, 1796 notation. As was illustrated in Riemann for Dummies 48, the complex exponential can be represented geometrically as a stereographic projection. Under this mapping, the arithmetic relationships of one surface are transformed into geometric relationships on another. As, for example, the lines of latitude and longitude on a sphere are transformed into exponentially spaced concentric circles and radial lines on a plane. (See Figure 12)

Figure 12

When this same complex exponential is mapped from one plane to another, it is expressed by the transformation of a grid of equally spaced lines into the same exponentially spaced concentric circles and radial lines of the stereographic projection. (See Figure 13)

Figure 13

WARNING TO THOSE INFECTED WITH CARTESIANISM: These mappings are not graphs. They indicate the transformation of one set of least action physical relationships into another by the introduction of a new physical principle.

Figure 14

Of particular importance for this discussion, is the periodicity of the complex exponential. (See Figure 14, Animation 4.)

Animation 4

From this standpoint, let’s unfold the hidden relationships among the hyperbola, circle and lemniscate. To do this, we adopt Gauss’ method of inversion. Instead of investigating the circular, hyperbolic and lemniscatic functions as derived from the relevant curves, we investigate how the curves are derived from the relevant functions.

Begin first with the circle. Think of the circle as that which is produced by a right triangle whose hypotenuse is constant, but whose angle varies. It is obvious that the circle is generated by two periodic functions: the sine and the cosine.

Now turn to the hyperbola. The hyperbola is generated by a rectangle whose sides change but whose area remains the same. (See Figure 15). This constant area determines two functions, the hyperbolic sine and cosine. (See Figure 16). However, it is evident from the figure, that unlike the circle, the hyperbolic sine and cosine are not periodic. The are always getting bigger and bigger.

Figure 15

Figure 16

Now look at the lemniscate. Gauss defined two transcendental functions for the lemniscate, called the lemniscatic sine and lemniscatic cosine (See Figure 17)

Figure 17

Something strange happens when we examine these functions. If we apply Gauss’ method of inversion, and derive the lemniscate from these functions, we see that to define the entire lemniscate, the sine appears to have two periods for each full rotation and the cosine appears to have one. (See Animation 5).

Animation 5

Keep this curiosity in mind.

As developed in the previous installments on hyperbolic functions (See Riemann for Anti-Dummies 33) the hyperbolic cosine is expressed physically by the catenary, (See Figures 18 and 19) and as such can be expressed by the exponential.

Figure 18

Figure 19

When this principle is expressed as a Gauss- Riemann complex mapping, the hyperbolic functions are formed from the midpoint of a line joining two of the circles formed by the exponential, that are rotating in opposite directions. (See Animation 6). From this standpoint, a periodicity emerges that otherwise had remained hidden.

Animation 6

Now, look again at the circle. As Leibniz indicated, the circular functions are also expressions of the complex exponential. (See Riemann for Anti-Dummies Parts 37 and 38.) (See Figures 20 and 21). When these functions are expressed as a Gauss-Riemann complex mapping, the full expression of their periodicity is brought forth. They too are formed by the countervailing rotations of the exponential circles, but these circles are formed from the mappings that are at right angles to the hyperbolic.

Figure 20

Figure 21

In sum, the circular and hyperbolic functions form the four possible variations of the mappings of the complex exponential. This brings out a unity between the hyperbola and circle that lies behind their visible appearance as conic sections, but on which that visible appearance is based.

In these examples we can see how the complex domain provides the means to create a synthetic visualization of the interaction between the visible domain and the unseen principles that lie behind it.

Now, look at the lemniscate. To unfold the lemniscate, Gauss applied the same principle of inversion illustrated above for the circle and hyperbola. A further elaboration of Gauss’ discovery will be developed in a future installment. But the characteristic of Gauss’ discovery is evident from the geometric characteristics that emerge when the lemniscate functions are expressed in the complex domain.

What Gauss discovered was that the lemniscate functions, and more generally, the elliptical functions are all doubly periodic. (See Figures 22, 23, and 24). This is no where evident from the relationships that appear in the visible domain. Furthermore, he demonstrated that the circular and hyperbolic functions were special cases of the higher class of elliptical transcendentals.

Figure 22

Figure 23

Figure 24

From this standpoint, Gauss established that the ellipse was governed by a higher type of transcendental which subsumed the circular, hyperbolic and exponential transcendentals. But these higher transcendentals were only expressible in the complex domain. In other words, the principles on which the planetary orbits really depended, were reflected by, but not visible in, the domain of sense-perception, or its representation. To passion to know these principles demanded the development of the complex domain.

Ironically, Kepler’s approach for calculating an approximate solution for the “Kepler Problem”, is essentially still the best approach for determining the position of a planet in its orbit. But Gauss’ determination to investigate the paradox it revealed, led to his discovery of those higher transcendentals on which the underlying principles of planetary orbits are based. This paved the way for even deeper investigations.

Riemann For Anti-Dummies Part 48

RIEMANN’S ROOTS

In December 1822, C.F. Gauss submitted a paper to the Royal Society of Science in Copenhagen titled, “General Solution of the Problem: To Map a Part of a Given Surface on another Given Surface so that the Image and the Original are Similar in their Smallest Parts”.

Notably, the paper contained the motto: “Ab his via sternitur ad maiora” (“These results pave the way to bigger things”).

They did. Nearly 30 years later (1851), B. Riemann submitted, to Gauss, his doctoral dissertation on functions of a complex variable, which, along with his 1857 works on Abelian functions and the Hypergeometric series, developed the further implications of the method Gauss had initiated. The deeper epistemological implications of these results, however, were only brought to light in our present time, through Lyndon LaRouche’s discoveries in the science of physical economy, as in his most recent work, “On Visualizing the Complex Domain”. Therein is established the highest vantage point from which to re-live the discoveries of Gauss and Riemann.

Initially, the Royal Society had posed a more limited subject for the so-called “Copenhagen Prize Essay” than the one ultimately addressed by Gauss. The original question was directed toward solving some particular problems involved in the production of geographical maps. At the suggestion of his collaborator, the astronomer H.C. Schumacher, Gauss proposed the more general question to the Royal Society. After several years without anyone providing a serious solution to the question he posed, Gauss submitted his own solution, and, of course, was awarded the prize.

Obviously, Gauss was less interested in winning the prize than taking the opportunity to present the more general results he had been developing his whole life, beginning with his earliest work on the division of the circle and the fundamental theorem of algebra. The Royal Society’s challenge afforded Gauss the opportunity to demonstrate the extension of Leibniz’ calculus, under the concept of the complex domain that Gauss had developed in those earlier works. While this application provided the solution to the more limited practical problem of producing accurate maps, as Gauss indicated, it was really about something much more fundamental: specifically, the improvement in the capacity of the mind to grasp and communicate truths concerning the unsensed principles that govern the universe.

Mapping the Sensorium

The roots of Gauss’ method go deep into the history of Mankind’s efforts to increase its power in and over nature, beginning with the earliest attempts to map sense perceptual space- time by the development of calendars and geographical and astronomical maps. These maps expressed not merely the visible changes of the heavenly bodies. The unsensed principles were reflected as well, in the form of anomalies, paradoxes, and distortions. Thus, each map already implies another map that lies “behind’, so to speak, the visible map.

Members of the LaRouche Youth Movement currently involved in observing the motion of Mars, are confronting the types of paradoxes that arise from the development of such maps. Go out and look at Mars rising in the eastern sky. The arrangement of the visual image of Mars and the surrounding stars on the inside of the celestial sphere gives rise to a mental image, i.e., map. Over the course of the night, the motion of Mars and the stars, relative to the observer, changes, forming a succession of mental images, which gives rise to a map of the changes in the night’s succession of maps, or, in other words, a map of maps. From night to night, the image of Mars changes its relationship with respect to the images of the other stars. This change gives rise to a higher map, formed from each night’s map of maps. This map of maps of maps gives rise to an even higher type of map, a map that reflects the underlying principles governing the changes among the maps of the sense perceptual images. This higher map only becomes fully intelligible from the standpoint of Gauss’ and Riemann’s complex domain.

While these investigations are most ancient, the roots of our modern knowledge begin with Thales’ (624-547 B.C.) effort to map the celestial sphere onto a plane by the gnomonic, or central projection. Thales’ method was to define on the plane, the image of every point on the sphere, by drawing a line from the center of the sphere, through the surface, until that line intersected the image plane. (See Figure 1.) (The same result can be demonstrated physically by using a light source located at the center of a transparent hemisphere to cast shadows of figures drawn on the sphere onto a flat wall.)

Figure 1

This projection immediately presents us with a crucial paradox. Certain relationships among the images on the sphere are different than the relationships among their images on the plane. For example, the image of a spherical triangle whose vertices are three stars on the celestial sphere, is a rectilinear triangle on the plane. Consequently, the angular relationship among the three stars cannot be preserved in the image, for the sum of the angles of a spherical triangle is always greater than 180 degrees, while the sum of the angles of a plane triangle is always equal to 180 degrees.

However, the measurable relationships among the images of the stars on the celestial sphere are determined by angular measurements, which are not accurately represented by Thales’ gnomonic projection. The first solution to this problem is attributed to Hipparchus (160-125 B.C.), to whom is attributed the discovery of the stereographic projection. In this projection, the sphere is mapped onto the plane from one of its poles. (See Figure 2.) (This can be demonstrated physically by moving the light source in the previous experiment from the center of the sphere to its pole. Notice the resulting change in the relationship among the shadows.)

Figure 2

In this stereographic projection, the lines of longitude of the sphere are transformed into radial lines on the plane. The latitude lines on the sphere are transformed into concentric circles. If we think of the point touching the plane as the “south” pole, and the point of projection as the “north” pole, the circles of latitude in the “southern” hemisphere all map to circles inside the circle which is the image of the equator. On the other hand, the circles of latitude in the “northern” hemisphere all map outside this circle. The radial lines on the plane make the same angles with each other, as the longitude lines of which they are images. However, the radii of the concentric circles that are images of the circles of latitude, increase exponentially, the farther the latitude circles they represent get from the “south” pole and the closer they get to the “north” pole.

In the case of the stereographic projection, the angles among the images are preserved.

Another projection that preserves angles was developed by Gerhard Kremer (1512- 1594), otherwise known as Mercator. On the Mercator projection, the image of the equator is a straight line, and the images of the circles of longitude are perpendicular lines spaced equally along it. The images of the circles of latitude are straight lines parallel to the equator, but the distance between them increases. (See Figure 3.) This is because on a sphere the longitude lines get closer together as they approach the poles. Consequently, the ratio of the distance along the surface of a sphere for a given angle of latitude, to the distance along the surface for the same angle of longitude, changes from the equator to the poles. (See Figure 4.) This change is reflected in the Mercator projection by increasing the distance between the lines of latitude so that the proportion between the lengths of latitude and longitude is the same as on the sphere.

Figure 3

Figure 4

The Mercator projection, although entirely different than the stereographic, also preserves angles. It also has the characteristic that the so-called rhumb line, or path that makes the same angle with all lines of longitude, is a straight line. This exemplifies the types of paradoxes that emerge, for the straight-line is the shortest path on the flat plane of the projection. But on the sphere, the shortest path is a great circle, and the rhumb line is a spiral path called a loxodrome, which is longer than the great circle arc. (See Figure 5.)

Figure 5

The Mapping of Principles

Thus, at first glance, the development of these angle-preserving projections (a characteristic that Gauss would later call, “conformal”), has a very important significance for the representation of the images of the Sensorium. Nevertheless, Gauss had something much more significant in mind. The distortions and paradoxes that result from these projections are not only due to the visual representation, but reflected something “behind” the visible. By developing a general means for transforming one surface into another conformally, Gauss paved the way for Riemann’s more general investigations into the nature of these transformations themselves, and their relationship to the underlying principles behind the Sensorium.

To do this, Gauss rejected the reactionary, a priori Euclideanism of Kant. For him a “surface” is not an object embedded in empty Euclidean space that is infinitely extended in three directions. Rather, a “surface” is generated by some physical action. For example, we don’t measure the celestial sphere by two angles. The celestial sphere is a physically determined idea, generated by the physical action of rotation with respect to the direction of the pull of gravity, and the direction around the horizon from some physically determined direction, such as the position of the rising (or setting) Sun.

In Riemann’s terms, these two acts of rotation are the physical modes of determination of the celestial sphere. Were these modes of determination created by some other physical action, they would produce a different surface. This is the basis for a physically determined geometry. The transformation of one surface into another, Gauss demonstrated, is accomplished by finding a function that transforms one pair of modes of determination into another.

This is exactly what takes place in a stereographic projection. The two modes of determination, represented as circles of latitude and longitude on the sphere, are transformed into concentric circles and radial lines, respectively, on the plane. In the Mercator projection, the same two modes of determination of the sphere, are transformed differently, that is, into straight lines. Nevertheless, both projections are conformal. Thus, the characteristic of conformality reflects a more general principle, not specific to a particular projection.

Gauss recognized that for a projection to be conformal, it must transform one surface into another equally in all directions. This is expressed geometrically by the fact that the images of circles are also circles. This can be illustrated physically in two ways. Think of an image on a stretchable, say, rubber, surface. If the surface is stretched proportionally in all directions, then the shape of the image will be the same, only larger. If it is stretched by a different amount in different directions, the image will be distorted. The former, represents a conformal transformation, the latter, non-conformal.

Another physical example has been pointed out in previous pedagogicals. Take a clear plastic hemisphere and tape to it circles of differing sizes, around what would represent a circle of latitude on a sphere. Shine a light from the center of the hemisphere. Look at the shadows. The circles are transformed into ellipses. Now, move the light to the pole. The shadows become more circular, with the shadows of the smaller circles being the most circular. This illustrates the difference between the gnomonic projection and the stereographic. In the former, the circles are stretched differently in different directions, thus producing ellipses. As such, this projection is non-conformal. In the latter, the circles are stretched into circles, i.e., they are conformal.

This characteristic of circular action, Gauss had already developed as the principle of complex numbers, as early as 1796, in his discovery of the division of the circle, and his 1797 fundamental theorem of algebra (published in 1799). That is, a complex number, Gauss showed, was not arbitrarily defined as the solution to an algebraic equation. Rather, the complex number expressed that rotational action, which transcended, and thus determined, all possible algebraic magnitudes. The transformation of one complex number into another, therefore, was the transformation of one rotational action into another rotational action, exactly the condition necessary for the mapping to be conformal.

For this reason, Gauss considered not the visible surface, but its representation in the complex domain. Each point on the surface corresponded to a complex number, which in turn was determined by the physical modes of determination of the surface. To transform one surface onto another, required transforming the modes of determination of one surface into the modes of determination of the other, which in turn transformed each complex number of the first surface into a definite complex number of the second. This is what Riemann would later call, “a function of a complex variable.”

Figure 6

This was illustrated above by the examples of the stereographic and Mercator projections. In the former case, the circles of latitude and longitude that were equally spaced around the sphere were transformed into exponentially spaced concentric circles and radial lines on the plane. Gauss notes that this corresponds geometrically to the transformation of the complex exponential. (See Figure 6.) The Mercator projection corresponds to a transformation of the stereographic projection. (See Figure 7.) To map a sphere onto a plane, Gauss applied these complex transformations, to the modes of determination of a sphere, that is, the two modes of rotation.

Figure 7

In his paper, Gauss demonstrated why these types of projections would be conformal. This can be illustrated geometrically, by looking at the behavior of a small square undergoing the desired transformation. As the accompanying animation illustrates, in order for the diagonals of the square to remain perpendicular, the sides of the square must change accordingly. Gauss expressed this geometric condition by a formula in the language of Leibniz’ calculus, which was restated by Riemann in his doctoral dissertation. It is one of the continuing frauds of modern mathematics, that this formula has become known as the “Cauchy-Riemann” formula, despite the fact that Augustin Louis Cauchy added nothing to its development. For the sake of historical accuracy, and mental health, this relationship should really be known as the Gauss-Riemann relationship.

Still Lurking Behind the Scenes

Within Gauss’ discovery, something is lurking behind the scenes, a spirit from the domain of unseen principles. In both the above examples, the projection becomes increasingly distorted– as the projection approaches the north pole in the case of the stereographic, and as the projection approaches both poles in the case of the Mercator. At the poles, the projection “blows up” completely and ceases to exist. Is this just a failing in the projection, or, is this an indication of some yet unknown, hidden principle?

In this question lay the “bigger things” for which Gauss paved the way.

Riemann for Anti-Dummies Part 47

DEFEATING I. KANT

In the opening of his Habilitation lecture, Bernhard Riemann proposed to establish the foundations of geometry on a rigorous basis:

“Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of general notions of quantity. From this it will result that a multiply extended magnitude is susceptible of various metric relations and that space accordingly constitutes only a particular case of a triply-extended magnitude. A necessary sequel to this is that the propositions of geometry are not derivable from general concepts of quantity, but that those properties by which space is distinguished from other conceivable triply-extended magnitudes can be gathered only from experience”.

Riemann’s program poses a paradox for those habituated to the doctrine of Immanuel Kant and its more extreme, modern form–existentialism. How can the propositions of geometry be determined by experience?

Kant had insisted that:

“Space is not an empirical concept which has been derived from outer experiences. For in order that certain sensations be referred to something outside me…the representation of space must be presupposed. The representation of space cannot, therefore, be empirically obtained from the relations of outer appearance. On the contrary, this outer experience is itself possible at all only through that representation.”…”Geometry is a science which determines the properties of space synthetically; and yet a priori. It must in its origin be intuition; for from a mere concept no propositions can be obtained which go beyond the concept as happens in geometry. For geometrical propositions are one and all apodeictic, that is, are bound up with the consciousness of their necessity; for instance that space has only three dimensions. Such propositions cannot be empirical or, in other words, judgments of experience, nor can they be derived from any such judgments.”

Kant was not very original. Nearly two centuries earlier, Johannes Kepler, through his discovery of universal gravitation, had already liberated science from similar Aristotelean dogmas that, from the murder of Archimedes until the Renaissance, had enslaved European civilization. Kant was deployed to put the chains back on. Those doctrines had taught that experience, (which, for them, was limited to sense perception), can tell us nothing about the physical world. For example, our experience of phenomena such as the motion of the planets and other heavenly bodies, is limited to the perceptions of the changes of position of points of light on the inside of a great sphere of unknown radius, whose center is always the location of the observer. For the Aristoteleans, the actual motions, as well as the principles that govern them, are inherently unknowable, and so they must be referred to some a priori determined set of propositions, such as those of Ptolemy, Copernicus or Brahe. These propositions, in turn, are ultimately derived from Euclidean-type axioms, postulates and definitions, which Kant insisted, are the only possible form by which we can conceive of space:

“Space is a necessary a priori representation, which underlies all outer intuitions. We can never represent to ourselves the absence of space, though we can quite well think it as empty of objects. It must therefore be regarded as the condition of the possibility of appearances, and not as a determination dependent upon them. It is an a priori representation, which necessarily underlies outer appearances.”

According to Kant: these propositions are not decidedly truthful; they make no judgement about the actual motions; they are the form by which the appearances must be represented; and nothing can happen that is not possible under these propositions.

When you begin to think about this, you come face to face with the fundamental question of science (and also, politics, history and art): What is experience? Is it sense perception? Therein lies Kant’s trickery, for if experience is limited to sense perception, then indeed, it can tell us nothing about the propositions of geometry. As Kant’s sophistry insists: “Were this representation of space a concept acquired a posteriori, and derived from outer experience in general, the first principles of mathematical determination would be nothing but perceptions. They would therefore all share in the contingent character of perception; that there should be only one straight line between two points would not be necessary, but only what experience always teaches. What is derived from experience has only comparative universality, namely, that which is obtained through induction. We should therefore only be able to say that, so far as hitherto observed, no space has been found which has more than three dimensions.”

However, Riemann had something far different in mind when he spoke of experience:

“There arises from this the problem of searching out the simplest facts by which the metric relations of space can be determined, a problem which in the nature of things is not quite definite; for several systems of simple facts can be stated which would suffice for determining the metric relations of space; the most important for present purposes is that laid down for foundations by Euclid. These facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses…”

For Riemann, as for all humans not self-degraded into Aristotelean/Kantian bestiality, experience is not sense-perception; it is the active interaction of the mind with the universe, of which it is a part. It is, as Plato insists, the formation of hypotheses, higher hypotheses, and hypothesizing the higher hypothesis. It is the investigator, investigating, how he is investigating what is being investigated. Or, as for Apollo, who sings in Percy Shelley’s Hymn, “I am the eye with which the Universe beholds itself and knows itself divine;”

The propositions of geometry can, and must, be derived from this type of experience and Riemann advanced the general methods for how this is done. He divides this task into two steps, both of which rested on foundations laid by Gauss. The first, as indicated above, is the determination of the general notion of multiply-extended magnitude. Here Riemann cites Gauss’s second treatise on bi-quadratic residues and his fundamental theorem of algebra. In those works, as well as other unpublished discussions, Gauss attacked Kant’s view as an “illusion,” and he advanced the concepts begun with the investigations of the Pythagoreans, Archytas and Plato, that physical action, not a priori intuition, gives rise to our concept of extension, as exemplified by the different principles, or powers, of physical action that extend a line, square, or cube.

In each type of action, the determination of the essential, distinguishing characteristic, Riemann noted, always leads back to “n” determinations of magnitude, in which “n” signifies the relative degree of the power governing the action. For example, a cube, which is determined by a triply-extended magnitude, cannot be determined by the doubly-extended square, nor a square by a simply-extended line.

However, there is still another consideration:

” …there follows as second of the problems proposed above, an investigation into the relations of measure that such a manifold is susceptible of, also into the conditions which suffice for determining these metric relations. These relations of measure can be investigated only in abstract notions of magnitude and can be exhibited connectedly only in formulae; upon certain assumptions, however, one is able to resolve them into relations which are separately capable of being represented geometrically, and by this means it becomes possible to express geometrically the results of the calculation. Therefore if one is to reach solid ground, an abstract investigation in formulae is indeed unavoidable, but its results will allow an exhibition in the clothing of geometry. For both parts the foundations are contained in the celebrated treatise of Privy Councillor Gauss upon curved surfaces.”

Thus, the question of discovering a physical geometry requires determining both the “n” determinations of magnitude, and their measure relations. Neither of these can be given a priori. How, then, can these matters be discovered from human cognitive, (as opposed to Kantian) experience?

The last three installments of this series, (Riemann for Anti-Dummies Parts 44, 45, and 46), explored the essential foundations of Gauss’ and Riemann’s approach. There we showed how Gauss, from his very earliest work under the tutelage of Kaestner and Zimmerman, recognized that only physical principles, not definitions, can lay the foundations for geometry, as, for example, the determination of what is a straight-line. From there, Gauss showed that these physical principles determine a characteristic curvature from which the measure relations of a surface are derived, and that these general principles of curvature are expressed, in the smallest parts, by the characteristics of the “shortest lines”, or geodesics, of the surface. What remains to discuss is this discovery’s inversion. How can the physical principles of the curvature of the surface be determined from the characteristics of the geodesics as measured by small changes in those geodesics?

Gauss’ work on this inverse problem is of crucial significance, as this is the form of investigation usually confronted in science, politics, history and art. We cannot know these physical principles by sense perception, but we can perceive their effects by some small measurable change, from which, by hypothesizing, we can determine the general principles that are determining that change. Kepler determined the general principles of curvature of the solar system as a whole, from small measured changes in the relationship between the orbits of Mars and Earth. Leibniz and Bernoulli determined the general principles of least- action from small measured changes in the shape of a hanging chain. Gauss determined the solar system’s harmonic dissonance that had been indicated by Kepler, from Piazzi’s very small measurements of Ceres’ arc. LaRouche determined the general direction of world history by measured changes in the cultural/mental outlook of the population following the death of FDR.

As Riemann indicated, an exploration of Gauss’ work in this direction is impossible without resort to abstract concepts expressible by formulae, but, these results are always capable of geometrical representation. For pedagogical purposes we will minimize the former and emphasize the latter, but limited reference to formulae are unavoidable, and will always be accompanied by the appropriate geometrical representation.

To begin to get a handle on the principles involved, take a simple case a line. When considered as a simply-extended magnitude, a line can be increased or decreased only by action along its length, that is, back and forth. Such changes can be measured only by increments of more or less, and expressed by rational numbers.

However, when that line is understood to be generated from a doubly-extended magnitude, such as the diagonal of a square or rectangle, its increases or decreases are measured by an entirely different set of relationships, as demonstrated by Plato in the Meno and Theatetus dialogues. In this case, the line is increased or decreased along its length, but only in a connected way, to changes in the lengths of the side of the square or rectangle. These changes cannot be measured by the simple ideas of more, or less, as expressed by rational numbers. Rather, they express the type of relationship that has come to be known by the “Pythagorean theorem”, which as Pythagoras and Plato emphasized, are incommensurable with simply-extended magnitudes.

Draw a rectangle and call the length of one side “x” and the length of the other side “y”. By the Pythagorean theorem, the length of the diagonal “s” can be measured as the square root of the sum of the squares of the two sides, or in shorthand, s=?(x²+y²). If the diagonal is extended by a small amount “ds”, the sides of the rectangle will be increased by proportional small amounts “dx” and “dy”. (“ds”, “dx”, and, “dy” are Leibniz’ notation for these infinitesimal increments, which he called differentials.) If this action is taking place on a Euclidean flat plane, then ds, dx and dy, will express the Pythagorean relationship, ds=?(dx²+dy²).

Thus, contrary to the textbook versions, the Pythagorean relationship is not an arbitrary formula; it expresses a characteristic relationship of a certain type of surface–a Euclidean flat plane. Inversely, if a physical process is measured by the Pythagorean relationship expressed above, that action is occurring in a Euclidean flat plane.

This measurable physical relationship, not Kant’s dictum of a priori certitude, is the only reality of a Euclidean flat plane. And, since real world physical measurements express a different relationship, the physical reality of a Euclidean flat plane is not only not necessary, it is illusory .

But, if our cognitive experience, i.e. physical measurement, determines that the Pythagorean relationship doesn’t hold, what relationship expresses a doubly-extended magnitude? A Kantian will fly into a fit of rage at this paradox. Kant insists that Euclidean space is the only possible way one can think about space, therefore, for the Kantian, Euclidean space must be the only space possible. And so, the Kantian will demand the world be treated as if it were Euclidean, even if physical measurements tell us otherwise. Fakers in the tradition of Gallileo’s deceitful attempt to curve-fit the catenary into a parabola, will have no problem with this. They will suggest limiting physical measurements to small enough regions, that the deviation from the Pythagorean relationship is below the errors of measurement. Such chicanery is, however, only self-deception, as the deviation from flatness, no matter how small it may seem, still exists, and, sooner or later, judgements made on that delusion will become impossible to ignore (as, for example, how the ongoing financial crisis was ignored by believers in such “New Economy” frauds as exemplified by the Winstar madness).

Gauss, of course, rejected such follies. He recognized that the Pythagorean relationship, as expressed in Euclidean geometry, was not sacrosanct. Rather, it was merely a special case of a more general principle. Rather than cling to the special case, Gauss discovered the foundations on which more the general principle was based.

To gain an understanding of Gauss’ discovery, it is pedagogically efficient to work through several examples, and then abstract from them the general principle at work.

Begin with the case of the physically determined celestial sphere. This surface is physically determined by the position of the observer and the direction of the pull of gravity. The former determines the center of the sphere and the latter determines the poles and the horizon. If we are bound by Kant’s constriction that we can only think of space as infinitely extended in three directions, then this sphere sits like a large object surrounded by empty space.

But, Gauss was free from Kantianism, and he understood the celestial sphere as a manifold of physical action, produced by two mutually inter-dependent angles: the angles around the horizon and the angles perpendicular to it. (For pedagogical simplicity, I will refer to these angles by the familiar names latitude and longitude respectively.)

All positions on the celestial sphere can be determined by these two physical parameters. (Thus, for the sphere, n=2, in Riemann’s terms of “n” determinations of magnitude.) From this relationship, Gauss constructed what is called a “parametric formula” in which the surface of the sphere is expressed entirely by these two parameters.

All the longitudinal circles have a common center, i.e. the observer. Positions along any one of these circles can be determined by a function of the angle from the horizon to the zenith. If this angle is called “p”, then any position along a circle of longitude can be determined as a function of the cosine and the sine of the angle “p”. (See Figure 1.) Each of these circles of longitude can be distinguished from one another, by another angle, as measured around the horizon. If that angle is called “q”, then positions around this circle can be determined as a function of the cosine and sine of angle “q”.

Figure 1

Thus, for all the positions along a single circle of longitude, q is constant while p varies, whereas for all positions along a single circle of latitude, p is constant while q varies. However, there is a significant difference between the two types of circles. The center of all the circles of longitude are the same as the center of the sphere, consequently, all circles of longitude are great circles. But, the circles of latitude all become smaller as they get farther from the horizon and closer to the poles. How much smaller they get, is function of angle p, that is, how close they are to the poles. From the geometry of the sphere, the radii of the circles of latitude are proportional to the cosine of angle p. (See figure 2.)

Figure 2

From this, all positions on a sphere can be reduced to determinations of the two parameters p and q, which reflect the physical curvature of the sphere. (footenote 1.) The empty box, which Kant insisted on, just disappeared, perhaps into the empty corners of his brain from whence it came.

Gauss now investigated the relationship between the length of an arbitrary geodesic of the sphere and the angles p and q, in order to determine a way of measuring the general principles of curvature of the surface from small measured changes in the geodesic.

To do this, Gauss established a more general form of the Pythagorean relationship. In the case of the sphere, a geodesic line can be thought of as a diagonal of a spherical rectangle, whose sides are circles of longitude and latitude. But, unlike the Pythagorean in the Euclidean flat plane, the relationship between the sides of the spherical rectangles are changing, depending on their position with respect to the horizon and the poles. Specifically, the latitudinal sides will get shorter, by a factor proportional to the cosine of angle p, as the longitudinal sides increase towards the poles. (See Figure 3.) Thus, the “spherical Pythagorean” must measure not only the relationship of the diagonal to the sides, but the also the change in this relationship as the relationship between the sides themselves changes. From this, Gauss showed, that the form of the “spherical Pythagorean”, is that the length of the geodesic “ds” =?((Cos[p])²dp² + dq²) where ds is the change in the length of the diagonal and dp and dq are the changes in longitude and latitude. The coefficient Cos[p]² expresses the shortening of the latitude lines as they get closer to the poles.

Figure 3

Consequently, if one is making physical measurements along what appears to be a “straight-line”, and the relationship measured corresponds to this “spherical Pythagorean”, then that “straight-line” is a geodesic on a sphere.

To further develop this idea, continue in this same vein to the two other examples used in Riemann for Anti-Dummies Part 46, the spheroid and the ellipsoid.

For the spheroid, a parametric formula can be constructed that expresses the geometrical relationship that the lines of latitude get shorter as they approach the poles, while the lines of longitude get longer. (See Figure 4.) In this case, the “spheroidal Pythagorean” must express the relationship between the length of the geodesic, “ds”, and the changing relationship between the lengths of the circles of latitude and the ellipses of longitude. This produces a somewhat more complicated formula for the “spheroidal Pythagorean”, but its geometrical representation can be gleaned from the accompanying figure.

Figure 4

By inversion, if physical measurements along a “straight- line” reflect the relationship expressed by the “spheroidal Pythagorean”, that “straight-line” must be a geodesic on a spheroid.

On the ellipsoid, as illustrated in the last installment, the relationship between the sides of the “ellipsoidal rectangles” changes as they move towards the poles and as they move around in the latitudinal direction as well. (See Figure 5.) Using Gauss’ method, it is possible to calculate a formula for this “ellipsoidal Pythagorean”, which enables us to make the same type of determinations about the general principles of curvature from small changes in the measurement of the “ellipsoidal geodesic”.

Figure 5

In his treatise on curved surfaces, cited by Riemann in the habilitation paper, Gauss developed a general form of the Pythagorean that expressed the relationship between the geodesic line and the two determinations of magnitude of that surface. For purely illustrative purposes, the form of Gauss’ generalized Pythagorean is, ds=?E²dp²+2Fdpdq+G²dq², where, E, F, and G are themselves functions which express the changing relationship of the two parameters of the surface. E expresses the function by which p changes with respect to q; G expresses the function by which q changes with respect to p; and F expresses the function by which the area of the rectangle changes relative to changes in p and q. For the Euclidean flat plane, Gauss’ Pythagorean reduces to the familiar form.

As Riemann indicated, this relationship in its most general form is expressed by the above formula, but, as in the above examples, its results can always be represented geometrically.

With this foundation laid by Gauss, Riemann went still further, to determine an even more general idea of a Pythagorean, not only for surfaces, but for manifolds of “n” determinations of magnitude. This leads into even more interesting areas, such as the one confronted in investigating physical processes in the very small and very large, or, in the biotic domain, where the characteristic of the geodesic is changing non-uniformly. To reach that point we must, however, first master the foundations laid by Gauss and Riemann, with some assistance of the insights gained from classical art, such as Beethoven’s late string quartets.

FOOTNOTE

1. For those who wish to know, a parametric formula for the sphere is Cos[q] Cos[p], Sin[q]Cos[p], Sin[p]. For the spheroid and ellipsoid, each direction is multiplied by a simple factor.

Riemann for Anti-Dummies Part 46

SOMETHING IS ROTTEN IN THE STATE OF GEOMETRY

When Gauss issued his 1799 doctoral dissertation on the fundamental theorem of algebra, he had much more in mind than just proving that particular theorem. He was creating the foundation for a mathematics that rested only on physical principles. He chose the domain of algebra, because that’s where his enemy was weakest. The algebraists, Euler, Lagrange, and D’Alembert, had boasted that they had “freed” mathematics from the paradoxes of geometry and, like their modern counterparts, the information theorists, had reduced even the most complicated problems down to a finite set of rules, definitions and procedures. This was the oligarch’s dream. The feudalist elite was constantly confronted with the dilemma that to stay in power they must rule over stupid people. But, the principles of physical economy would always intervene to bring about the destruction of any society that failed to value human creativity. The method of Newton, Euler, D’Alembert and Lagrange, a variation of an old Babylonian trick, offered the prospect of maintaining the world in a state of perpetual stupidity, while a small group of magicians (algebraists) kept things orderly.

Gauss showed that what Euler, Lagrange and D’Alembert considered their strength–the ability to plug holes in their system with a new definition based solely on their authority–was its weakness. Just as his teacher Kaestner had done with respect to the parallel postulate of Euclid, Gauss demonstrated that the square root of minus 1, was not the imaginary fiction Euler had defined it to be, but was the spirit of the physical universe come to haunt the algebraic system. Algebra, like the oligarchical system, could not solve its fundamental problem. It must yield to physical geometry, and the algebraists, to the creative scientist.

By the time Gauss was writing his 1799 dissertation, he had already begun the process of constructing a physical geometry based on measurement. As soon as he arrived at Goettingen in 1795, he borrowed one of the university’s theodolites and spent many hours measuring out triangles on the Earth. Some of his early notebook entries show him constructing a physical geometry from these measurements without resort to the axioms, definitions and postulates of Euclidean geometry. A quarter century later, when Gauss undertook to survey the entire Kingdom of Hannover, many of his colleagues were shocked that someone of his stature would undertake a project they considered pedestrian. Yet, Gauss saw in this undertaking, the chance to further his youthful efforts to construct a physical geometry based on measurement. This work was brought to fruition in his published papers on curvature, mapping and geodesy, the which provided the foundation for Riemann’s development of complex functions.

The past two segments of this series have dealt with some of the ideas developed by Gauss in his “General Investigations of Curved Surfaces” ,which established the principles of what, today, is called, “differential geometry”. Before turning to the more universal applications of Gauss’ geodetic investigations, it is advisable to review a principle of Leibniz’ calculus through the example of the catenary.

The determination of the catenary, as the shape formed by a hanging chain, required the discovery of a physical principle that, like the square root of minus 1, was outside the rules, definitions and procedures of Euclidean geometry, and outside the domain of sense-perception. Leibniz and Bernoulli determined that, while that physical principle could not be known by some a priori rule, it could be discovered from the way it expressed itself in the smallest parts of the chain.

To illustrate this, the reader should perform the pedagogical experiment described by Bernoulli in his text on the integral calculus. Take a string and tie a light weight to the middle of it. Take one end of the string in each hand and let the weight hang freely between them. As you pull your hands apart, you will feel an increase of force exerted on your hands by the weight. If you lift one hand higher than the other, the force on the raised hand increases, while the force on the lower hand decreases.

To simulate the action that produces the catenary, now hold the string in each hand very close to the weight. Move one hand so that it pulls the weight away from the other, while allowing the string to slide through both hands. As you do this, keep the segment of the string that connects the weight with the stationary hand horizontal. (This simulates the lowest point on the catenary.) To do this, you will have to constantly raise the moving hand. The farther your hands are separated, the faster the moving hand must be raised in order for the opposite segment to remain horizontal. The curve traced out by the moving hand will be the catenary.

The reason for performing the above described experiment, is to realize that the catenary curve is physically determined. In order to keep one of the string segments horizontal, the moving hand is compelled, by a physical principle, to follow the catenary curve. The curve is not seen, but its effects are “felt” by the moving hand, at every small interval of action. That infinitesimal expression of the catenary principle, Leibniz called the “differential”.

These effects can be measured by the increasing length of the curve for equal amounts of horizontal motion. In other words, when the hands first start to move apart, the moving hand only has to be raised a little to counteract the force of the weight. But as the moving hand moves further out, the amount of vertical “lift” for small horizontal increments, increases, which in turn, increases the length of the curve for corresponding horizontal motion. (See Figure 1.)

Figure 1

(Warning to those indoctrinated in Cartesian geometry: The horizontal and vertical here are not Cartesian axes, but directions of motion, physically determined with respect to the direction of the pull of gravity.)

In the hanging chain this action happens all at once. The horizontal and vertical are not simply directions of motion, they are the physically determined boundaries at which the catenary ceases to exist. The curve of the catenary unfolds as the path of least action between these two extremes. Its length per unit of action increases as it nears the vertical extreme, and decreases towards the horizontal. Since the length of the curve is a direct function of the physical principle governing the chain’s action, it is an appropriate measure of that principle.

Gauss’ geodetical investigations led him to extend Leibniz’ calculus into a higher domain.

All measurements of the Heavens and the Earth are made with respect to a physically determined direction, as indicated by the direction of a free hanging weight on a string, called a plumb bob. The surface of the Earth is that surface that is everywhere perpendicular to the direction of the plumb bob. The question Gauss investigated in his geodesy, is, ” What is the nature of this surface?” Since it were impossible to know the answer from sense perception, Gauss determined the overall nature of the surface of the Earth from small (differential) changes in action measured on it.

To begin to grasp Gauss’ idea, begin with the simpler case of the celestial sphere. This sphere can be entirely determined by two angles, one around the circle of the horizon, and one from the horizon to the pole. These two angles define an orthogonal network of circles, which, for pedagogical purposes, we will call latitude and longitude. The longitudinal circles are great circles, which are what Gauss called “geodesics” or shortest lines, while the latitudinal circles are not. (See Figure 2.)

Figure 2

While these circles are always orthogonal to each other, nevertheless, their relationships change, depending on where they are on the sphere. As the longitudinal lines get closer to the poles, they get closer together. Thus, the length of an arc of a circle of latitude between two determined arcs of longitude changes, depending on its position with respect to the poles. For example, the distance along the arcs of latitude between two arcs of longitude separated by 10 degrees, will decrease, as the latitudes get closer to the poles.

Now, compare that with a spheroid. Here, the lines of longitude are elliptical, and the lines of latitude are circular. In this case, the length of the arcs of latitude between two determined arcs of longitude still get smaller as they approach the poles, but also, the length of the arcs of longitude between any two determined arcs of latitude get {longer} as they approach the poles. (See Figure 3.)

Figure 3

This characteristic can be measured in a geodetic survey, by measuring the lines of latitude as angular changes in the inclination of the north star. If the Earth were a sphere, equal angular changes will correspond to equal changes in length of the geodesic longitudinal lines. If these geodesic lines get longer, with equal angular inclinations of the north star, then the Earth is spheroidal. What the specific measurements of that spheroid are, cannot be known by a priori mathematical methods, but require more refined physical measurements, as developed by Gauss.

Now, look at an ellipsoid. Here the lengths of the lines of latitude between any two determined lines of longitude, change, both as they approach the poles, and as they move around in the “equatorial” direction as well. Additionally, the lengths of the lines of longitude between any determined lines of latitude, increase as they approach the poles. (See Figure 4.)

Figure 4

(Note: The accompanying computer generated graphics are supplied merely to illustrate the text. The reader is strongly encouraged to make physical demonstrations on real surfaces approximating these shapes. The reader is also encouraged to experiment with wildly irregular shapes as well.)

Gauss recognized this characteristic of change as a new type of “differential”, which, for pedagogical purposes, I will call “surface differentials”. Like Leibniz’ differentials, these surface differentials express how the overall principle of action of the surface is manifest in every small part. However, instead of directly characterizing the least-action pathways, i.e. geodesics, these “surface differentials” characterize the changing nature of the principles through which the least- action pathways unfold. In other words, the surface differential expresses the characteristic of change of the principles that determine the characteristics of change of all possible least-action pathways, i.e. geodesics, on that surface.

To get an intuitive sense of this idea, think of these surface differentials being approximated by small “rectangular” patches of the surface. (See Figure 5, Figure 6, and Figure 7). Notice how the shape of these patches changes, as their positions change on the different surfaces.

Figure 5

Figure 6

Figure 7

Gauss determined that the relationship between these surface differentials and the characteristic geodesic lines of the surface could be measured, because even though these geodesic lines are always the shortest distance between two points on the surface, the length of the geodesic with respect to the surface differential changes, according to the overall curvature of the surface.

Gauss established the relationship between the surface differential and the characteristic curvature of the changing geodesic line, as a generalization of the Pythagorean relationship between the diagonal and the side of the square or rectangle. In the case of a flat plane, (or surface of zero-curvature, as Gauss would see it), the relationship of the diagonal to the side of a square (or rectangle), expresses the power that generates areas, as distinct from the power that generates lines. Thus, the line that forms the diagonal of the square is a different type of line than that line which forms the side of the square, because it is generated by a higher power. This relationship can be measured by the relationship of the length of the diagonal, to the lengths of the sides of the square or rectangle. The common expression for this “Pythagorean” relationship is that the length of the diagonal is equal to the square root of the sum of the squares of the sides of the square or rectangle.

This “Pythagorean” relationship, Gauss showed, was just a special case of a more general principle. On a curved surface, the sides of the square are the constantly changing “sides” of the surface differential, and the diagonal is the geodesic. The principle that governs the constantly changing lengths of the “sides” of the surface differential is a function of the curvature of the surface, which, in turn, is reflected in the changing length of the geodesic diagonal. Consequently, the overall curvature of the surface is reflected in the smallest parts of the geodesic. From this Gauss devised a more general idea of the “Pythagorean”, in which the lengths of the “sides” of the surface differential are multiplied by a function that characterizes the physical curvature of the surface. As the surface differential changes according to the curvature of the surface, the length of the geodesic diagonal changes accordingly. (A future pedagogical will illustrate, geometrically, Gauss’ idea.)

To get an intuitive sense of this concept, look again at figures 5, 6, and 7. Imagine the diagonals of each surface differential. Imagine how the lengths of these diagonals change with the position of the surface differential. Now conduct a similar investigation on the physical surfaces you experimented with earlier. Draw on these surfaces geodesic diagonals to the orthogonal curves you previously drew. This can be done by holding a string taught between opposite corners of each “rectangular” patch, and tracing the string path with a marker. Notice the changes in length and direction of these diagonal’s geodesic as the curvature of the surface changes.

Now, look at a concrete example with respect to the physics of navigation. On a flat surface draw a grid of orthogonal lines. Draw a diagonal line that cuts all the vertical lines at the same angle. This will produce a straight line. (See Figure 8.) Now try the same thing on a sphere. That is draw a series of geodesic lines that cut the lines of longitude at a constant angle. The result is not a straight line, but a spiral like curve called a “loxodrome” (See Figure 9.) A navigator who has not mastered the principles of curvature, will find himself getting farther and farther from his destination, and closer and closer to the north pole!

Figure 8

Figure 9

(Figure 10 illustrates a similar process for the spheroid and ellipsoid. Notice the difference between them and the sphere.)

Figure 10

From this relationship, Gauss showed that it were possible to discover the surface differential, and thus the characteristic curvature of the surface, by measuring small variations in the length of geodesic lines. For example, in determining the length of the geodesic line that connected his observatory in Goettingen with Schumacher’s in Altona, Gauss measured a 16 seconds of an arc deviation from what that length should be if the Earth were a spheroid. That led Gauss to prove that the shape of the Earth could not conform to any a priori geometric shape, but was being determined by the physical characteristics of the Earth’s matter and its motion.

Riemann for Anti-Dummies Part 45

THE MAKING OF A STRAIGHT LINE

Straight lines are not defined, they are made.

The above statement might seem jarring to one fed a steady diet of neo-Aristotelean dogma from their primary, secondary and university teachers, but it is the standpoint adopted by C.F. Gauss by the time he was 15 years old. In July 1797, at the age of 20, Gauss wrote in his notebook, “Plani possibiliatem demonstravi,” (The Possibility of the Plane Proven). He later elaborated on this idea in a January 1829 letter to Bessel, where he spoke of his conviction, “for nearly 40 years,” that “it were impossible to establish the foundations of geometry a priori.” Gauss gives as an example, the Euclidean definition of a plane, as a “surface that lies evenly with the straight lines on itself” (Euc. I, def.7.) “This definition,” Gauss wrote, “contains more than is necessary to determine the surface, and involves tacitly a theorem which first must be proven.”

An individual subjected to the aversive conditioning of today’s information society education, might think Gauss was making some esoteric quibble, of interest only to the arcane curiosity of certain specialists. In fact, the epistemological issue Gauss is addressing, is exactly the one that is the cause of much of today’s mass psychosis, upon whose successful treatment the future of civilization depends.

To grasp the point, consider the following illustration, along with Euclid’s concomitant definition of a line as “breadthless length” (Euc. I,def. 2.) and a “straight line” as, “a line that lies evenly with the points on itself.” (Euc I, def. 4.) Imagine an octahedron, or some other solid, and think of the line connecting two of its vertices. Under Euclid’s definition, this line would be straight, as it lies evenly with the vertices, and the face of the solid would be a plane because it lies evenly with the lines that form its edges. But, from the standpoint of construction, the solid is generated from a sphere. Those same vertices also lie evenly with great circle arcs along the surface of the sphere, which themselves lie evenly with the spherical surface. (See Figure of inscribed octahedron.) How can one distinguish which of the two surfaces, spherical or planar, and which set of lines, circular or linear, are the “straight” ones, by Euclid’s definition?

Inscribed Octahedron

Euclid’s definition applies equally to both types of lines and surfaces, as well as to an infinite number of other possible surfaces and lines that could conceivably lie even with the vertices of a solid. A definition alone is insufficient to distinguish one from the other, because, as Gauss says, the definition assumes a theorem concerning the physical characteristics of the surface and the lines contained in it. Such characteristics must be proven, or, in the domain of physical action, measured, by what Riemann called a “unique experiment.” In the tradition of Leibniz, Gauss called these characteristics, “curvature.” The difficulty today’s victims of information education have in grasping Gauss’ point, is that they’ve become accustomed to thinking of “straight” and “flat” in a certain pre-defined way, which, in the minds of the victim, carry the authority of “Roman law.” As long as this “rule of law” holds its sway over the victim’s mind, the afflicted person will cringe at the thought that the physical universe might disobey this defined law of straightness. But, whatever the authority with which this rule of law is pronounced, the universe decides what is straight as a matter of principle. This produces a psychological crisis that intensifies as long as the victim cowers under the arbitrary dictate of definitional straightness.

A baby and a drunk both walk a crooked path. The baby because it’s trying to discover the multiply-connected principles of physics, biology and cognition that determine its intended path. The drunk because, his damaged state prevents him from recognizing the principles he once knew, and he responds to whatever definition of straightness his inebriated impulses conjure up. The baby’s frustration, when it falls upon reaching the limit of each temporary hypothesis, is transformed into joy, when the discovery of the missing principle increases its power to proceed on its way. The stumbling frustration of the drunk, oligarch, lackey, or victim of Straussian brainwashing, stews into bi-polar rage at his loss of control, screaming, like Shelley’s Ozymandias, “Look upon my works ye mighty and despair….”

As long as one’s mental powers are impaired by arbitrary definitions whose only force is the arbitrary authority with which they’re uttered, one remains in a stupor, either intoxicated with the power to wield such authority, or, the depression brought on by submitting to it. To free the victim and restore those inherently human powers he or she once experienced, the sobering balm of classical art and science must be applied. Hence, the importance of pedagogical exercises.

The Determination of Curvature

While Gauss’ youthful insight was at odds with such contemporary authorities as Leonard Euler and his protege, I. Kant, who insisted that the principles of geometry could only be given by a priori definitions, its roots were quite ancient. Euclid’s “axiomization” of geometry was itself at odds with the very process by which the geometrical principles contained in it were discovered. As the solutions of Archytas and Meneachmus for the problem of doubling of the cube, and more generally, the construction and uniqueness of the five Platonic solids from spherical action indicate, the relationships of rectilinear geometry are derived from non- rectilinear physical action. The investigations into this type of physical action are further exemplified by the works of Apollonius, Archimedes and Eratosthenes, as well as the Pythagoreans’ demonstration that the relationships among musical tones are generated by a higher principle than the linear divisions of a string.

It was this “anti-Euclidean” approach that was adopted by Cusa, Kepler, Fermat, and Leibniz, who replaced the sophistry of an arbitrary definition of a straight line, with the idea of least-action pathway, or what later became known as a “geodesic.” The geodesic is the straightest and shortest line, whose nature is determined by the physical properties of the surface, or, from Riemann’s more general standpoint, the “n-dimensional manifold” or phase-space, in which it occurs.

For example, Kepler’s planetary orbit is the least-action pathway created by the harmonic principles of the solar system, the which “define” the elliptical orbit as its “straight line.” Similarly the principles of reflection of light “define” the shortest distance as its “straight line” while the principles of refraction “define” shortest time as straightness. The introduction of the principle of the changing velocity of light under refraction, “re-defines” the straight line, from the path of shortest-distance to the path of least-time. Or, conversely, the change in what is straight, indicates the presence of a new principle.

In each case, the definition of a straight-line is not given by some arbitrary authority, but by a set of measurable physical principles. The question for science, politics, economics and history, is how to determine those governing physical principles, from what amount to small pieces of the “straight” lines determined by them. This entails being able to discover the principles from the “curvature” of the line, and to discover new principles by measuring changes in that characteristic curvature.

Exemplary is Leibniz’ and Bernoulli’s determination of the curvature of the hanging chain. Unlike Galileo, both Leibniz and Bernoulli recognized that the chain’s curvature was determined by a physical principle. This principle does not exist in an empty, infinitely-extended Euclidean-type plane, but is produced in a physical manifold with a characteristic curvature. This manifold is bounded by physically-determined extremes, expressed by the relationship of the lowest point of the chain to the hanging points. If the hanging points coincide, there is no tension, and the chain has no curvature. If the hanging points are pulled apart, at some point the chain will break. At these extreme conditions of maximum and minimum tension, there is no curvature, and no stable “orbit” for the chain. The common-sense notion of straight, that is not-curved, only exists outside the physical manifold in which the chain hangs. In that manifold the catenary curve is the only possible “straight” line. Thus, “straightness” for the chain, is a curve–a curve determined by a measurable physical characteristics. At every small interval along the chain, the links steer a course which is constantly changing, but changing according to a measurable principle. As developed in previous installments, Leibniz and Bernoullli demonstrated that this characteristic changed according to a principle that Leibniz called, “logarithmic.”

A similar relevant case is Gauss’ earth-shaking determination of the orbit of the asteroid Ceres, from that infinitesimal portion of its orbit represented by Piazzi’s observations. All the established scientific authorities were stymied in their efforts to find Ceres’ orbit, hampered by their insistence that Ceres’ orbit was moving in an empty box and its orbit was a deviation from the definition of straight-line action that Galileo and Newton had re-imposed on physics, after Kepler’s liberation of science from such Aristoteleanism. Gauss, as a student of Kaestner, was guided by the knowledge that Ceres was following a least-action pathway in a solar system governed by the harmonic physical principles that Kepler described, and that those harmonic principles indicated a discontinuity in the region between Mars and Jupiter. Unlike his competitors, Gauss knew that the Galilean/Newtonian straight line did not exist in the physical manifold in which Ceres and the Earth were moving. So, while everyone else was looking for a path among an infinite number of possible pathways, in a manifold that did not exist, Gauss was looking for the unique least-action pathway that Kepler’s solar system would produce. His successful approach was focused on determining how those principles would be expressed in the small portion of the orbit that Piazzi had observed. (See “How Gauss Determined the Orbit of Ceres,” Summer 1998 Fidelio.)

A crucial distinction occurs when one compares the case of the catenary with the case of the Ceres orbit. The discovery of the catenary principle required the determination of a single pathway. The discovery of Ceres’ orbit involved the relationship between two different pathways, Ceres’ and Earth’s as these pathways were viewed as projections on the inside of the celestial sphere. These two pathways, though different, are both least-action, i.e. “straight,” paths within Kepler’s solar system. Thus, the solar system produces “straight-lines” of different curvatures in different parts.

Gauss found a similar situation in his geodetic measurements, where he measured a variation in the direction of the pull of gravity from place to place on the Earth. As Gauss moved north, the angle of inclination of the north star increased, but non-uniformly with the distance traveled along the Earth’s surface. But, Gauss also determined that the direction of the pull of gravity varied as he moved east to eest or some other intermediate direction. The question for Gauss was how to determine the shape of the Earth, from these variations along small parts of its surface? Or, in other words, how is the overall curvature of the Earth, and its local variations, reflected in every small part of its surface, in the same way that the physical principle of the catenary is reflected in every small part of the chain?

The Making of Curvature

These types of considerations gave raise to Gauss’ theory of curved surfaces. As illustrated in the previous installment, Gauss measured the “total” or, “integral” curvature of a surface by mapping the changes in direction of the normals onto an auxiliary sphere. Following the direction of Leibniz’ infinitesimal calculus, Gauss showed how this overall curvature was related to the curvature at every infinitesimal surface element:

“The comparison of the areas of two corresponding parts of the curved surface and of the sphere leads now (in the same manner as e.g. from the comparison of volume and mass springs the idea of density) to a new idea. The author designates as “measure of curvature” at a point of the curved surface the value of the fraction whose denominator is the area of the infinitely small part of the curved surface at this point and whose numerator is the area of the corresponding part of the surface of the auxiliary sphere, or, the integral curvature of that element. It is clear that, according to the idea of the author, integral curvature and measure of curvature in the case of curved surfaces are analogous to what, in the case of curved lines, are called respectively amplitude and curvature simply. He hestates to apply to curved surfaces the latter expressions, which have been accepted more from custom than on account of fitness. Moreover, less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.”

Gauss goes on to develop the methods by which to measure what has now become known as “Gaussian curvature.” If, following the tradition of Euler, the surface is considered as the boundary of a three-dimensional solid object, then this curvature could be measured by cutting the surface at the point by two planes, normal to the surface and perpendicular to each other. The curves formed by the intersection of these planes with the object express the curves of minimum and maximum curvature at that point.

To illustrate this, cut an egg, apple, or some other curved solid in half. Then cut a similar shaped object in half at a 90 degree angle to the first cut. Compare the curves formed by these cuts. Cut another similarly shaped object at another angle. Compare the curvature of the three types of curves.

This method is totally useless for a real physical problem such as measuring the surface of the Earth, for it is obviously impossible to make orthogonal cuts in the Earth at every point and measure the curvature of the resulting curves.

To solve this problem, Gauss conceived of a curved surface as a two-dimensional object. Thought of in this way, the curvature could be determined by measuring the behavior of the “shortest” lines, i.e. geodesics, emanating from that point.

For example, the surface of a sphere can be entirely determined by a system of two sets of orthogonal circles, akin to “lines” of longitude and latitude, the former being “geodesics” and the latter not. In a sphereoid, the lines of latitude remain circular, while the longitudinal ones become elliptical. In an ellipsoid, both sets of curves are elliptical. Other examples are a psuedosphere, where the one set of curves are circles and the other tractrices, or, the catneoid, where the curves are circles and catnaries. For more irregular surfaces, the curves are irregular, but such an orthogonal system can always be developed.

From this standpoint, the common-sense notion of a “flat” Euclidean plane is just a special type of surface, with no particular, a priori, “legal” authority. The common sense notion of “straight” line becomes simply the “geodesic,” characteristic of this type of surface.

Gauss showed that the behavior of the shortest lines emanating from any point on a surface could be measured with respect to these systems of orthogonal curves by extending the method of Leibniz’ infinitesimal calculus. And, more importantly for physical science, the nature of these orthogonal curves, and consequently the curvature of the surface, could become known by the measured changes in these geodesic lines.

In the next installment we will delve into Gauss’ method more directly. For now, we supply an intuitive introduction through the accompanying animations. Here you can compare the behavior of geodesic lines emanating from a point on different surfaces, namely: a sphere, spheroid, ellipsoid, monkeysaddle, and torus.

Sphere

Sphereoid

Ellipsoid

Monkeysaddle

Torus

Notice the relationship between the behavior of these lines and the system of orthogonal curves drawn on the surface. Notice how the shape of these geodesics depends only on the nature of the surface and their direction. Also, compare the behavior of these geodesics with the integral curvature of these surfaces illustrated in the Gauss mappings that accompanied Riemann for Anti-Dummies Part 44.

The first step is to develop the power to measure the physical characteristics of the surface from the “straight” lines that surface produces. But humans possess a greater power–to discover and apply new principles, thus changing the curvature, and making “straight” lines.

Riemann for Anti-Dummies Part 44

PRINCIPLES AND POWERS

Rembrandt van Rijn’s masterpiece, ?Aristotle Contemplating a Bust of Homer?, conveys a principle that leads directly into the deeper implications of Gauss’ and Riemann’s complex domain. In the painting, the eyes of both figures are fixed directly before them, yet, Aristotle’s gaze is insufficient to guide him. To find his way, he reaches forward to touch the likeness of the poet, who, though blind in life, leads the blocked philosopher in a direction he would otherwise be incapable of finding.

Like the navigators of ancient maritime civilizations, Rembrandt’s Homer knows that straight-ahead is not necessarily where your eyes point. When following a course across some wide expanse, these discoverers would mark their passage by noting the motions of celestial bodies, the which were charted as changes of positions on the inside of the sphere whose center was the eyes of the observer. When the observer’s position changed, so did the positions of everything on the sphere, but the manifold of vision remained a sphere and the eyes of the observer remained at its center. A stationary observer would note certain changes in the positions of celestial bodies over the course of a night and from night to night. An observer moving on the Earth noted these changes, plus the changes in these changes resulting from his own motion. These changes, and changes of changes, formed a map in the mind of the explorer — not a static map, but a map of the principles that caused the map to change. It is the map of principles on which all explorers, from those days to this, place their trust.

While a map, such as one of positions of celestial bodies on the inside of a sphere, can be represented directly to our senses, a map of principles can only be represented by the methods exemplified by Rembrandt’s painting. Principles do not appear as objects in the picture, but as ironies that evoke the formation of their corresponding ideas in the imagination of the viewer. The scientist in pursuit of unknown principles, must master the art of recognizing the ironies that appear, not only from known principles, but those still to be discovered; the latter emerging as paradoxes. In the case of physical principles investigated by mathematical images, these paradoxes present themselves as anomalies, as for example, the emergence of “-1, within the domain of algebraic equations. The poetic scientist takes the existence of such anomalies as evidence of a principle yet to be discovered, and re-thinks how his map must change to include this new principle. C.F. Gauss measured this type of transformation as a change in curvature. This work was extended by B. Riemann through his theory of complex functions, most notably in his major works on the hypergeometric and abelian functions.

What has failed Rembrandt’s Aristotle, is not his eyes, but his map. A map which has changed by a principle, that on principle, he insists doesn’t exist and cannot be known if it did. Disoriented he’s left to grope in the only direction he knows straight ahead. Fortunately for him, straight ahead stands the lifeless image of Homer, possessed with the power to light his way.

Curvature and Power

This method of discovery is already evident in the work of Archytas, who taught that the physics of the universe could be discovered through investigations of the paradoxes arising in arithmetic, geometry, spheric (astronomy) and music. His collaborator, Plato, prescribed mastery of these four branches of one science, as essential for the development of political leadership.

The solution Archytas provided for doubling the cube exemplifies the principle. As it was developed more fully in previous pedagogical discussions (see Riemann for Anti-Dummies Part 42) the problems of doubling the line, square and cube, presents us with the existence of magnitudes of successively higher powers, each of which is associated with a distinct principle. The Pythagoreans called the power that doubles the line, arithmetic, and the power that doubles the square geometric, which they associated with musical intervals as well as mathematical ones. In their most general form, the arithmetic is associated with a division of a line, while the geometric is associated with a division of a circle. (See figure 1 and figure 2). From Gauss’ standpoint, the change in power from the arithmetic to geometric is associated with a change in curvature from rectilinear to circular.

Figure 1

Figure 2

As Hippocrates of Cios indicated, to double the cube requires placing two geometric means between two extremes. At first blush, this can be accomplished within the domain of circular action by connecting two circles to each other. (See figure 3.) Thus, while the difference between the arithmetic and geometric presents clearly a change in curvature, the power associated with generating two geometric means, in first approximation, seems to require only another circle, hence, no change in curvature.

Figure 3

Yet, when the specific physical problem of doubling the cube is posed, that is, to find two geometric means between two determined extremes, (in this case 1 and 2) the existence of the higher power emerges into the map, as a new type of curvature. (See animation 1.) As can be seen in the diagram, to find two geometric means between 1 and 2, we must find the place along the circumference of the circle for point P so that the line OB is one-half of OA. This will occur somewhere along the pathway traveled by B as P moves around the circle from O to A. But, as the dotted line which traces that path indicates, this curve is not a circular arc, and is, in fact, non-uniform with respect to the circle. Thus, the existence of the yet to be discovered principle, emerges through the presence of an anomalous change in curvature in our map.

Animation 1

This anomaly takes on an entirely different characteristic in Archytas’ construction using the torus, cylinder and cone. (See figure 4 andanimation 2.) When the torus and cylinder are generated by rotating one circle orthogonally around another, the motion of point P is now simultaneously on two different curves: the circle and the curve formed by the intersection of the torus and the cylinder. An observer facing the rotating circle, and who was rotating at exactly the same speed as the circle, would only see point P move around the circumference of the circle and would adequately conclude that one geometric mean between two extremes is a function of circular action alone. But, as indicated above, the emergence of the non-circular curvature of the path of point B, would indicate to such an observer, the existence of a new principle causing the motion of P around the circle. Archytas’ construction takes that new principle into account, by determining the motion of P around the circle as a function of P’s motion along the curve formed by the intersection of the torus and cylinder. In other words, the circular rotation of P is only a shadow of a higher form of curvature. That latter curve expresses both the power to produce one geometric mean between two extremes, and also, when combined with a cone, to produce two. (See figure 4.)

Figure 4

Animation 2

Two other examples, presented summarily, will help illustrate the point. Kepler, like all astronomers before and since, observed the motions of the planets as circular arcs on the inside of a sphere. His discovery of the elliptical nature of these orbits occurred, not by suddenly seeing an ellipse, but by his recognition that the 8′ of an arc deviation between the circular image of the planet’s orbit on the celestial sphere, and the circular image of the Earth’s motion (as reflected in the motion of the fixed stars on that same celestial sphere), was evidence of a new principle of planetary motion. The new principle manifested itself as a change in curvature within his map of principles. He measured that change in curvature by measuring equal areas instead of equal arcs, and measuring eccentricities by the proportions that correspond to musical harmonics.

Similarly, Leibniz and Bernoulli determined that the catenary was not the parabola that Gallileo wrongly considered it to be, by showing that the slight deviation of the curvature of the physical hanging chain, from the curvature of the parabola, was evidence that the chain was being governed by a different principle than the one Gallileo assumed. Gallileo demanded, as if in a bi-polar rage, that the chain conform to a parabolic shape, because he was obsessed with his mathematical formula that the velocity of a falling body varies according to the square root of distance fallen. Leibniz and Bernoulli, demonstrated that the chain was, in truth, obeying a higher principle, the non-algebraic, transcendental principle associated with Leibniz’ discovery of natural logarithms. A principle that the enraged Gallileo was incapable of conceiving.

Gaussian Curvature

To proceed further it is important to distinguish between commonplace sense-certainty notions of curvature and the rigorous understanding of that idea associated with Gauss. The commonplace notion, associated with the doctrines of Gallileo, Newton, Euler, et al. is that curvature is a deviation from the straight. But, from the standpoint of the planet, for example, ?straight?, is a unique elliptical path; or, from the standpoint of a link in a chain, ?straight? is the catenary curve. It is only a self-deluded fool who thinks that ?straight? can be determined by some arbitrary, abstract dictate. Rather, ?straight? is a function of the set of principles that are determining the action. The addition of a new principle will change the direction of ?straight?. That change in principle is measured as a change in curvature.

This is the basis from which Gauss developed his, ?General Investigations of Curved Surfaces?. He considered a curved surface to be a set of invariable principles that determined the nature of action on that surface. As long as that set of principles was not changed, the nature of the action did not change, even if the surface was bent or stretched. The nature of the action could only be changed, by a change in the set of principles that defined the surface. Gauss measured such a change in principle as a change in curvature, which in turn, determined what is ?straight? with respect to that set of principles.

Furthermore, Gauss showed, as Leibniz did for curves, that this set of invariable principles were expressed in the smallest elements of the surface. Consequently, from the smallest pieces of ?straight? curves (geodesics) on the surface, and their directions, the curvature of the surface could be determined. (This aspect of Gauss’ work will be developed in a future pedagogical.)

The method Gauss developed to measure curvature had its roots in Kepler’s method for measuring the elliptical nature of a planetary orbit, the which was generalized by Leibniz into his development of the calculus. Confronting the difficulty of measuring the planet’s non-uniform elliptical motion directly, Kepler mapped the constantly changing speed and direction of the planet onto a circular path, and measured the planet’s action by the relationships among the three anomalies (eccentric, mean and true) that appeared in the circular map. (See Summer 1998 Fidelio pp. 29-34).

To measure the curvature of a surface, Gauss extended Kepler’s method from the mapping of a curve onto a circle, to the mapping of a surface onto a sphere, a method he likened to the ancient use of the celestial sphere in astronomy. In that case, the motion of celestial bodies are mapped by the changing directions of lines from the observer to the body’s image on the inside of the celestial sphere. Since whatever principle is governing the body’s motion, is governing the changes in direction of those lines, measuring the map of those changes in direction is an indirect measurement of the governing principle.

Gauss recognized that the invariable principles governing a surface could be expressed by the changing direction of the lines perpendicular to the surface at every point, called ?normals?. While at any point of a surface there are an infinite number of tangents (See Figure 4) there is a unique tangent plane for each point, which in turn defines a unique normal that is perpendicular direction to this tangent plane. The direction of the normal is a function of the curvature of the surface. (This is a principle of physical geometry, as exemplified by the determination of the physical horizon as that direction that is perpendicular to the pull of gravity.)

The sphere has the unique characteristic that all its normals are also radial lines. Using this property, every normal to a surface will correspond to a radial line of a sphere that is pointing in the same direction. As a normal moves around on a surface, its direction changes. If a radial line of the sphere is made to change its direction in the same way as the normal to the surface, it will trace out a curve on the surface of the sphere, that will reflect the principle governing the changes in direction of the normal on the surface.

This is illustrated by example in animation 3 and animation 4. In these examples, the part of the ellipsoid marked out by the yellow curve, is mapped onto a sphere. As the red stick moves around the ellipsoid, its changing direction is determined by the changing curvature of the surface. These changes are mapped onto a sphere, by the motion of the blue stick, which emanates from the center of the sphere and is always pointing in the same direction as the red stick. Gauss called the area marked out by the blue stick on the sphere the, ?total or integral curvature? of the surface. If the red stick were moving along a plane, its direction would not change, and the blue stick would not move. Since this would obviously mark out no area, Gauss defined a plane as a surface of zero curvature. The greater the area marked out on the sphere, the greater the curvature of the surface being mapped.

Animation 3

Animation 4

This can be seen from the above two examples. In animation 3 the red stick is moving around a large area of the ellipsoid, but because that region is less curved, its direction doesn’t change very much, and the corresponding area on the sphere is small. Whereas, in animation 4, the area on the ellipsoid is small, but very curved, so the area marked out on the sphere is larger.

This total curvature does not change even if the surface is deformed. For example, try and determine the spherical map of a part of a cone or a cylinder.

Using this method, Gauss was able to not only measure the ?amount? of curvature, but he was also able to distinguish types of curvature that are determined by different sets of principles. For example, animation 5 illustrates the mapping of a surface called a ?monkey saddle?. (This type of surface should be familiar to those who have been working on Gauss’ 1799 proof of the fundamental theorem of algebra.) In this mapping the curvature of the area denoted by the yellow curve on the monkey saddle is mapped to the sphere. As the red stick moves once around the area on the monkey saddle, the blue stick marks out the spherical area twice. This double covering of the spherical area indicates that the curvature of the monkey saddle embodies a different set of principles than the curvature of the ellipsoid.

Animation 5

A still different type of curvature emerges when Gauss’ mapping is applied to a torus as in animation 6, and animation 7. In animation 6, a part of the outside of the torus is mapped producing a corresponding area on the sphere, similar to what happened on the ellipsoid. But, in animation 7, the area of the torus is situated on both the inner and outer part. The mapping of these directions produces a figure eight type of curve on the sphere that crosses itself at the north pole. Each time the red stick crosses the circle that forms the boundary between the inner and outer part of the torus, the blue stick crosses the north pole of the sphere, with one loop of the figure eight corresponding to the inner part of torus, and the other loop the outer part. The area on the torus is bounded by a non-intersecting curve, while its map on the sphere is bounded by an intersecting one. The presence of this singularity on the spherical map indicates that the boundary between the inner and outer parts of the torus is a transition from one type of curvature to another. Consequently, the torus must be governed by a different set of principles than the ellipsoid or monkey saddle a set of principles that encompass a transition between two different types of curvature.

Animation 6

Animation 7

To summarize: for the ellipsoid, the Gaussian mapping produced a simple area whose size varied with the curvature of the surface. The mapping of the monkey saddle produced an area that was double covered. The mapping of the torus, produced a singularity. These mappings not only measure the ?amount? of total curvature of the part of the surface mapped, but the appearance of anomalies and singularities in the mapping, indicate the presence of additional principles of curvature as well.

Like the Chorus in Shakespeare’s Henry V, who, alone on an empty stage, summons the imagination of the audience to envision the real principles of history and statecraft that are to be depicted, these anomalies and singularities call the attention of the scientist to imagine the set of principles that produced them. Therein, is where

Riemann for Anti-Dummies Part 42

ARCHYTAS FROM THE STANDPOINT OF CUSA, GAUSS, AND RIEMANN

A citizen in 2003 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.

The summary of the relevant concept, spoken by the then 28 year old Riemann acting under the tutelage of the 77 year old C.F. Gauss, focuses on the relationship of the idea to that which creates it:

“Accordingly, I have proposed to myself at first the problem of constructing the concept of a multiply extended magnitude out of general notions of quantity. From this it will result that a multiply extended magnitude is susceptible of various metric relations and that space accordingly constitutes only a particular case of a triply-extended magnitude. A necessary sequel of this is that the propositions of geometry are not derivable from general concepts of quantity, but that those properties by which space is distinguished from other conceivable triply extended magnitudes can be gathered only from experience. There arises from this the problem of searching out the simplest facts by which the metric relations of space can be determined, a problem which in nature of things is not quite definite; for several systems of simple facts can be stated which would suffice for determining the metric relations of space; the most important for present purposes is that laid down for foundations by Euclid. These facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses; one may therefore inquire into their probability, which is truly very great within the bounds of observation, and thereafter decide concerning the admissibility of protracting them outside the limits of observation, not only toward the immeasurably large, but also toward the immeasurably small.”

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.

Archytas’ magnificent composition could only have been produced by a mind closer to Riemann’s than Euclid’s, and, having lived and worked nearly a century earlier, we can be assured that his was not constricted by the deductive method of the Elements, let alone its later Aristotelean transmogrifications. The few extant fragments of Archytas indicate his concept of mathematics was far different from today’s formalists:

“Mathematicians seem to me to have excellent discernment, and it is not at all strange that they should think correctly about the particulars that are; for inasmuch as they can discern excellently about the physics of the universe, they are also likely to have excellent perspective on the particulars that are. Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music. These seem to be sister sciences, for they concern themselves with the first two related forms of being number and magnitude.”

As with his collaborator Plato, Archytas understood that the betterment of humanity depended on the improvement of the cognitive powers of the mind through pedagogical exercises:

“When mathematical reasoning has been found, it checks political faction and increases concord, for there is no unfair advantage in its presence, and equality reigns. With mathematical reasoning we smooth out differences in our dealings with each other. Through it the poor take from the powerful, and the rich give to the needy, both trusting in it to obtain an equal share.”

However, today’s responsible citizen need not rely merely on fragmentary quotes to bring Archytas’ method into his or her mind. A more secure approach is directly accessible–coming to know for oneself, Archytas’ solution for the problem of doubling the cube a method 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.”

The Polyphonic Determination of Mean Proportionals

To begin to obtain knowledge of Archytas’ method, one must throw out all axiomatic/deductive approaches and adopt the physical/geometrical approach demonstrated by Jonathan Tennenbaum in his quite notable pedagogical exercise: “A Note: Why Modern Mathematicians Can’t Understand Archytas” That discussion should be throughly worked through and forms an excellent basis for the pedagogical workshops now taking place among youth organizers and others. That argument is summarized here with the aid of some graphics and animations.

To double the cube, Archytas focused on the more general form of the problem posed by Hippocrates of Cios, that of placing two means between two extremes. Hippocrates had recognized that when doubling the square, the areas of the squares produced, as well as the corresponding sides, were all in what the Pythagoreans called “geometric” proportion. A similar relationship held when doubling cubes, but with one notable exception. Instead of one geometric mean between each action, the cube required two.

For example, if we take a square and double its side, the area of the square so produced quadruples. Thus, the square whose area is double, is the geometric mean between the original square and the square produced by doubling its side. That proportion expressed in numbers is 1:2::2:4. This same proportion is reflected among the sides of the three squares. That is, the side of the square of 1 is to the side of the square of two, as the side of the square of two is to the side of the square of four. Or, in numbers, 1:\/2::\/2:2. Thus, doubling a square amounts to constructing the geometric mean between 1 and 2.

However, when the edge of a cube is doubled, the volume of the cube increases from 1 to 8. From the standpoint of whole numbers, there are two geometric means between 1 and 8, specifically 2 and 4. Hippocrates noted that a similar proportionality exists between the edges of these cubes. The problem, as Riemann would recognize it, is extending this relationship outside the realm of visible observation between 1 and 8, into the smaller interval between 1 and 2. The discrepancy between what can be done in the large and with what can be done in the small, is analogous to the idea of a changing geodesic from the macro-physical to the micro-physical domain, and indicates that the doubling of cube is an action of a higher power, as Plato defines power.

The relationship between the problem of doubling the cube and the problem of finding two geometric means between two extremes, can prove a stubborn one to master for those stuck in logical/deductive habits. The root of this mental block is in large part due to the false, but persistent, habit of trying to understand things from the bottom up. While in the domain of squares, the geometric means from one to eight are created one at a time, one, then two, then four, then eight; in the domain of cubes, these two geometric means, must be created all at once from the top down.

Archytas, who understood music, geometry, number and sphaerics (astronomy) as one, would have no more problem with this than J.S. Bach, Mozart, Beethoven, Schubert, Schumann or Brahms. For the well-tempered system of bel-canto polyphony is not produced from the bottom up, note by note. Rather, each note is determined from the relationships that arise from bel-canto sung polyphony. Thus, think of the placing of two means between two extremes in the same epistemological light as a musical composition. The composer’s idea of the composition determines the relationship among the notes. The question Archytas solved, was, “What composition produces two geometric means between two extremes?”

As demonstrated in the above cited pedagogical discussion by Jonathan Tennenbaum, the placing of one geometric mean is a function of circular rotation, not the generation of squares. While the doubling of the square requires finding the geometric mean between 1 and 2, the circle expresses the placement of a geometric mean between any two extremes. (See figure 1.) In the figure, triangle OPA is inscribed in a semi-circle making the angle at P a right angle.(fn.1.) A line drawn from P perpendicular to diameter at Q, forms a whole set of geometric proportions. The one that most concerns us here is OQ:OP::OP::OA. As P rotates around the circle from A to O, OP remains the geometric mean between OQ and OA, even though the proportion between OQ and OA changes from 1:1 to 1:0.(fn2.) (See animation 1.)

Figure 1

Animation 1

As indicated in Jonathan’s pedagogical, a second set of geometric proportions can be linked to this first set, by drawing a circle around right angle OPQ and finding B as a projection of Q onto OP. (See figure 2.) This produces the double set of geometric proportions OB:OQ::OQ:OP::OP::OA. Now, as P moves from A to O, not only do the geometric relationships between OQ, OP and OA change, but so do the geometric relationships between OQ, OB and OP. (See animation 2.)

Figure 2

Animation 2

Think of these sets of relationships polyphonically. OQ:OP::OP:OA is one voice. OB:OQ::OQ::OP is a second voice. The internal relationships within each voice are the same, but together they form a relationship between two sets of relationships, akin to the voices in a two part fugue.

The problem Archytas confronted is expressed by the non-uniform nature of the curve traced by B as P moves around the larger circle. In order to double the cube, there must be some definite way to determine a ratio between OB and OA of 1:2.

Archytas recognized that this could not be determined in what Riemann called a doubly-extended manifold, but rather, was derived from the higher powers associated with a triply-extended manifold.

Archytas effected this by generating the first set of geometric means from rotating one circle (with diameter AO) perpendicular to another (with diameter OD). (See animation 3.) A point P on circle AO connected perpendicularly to a point Q on the circumference of circle OD, produces the geometric relationship, OQ:OP::OP::OA. The action of the rotating circle AO produces a torus. (See animation 4.)

Animation 3

Animation 4

The rotation of PQ produces a cylinder (See animation 5.) The intersection of the torus and cylinder forms a curve of double curvature, which expresses, from the standpoint of a higher power, the geometric relationships between OQ, OP and OA. Now, when Q is projected to a point B on OP the relationship OB:OQ::OQ:OP::OP:OA is produced. (See Figure 3.)

Animation 5

Figure 3

From this, the magnitude that doubles the cube can be found by constructing OM as a chord of circle OQD so that it is of diameter OD. (See Figure 4.) Rotate chord OM around the axis OD until it coincides with line OP. This action produces a cone with apex at O and axis OD. P now lies at the intersection of a cone, torus and cylinder. After this rotation, M will coincide with B and the length of OB will equal OM. Consequently, if OB=1, OQ is the edge of the cube whose volume is 2; OP is the edge of the cube whose volume is 4; and OA is the edge of the cube whose volume is 8.

Figure 4

The Maximum and Minimum

With the just completed work fresh in mind, step back and ask an elementary question. What universal principle is expressed by the physical characteristic that a doubly extended manifold expresses one geometric mean between two extremes, while a triply extended one requires two? As Plato states in the Timaeus, this is not an abstract question, but one necessary to know in order to understand the real world:

“Now if the body of the One had to come into existence as a plane surface, having no depth, one mean would have sufficed to bind together both itself and its fellow terms; but now it is otherwise; for it behoved it to be solid of shape and what brings solids into unison is never one mean but always two.”

A crucial insight can be gained when this question is examined from the standpoint of Nicholas of Cusa’s principle of the Maximum and Minimum.

Look back at our initial exploration of geometric means. We discovered that this relationship is expressed by a right angle in a circle. But, there was an underlying assumption that we didn’t investigate. What is a right angle? Should we accept some definition, such as Euclid’s, that a right angle is that angle formed when two lines that intersect form two angles that are equal? Should we accept the definition of right angle as that angle inscribed in a semi-circle, when such a definition assumes that the sum of the angles of a triangle equals two right angles? As Kaestner and Gauss would later demonstrate, such definitions already assume that the manifold in which the action occurs is characterized by infinite flatness.

Even before Kaestner and Gauss raised these questions, Cusa recognized that such formal definitions never lead to the truth. For Cusa, the truth is the unqualifiedly Maximum, which, in the Absolute, coincides with the unqualifiedly Minimum.

“Therefore, the truth, which is itself the measure of things, is not comprehensible except through itself. And one sees that in the coincidence of the measure and the measured. Indeed, in everything this side of the infinite, the measure and the measured differ according to more or less. In God, however, they coincide. The coincidence of opposites is therefore like the periphery of the infinite circle. The distance of opposites is as the periphery of the finite polygon. Therefore, in theological figures the complement of that which can be known is to know this, namely, that in the infinite the difference of the measure and the measured is in God equality or coincidence. Hence, the measuring is there infinite rectitude. And the infinite circular line is measurable through infinite rectitude. And the measuring itself is the unity or the connection of both.”

This method of the maximum and minimum is already implicit in the problem addressed by Archytas. The maximum and minimum coincide in God, but “this side of the infinite” they’re opposites. Physical action, can be thought of as least-action, or a mean between the extremes of the minimum and maximum.

As Cusa notes, the circle is generated as the action that encompasses the maximum area by the minimum circumference, otherwise known as “isoperimetric”. This action of circular rotation, itself contains a maximum and minimum. If the rotation begins from a point A, then there is some point O at which the rotating point has reached a maximum divergence from A and begins to converge back towards A. Thus, the circle, which is itself a maximum/minimum, contains within it, another maximum/minimum, expressed by the ends of the diameter as points of maximum divergence.

This principle of maximum divergence within a circle, also contains within it, a maximum and minimum. The angle formed by a line pivoting about the center of the circle reaches an angle of maximum divergence, which now defines a right angle, which Cusa calls the maximum acute angle and the minimum obtuse angle. In other words, instead of defining a right angle as a thing, think of it as a maximum/minimum within a circle, which is itself a maximum/minimum within a doubly-extended manifold. It is here, in the nature of the doubly- extended manifold, that the minimum/maximum characteristics expressed by the circle, and in turn, the right angle, are determined.

Thought of from this standpoint, the geometric mean expresses this maximum/minimum relationship of what Riemann would call a doubly-extended manifold. In sum, the circle expresses the maximum/minimum characteristic of a doubly-extended manifold; the right angle, in turn, expresses the maximum/minimum characteristic of a circle. From Cusa’s standpoint, the geometric mean can be thought of as the mean between the maximum and minimum right angle, (as illustrated in the in the first animation.)

Thought of in this way, the geometric mean is not produced by squares. Rather, the squares are artifacts, generated by the placing of one geometric mean between two extremes, which is the characteristic least action principle of a doubly-extended manifold of zero-curvature.

Now on to the real world of solids as exemplified by cubes. Cubes express two geometric means between two extremes. From the standpoint of Cusa’s maximum/minimum, this relationship must express a mean between the minimum and maximum in what Riemann would call a triply-extended manifold. The expression of the maximum/minimum in a triply- extended manifold is spherical action, which encloses the maximum volume in the minimum surface, and produces two orthogonal degrees of circular action. Each circle expresses a complete set of geometric means. But, the two means that produce solid bodies, such as cubes, are only generated when the spherical action is “unfolded” as in Archytas’ construction.

As in the case of the square, the cube does not produce two means between two extremes. Cubes are artifacts generated when two geometric means are placed between two extremes of spherical action, when that action is “unfolded” as in Archytas’ composition. This is a characteristic least-action principle of a triply-extended manifold.

Figure 4

Kepler’s Orbits and the Catenary

Cusa’s method of maximum/minimum led to a revolution in scientific thinking as typified by Kepler’s determination of the harmonic ordering of planetary motion. The position of any planet within its orbit, Kepler demonstrated, is determined by the minimum and maximum speeds of that orbit. From aphelion the planet increases its speed non-uniformly as determined by the speed it intends to be at when it reaches perihelion, and inversely, from perihelion back to aphelion. Thus, each planet’s orbit is a type of mean between these two extremes. Kepler further showed that these intra-orbital extremes when thought of together, form a set of means, as expressed by the musical harmonic relationships among them.

Similarly for Leibniz’ principle of the catenary. As both he and Bernoulli showed, the catenary unfolds its “orbital” pathway from its lowest point, which is the only point that holds up no chain. From a physical standpoint, the catenary is a mean, a least-action pathway, between the maximum potential force and minimum potential force, which are exerted at right angles to each other. This example will rankle anyone stuck in a Cartesian coordinate system, where the horizontal and vertical axes are straight-lines, and the mean between them is just another straight-line at a 45 degree angle to both. In Leibniz’ physical geometry, the vertical and horizontal are the potential maximum and minimum of physical action, and the catenary is the least action pathway between these two extremes.

Freed, now, from the “ivory tower” notions of Euclidean geometry, as Archytas was, we can gain a greater comprehension of the expression of Cusa’s principle of the maximum/minimum in the Gauss/Riemann complex domain. This is where we’ll begin in the next installment.

FOOTNOTES

1. This should not be taken for granted, and so the reader is encouraged to discover a demonstration that an angle inscribed in a semi-circle is a right angle. The original proof of this is attributed to Thales. The proof in Euclid, III. 31, is based on Euclid’s earlier proof that the sum of the angles of a triangle is equal to two right angles, which Kaestner, Gauss and Riemann would later note depends on the parallel postulate, which in turn depends on an assumption about the nature of the curvature of space. As, Riemann noted in his 1854 habilitation lecture, “these facts are, like all facts, not necessary but of a merely empirical certainty; they are hypotheses;”

2. Here we confront another paradox of abstract mathematics between the expression of the geometric mean in numerical terms, and its physical geometric expression.

DIE WIDMUNG

The following pedagogy is dedicated to the celebration of Lyn and Helga’s silver wedding anniversary on Dec. 29, which is an occasion for joy, not only for said happy couple, but for all people around the world to whom this marriage has contributed such happiness over the past quarter-century.

The Long Life of the Catenary: From Brunelleschi to LaRouche

The shift from a consumer society to a producer one is, fundamentally, a question of understanding power. For consumers, the world is a universe of magical powers, from which the apparent requirements of life are obtained through the intercession of high priests. Such people, when confronted with a turn of events, such as the present, in which the powers on which they’ve relied no longer deliver, become frightened. They demand action from their increasingly impotent priesthood, who, despite all boasts to the contrary, fail to produce results. They then take matters into their own hands, adopting ever increasing desperate rituals designed to appeal directly to the powers on which their hopes are pinned. The high priests, seeking to restore “consumer confidence” and regain their positions of lofty authority, suggest to the frightened populace, that new rituals be performed and new incantations be recited. Each desperate effort fails to bring about any respite from the crisis. Suspicions mount that the unseen potencies have either gone deaf or departed the scene. Dead or deaf, the thought that such powers might never have existed is their ultimate terror. Even if these forces are now believed gone, the idea of their previous existence not only persists, but continues to govern the thoughts and actions of the populari, feeding the hopeless pessimism that leads such unfortunate creatures into a contorted dance reminiscent of those depicted in a Breugel masterpiece.

There was a similar state of affairs in 14th century Florence, when in the aftermath of the Black Death in which nearly 80% of the Florentine population perished, a group of artists succeeded in launching the project to build a Dome with a diameter of 42 meters over the church of Santa Maria del Fiore. (See Figure 1.)

Figure 1

When the decision to undertake that project was made in 1367, the man who would ultimately execute it, Filippo Brunelleshci, was not even born, but the intention that would guide him was already embedded in the proposed size of the structure, and the requirements of its design. The Dome was to be equal in size to the Roman Pantheon, that temple to the mystical powers on whose behalf Roman popular opinion ruled. (See Figure 2a and Figure 2b.)

Figure 2a

Figure 2b

From the time of its construction in 128 A.D. under Emperor Hadrian, the Pantheon had remained the largest domed structure in the world. But, unlike the Pantheon, the Florentine Dome was to be beautiful from both the inside and out. It was to be a complete break with the pantheonic culture that, even though the Emperors had long since ceased to rule directly under its name, had continued to persist for more than 13 centuries and had brought about the very recent calamity from which Europe was still reeling.

The Dome was a project of bold optimism. It would not only span such a large structure, but by being self-supporting and free standing, it would demonstrate a principle that would transform the entire surrounding region, and through the minds of travelers and the imaginations of those with whom they would speak, the entire world.

The full implications of the principles necessary to construct the Dome were not known to the Dome’s original designers, but, to accomplish the feat, Brunelleschi would have to discover, apply, and communicate a form of the universal principle of least-action whose further elaboration would unfold over the ensuing 500 years. The crucial development occurred in 1988, when Lyndon LaRouche, faced with imminent unjust imprisonment, visited the Dome and recognized the implications of Brunelleschi’s discoveries for the subsequent breakthroughs of Leibniz, Gauss and Riemann, and for the future, development of a new physical science.(See 1989 issue of 21st Century with Dome on the cover.)

The Dome and Anti-Euclidean Geometry

Imagine yourself in 1420 looking at the octagonal drum of Santa Maria del Fiore without the Dome. What do you see? Empty space? If so, you would never envision, let alone build, the Dome. The construction of the Dome required a mastery of principles not visible to the eye. Not the invisible mystical powers of the Pantheon, but the universal physical principles, which, though unseen, are known clearly through the imagination. For the scientist, like the artist, there is no empty space, no empty canvass, no blank slate. There is a manifold of physical principles characterized by a set of relationships whose expression takes the ultimate form of the work of art. To visualize the unbuilt Dome, as the artist Brunelleschi would, imagine the physical principles, and the bricks and mortar take the required form.

This is the basis on which to begin to construct a physical geometry from the standpoint of Brunelleschi, Leibniz, Gauss, Riemann and LaRouche.

To grasp this, think of the difference between abstract geometrical shapes, and the physical geometry of bricks and mortar.

Start with a vertical column. An abstract geometrical line, according to Euclid, is that extension in empty space which has only length. No matter how long the line, its width, is always the same, namely, nil. However, when building a vertical column (line) of bricks, the higher the column, the greater the pressure on the lower bricks. To build the column higher requires strengthening the lower portions of the column, by increasing its width, or by some other means.

Extend this thought to an area. From the standpoint of empty Euclidean space, an area is that which has length and width. A bounded area is enclosed by a line either straight or curved. A physical area, however, is enclosed by a physical structure, the shape of which is determined by physical principles. One approach to enclosing a physical area, would be to build two vertical columns and span those columns with a flat roof. But, this is a relatively weak structure, as the roof is only strong near where it is supported by the columns. The farther apart the columns are, the weaker is the roof. (See Figure 3.)

Figure 3

A far more stable structure for vaulting a vertical area is an arch. The circle is at first thought the simplest type of arch, because the circular boundary encloses the largest area by the least perimeter. If the arch is designed so that all the bricks point to the center of the circle, the arch will be relatively stable upon completion. (See Figure 4.) But, while under construction, the arch cannot support itself, requiring the use of a temporary scaffolding to support the arch under construction. Thus, the arch is self-supporting as a whole, but not in its parts.

Figure 4

The circular arch poses another problem. Even though it encloses the greatest area with the least perimeter, its height is a function of its width and the line of pressure does not conform to the circular curve. (See Figure 5.)

Figure 5

To enclose a taller area requires a wider arch which in turn decreases the overall stability of the structure because the downward pressure from the upper bricks pushes the sides of the arch outward. Thus, even though, from the standpoint of abstract geometry, the circle is isoperimetric, from the standpoint of physical geometry, some other shape provides the greatest stability for the tallest area. The shape that achieves this is a pointed arch in which the two arcs that make up the arch are circular arcs with different centers. (See Figure 6.) The pointed arch, thus, not only encloses a taller area, but is more stable because the curvature of the arch conforms more closely to the physical line of stresses in the structure. In other words, the shape of the arch is determined not by abstract geometrical characteristics but by physical ones.

Figure 6

Now, to the problem facing Brunelleschi enclosing a volume. Geometrically, a volume is enclosed by a surface, which is produced when a curve is moved. For example, from the famous construction of Archytas, when a circle is moved along a line, a cylinder is produced; when rotated around a point, a torus is produced; and when rotated around a line, a sphere is produced. And so, a dome can be produced by rotating an arch, either circular or pointed around an axis. (See Figure 7.)

Figure 7

But, a physical surface, such as a dome, is not simply a rotated arch, because a new set of stresses occurs in the dome that does not occur in the arch. In addition to the stresses along the arch, (from top to bottom, i.e., longitudinal), there are stresses around the dome (circumferential or hoop).

So, the problem Brunelleschi faced in building the Dome of the required size was to design a structure whose shape would balance these stresses without requiring external buttressing which would diminish the Dome’s beauty and undermine its effectiveness for changing society by changing the minds of the population.

Additionally, a dome, like an arch, generally requires a supporting scaffold, or centering, to hold it up until its completion. Here, Brunelleschi faced his most formidable obstacle. The Dome over Santa Maria del Fiore was so big that there was not enough wood available to build such a large scaffolding. Consequently, Brunelleschi took the bold step of building the Dome without centering, requiring him to construct a dome that was self-supporting in its whole and its parts. Such a shape could not be determined by the methods associated with Euclidean geometry. The shape Brunelleschi required was determined only by physical principles.

To do this, Brunelleschi decided to construct two domes, one inside the other, with a stairway between them. Both would conform to the pointed arch form called for in the original design. However, according to the architect Bartoli, (see 1989 21st Century) the curve of the inner dome was a based on a circle whose diameter was three fourths the inside diameter of the octagonal drum (pointed fourth) while the outer dome’s curvature was four-fifths the outer diameter (a pointed fifth) (See Figure 8.)

Figure 8

Since the use of centering had to be avoided, Brunelleschi had to control the curvature of both domes very carefully as the were being constructed. This entailed controlling three different curvatures: the longitudinal curvature; the circumferential curvature; and the curvature inward towards the center of the Dome. If all three curvatures could be controlled during all phases of construction, the Dome would not only be stable upon completion, but each stage would be stable enough to be a platform from which the next stage would be built. Thus, the Dome had to conform to a shape that would be self-supporting in its whole and its parts.

Brunelleschi had to solve a multitude of problems, each requiring revolutionary new ideas to accomplish the task. But, the discovery most central to his success, the one most relevant for the future development of the anti-Euclidean physical geometry of Kepler, Fermat, Leibniz, Gauss and Riemann, is the one identified by LaRouche. Brunelleschi used a hanging chain to guide the development of the curvature of the dome at each stage of construction. Thus, the overall shape of the Dome was determined, not by a curvature defined by abstract mathematics, but by a physically defined principle. Just as a hanging chain is self-supporting in its whole and its parts, the Dome, whose curvature is guided by the curvature of the hanging chain, is, likewise, self-supporting surface, in its whole and its parts.

The beauty of the Dome demonstrates the truth of Brunelleschi’s discovery, but, it would take the discoveries of Kepler, Fermat, Leibniz, Gauss, Riemann and LaRouche to fully grasp the underlying principle.

The Development of the Physical Idea of Shape

The success of Brunelleschi’s Dome demonstrated that the architectural principles of physical geometry on which it was based were universal. This view was expressed by Johannes Kepler, who approximately 150 years later wrote concerning the construction of the solar system in his Mysterium Cosmographicum, “We perceive how God, like one of our own architects, approached the task of constructing the universe with order and pattern, and laid out the individual parts accordingly, as if it were not art which imitated Nature, but God himself had looked to the mode of building of Man who was to be.”

Kepler went on to develop, in that work and in his subsequent New Astronomy and Harmonies of the World, that the shape of the solar system, like the Dome, was determined not by considerations of abstract mathematics, (which would have indicated perfectly circular orbits) but by physically determined harmonic principles. Thus, the elliptical planetary orbits, like Brunelleschi’s Dome, were the size and shape that they had to be in order to express the harmonic relationships of those physical principles.

This physically determined idea of shape took another step in its development with Fermat’s determination that the shape of the pathway of light was determined by the principle of shortest-time:

“Our demonstration is based on the single postulate, that Nature operates by the most easy and convenient methods and pathways — as it is in this way that we think the postulate should be stated, and not, as usually is done, by saying that Nature always operates by the shortest lines … We do not look for the shortest spaces or lines, but rather those that can be traversed in the easiest way, most conveniently and in the shortest time.”

Leibniz, following up on the discoveries of Kepler and Fermat, generalized these discoveries as a universal principle of least-action:

“…the Architect of all things created light in such a way that this most beautiful result is born from its very nature. That is the reason why those who, like Descartes, reject the existence of Final Causes in Physics, commit a very big mistake, to say the least; because aside from revealing the wonders of divine wisdom, such final causes make us discover a very beautiful principle, along with the properties of such things whose intimate nature is not yet that clearly perceived by us, that we can have the power to explain them, and make use of their efficient causes, along with their artifacts, such as the Creator employed them in order to produce their results, and to determine their ends. It must be further understood from this that the meditations of the ancients on such matters are not to be taken lightly, as certain people think nowadays.”

Leibniz’ most far reaching discovery of this principle of least-action, made in collaboration with Johann Bernoulli, demonstrated that the catenary, the guiding principle of Brunelleschi’s Dome, embodied the most universal expression of least-action. As we’ve developed in other locations, the physical characteristic by which the hanging chain supports itself, expresses all the elementary transcendental relationships of geometry: the circular, hyperbolic, and the powers associated with lines, surfaces and volumes. (See Figure 9.)

Figure 9

Look back at our earlier comparison of the difference between abstract geometrical notions of line, area and volume, with the physical requirements of constructing a column, arch and dome. As is already implicit in the concept of powers developed by Pythagoras, Archytas, Plato, et al., even the purely geometrical concepts of line, area and volume are ultimately determined by the physical principles which Leibniz demonstrated are expressed by the catenary. The idea of lines, areas and volumes, separated from this idea of power as universal physical principles, is as chimerical as the mystical powers of the Roman Pantheon.

From Pathways to Surfaces

Brunelleschi’s Dome points the way to a still further development of the universal principle of least-action. Planetary orbits, the curvature of light, and catenaries are all pathways, i.e. curves. Brunelleschi’s Dome is a least-action surface.

The concepts to understand this distinction were developed by Gauss who, looking back as we’ve been doing, on the discoveries of Kepler, Fermat and Leibniz, developed the foundations of a physical theory of surfaces.

The context for Gauss’ discovery was his measurement of the surface of the Earth, which, because it is physically determined, must, in keeping with Leibniz’ principle, be a least-action surface. Over a more than 20 year period, Gauss made careful astronomical and geodetical measurements of the Earth. Abstract geometrical considerations would suggest that the Earth would be a perfect sphere, because the sphere enclosed the largest volume inside the smallest surface. But, because the Earth is a rotating body in the solar system, its physical shape is not spherical, but ellipsoidal.

As we have developed in earlier pedagogicals, Gauss’ measurements led him to discover that the physical shape of the Earth was not ellipsoidal, but something more irregular. He identified the, “geometrical shape of the Earth, as that shape which is everywhere perpendicular to the pull of gravity.” In other words, Gauss did not try to fit the Earth into a shape pulled from the text books of abstract mathematics, rather, he invented a new geometry that conformed to the physical characteristics of the rotating Earth.

Gauss reported the generalization of his discoveries in his 1822 Copenhagen Prize Essay, on conformal representation and his 1827, “General Investigations of Curved Surfaces”. Future pedagogicals will develop these concepts in greater detail, while here we focus on the general ideas most relevant to this discussion.

For Gauss, all surfaces had a characteristic curvature, which in turn determined certain least-action pathways, that he later called, “geodesics”. For example, in a plane, the geodesic is a straight-line, while on a sphere, the geodesic is a great circle. In these two cases the curvature is uniform and so the geodesic is the same every where on the surface. In contrast, an ellipsoid, for example, is a surface of non-uniform curvature. Consequently, the geodesic is different depending on its direction and position on the surface. (See Figure 10.)

Figure 10

To illustrate this, the reader is encouraged to do some physical experiments. Take a plane, sphere, spaghetti squash or other irregular shaped object. Mark two points at different places on the surface and stretch a thread between them so that the thread is taught. The thread will conform approximately to the geodesic between those two points. Notice that on the plane, the geodesic is always a straight line, while on the sphere it is always a great circle, while on the squash, the geodesic changes from place to place, and direction to direction.

There is a further distinction between the plane and the sphere or ellipsoid. On the plane there are an infinite number of pathways between any two points, but only one of these paths is a geodesic, i.e. least-action. This is also true on a sphere or ellipsoid, except, if the two points are at the poles. Then there are an infinite number of geodesics between these two points. Thus, the bounded nature of the sphere and ellipsoid, produce a singularity with respect to the nature of the geodesics. The significance of this distinction will become more clear as we develop more of Riemann’s geometry in future pedagogicals.

What Gauss investigated was the general principles by which the curvature of the surface determined the characteristic of the geodesic. Of immediate relevance for this discussion is Gauss’ determination of a means to measure the curvature of the surface at any point. It is sufficient for our purposes here to illustrate this by a physical demonstration. On the squash, draw a circle by tying a marker to one end of the thread and rotating it while holding the other end of the thread in a fixed position. The radii of this circle are all geodesics in different directions. Now examine the curvature of each geodesic, which will vary for each direction. However, one geodesic will be the least curved, while another will be the most curved. Now, try this on a different type of surface, such as a butternut squash shaped like a dumbbell. The part of the round ends of the butternut squash have the same characteristic as the spaghetti squash, in that the center of curvature is always inside the squash. But, in the middle of the squash something different happens. Here the center of curvature is either inside or outside the squash, depending on the direction of the geodesic. This characteristic Gauss called, “negative curvature” and is the characteristic of curvature expressed by a surface formed by a rotated catenary called a catenoid. (See Figure 11.)

Figure 11

Brunelleschi’s Dome expresses this characteristic of negative curvature.

Furthermore, Gauss proved that on any surface, no matter how irregularly it was curved, the geodesics of maximum and minimum curvature would always be at right angles to each other!

Thus, at any place on a surface, the curvature of the surface expresses a physical principle that in turn determines the geodesic, or least-action pathway within that surface.

From Surfaces to Manifolds

Working from Gauss’ discovery, Riemann generalized this concept still further to the idea of a geodesic within a manifold of universal physical principles. The manifolds cannot be directly visualized but the characteristics of that manifold can be directly determined by a change in geodesic.

For example, light under reflection and refraction follows a pathway within a surface, but each type of action expresses a different pathway because the physical manifold of refraction includes a principle, changing speed of light, that does not exist within the manifold of reflection. The addition of this new principle to the manifold of action, changes the geodesic.

Riemann developed the means to represent these higher manifolds by complex functions. For example, as was developed in the previous pedagogical, the conic section orbits and catenary are both least-action pathways with respect to the manifold of universal gravitation. In other words, each represents a changing geodesic with the manifold of universal gravitation. But, when the catenary is expressed as a function in the Gauss/Riemann complex domain, the conic section orbits are seen as a subsumed geodesic within the higher principle represented by the catenary. (See Riemann for Anti-Dummies Part 40.)

More general examples are illustrated in the accompanying animations. These illustrate how the same action, when carried out in different manifolds, is changed by the characteristics of the manifold. Think of the orthogonal nets in each figure as the minimum and maximum geodesics in each manifold. The loopy curve maintains the same angular orientation with respect to these geodesics in each case. But, because the geodesics change, from manifold to manifold, the action changes. Thus, a change in the principles that determine the manifold, change the geodesics, which in turn change all action within that manifold. Conversely, to effect a physical change in any action, one must act to change the characteristics of the manifold.

z2

z3

ez

1/z

Catenary

Now look at Brunelleschi’s Dome from this standpoint. The Dome is a surface whose geodesic, in principle, conforms to the catenary. As a least-action surface, it expresses a geodesic with respect to the principle of universal gravitation. With respect to the manifold of universal history, building the Dome was the geodesic from that dying culture of the Roman Empire to the Renaissance.

At our present place in the manifold of universal history, building LaRouche’s youth movement combat university on wheels and making LaRouche President of the United States, is for us, Brunelleschi’s Dome– the geodesic from this dark age to our Renaissance.

Riemann for Anti-Dummies Part 40

COGNITIVE LEAST ACTION

Throughout his various works on complex functions, Riemann notes that the hidden harmonies of the complex domain, not calculations, are the least action pathways for the discovery of truth.

Riemann’s concept is in keeping with the tradition from Plato to Gauss, as exemplified by Gauss’ determination of the orbit of Ceres. In that case, every leading astronomer attempted to deduce Ceres’ orbit by an ever increasing level of calculations upon the few data points of observations supplied by Piazzi. All failed. Gauss, on the other hand, focused on Kepler’s harmonic principles, finding Ceres’ orbit as a consequence of them. Where the failures calculated unknown data from known data, Gauss sought the principles that determined both the known and unknown data. Gauss later distinguished his method from the failed ones with reference to Euler’s famous attempt to deduce the orbit of a comet by detailed calculation, an effort that left Euler blind in one eye. “I too would have gone blind,” Gauss is reported to have said, “If I calculated like Euler did.”

As urgent and necessary as it is for political leaders to grasp Gauss’ and Riemann’s method, it nonetheless presents serious psychological difficulties for those reared in the fantasy world of the post-1966 consumer society. As noted in last week’s installment, consumers only know objects and how to manipulate them. Acting on the world is limited to manipulating those objects according to a set of authoritative rules. Consumers look in awe to those other consumers who appear to manipulate the greatest number of objects. (Modern academics call this objective science.) It is no wonder that under conditions of systemic collapse, such consumers become fear-driven pessimists.

Yet, to act on the world as Gauss and Riemann did, requires a capacity of mind to grasp not things, but relationships. Not merely relations among things, but relationships among sets of relations. This, as Gauss and Riemann demonstrated, is the province of the complex domain.

To further illustrate this point, look at the principle of universal gravitation. As a principle, universal gravitation is not expressible by a number, such as 32/feet/second/second. Nor is it expressible by an algebraic formula such as the inverse square law. Principles, to be truly comprehended, can only be known by how they act on other principles. As such, principles must be thought of only by Riemann’s concept of a manifold. Under Riemann’s idea of a complex function, the mind acts, not on things, but on manifolds of principles.

This concept is well developed by Kepler’s succession of discoveries with regard to planetary motion, from the astronomical significance of the Platonic solids, to the elliptical orbits, to the harmonic proportions among the orbits. Each of the above principles expresses a set of relations. Universal gravitation expresses the relationship among these sets of relations.

The Leibniz/Bernoulli principle of the catenary similarly expresses a set of relations, ordered by the principle of universal gravitation. As will be illustrated below, the complex domain affords us the power to comprehend the unifying relationship of universal gravitation between these two sets of seemingly different relations.

Before exploring more specifically the manifold of universal gravitation, it were beneficial to engage in a short warm-up exercise. Look again at a simple example of a complex function, for example, the complex function that corresponds to the general principle of doubling the square. (See figure 1a, figure 1b, and figure 1c.)

Figure 1a

Figure 1b

Figure 1c

Use this example to begin to wrench your mind away from the consumerist fixation on things, so as to be able to better grasp the physical example that will follow. In this example, think of the orthogonal grid of lines depicted on the left side of 1b not as a collection of lines and points, but as a metaphorical representation of a set of ideas related to each other in that way. Each idea is represented by a complex number, which denotes a unique action. (That action is always with respect to some origin, which is always defined by some physical singularity.) The lines connecting the points can be thought of, metaphorically, as least action pathways, i.e. geodesics, with respect to the principle of organization of the manifold of ideas represented.

Now, a new principle is introduced, in this example the principle of doubling the square. The introduction of that principle transforms all the straight-line relationships into parabolic pathways. The introduction of such a new principle, transforms all the relationships of the manifold, all at once, without, so to speak, calculation. To properly grasp this conception, resist the tendency to think of the grid of lines as a thing and the grid of parabolic pathways as another thing. Rather, think of one manifold– a polyphony evoked by the introduction of the new principle of squaring, which changes the “geodesic” within each set of relations. (See animation 1.)

Animation 1

Armed with this warm up, move on, to a physical example:

What is the connection, with respect to the principle of universal gravitation, between the least action pathway expressed by the catenary and the least action pathways expressed by the conic section planetary orbits of Kepler and Gauss? This can only be grasped from the standpoint of the complex domain.

Begin with Leibniz’ discovery that the catenary expresses the arithmetic mean between two exponential (logarithmic) functions. (See figure 2.)

This is already a precursor of Riemann’s idea of a complex function. Think of each exponential not as a “graph” as you were taught in school, but, as, Leibniz did, as a relationship of two relationships, the arithmetic and the geometric. (See figure 3.) Thus, the catenary expresses a relationship between two sets of arithmetic-geometric (exponential/logarithmic) relations.

Figure 3

But, as Riemann noted, when this relationship among sets of relations is viewed from the standpoint of the complex domain, “a regularity and harmony emerge that otherwise remain hidden.”

Look, first, at the exponential/logarithmic relationship in the complex domain. To do this, we must think, as Gauss did, of the complex domain as represented by a doubly-extended surface, with a physically determined origin, in this case, the lowest point of the catenary. Each point on the surface represents a complex number which itself denotes a unique exponential action. (See figure 4.)

Figure 4

Complex numbers represented by points equally spaced along a line are related to each other arithmetically. Complex numbers represented by points along a spiral, are related to each other geometrically. Thought of in this way, a grid of lines expresses an arithmetic relationship between complex exponentials. (See figure 5.)

Figure 5

The points along any one line are arithmetically related complex numbers. Lines that are equally spaced to other lines, (such as those in the grid) are arithmetically related sets of arithmetically related complex numbers.

Now, take this relationship so described, and use that as a principle of transformation of the complex domain itself. (See animation 2.)

Animation 2

This transformation expresses, from the higher standpoint of the complex domain., the relationship between the arithmetic and geometric that was otherwise previously expressed by Leibniz, et al. with respect to exponential, circular and hyperbolic functions. The complex exponential transforms the arithmetic relationships into geometric ones, creating a sort of arithmetic-geometric relationship among arithmetic-geometric relationships. Here, the arithmetically related (equal spaced), vertical lines become circles whose radii are geometrically related. The points along each line that are arithmetically related (equally spaced) become equally spaced points along the arcs of the circles. Inversely, the arithmetically related (equally spaced) horizontal lines become equally spaced radii, while the arithmetically related (equally spaced) points along each horizontal line, become geometrically related points along each radii. (See figure 6.)

Figure 6

When one considers the circles as special cases of spirals, the equally spaced radii and equally spaced points along the arcs are also in geometric relationship to each other as well.

Now take the next step, and apply Leibniz’ principle of the catenary as the arithmetic mean between two exponentials in this complex domain. The arithmetic mean between two inversely related circles forms an ellipse. (See animation 3.)

Animation 3

The arithmetic mean between two inversely related radii forms an hyperbola. ( See animation 4.)

Animation 4

The totality of this transformation forms an orthogonal array of ellipses and hyperbolas which correspond to the least-action pathways of the catenary in the complex domain. (See figure 7, and animation 5.)

Figure 7

Animation 5

Thus, Leibniz’ catenary principle, expressed in the complex domain of Gauss and Riemann expresses a manifold of elliptical and hyperbolic least-action pathways. In other words, the complex domain, as that domain in which we can represent as shadows, the manifold of relationships among sets relationships, increases the cognitive power of the mind, such that we become able to comprehend universal gravitation, as a principle of unity of least action among the catenary and conic section planetary orbits, and, of course, much, much, more.

Riemann For Anti-Dummies Part 39

To paraphrase Nicholas of Cusa, consumers, like animals, don’t count. They have no concept of number. Their mental world is made up only of the things they consume. How those things are produced, what power generates such things, is beyond their ken. Numbers, for them, are mere symbols, that, when manipulated according to a learned set of authoritative rules, (such as the rules of money), have some unexplainable, but psychologically powerful, connection to the things they consume.Such numbers don’t count.

“Number”, according to Cusa, is, “unfolded reason … theprime exemplar of the mind”. In the Pythagorean tradition of Plato, Cusa recognized that numbers don’t count things. Number is the means by which the mind comprehends relations. Not simple relationships among things, but the relationships among relationships. “We conjecture metaphorically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

This capacity is a unique human characteristic and is bound up with man’s capacity to increase his power in and over the universe. As Prometheus states in Aeschylus’ drama, Man was, “in an ignorant condition, living in holes and– … utterly without knowledge, until I taught them to discern the rising of the stars and their settings, which are difficult to distinguish. Yes, and numbers, too, chiefest of sciences, I invented for them, and the combining of letters, creative mother of the Muses’ arts, with which to hold all things in memory….”

This is the concept of number specific to a healthy producer society. For the consumer milk comes from the supermarket. There is no concept of number associated with this idea other than the symbol appearing on the display of the cash register in response to a computer scanning of a bar code. Contrast this to the numbers that express the relationship of that milk in the supermarket to the dairy farm and transportation system that produced the milk and transported it to the store. Now, think of the higher concept of number associated with the multiply-connected manifold of abiotic, biotic, and cognitive processes that comprise the physical economic relationships of the above transaction. Think still higher of the numbers associated with the multi-generational trajectory in which that multiply-connected physical economic manifold resides. And, still higher to the numbers associated with the manifold of universal principles — those discovered and those still yet to be discovered — which have the power to change that trajectory and all that follows from it.

The productive power of the mind counts, and it counts with complex numbers.

To grasp the concept of complex numbers requires the mind be wrenched away from those neurotic, consumerist tendencies that associate numbers with sensible objects. For that reason, a careful study of Gauss’ “Disquisitiones Arithmeticae” is highly recommended, as anyone who has had the pleasure of unfolding its wisdom will attest. By the time Gauss composed this opus, at the age of 22, he had already a fully-developed concept of the complex domain, as indicated in the climactic seventh section on the division of the circle. That section, although last in the book, was written first, and the principles expressed there unfold from the opening motivic idea of the work Gauss’ concept of congruence.

Gauss generalizes Kepler’s notion of harmony by establishing numbers with respect to their intervallic relationship, as distinct from the sense-certainty concept of things. Illustrate this idea (as we have in past pedagogicals) with a simple example. Draw five dots in roughly a circle. Label these dots 0, 1, 2, 3, 4, respectively. Now, connect these dots in sequence. Call that sequence 1. Next, connect every other dot. Call that sequence 2. Next, connect every third dot. Call that sequence 3. The same with every fourth dot. Connecting every fifth dot leads nowhere. Now, try every sixth, seventh, etc.

Calling five a modulus, Gauss’ concept of congruence establishes each number by its relationship to all other numbers with respect to modulus five. You should have noticed that sequence 1 and 6, 2 and 7, 3 and 8, etc. produced the same results. Such relationships were called by Gauss “congruent with respect to modulus five”. (The difference between two congruent numbers is equal to the modulus.) Also, sequence 2 produced the same result as sequence 3, except in opposite direction. A similar relationship was produced for 1 and 4. Introducing “-“to denote direction, Gauss’ congruence is also expressed by the relationships of 1 to -4; 4 to -1; 2 to -3; and 3 to -2.

Think back over this simple exercise. The numbers0, 1, 2, 3, 4, -1, -2, -3, -4, always denote relationships, not things. The initial numbers, 0, 1, 2, 3, 4, denote the relationship of each individual dot to all five. The sequence numbers, both positive and negative, denote relationships among these relationships.

Thus, the modulus, in this case five, denotes not five things, but a type of intervallic relationship among relationships. Inversely, the domain of such relationships denotes five as a prime number, as it establishes a total number of relationships equal to 5 minus 1. (It is suggested the reader try the above experiment with moduli 6, 7, 8, 9, 10, 11 to establish this concept more fully in your mind.)

Underlying these relationships is the “dimensionality” of the domain in which they arise. Each number and each sequence followed its predecessor by the same difference by which it was followed by its successor, i.e., arithmetically. Now, extend this investigation into a doubly-connected, i.e., geometric, domain. To do this think of 1,2,3,4 as sides of squares. 1 and4 produce squares of area 1 and 16 respectively, which, with respect to modulus five are both congruent to 1. 2 and 3 produce squares of 4 and 9 respectively, which with respect to modulus 5are congruent to -1. Thus, with respect to modulus 5, both 1 and-1 are squares! 1 and 4 are ?1, and 2 and 3 are ?-1 modulo 5! For this and related reasons, Gauss insisted that ?-1 be given “full civil rights” with all other numbers, and the domain of numbers should be extended to include them.

“The mathematician always abstracts from the constitution of objects and the content of their relations. He is only concerned with counting and comparing these relations; in this sense he is entitled to extend the characteristic of similarity which he ascribes to the relations denoted by +1 and -1 to all four elements +1, -1, i (?-1) and -i (-?-1),” Gauss wrote in the announcement to his second treatise on biquadratic residues.

Gauss insisted, and repeatedly stated, that this, “complex domain” was a higher concept that existed outside the domain of the senses, but, “the true metaphysics of these magnitudes could be placed in a most excellent light”, by representing these relationships as a quadratic array on a doubly extended surface:

“To form a concrete picture of these relationships it is necessary to construct a spatial representation, and the simplest case is, where no reason exists for ordering the symbols for the objects in any other way that in a quadratic array, to divide an unbounded plane into squares by two systems of parallel lines, and chose as symbols the intersection points of the lines. Every such point A has four neighbors, and if the relation of A to one of the neighboring points is denoted by +1, then the point corresponding to -1 is automatically determined, while we are free to choose either one of the remaining two neighboring points, to the left or to the right, as defining the relation to be denoted by + i. This distinction between right and left is, once one has arbitrarily chosen forwards and backwards in the plane, and upward and downward in relation to the two sides of the plane, in and of itself completely determined, even though we are able to communicate our concept of this distinction to other persons only by referring to actually existing material objects.*

[* Kant already had made both of these remarks, but we cannot understand how this sharp-witted philosopher could have seen in the first remark a proof of his opinion, that space is only a form of our external perception, when in fact the second remark proves the opposite, namely that space must have a real meaning outside of our mode of perception.]

“The difficulty, one has believed, that surrounds the theory of imaginary magnitudes, is based in large part to that not so appropriate designation (it has even had the discordant name impossible magnitude imposed on it). Had one started from the idea to present a manifold of two dimensions (which presents the conception of space with greater clarity), the positive magnitudes would have been called direct, the negative inverse, and the imaginary lateral, so there would be simplicity instead of confusion, clarity instead of darkness,” Gauss wrote in his second treatise on biquadratic residues.

Thought of in this way, each complex number denotes a two-fold complex of direct and lateral action, or, rotation and extension. The physical example Gauss gave for this relationship was the motion of the bubble in a plane leveller. The bubble could only move back and forth if the ends of the level moved up and down.

Gauss’ idea of complex numbers extends his concept of congruence, by which the relationships among numbers are grasped, to the complex domain, by which relationships among a set of relations can be comprehended. Just as his earlier concept of a modulus expressed a simple intervallic relationship among numbers, the complex modulus expressed a complex interval. (See figure 1a, and figure 1b.)

Riemann for Anti-Dummies Part 38

YOU ARE NOT IMPOSSIBLE

When Gauss set about writing his 1799 dissertation on what he called, “The Fundamental Theorem of Algebra,” he had already in his mind a fully developed concept of the complex domain as the idea that penetrated most deeply into the metaphysics of space, and he would spend the rest of his life unfolding the implications of that youthful discovery. But, in order to achieve what he would later call, “full civil rights for complex numbers,” he first had to root out the source of their oppression: the popular acceptance of Euler’s diktat that such numbers were “impossible.”

What one considers “impossible” is, fundamentally, a function of one’s concept of what is “possible.” Think of the foolishness today of those who insist that what Lyndon LaRouche says must be done, (most emphatically his electability as President of the United States) is “impossible.” Their tragic mistake flows not from any reasoned, scientific assessment of the matter. They assert its impossibility because they don’t want to face the possibility that their continued existence is possible only if what they think is “impossible” actually happens.

In one of his epistemological fragments, Bernhard Riemann spoke of the significance of the possible for science:

“Natural science is the attempt to understand nature by means of exact concepts.

“According to the concepts through which we comprehend nature our perceptions are supplemented and filled in, not simply at each moment, but also future perceptions are seen as necessary. Or, to the degree that the conceptual system is not fully sufficient, future perceptions are determined beforehand as probable; according to the concepts, what is “possible” is determined (thus what is `necessary’ and conversely, impossible). And the degree of possibility (of `probability’) of each individual event which is seen as possible, in light of these concepts, can be mathematically determined, if the concepts are precise enough.

“To the extent that what is necessary or probable, according to these concepts, takes place, then this confirms the concepts, and the trust that we place in these concepts rests on this confirmation through experience. But, if something takes place that is unexpected according to our existing assumptions, i.e. that is impossible or improbable according to them, then the task arises of completing them or, if necessary reworking the axioms, so that what is perceived ceases to be impossible or, improbable. The completion or improvement of the conceptual system forms the `explanation’ of the unexpected perception. Our comprehension of nature gradually becomes more and more complete and correct through this process, simultaneously penetrating more and more behind the surface of appearances.

“The history of causal natural science, in so far as we can trace it back, shows that this is, in fact, the way our knowledge of nature advances. The conceptual systems that are now the basis for the natural sciences, arose through a gradual transformation of older conceptual systems, and the reasons that drove us to new modes of explanation can always be traced back to contradictions and improbabilities that emerged from the older modes of explanation.”

By maintaining the “impossibility” of complex numbers, Euler, (whose patrons were the enemies of the American Revolution), along with J.L. Lagrange, (Napoleon’s favorite mathematician), sought not merely to exclude such magnitudes from mathematical calculations. Both Euler and Lagrange made liberal use of these “impossible” magnitudes in formal calculations. Rather, Euler et al. sought to exclude the possibility that the human mind could penetrate beneath the surface of appearances into the deeper domain of, what Plato called “powers,” where complex numbers arise.

In his 1799 dissertation Gauss attacked Euler’s method directly:

“If imaginary quantities are to be retained in analysis at all (which seems for several reasons more advisable than to abolish them, once they are established in a solid manner), then they must necessarily be considered equally possible as real quantities; for which reason I would like to comprise the reals and the imaginaries under the common denomination of {possible quantities}: Against which I would call {impossible} a quantity that would have to fulfill conditions that could not even be fulfilled by allowing imaginaries.”

The existence of complex numbers was not only possible: It was necessary to comprehend what ultimately made all numbers possible.

To establish this, Gauss tapped into the deep vein of investigations that goes all the way back to the Pythagoreans, who understood number as the means by which the mind expresses the harmonic principles that lie beneath the shadow of sense perception.

Writing in “On Learned Ignorance,” Nicholas of Cusa described this concept of number this way:

“All those who investigate, judge the uncertain by comparing it to a supposed by a system of proportions…. But the proportion which expresses agreement in one aspect and difference in another, cannot be understood without number. That is why number embraces everything which is susceptible of proportions. Thus, it not only creates proportion in quantity, but in every respect through which, by substance or accident, (two things) might agree and disagree. Thus, Pythagoras rigorously concluded that everything is constituted and comprehended through the power of numbers.”

Number has the power to express powers through proportions. Complex numbers express proportions among powers.

For example, the power to double a square is expressed through the geometric proportion 1, 2, 4, 8, 16, 32, etc., even though the magnitude that doubles the square is incommensurable to all these numbers. Furthermore, the power that doubles the cube is also expressed, but in a different way, by the same proportion, (as two geometric means between two extremes instead of one), even though the magnitude that doubles the cube is also incommensurable to all those numbers.

From this standpoint, all numbers can be generated by a succession of powers, and this is what is meant by the term, “logarithm”–a term coined by John Napier in 1594 from the Greek words “logos” and “arithmos.”

The most general form of this concept is expressed by Jakob Bernoulli’s equiangular spiral and Huygens’ hyperbola. In the former, all possible magnitudes are expressed by the radii whose lengths are a function of an angle of rotation. Proportional lengths correspond to equal angles (see Figure 1).

Figure 1

In the latter, all possible magnitudes are expressed by lengths along the asymptote that correspond to equal areas (see Figure2).

Figure 2

In both cases all possible positive quantities are expressed, inversely, as a function of a power, expressed as either an angle (spiral) or an area (hyperbola).

In both cases, adding the logarithm (angle or area) produces proportional changes in length.

As discussed in the previous installment of this series, Leibniz brought a crucial contradiction to light by posing the question, “What has the power to produce a negative number?” This provoked a dispute with his collaborator Johann Bernoulli, who insisted that negative numbers were produced by the same powers as positive numbers. Leibniz, on the other hand, correctly disagreed. For Leibniz the very existence of negative numbers (which had been called “false” numbers) demanded a higher principle, which Gauss later discovered as the complex domain.

For Gauss, negative numbers were not absolute quantities. They were physically determined. In numerous locations, Gauss repeatedly polemicized, (against I. Kant) that the difference between positive and negative, right and left, up and down, could not be determined by mathematics but only with reference to physical action.

Look at this from the standpoint of the above illustrations of the spiral and hyperbola. Both generate all possible magnitudes as a function of powers. But, in both cases, the exact same result can be obtained, but in exactly the opposite orientation (see Figure 3 and Figure 4). In Figure 3, you can see two spirals that produce the same magnitudes, but in different directions.

Figure 3

In Figure 4 you can see two branches of an hyperbola that produce the same magnitudes, but in different directions.

Figure 4

In each case, if one set of magnitudes is denoted positive numbers the other set can be denoted negative. But, as Gauss pointed out, there is no {a priori} way to distinguish one from the other. Only when presented with both, is the existence of positive and negative established.

But, there is a still deeper, much more profound principle embedded in this. Look at the transition between the positive and negative hyperbola. The vertical asymptote is an “infinite” boundary separating positive from negative. Similarly, for the spirals. The transition from the positive to the negative spiral is the point of the change in direction, which each spiral approaches, but never crosses.

Thus, the domain in which both positive and negative numbers exist together must be of a higher power, where the powers that generate powers reside. It comprises Gauss’ domain of all possible (complex) quantities.

Like all ideas, Gauss recognized that this domain could not be seen directly, but, it, nevertheless, was susceptible of metaphorical representation. Since it was the domain of powers, it could not be represented by simple proportions among things, but as a proportion between proportions. Consequently, each complex number represented a proportion, not a quantity. The manifold of complex numbers, Gauss said, could only be represented on a surface extended in two directions. The physical example Gauss used was the geodesist’s plane leveller. The position of the bubble at rest is determined by both the axis of the tube and the direction of the pull of gravity, which is perpendicular to it.

On Gauss’ surface each point represented a power that was denoted by a complex number. Using some physically determined point and line as a reference, each power, i.e., each complex number is generated by a spiral action rotation and extension (see Figure 5).

Figure 5

In this way, proportions between proportions could be represented.

For example, a power acting on a power. In Figure 6 and Animation 1, complex number a+bi represents a power produced by a combination of rotation and extension. When that power acts on itself, it produces (a+bi)². When it acts on itself again it produces (a+bi)³, etc. What results is a series of similar triangles conforming to an equiangular spiral. While this spiral looks similar to Bernoulli’s spiral, it is different. Bernoulli’s spiral represents a succession of powers that produce simple magnitudes. Gauss’ complex spiral represents a higher power that produces a succession of powers, not simple magnitudes.

Figure 6

Animation 1

Figure 7, illustrates the more general case of the proportion between two different powers, or, what is commonly known as multiplication. Here the complex number 2 + i is multiplied by 1 + 2i to form 5i. In this case the 2 + i forms the red triangle with vertices 0, 1, 2 + i. The product is the point (5i) which forms the similar triangle with 0, 1 + 2i as its base. (The schoolbook arithmetic idea of multiplication as a set of rules is brainwashing. As Gauss emphasized, multiplication is a proportion such that 1: a :: b: a x b. You, the reader, are left to confirm this for yourself experimentally. Try experimenting with Theatetus’ squares and rectangles.)

Figure 7

As in the previous examples of the hyperbola and spiral, the proportional changes in extension are “connected” by adding the angles (logarithm) and multiplying the lengths.

RIEMANN FOR ANTI-DUMMIES PART 37

THE DOMAIN OF POSSIBILITY

Plato, speaking in the Laws through the voice of an Athenian stranger, holds it indispensable for leaders of society to possess elaborate knowledge of arithmetic, astronomy and the mensuration of lines, surfaces and solids. He also considers it a disgrace for any common man to lack a basic understanding of these same subjects.

It is altogether fitting that these words should issue from someone far from home, for, as Helga Zepp-LaRouche so artfully demonstrated in her presentation to the ICLC Labor Day Conference, by the time of Plato’s writing ,Athenian culture had estranged itself from these concerns and embarked on that chain of events which led to the disastrous Peloponnesian Wars. It is further fitting that Plato speaks here as a stranger, as all three subjects share a common focus on the exploration of those universal principles that govern, but don’t reside, in the domain of objects and sense perception. To one trapped in the domain of the senses, those principles appear to come from some foreign land “over the horizon”, or, “beyond the finite” . But, to one willing to ascend to its not too distant shores, that place is the province from which come the common principles, that make possible such diverse discoveries, as the founding ideas of the American Republic, the Gauss-Riemann concept of the complex domain and Beethoven’s late string quartets.

The Universe is a wondrous, but not a strange place. As Nicholas of Cusa and G.W. Leibniz repeatedly emphasized, nothing exists or happens in the Universe that is not possible. It is the province of science, therefore, to discover what makes things possible. In so doing, the mind discovers not only the possibility of a particular thing, but it also discovers, and changes, the possibility of what it can discover about what is possible. Hence, Plato’s emphasis on the study of the above mentioned subjects. It is both a means to discover what makes these things possible and a pathway for the mind to discover how it is possible for it to discover what is possible, thereby increasing its power.

Proceed through the example of the mensuration of the line, surface and solid. As presented in earlier locations, each object is made possible by a principle that possess the power to produce it. The power to generate the line is different, and incommensurable with, the power that generates a square, which, in turn, is different and incommensurable with, the power to generate a solid. This, in itself, is a crucial discovery. But, the more important discovery comes when the next question is posed. Since all three distinct powers exist in the one Universe, what is it about the Universe that makes possible these three distinct powers?

The answer to this type of question does not lie in the particular nature of each discovery, but in the paradox that one universe produces all three. As Cusa put it in “On Learned Ignorance”:

“All our wisest and most divine teachers agree that visible things are truly images of invisible things and that from created things the Creator can be knowably seen as in a mirror and a metaphor. But the fact that spiritual matters (which are unattainable by us in themselves) are investigated metaphorically has its basis in what was said earlier. For all things have a certain comparative relation to one another, a relation which is nonetheless, hidden from us and incomprehensible to us), so that from out of all things there arises one universe and in this one maximum all things are this one. And although every image seems to be like its exemplar, nevertheless except for the Maximal Image (which is, in oneness of nature, the very thing which its Exemplar is) no image is so similar or equal to its exemplar that it cannot be infinitely more similar and equal…”

The square is bounded by lines, but those lines can only be produced from squares, not from lines alone. The action of doubling a square, as the Pythagoreans discovered, produces a certain harmony which they called geometric. Contained within that geometric series is a reflection of the harmony that doubles the cube, expressed as two geometric means between two extremes, instead of the one geometric mean expressed by the square. But, as the discoveries of Archytus and Menaechmus demonstrate, that “cubic” harmony, although reflected in the process of doubling the square, can only be constructed by a completely different process, that associated with the conic sections. Since the cube can generate a square and a square can generate a line, but not vice versa, all three powers can be understood as flowing from the higher principle of generation expressed by the conic sections. In other words, what makes all three powers possible is not manifest, sensually, in any of them. What makes them all possible is manifest only outside all lines, squares and cubes, in the principle of action exhibited by the conic sections.

Now, begins more fun. What makes the conic sections possible? To answer this question, one must first ferret out the contradictions within the domain of the conic sections. This will take us directly to Gauss’ discovery of the complex domain.

While Greek culture made significant advances in this direction, as exemplified by Apollonius’ Conics, the most significant advance was made by Kepler’s discovery of the projective relationship among the conic sections.

To grasp this, first think of the conic sections, as Apollonius did, as the curves produced by a plane cutting a set of cones joined at their apexes. A plane cutting the lower cone perpendicular to its axis will generate a circle. With the slightest tilt, that circle becomes an ellipse. As the plane’s tilt becomes parallel to the side of the cone, that ellipse becomes a parabola. With the slightest additional tilt, the plane now intersects both cones, forming an hyperbola. The top cone was sitting there all along, but didn’t come into play, until the hyperbola was formed.

From the standpoint of the visual appearance of the cone, all four conic sections are formed by one continuous motion of a plane intersecting with the cones. However, nothing can be discovered from this about what makes this conic manifold possible, unless the cognitive paradoxes, that reside “beyond the finite” are brought more sharply into view, metaphorically.

To do this, Kepler applied the method Cusa states in “On Learned Ignorance”:

“But since from the preceding points it is evident that the unqualifiedly Maximum cannot be any of the things which we either know or conceive: when we set out to investigate the Maximum metaphorically, we must leap beyond simple likeness. For since all mathematicals are finite and otherwise could not even be imagined; if we want to use finite things as a way for ascending to the unqualifiedly Maximum, we must first consider finite mathematical figures together with their characteristics and relations. Next, we must apply these relations, in a transformed way, to corresponding infinite mathematical figures. Thirdly, we must thereafter in a still more highly transformed way, apply the relations of these infinite figures to the simple Infinite, which is altogether independent even of all figures. At this point our ignorance will be taught incomprehensibly how we are to think more correctly and truly about the Most High as we grope by means of metaphor.”

Kepler’s interest in discovering the generating principle of the conic sections was not a matter of mathematical curiosity. His demonstration of the elliptical nature of the planetary orbits demanded a higher comprehension, beyond the simple mathematical relationships within and among the specific curves, of that universal principle (power) which made conic sections possible.

This required him, as Cusa indicated, to consider the finite relationships within and among the conic sections, from the standpoint of the infinite. By projecting the above cited process of a plane cutting a pair of cones onto one flat plane, Kepler brought out the infinite divide between the circle and the hyperbola. From the standpoint of Kepler’s projection, the hyperbola and circle were on opposite sides of the infinite. (See Figure 1.)

Figure 1

With this contradiction brought into view, the stage was set for Fermat, Huygens, Jakob and Johann Bernoulli and Leibniz to bring this paradox up to the point which demanded the discovery the complex domain by Gauss.

This was accomplished by focusing on the significance of this infinite divide between the circle and the hyperbola from the standpoint of the generation of Plato’s powers. On the one side, Jakob Bernoulli demonstrated that the circle, as a special case of an equiangular spiral, expressed the transcendental principle that generated all the so-called algebraic powers as a function of rotation. (See Figure 2.)

Figure 2

On the other side, Huygens demonstrated that the hyperbola expressed that same principle as a function of area and length. (See Figure 3.)

Figure 3

In this contradiction, Leibniz discovered something additional. While the circular principle expressed the transcendental number Pi, the exponential embodied by the hyperbola expressed a different transcendental number, that he called “b”, (later called “e” by Euler). (See Figure 4a and Figure 4b.)

Figure 4a

Figure 4b

Thus, both the circle and the hyperbola expressed, in different ways, a principle that had the power to produce all algebraic powers. Each expressed that power with respect to a different transcendental magnitude. An infinite gap lay between them. The question now posed anew was, what universal principle embodied the higher power that had the potential to generate both distinct transcendentals?

For Leibniz, as for Kepler earlier, this question was not posed as a formal mathematical curiosity. His and Johann Bernoulli’s joint discovery of the catenary demonstrated that the “frozen motion” of the hanging chain, expressed as a physical principle the simultaneous unity of both the trigonometric and exponential transcendentals. (See Figure 5a and Figure 5b.) The catenary, therefore, was the physical expression of a still yet undiscovered domain, that possessed the potential to generate all such transcendental magnitudes.

Figure 5a

Figure 5b

Leibniz understood that this higher domain existed outside the boundaries of the senses. Like all universal principles it could only be known with the mind, and so he referred to it as “imaginary” (not, as Euler would later say, “impossible”). This domain produced artifacts such as the ?-1, which posed a paradox because nothing within the known world could produce a magnitude, which when squared produced -1. Leibniz called ?-1, “a fine and wonderful recourse of the divine spirit, almost an amphibian, somewhere between being and non-being.”

The paradox remains regardless of whether one generates the powers by the spiral or the hyperbola. In the case of the spiral, successive angular rotation produces corresponding increases in the length of the radii of the spiral. The lengths increase by the power that corresponds to the how much the angle of rotation is increased. For example, if the rotation doubles, the length of radius is squared. If the rotation is tripled, the length of the radius is cubed. If the direction of the rotation is reversed, the length of the radii decrease, by the power equivalent to amount of rotation.

Similarly with the hyperbola. Equal areas between the hyperbola and the asymptote correspond to geometric increases in length along the asymptote. Thus, if the area is increased by two, the corresponding length along the asymptote is squared. If the area is increased by three, the corresponding length is cubed, etc. If the area is reduced by half, the corresponding length is reduced by the square root, etc.

So, in the hyperbola the areas change arithmetically while the lengths change geometrically. For the spiral, the angles change arithmetically while the lengths change geometrically. The angles of the spiral, and the areas for the hyperbola, were called by Huygens, Leibniz, and Bernoulli, logarithms.

The paradox posed by Leibniz was this: Since increases or decreases in the logarithms always produce a positive length, “what is the logarithm of a negative number?” or, in other words, what has the power to produce the ?-1.

This provoked a dispute with Johann Bernoulli. Bernoulli maintained that the logarithms of negative numbers were the same as the logarithms of positive numbers. For example, he considered 0 to be the logarithm of 1 and -1, just as 1 and -1 are both square roots of 1. Leibniz, on the other hand, recognized that the same action could not produce 1 and -1.

For Leibniz, this matter could not be resolved within the existing domain of accepted mathematical formalism, just as Gauss would demonstrate in his investigation of the fundamental theorem of algebra. The logarithms of negative numbers, Leibniz insisted, had to exist in a domain beyond the visible, i.e., the “imaginary” (not “impossible”). However, Leibniz was unable to complete this work and it wasn’t until Gauss developed his concept of the complex domain, that the full implications of Leibniz’ conjecture were resolved.

In the intervening period, Euler, commenting on the dispute between Leibniz and Bernoulli, developed a formal demonstration that indicated Leibniz, not Bernoulli was correct concerning the logarithms of negative numbers. Out of this came Euler’s famous identity, e^Pi?-1 – 1 = 0, a formula that has been used to torture students and brainwash potential thinkers ever since. For Euler, this was merely a formalism that has no real meaning other than the successful manipulation of symbols according to a regular set of rules. Countless victims have been brainwashed trying to find a meaning in this formalism within the formal mathematical domain. This is not possible, because the ?-1 is not possible in the formal mathematical domain of Euler, no matter how many times he refers to it.

Nevertheless, if looked at from the standpoint of Leibniz, Huygens, and Gauss, we can remove the mysticism associated with Euler’s identity and, using Cusa’s method of transforming the finite into the infinite, bring the matter clearly into view.

In the accompanying figure, the alternating areas of blue and yellow are all unit areas. Moving to the right, equal areas correspond to the logarithms that produce increasing powers of “e”. For example, beginning at 1, (where the logarithm is 0 because no area has been swept out) moving one unit area to the right increases the length from 1 to e. Moving another unit area increases the logarithm of 1 to 2 and the length from e to e². Halving the area between 1 and e produces the length ?e or e^1/2.

When moving left from 1, the principle of equal areas is maintained, but in the opposite direction. The lengths produced by these areas are the inverses of those produced by moving to the right. For example, moving one unit area to the left produces the length 1/e. Moving two unit areas left produces 1/e², etc.

The paradox to which Leibniz referred emerges when one tries to think how the hyperbola can produce a logarithm of a negative number. As is evident from the diagram, moving to right increases the lengths geometrically, while moving to the left decreases them. But because of the asymptotic nature of the hyperbola, the areas can never produce a length on the other side of 0.

As can be seen from the diagram, -1 is accessible, but only if the action detaches from the hyperbola and moves along the pathway around the circle. That is, to produce a logarithm of a negative number, we have to cross Kepler’s infinite boundary between the hyperbola and the circle! Half way around the circle will produce a length of -1. Dividing that action in half will, therefore, produce the action that corresponds to ?-1. Thus the logarithm of -1 can be thought of as a function of Pi and ?-1.

This matter cannot be resolved except from the standpoint of Gauss’ concept of the complex domain. Moving left and right within the domain of the hyperbola yields negative logarithms, but not the logarithms of negative numbers. Consequently, a higher concept that goes beyond simple back and forth action is required. This is exactly what Gauss specified as his complex domain.

As stated in his second treatise on biquadratic residues,: “Positive and negative numbers can be used only where the entity counted possesses an opposite, such that the unification of the two can be considered as equivalent to their dissolution. Judged precisely, this precondition is fulfilled only where relations between pairs of objects are the things counted, rather than substances (i.e. individually conceived objects). In this way we postulate that objects are ordered in some definite way into a series, for example A, B, C, D, … where the relation of A to B can be considered as identical to the relation of B to C and so forth. Here the concept of opposite consists of nothing else but interchanging the members of the relation, so that if the relation of (or transition from) A to B is taken as +1, then the relation of B to A must be represented by -1. Insofar as the series is unbounded in both directions, each real whole number represents the relation of an arbitrarily chosen member, taken as origin, to some determinate other member in the series.

“Suppose however the objects are of such a nature that they cannot be ordered in a single series, even if unbounded in both directions, but can only be ordered in a series of series, or in other words form a manifold of two dimensions; if the relation of one series to another or the transition from one series to another occurs in a similar manner as we earlier described for the transition from a member of one series to another member of the same series, then in order to measure the transition from one member of the system to another we shall require in addition to the already introduced units +1 and -1 two additional, opposite units +i and -i. Clearly we must also postulate that the unit i always signifies the transition from a given member to a determined member of the immediately adjacent series. In this manner the system will be doubly ordered into a series of series.”

In our diagram, the hyperbola is determining the action in the domain of logarithms of positive numbers, while the circle is generating action in the “imaginary” domain where the logarithms of negative numbers reside. If one mentally rotates the hyperbola perpendicular to the circle, as is its orientation within the cone, it would no longer be visible in our diagram. From this view, the hyperbola becomes, “imaginary” and the circle, “real”.

The complex domain is neither the domain of the “imaginary”, nor the “real”. It is the domain of possibility ( potential or power). As Riemann noted it is the efficient metaphor from which emerge, “a harmony and regularity that otherwise would remain hidden.”

To see this, look again at the harmony presented in the last installment. (See Figure 6, 7, 8, 9.)

When the catenary is expressed in the complex domain, the hyperbola and the circle (ellipses) are not on opposite sides of the infinite, but reside together, as a unified network of orthogonal least- action pathways within the complex domain.

If you listen carefully, you just might here, in these hidden harmonies, echos of Beethoven’s late string quartets.

TRANSCENDENTAL HARMONIES

Discoveries indicating the existence of what Gauss would later call the complex domain began with Pythagoras and his followers in the 6th Century B.C, These discoveries, which include the ratios of musical intervals, the doubling of the line, square and cube, the five regular solids, and many others, demonstrated that universal principles expressed themselves in the shadow world of the senses by harmonic proportions. Yet, in all cases, this harmony was never complete. There was always some small discrepancy, some paradoxical dissonance, that indicated a still undiscovered principle. This is why Pythagoras called geometry “science or inquiry”, and, according to Proclus, he thought that each discovery, “sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among sensible objects and so become subservient to the common needs of this mortal life.”

The most persistent of the dissonances recognized by the Pythagoreans were not resolved until nearly 2500 years later, when, Bernhard Riemann, in his 1851 doctoral dissertation, noted that in Gauss’ domain of complex magnitudes, “a harmony and regularity emerge that otherwise remains hidden.”

To discover these otherwise hidden harmonies, we must first take a closer look at some paradigmatic discoveries in which the dissonances arise:

Pythagoras discovered that the concordant musical intervals corresponded to the proportions, 2:1, 3:2, 4:3, if produced by a straight vibrating string. These ratios produced the intervals known as the octave, fifth and fourth, respectively. More importantly, the Pythagoreans also discovered that if these proportions are simply extended, a discrepancy emerged called the Pythagorean comma.(See Fred Haight Pedagogy.) However, ideas are conveyed by the human voice singing poetry, not vibrating strings. The comma, therefore, is not a mere deficiency. It is an indication that a higher principle exists, a principle that actually governs musical harmonies, but which cannot be derived from the manifold of vibrating strings. It can only be derived from the manifold that has come to be known as the well-tempered system of bel canto polyphony, to which many analogies can be drawn to the complex domain. (fn. 1.)

Similar harmonic proportions are expressed by the principles governing the extension of a line, square and cube. Extension of a line produces relationships that the Pythagoreans called “arithmetic”, which correspond to the musical interval of a fifth. Extension of a square produces relationships called by the Pythagoreans, “geometric”, which correspond to the Lydian musical interval. While the arithmetic and geometric are harmonic within their own individual domain, together they form a dissonance, expressed as the incommensurability between arithmetic and geometric magnitudes. That dissonance indicates, as Plato noted, that the line and square were produced by principles of different “powers”.

The extension of the cube produces a third, higher, power, that cannot be generated by the line or square. Nevertheless, this power is expressed in the lower domain of squares by two geometric means between two extremes. But, as the discoveries of, most notably, Archytus and Menaechmus, showed, the construction of the magnitudes of this third power, cannot be generated by the squares among whose shadows it dwells. This cubic power is only generated by a higher form of curvature, such as that associated with conic sections and the torus.

Plato understood that the extension of line, square and cube denoted a succession of distinct higher powers. Leibniz would later discover an even higher principle that transcended all such powers. He called this transcendental principle, “exponential” or, inversely, “logarithmic”, the significance of which will be made more clear below.

Another class of harmonic proportions investigated by Pythagoras and his followers was associated with the five regular solids and the constructability of the regular polygons. The regular solids and constructable polygons were artifacts produced by the harmonic divisions of the sphere and circle. However, these harmonic divisions are bounded. There are only five regular divisions of the sphere, and, at least as far as the Pythagoreans were concerned, the constructable polygons were limited to the triangle, square, pentagon and certain combinations of the same. (fn.2.) The boundaries confronted by the divisions of the sphere and circle express a dissonance with respect to the harmonies governing those divisions.

This general class of principles, that is those associated with the divisions of the sphere and circle also comprise a class of transcendentals called “trigonometric”.

The unity between these two classes of transcendentals exemplifies the otherwise hidden harmony to which Riemann refers in his dissertation.

The first step toward the elaboration of this unity was taken by Nicholas of Cusa, who, citing Pythagoras, recognized that all universal principles expressed themselves harmonically in the domain of the senses. But, Cusa emphasized that these harmonies could only be expressed by the transcendental magnitudes typified by the dissonances identified in the above examples. Cusa, thus presented, the paradoxical proposition that the art of science is to seek out the dissonances and discover the transcendental principle that harmonizes them.

Johannes Kepler, applying Cusa’s insight, provided the first crucial experimental demonstration that physical principles could only be known through this transcendental harmony. This begins with his discovery of the harmonic correspondence between the five regular solids and the approximate orbits of the six visible planets, the discovery of which, Kepler states, depended on Cusa’s emphasis of the dissonance between the curved (spherical) and the straight (planar). Kepler’s further discovery of the eccentricity of the planetary orbits expressed another harmony through dissonance. Unlike a circular orbit, the regular divisions of an eccentric are dependent not on the angle, but on the sine of the angle, which is transcendental to the angle. Additionally, Kepler showed that the harmonic relationships among the orbital eccentricities of all the planets are dependent, not on the simple harmonies of the vibrating string, but on the dissonances indicated by the Pythagorean comma. (See “How Gauss Determined the Orbit of Ceres”, Summer 1998 Fidelio, and earlier installments of “Riemann for Anti-Dummies”.)

Fermat’s proof that the principle of least-time, not shortest distance, governed the propagation of light, is another experimental demonstration of physical action that is dependent not on the equality of angles, but on the proportionality of the sine.

In sum, the discoveries of Kepler and Fermat demonstrate that harmonic relationships in the physical universe are, as Cusa indicated, not expressible by precisely calculable numbers, but only by transcendental quantities a polyphony of dissonances.

The Leibniz-Bernoulli collaborative investigations into the principle governing the hanging chain, provide the crucial step to Riemann’s assertion.

As detailed in other locations, Bernoulli applying the principles of Leibniz’ calculus, demonstrated that the physical principle that determined the shape of the hanging chain was expressed by a proportionality of the sines of the angles formed by the chain and the physical singularity located at the chain’s lowest point. (See figure 1.)

On the other hand, Leibniz demonstrated that this same physical principle was also expressed as an exponential function. (See figure 2.)

Thus, the catenary expresses a unifying physical principle between what had appeared to be two different classes of transcendentals: the trigonometric and exponential. That unity, as Riemann indicates, only fully emerges when seen from the standpoint of Gauss’ complex domain.

The means to discover that harmonic unity, as in a musical composition, is by inversion.

Remember that the exponential and trigonometric functions first emerged as dissonances embedded in the harmonic relationships among objects in the visible domain. Now, think of those objects as artifacts of the dissonances, instead of the dissonances as artifacts of the objects.

For example, think of the circle as an artifact of the trigonometric transcendentals, and the line, square and cube, as artifacts of the transcendental exponential function. (See animation 1 and animation 2.)

This poses the difficulty of forcing the mind, as Cusa insists, away from the simple harmonic proportions among objects of visible space, to the transcendental harmonic proportions among the principles that generate them.

If we use the principle of the catenary as a pivot, we can present, at least in an intuitive form, the harmony of which Riemann speaks. A more complete demonstration will be left to future pedagogicals and to the oral discussions that this installment will undoubtedly provoke.

As previously noted, the catenary expresses both the trigonometric and the exponential functions. Thus, the catenary as the principle of physical least-action, subsumes both the principle of constant length (circle) and constant area (hyperbolic). (See figure 4.)

To this Leibniz added a new crucial conception: the exponential is the curve that embodies the principle of self-similar change. (See figure 5.) This led Leibniz to discover a new transcendental number that he denoted by the letter “b”. (Euler later derived the same quantity from formal algebra and denoted it by the letter “e” which is used today. It is typical of today’s academic frauds that this discovery is attributed to Euler’s formalism, instead of Leibniz’ Socratic idea.)

Figure 5

We have already seen how the hyperbola is generated by the exponential functions derived from the catenary. But, the exponential also generates the circle when the circle is thought of, as it should be, as a special case of an exponential spiral. Keep in mind Kepler’s projective relationship among the conic sections. (See Riemann for Anti-Dummies Part 33.) For Kepler the circle and the hyperbola were at opposite extremes of one manifold, and as such embody a common principle of generation. But, in that projective relationship, there was a discontinuous gap, a dissonance, between the hyperbola and the circle, giving the appearance that the hyperbola was on the “other side of the infinite” from the circle. Only in the complex domain of Gauss and Riemann does that gap disappear and that common generating principle harmonically expressed.

Since both the circle and the hyperbola are generated by the common principle expressed by the exponential, the trigonometric and hyperbolic functions can be represented as complex functions. Riemann created a concept of complex functions as transformations that produce manifolds of action, which in turn produce least-action pathways within that manifold. The study of complex functions formed the basis of Riemann’s work on algebraic, hypergeometric and abelian functions, which will be elaborated in future installments. As a precondition to that deeper study, we provide the reader with an intuitive view of the “otherwise hidden harmony and regularity” that emerges there.

Figures 6, 7, 8, 9, illustrate the complex mappings of the sine cosine, hyperbolic sine and hyperbolic cosine. As can be seen, all four functions express as artifacts, not one hyperbola or circle, but a system of orthogonal hyperbolas and circles.

Figures 10 and 11, and figures 12 and 13 illustrate surfaces constructed by the complex sine, cosine, hyperbolic sine and hyperbolic cosine. In the visible domain the circle is closed and periodic, while the hyperbola is infinite. Yet, when viewed from the standpoint of the complex domain, both are periodic. The shape of the curves rising from the surface, in both cases, are catenaries!

And, this is only the beginning.

NOTES

1. The analogy between well-tempered polyphony and the complex domain is most directly seen in the late quartets of Beethoven. There the characteristic half-step boundaries between neighboring keys and modes are transformed. Just as a solid is bounded by surfaces and a surface is bounded by lines, Beethoven transforms the keys and modes from the bounded to the boundaries of a “musical solid”.

2. It was one of Gauss’ earliest discoveries of the complex domain that the constructable polygons included the 17-gon and all polygons with the prime number of sides of the form 2^{2^n} + 1.

3. For generations students have been brainwashed by the Euler’s mystical algebraic derivation of the unity between the exponential and the trigonometric. The algebraic form of the circle as the curve of constant length is x² + y² = 1, where x and y are the legs of a right triangle. The algebraic expression of the hyperbola is x² – y² = 1. When factored algebraically the circle yields, (x + y?-1)(x – y?-1), while the hyperbola yields (x + y)(x – y).