december 2006 - larouchepac · 2018. 4. 30. · Δυναµις vol. 1, no. 2 december 2006 needed...

December 2006 Vol. 1 No. 2

www.seattlelym.com/dynamis

EDITORS IN CHIEF Peter Martinson Riana St. Classis

ASSISTANT EDITOR

Jason Ross

ART DIRECTOR Chris Jadatz

LaRouche Youth Movement

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On the Cover Myron, 5th century B.C. Lancelotti Diskobolos.

The infinitesimal, caught in the act.

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4

15

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From the Editors Mapping What’s Invisible By Aaron Halevy On How to Do a Stereographic Projection By Ricardo Lopez Are You Living An Imaginary Existence? By Rachel Brown A General Theory of Earth’s Magnetism By Carl Friedrich Gauss

“…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.”

Johannes Kepler Mysterium Cosmographicum

Δυναµις Vol. 1, No. 2 December 2006

From the Editors This issue of Δυναµις goes to print during one of the most dynamic periods in the political history of the United States. On November 7, the Democratic Party received an unexpected landslide victory in the mid-term congressional elections. Observing how the Democratic national leadership campaigned, milquetoast, in the period leading up to the pivotal vote, one must wonder what generated such a sweeping victory. As Lyndon LaRouche characterizes it in his breakthrough paper, The New Politics, “What was done by the LPAC’s LYM in catalyzing the ’18-35’ surge among young Americans in crucial sections of the voting population, was to employ the dynamic method of organizing creatively, around ideas, in such a way that a relative handful of the population was able to evoke a mass effect, consistent with Shelley’s principle – the Renaissance principle – within significant regional clusters of the generation between 18 and 35. These ideas, so put into circulation in a dynamic way, led to the simple decision to go and vote your conscience.”1 LaRouche’s Youth Movement takes full credit for that victory. In the few months prior to the election, the LYM engaged itself in several key activities: First, they targeted the nation’s campuses with an LaRouche Political Action Committee pamphlet, exposing the original “Get LaRouche” taskforce of John Train, who paired up with Lynne “Big Sister” Cheney, in their role of creating a campus Gestapo, which actively prevented serious discussion of politics among university students and professors.2 Second, they developed their improving mastery over Bel Canto choral singing, and used it as a sharp political organizing weapon. And, third, they launched into a focused study of the world’s first experimental astrophysicist Johannes Kepler’s major discoveries, which was spearheaded by a small group of pioneers, who spent several months cracking open Kepler’s New Astronomy, and who then produced the first-ever series of pedagogical animations to illustrate the most important parts of the work. These activities were the tools the LYM used, to spark the creative discussions on the campuses, breaking the students and professors from Cheney’s new campus McCarthyism. A bit more on the second point. Lyndon LaRouche has stated that, the most important work of the LYM is their Florentine Bel Canto voice training, through daily vocal warm-up exercises and a study of the emergence of the Pythagorean Comma in, particularly, Bach’s “Jesu, Meine Freude” motet, and Mozart’s short piece “Ave Verum Corpus.” The residue of the oligarchical attempt to quiet the fire of human cognition, in the form of the separation of Art from Science, soils our society today. Science is thus regarded as the cold, logical investigation, by computers, of how an imaginary, non-frictional universe works, while Art is the existentially lawless expression of the most embarrassing emotions. Of course, there are variations of both views, and there have also been strange attempts to remove the separation, such as contraptions like

“math rock.” But, true Science and Art are really no longer studied. LaRouche offers the correction, that what should be termed “physical science” is the study of the physical principles of the universe, while “classical artistic composition” is the study of the physical principles of human social processes. Both must be studied for a complete understanding of either. Singing is the study of how people communicate those ideas of social organization that correspond to Universal Physical Principles. Thus, singing is one of the most powerful weapons of mass political organizing, as we saw, for example, in our San Antonio special election campaign, where we won yet another race that “Coward” Dean and the DNC tried to ignore. Singing beautifully catches people’s attention and sparks the curiosity. What are they singing about? When someone walks away from the LYM, having heard the singing, they walk away with a higher part of themselves activated. But, there’s more. Recently, LaRouche explained the importance of the experience of the Pythagorean Comma in the Bel Canto choral work: “…instead of having a broad sense of ‘this, plus this, plus this, plus this,’ it's better to have a narrow focus [in physical science]. Because you want the individual human mind to focus on the subject. And you will not get that individual human mind to focus unless you also have the kind of intensity of work on choral music that we have with the ‘Jesu, Meine Freude.’ Because, unless you force people to recognize that the comma corresponds to the same thing as the infinitesimal in physics, the comma is the social equivalent of the infinitesimal in the mathematic domain. And therefore, if you cannot feel the idea, you don't have it! And the big problem, is people who can explain the idea, but they don't feel it.” This brings us to the third key LYM activity – the work on Kepler. As the LYM was launched, during the early onset of the Bush Administration disaster, LaRouche got them started by establishing an educational curriculum centered on Carl Gauss’ doctoral dissertation of 1799. Mastering this paper unleashed a science driver process, and the studies broadened to include the elements of Pythagorean sphaerics (emphatically, Archytas’ construction of the doubled cube), the origins of Algebra, the work of Leibniz and Bernouilli on the Catenary, and branched out into studies of Gaussian curvature, and the beginnings of Riemann’s work on Abelian functions and acoustical shockwaves. The intent, always, was not to pump out a bunch of know-it-alls, but to create a cadre of educators, who could do the work of reviving a truly scientific culture. This work acted like a “yank by the bootstraps” of the 18-25 generation, which now has a small hard core of emerging scientific leaders. In beginning the next phase of the education, the application of Riemannian hypergeometries to economics, in the form of economic animations, it became clear that the educational work


needed some reorganization. “…[O]ne of the ways this thing got into motion,” LaRouche said, “is, I just saw the animations work was not getting to the point of animations. It was too much illustration, and not enough presentation of ideas. And so therefore, we had to get this question of ideas clarified, what is a scientific discovery, how do you make it? What's the difference between that and just simply an illustration?” Gauss was rarely explicit in how he made his discoveries, instead presenting his discoveries as finished products. Therefore, LaRouche reoriented all focus to Kepler, for a true insight into how a human mind functions during scientific investigation. “Now, with the work of Kepler, that's your foundation of all modern science – Kepler. And therefore, this work on Kepler has to be the central focus of all work on science. And you have everything leading into that. There is no writer, there is no scientific mind in modern history, who is more thorough in exposing exactly what his mind is doing, in making the discoveries, than Kepler. So that the training in the mind, by Kepler, exceeds that of Gauss.” This “training of the mind” was the crucial factor in these elections. First, the study of how Universal Principles organize the universe, and, second, applying Universal Principles to organize social processes. The large, unexpected turnout of the youth to vote on November 7, was the result of a crucial experiment run by LaRouche and the LYM. Now, the revolution has begun. The LYM is now on a focused path in science. The current track was launched by the initial mastery of Kepler’s discovery of the principle that generates the orbit of Mars, Universal Gravitation, in his New Astronomy, which is currently being followed up by a study of how this principle expresses itself in the harmonic organization of the Solar System, in Kepler’s Harmony of the World. Next, there will be work animating Carl Gauss’ discovery of the orbit of the asteroid Ceres, and his study of the perturbations of the second asteroid Pallas. The subsequent phase of work will be the capstone, with Bernhard Riemann’s discoveries in physical hypergeometry. Thus, Δυναµις is issuing a call for material. The next few issues will focus exclusively on the work of the LYM in reliving Kepler’s discoveries. Read LaRouche’s A Design for Legislation: Saving the U.S. Economy3 for a look at what ideas, from Kepler, are necessary for the next 50 years of economic development. Every discovery, in this context, that the LYM required for understanding how Kepler thought should be reported. What were the crucial concepts Kepler wrestled with in making his breakthroughs? For example, we could use some good demonstrations of LaRouche’s attacks on statistical mechanics, using material from Kepler’s work. How is the principle of dynamics used by Kepler? How does Kepler develop the idea of physical cycles? How does Kepler crush the Cartesians and future Newtonians? Or, what is the relationship between the actual principle, and what the senses perceive, as demonstrated by Kepler? How does Kepler discover the

infinitesimal? How does Kepler lay the foundations for an Anti-Euclidean geometry? What exactly is the Kepler Problem? It is this kind of discussion that must take place, so the movement can rapidly hone in on Kepler’s discovery of Universal Gravitation, and then carry this over into the domain of Physical Economics, and the crafting of National Economic policies. If your insight is useful, submit it to Δυναµις. This issue of Δυναµις presents material from the work the LYM has done on Gauss, particularly his work on physical curvature. The articles should be read as observations, by the LYM, into the mind of a scientist who was steeped in Kepler. Gauss’ teacher, Abraham Kästner, in defending Gottfried Leibniz from the attacks unleashed in Europe following his death, revived Kepler’s method, explicitly, in launching the epistemological battle around “Anti-Euclidean Geometry.” Gauss, the product of Kästner and his circles, thence laid the foundations from which Riemann established the first truly anti-Euclidean geometry, freed from all assumptions. The first article, by Aaron Halevy, is an account of the history of mapping, leading up to the work of Gauss, and puts it into the context of LaRouche’s work on Physical Economy. (This was a response to a challenge question from the Los Angeles LYM’s Curvature Group.) The second, supplementary article, by Ricardo Lopez, is an example of a particular conformal map, the Stereographic projection, which figures heavily in Gauss’ work. The third article, by Rachel Brown, develops Gauss’ concept of the Complex Domain, how it was elaborated by Riemann, and gives an application to economics. And, finally, Tarrajna Dorsey issues a translation of the opening paragraphs of Gauss’ General Theory of Earth’s Magnetism. This work formed a basis for Gauss’ concept of physical potential, which would lead into the work of Riemann on Abelian functions and physical hypergeometry. As in all useful scientific work, there are leaps of development. This issue of Δυναµις is a snapshot in time of the beginnings of a scientific renaissance. It will be looked back upon from a future organization of society, which was thus shaped by a youth generation who had been led onto the stage of history by a growing cadre of true scientists.

Peter Martinson Riana St. Classis

Editors 1 Lyndon H. LaRouche, Jr., “The New Politics,” EIR, December 8, 2006 2 “Is Joseph Goebbels on Your Campus?” LaRouchePAC pamphlet, October 2006. www.LaRouchePAC.com 3 Lyndon H. LaRouche, Jr., “A Design for Legislation: Saving the U. S. Economy,” EIR, November 24, 2006

Mapping What’s Invisible Halevy


Mapping What’s Invisible Aaron Halevy1 Not too many years into the 1700s, the Ptolemaic clouds once again descended on science. The formalists Euler, LaGrange, D’Alembert, et.al., who were haters of the great genius and universal humanist, G.W. Leibniz, had, through Venetian tricks, taken science into the Ivory Tower, where only the elites were allowed. They were holding her in a confinement cell, with no trial, somewhere in Europe, to torture out of her, Dick Cheney style, any of the hope she had left. These analysts and their like had brushed over almost all scientific investigation to that time, with a Euclidean-smelling style of formalism, a formulism that reduced great scientific discoveries to mathematical equations, and reduced man’s creative spark to smoldering ash. This sinister attack on science reeked across Europe and froze all real progress in its wake.2 This was not a coincidence of events - it was a deliberate conspiracy from the anti-human Oligarchs of Europe collaborating with the remnants of the Venetian Empire. The intention of this was to keep from the masses, or as they thought of them, human cattle, all real science. This attack had been waged in the schools and intellectual circles, from 1714-1853 for the purpose of eliminating the positive classical humanist renaissance in thinking and culture that had been sparked by the great minds of Cusa, Kepler and Leibniz, a renaissance whose foundations lay in the concept that each human is a creative force who can know, through rigorous scientific and philosophical investigation, the powers that govern the universe, the truths which lie beyond the senses, and with this knowledge have the power to willfully change the entire universe. This very same attack on the minds of the people generally, has been the method of the ruling elites since the time of ancient Greece, as proven in Aeschylus’s story of Prometheus Bound, where Zeus has Prometheus chained to the rocks to be tortured by the elements and have his liver picked out by a vulture every day for eternity. Why? Because Prometheus gave man fire, or, more precisely, knowledge. This battle rages on still. Any honest person’s investigation into the operation, of both the 1960’s Congress for Cultural Freedom and the Lynne Cheney-led takeover of the American educational institutions, will show that this is the reality of today. At this moment, Lynne Cheney and her cohorts are trying to vector the development of young Americans into becoming total Neo-Con zombies, zombies who not only cannot argue against war, torture, and genocide, but even logically accept it and promote it in some cases.3 It is therefore crucial, from our standpoint today in modern history, that we gain the weapons of truth and scientific rigor, to fight this old dragon of intellectual suppression of the youth. It is necessary that we engage in an honest discussion with our

peers and younger siblings, on necessary questions of worldwide importance. The current crisis, that of dealing with the mortgage collapse and the derivatives market breakdown, in the US and therefore the global economy, requires us today to deal with the question, What is economics science, or scientific economics? What is the way to think about the organization of policy, such that we do not end up with a potential dark age on our doorstep as we have today? This essay is devoted to such questions.

What is the Universe that Humans Can Know It?

Imagine an Ancient poet, before the Oligarchy had brainwashed him to assume that everything came from myths or self evidently true textbooks. Imagine the wonder at his first observations of the nighttime sky, not looking up at the stars in a romantic fantasy, but imagining the causes, and trying to discover what was the reason he was there. At first, it looks like a million beautiful sparks or dots on a black background. But the more time he spends watching those dots, the more he begins to develop an idea in his mind about the changes in the positions of those dots. After one full night of observing and contemplating these dots, he realizes that all of them are moving. And after many days of this observation, he sees that some of these dots are moving differently over time in the opposite direction of the nightly motion. And then, after the winter, summer and back to winter, his observations bring him to understand whole cycles of these “backwards moving dots.” It becomes for him a set of very complex motions, some slow, some fast, but definitely now, he sees a lot more, in his mind, than he thought he could see before. This is described in Plato’s Timaeus, “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…”4 Thus, the stars and their motions aid the mind in the study of these invisible realities. You observe the phenomenon with your senses, and develop a concept of the causes of these changes, never being able to see the cause of this phenomenon, but knowing it to be more real than the former. And so, our poet, thinking like every creative man has before and after him, realizes that he needs to communicate this to other people, and to people yet unborn. He will need something more than just a mental picture; he can’t do too much with his vivid memories. He needs a means of describing it, a way



anyone can understand what he sees in his mind to be clear. He needs a map. As Lyndon LaRouche puts it, “The principle problem posed by this…is, broadly speaking, whether or not it is possible to bridge the projection of the real world upon the perceptual world, since our sense- perceptions know only the latter? In other words, putting the same point in a more fruitful form, can we understand the bounding-principle of visible space adequately? Can we understand that bounding principle sufficiently well, that we can ‘translate’ appearances back into the real form of the occurrences which cause the appearances? Once we have succeeded in making such a translation, can we prove that such a translation affords us an improved mastery of nature, a quality of improved mastery not achievable without aid of that particular choice of translation? How, then, can we define the bounding principle itself, in a manner adequate to the objective we have just stated?” 5 It is this method, embedded in all great scientific discoveries, which must become now a self conscious process in the mind of every student and every economic policy maker, if we’re to survive this historical crisis. If you’re outside at night and you point to a star and call out to your friend, “look at that one,” it’s not so obvious which one you are pointing to, because your friend is standing somewhere else. If you move over to him, and stand where he is, the star is still in the same spot it was before; if you’re in a car while driving along a unswerving road, unlike the trees and mountains around you, the stars are staying in the exact same place; somehow you seem to be at the center of this sparkling tapestry wherever you go. So why can’t your friend see which star you’re talking about when you point it out? You must describe to him where this star is in a set of actions; beginning from the spot you both independently share, the zenith (directly above you). While you both face the same direction, the first motion of your arm, is coming from the top-down, to the same height as the star; secondly, you move your arm, now, right or left, depending on the direction that that star is. He sees it! Now, what is this double motion you have done with your arms? What could you represent this as? We know from before that it is irrelevant where you stand to see these stars, and also to describe any of the star’s positions, you need these two angular directions. So what is this? How can you communicate it as a concept? Instead of a memorized answer, you have discovered something new, something you from now on can call a sphere. To map this out you immediately have a few problems. One, you don’t have a big sphere to carry around with you, and two, it’s not so easy to make a perfect sphere of infinite radius. You must discover a way to make sure that all is correctly translated from the sphere onto the plane. In ‘your celestial sphere,’ you’re at the center, above is your zenith, and every star is at an intersection of the two types of angles (see Figure 1). These co-ordinates can be thought of generally, as a net of circles all over the sphere that intersect at 90º. Call these circles longitude and latitude. The latitude circles are a series of circles parallel with

the horizon that become smaller toward the top, but the longitude circles are all the same length. These longitude circles also all pass through the same two poles, top and bottom; the center of each of these types of circles is at the center of our sphere. Thus, because you are at the center, when you draw a line on your celestial sphere to connect any two stars, that line is actually a part of one of these types of circles, one can call these circles great circles. To have a good map, it must maintain the angles that are on the sphere, and the distances must be proportional. Since you are at the center of your own “celestial sphere,” your first attempt to map this might be, to draw a line from you, through the sphere, to an imaginary plane sitting above the sphere, tangent to the zenith point.6 This map is called a “Gnomonic Projection” (see Fig. 2). Looks good, are there any problems with this?

Figure 1



Figure 2 – The Gnomonic Projection

What must we map from the nighttime sky? The pole, the horizon and everything in between. As the night goes on, unless you’re at the North Pole, you’re going to see some stars move out of your visible sphere, and new ones come in. If we think of a thin spherical sheet on top of our celestial sphere with all the stars painted on it, throughout the night it will look as if this sheet is spinning, pivoting around a single point, the north star Polaris.7 This pivot point, of our celestial sphere, which differs to the observer at different latitudes, is for us here, at 35º above the horizon. You notice also, night after night, all the planets are rising in the same place along the horizon, and later the sun and the moon rise near there too. Think of this as a belt around your celestial sphere -it is called the ecliptic pathway.8 Look back to our first map, the Gnomonic Projection. Where is the horizon in this projection? Is it placed far outward on the plane? No, even further, this procedure shoots the horizon circle to infinity –not very useful.

Figure 3 – The Second Projection

Try another map. This time we will project from the zenith, through the latitude circles, on to the plane. This projected map on the plane gives you nothing in the middle, an exact mapping of the horizon, and all coordinates above this horizon are going to grow as you move further and further out. Look at the intersection of the projected horizon with the lines coming from the center; they intersect at 90º, just as they do on the sphere,

when these lines come down the side to touch the equator at 90º. Or again, if you imagine a triangle on the sphere, drawn from the latitude line as the base, and the equator as the point of intersection; the spherical triangle, whose internal angles are more than 180º becomes a triangle, on the plane, made of curved lines, equaling also, more than 180º. This quality is called equal angular projection, something very important for a map.9 But, this time your zenith point will be at the infinity of your plane (Fig. 3). Again, not a useful map. The first projection gave us the zenith and all coordinates around it accurately, but, it did not include the horizon. The last projection gave us a map with equal angles, the projected horizon was exactly the same size as it was on the sphere, but the zenith was unmappable. Because we are on a sphere, it’s safe to say there is another half on the other side of our horizon. Why not project from the exact opposite zenith (Nadir) to the horizon plane? (Fig.4)

This particular type of projection is called a “stereographic” projection, in Greek, stereo - solid and graphic- to write.10 This gives us all we need! The zenith becomes the center, the horizon is the same size as in the sphere, and the ecliptic, with all it encompasses, can also be mapped onto our plane. This is the one we want for our map. Now construct it geometrically! Take a slice of your sphere and look at it from the side (Fig. 5). In this projected slice, you see the equator/horizon of the sphere and the ecliptic both are transformed into lines (they are great circles in fact, so they become lines on this plane). Now draw several lines, each starting from point N, extending to the zenith, the pole, the ends of the horizon, and the ends of the ecliptic. And make sure every line is followed till it touches the horizon line. What you’ve now done, is translated the lines and circles to be proportional. Some have shrunk in size and others have grown, but now they are all relative to your horizon plane. The next step is to take those points and use them to draw a new map, below this one.

Figure 4 – The Stereographic Projection



Figure 5 – Projection Construction of Stereographic Map

In the picture, you see that all the lines, in the original map, became circles in the new one, and the circle became a line. We are looking at the original map now, from the top-down. Map the projected horizon, zenith, pole star and ecliptic pathways onto this new map. Compare the two in your mind, do you see the original sphere as implicit?

Figure 6 – Side View of Stereographic Projection

Now continue this same process for all the sphere’s latitude circles, from our horizon downward to the 40º latitude circle, and again upward to the Zenith (Fig. 6).11 Once you have completed that draw all the lines from the zenith in 20º increments to represent the longitudinal lines. This is the

stereographic projection; it is a projection without any angular distortions.12 All you have to do now, is take measurements of the night sky with some instrument, and place those stars, in their places, on your map.13 There is no trace of who actually discovered this map. Thales is sometimes named the discoverer of the method of the Gnomonic Projection -but not the stereographic projection. A clear picture of who discovered the stereographic projection, is contested among many different sources in academia. Most likely, it is attributed to Hipparchus (160-125 B.C.), while others cite the illustrious Ptolemy. What is even less discussed, but definitely more important, on this topic of maps, is the question of why this projection was developed. What was the culture, philosophy or the political state of mankind at this time, that this conceptual breakthrough was possible? As all tools are created for some acting purpose, so too must it be with this projection. It survives today with us, only as a “noetic fossil,” or a cognitive artifact. The sphere and the plane are two totally different species of surfaces, this stereographic projection gives you a truthful map of their relationships. This new map doesn’t look like a sphere anymore, but it effectively is a sphere, in terms of its relationships and actions. As you projected the sphere to the plane, so does the universe project its actions to your visual sphere. The task now, for science and economics, is to create a map that reflects the invisible. Think of Eratosthenes, who was alive in Greece 200B.C., measuring the angles created by the sun’s rays, on the same day (the summer solstice) at two cities, 200 miles apart. This is how he discovered his highly accurate measurement of the circumference of the entire earth, proving it to be spherical. He calculated the circumference of the earth, to be 25,000 miles; modern measurements reveal this circumference to be 24,860 miles. Eratosthenes efficiently mapped the invisible. Until the 1960’s U.S. Apollo project, no one in mankind had “seen” the spherical shape of the earth, yet Eratosthenes, 2100 years earlier, saw it with his mind!

Discovery vs. Definition In circa 260 B.C., the Greek sophist Euclid, a librarian at the library of Alexandria, had taken all the discoveries out of science and put all the answers in his Elements. For example, keeping in mind the way we had discovered the sphere before, here is how Euclid defines it, in the 11th book of his Elements, “Definition 14: When a semicircle with fixed diameter is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.” It was these Thirteen Books of castrated geometry, which became the fatal cause for the blocked outlook of so many scientists, philosophers, and people generally up until today. A method of thinking without the thinking: an assumed set of “self-evident



axioms” to be taken as truth no matter what the consequences. Much like a University textbook today. Science became, then, chained in the prisons of formulas not to be fully released for at least 1200 years. This degeneracy in thinking expressed itself during the dark ages; governed by the Oligarchy controlled church, science had become more about accepted belief and faith. It was as if people had thrown away the scientific method, and began to believe in magic once again; they gave the causes of the phenomenon they observed to some unknowable force, never to be explained and not worth thinking about. For example, people believed that the Earth was at the center of the universe, and all the phenomenon, the motions of stars and planets in the heavens, could be explained by the “self evident truth” that the Earth was at the center and everything moved around it. Another example is from Cosmas and Isidore of Seville, who shaped their geographic maps on their strict accordance to the bible, and made the earth out to be completely flat.14 Maps became the mere entertainment of Oligarchs and paranoid idiots alike, who would, for the lack of knowledge of the content of this or that area on the map, would simply place a big ass sea dragon, or a monster. If you wanted a very detailed map of a town or state, and you wanted it as accurate as possible, how would you do it? Here also, we are dealing with the same principles of projection from before; to map all the land and water the same as in reality, you would need a giant sphere. Most people fudged it a little here and there, which is possible when you’re dealing with smaller areas, but if you want to map the whole world, you begin to have some serious difficulties. Thus there is a need again for a map similar in smallest parts15. The requirements for making a map conformal are two; equal angles, as on the sphere, and lines mapped proportionally. The stereographic map we made above is a conformal map, but there are many other types that were developed after it. For example, if you want to map the sphere on to a plane, you can map the sphere to something else first, and then map that to the plane. As Gauss proves in his General Investigations of Curved Surfaces, the plane has the same amount of curvature as the cylinder and the cone. Take a piece of paper and draw anything on it, a face or a line, and then roll it up into a cone, then into a cylinder. What happens to the picture? Is it stretched or distorted in any way? This shows, that as far as curvature is concerned, there is no difference between these surfaces. With another type of projection, called “conical projections,” developed by the Greeks, you can take a cone and place it on top of the sphere and project the lines of latitude into circles along the cone, and the longitudinal arcs become straight lines (Fig. 7).16

Gerardus Mercator (1512 -1594), once a young member of the Brotherhood of the Common Life, became one of the main contributors to the developments of mapmaking. His map of 1569 is titled, “New and Improved description of the lands of the world, amended and intended for the use of navigators.” This map has the property of conformal shape, different from conformal angles.

Figure 7 – Conical Projection

His map deals with a special type of cylindrical projection (Fig. 8). To understand the difference, between the cylindrical projection and Mercator’s projection, first, imagine a sphere, with the point of projection at the center, projecting outward onto a cylinder. With the cylinder touching the sphere, only on the equator, this makes the projected equator the only part that is exactly the same in scale to the sphere. Right away you can see that the North Pole of our globe here is going to be projected to infinity.

Now, examine the limits within this method of projection. On the sphere, all the meridian lines are equally spaced out at their intersection of the equator, but they all come together to a point at the North Pole. Not so in our projection, all the meridians stay an equal space from each other; staying parallel from the equator up. On the sphere, the latitude circles are getting proportionally smaller as you go up to the

North Pole; on our projection the latitude circles all become lines, equal in length to the equator, and the distance between them is geometrically increasing as you tend towards the projected pole at infinity.

Figure 8 – Cylindrical Projection



Mercator’s map (Fig. 9) is not exactly this cylindrical projection, for he saw the same problems that we just did and corrected for them. He made it such that the degrees of latitude and longitude in the map maintain the same proportions of those on the surface of the globe; that is, to make the relationship of the ascending space between the latitudinal lines proportional to the change of the meridian lines.17

Figure 9 – Mercator’s Projection

But, you still have a problem with the scale, your equator is the same length as your 60 º latitude line. Now examine the map some more, did you know that Greenland is as wide as Africa? To account for this Mercator created a changing scale: the further you went up, the longer the latitudinal lines became, (i.e. 1 in. = 1000 miles at the equator and 4 in. = 1000 miles at the 70º latitude.) So what is the use of this map, if the area is so skewed? Mercator wanted to produce a chart with which a navigator could draw a straight line between two points and immediately determine the constant course he must steer in sailing between those two points. This was the property of Mercator’s map: to get from one city to another, you draw a line with a ruler on the map, and if you make sure that every time you pass a longitudinal line with your boat, you are at an equal angle as before, you will always get to the destination. This creates what was called a “Rumb line.” On the map draw a line from South Africa to New York, if you extend this line in the same direction it will go to the top of the map. On the globe, the line you just drew looks like a small pitched spiral. This is called on the sphere a loxodrome, and any line on the map will, on the sphere, spiral to one of the poles every time. Mercator sought to convert the curving rumb line to a more easily plotted straight line.18

A neighbor of Mercator’s said, “Mercator set out to describe the world, most accurately, projecting the globe onto a flat plane by a new and convenient device, which corresponded so closely to the squaring of the circle that nothing, as I have often heard from his own mouth, seemed to be lacking except formal proof.”19 The problem is that scientific geniuses did not spring from Mercator’s method. He had no explicit investigation into the nature of the human mind in the universe, or the individual human’s place as a creative force. He did come up with a useful discovery, but he would not be able to tell you how he made that discovery. For this reinvestigation into science as a science, we must look toward a new generation 200 years after Mercator.

Shadows vs. Substance In 1777 Carl Fredrick Gauss was born to two peasant parents in the small German town of Brunswick. At the age of 14, Gauss was swept up into the humanist networks of Duke Carl Wilhelm Ferdinand, who was building an army of intellectual youth, in the tradition and intention of G.W.Leibniz before him.20 He went to several schools in Germany which had been set up by Leibniz and his collaborators, such as the Volksschule, the Collegium Carolinum and finally Göttingen University, where Gauss and other students were taught by leading German intellectuals, such as E.A.W. Zimmerman and Abraham Kästner, who “…regard[ed] themselves as carriers of a new culture, of the freer and nobler education of the taste and the heart.” 21 Gauss had made many discoveries from his youth to his mid 20’s, ranging from geometry, to mathematics, to astronomy and physics. He was vigorously in pursuit of not just a non-Euclidean mathematics, but an anti-Euclidean mathematics, with no-assumptions, but completely based on knowable truths. A mathematics derived from physics so to speak; a method for a mathematics of change, with which man could measure and map the changes in that universe. In his hunt, Gauss took up the challenge of Geodesy in 1822-1832. In a letter to his friend Olbers, Gauss wrote of Geodesy, “The most refined geometer and the perfect astronomer; these are the two separate titles which I highly esteem with all my heart, and which I worship with passionate warmth whenever they are united.” Looking back to the question of conformal mapping, Gauss turned the question on its head in his Copenhagen Prize Essay. The opening paragraph says, “General Solution of the Problem to so represent the Parts of one Given Surface upon another Given Surface the representation shall be similar, in its Smallest Parts, to the Surface represented.”22 Here he did what we did above with the sphere and plane, but now with any two surfaces, transforming them mathematically with complex numbers. In the paper, he focuses on the development of curvature as a more general method, i.e. less on the shapes themselves and more towards the measure of the curvature of space, in and of itself. Trying to map any surface boils down to this concept: the



lengths of all shortest lines, 23 emanating from a point on the first surface, must be proportional to the corresponding lines in the second surface, and that the lines that connect these lines will have equal angels of intersection on both surfaces. Remember, this is exactly like our stereographic projection; the lengths on the plane were proportional to the sphere and the angles of intersection of the projected lines were also equal. On our celestial sphere, any star can be located on our sphere, with two angles -one for latitude and one for longitude - call these angles t and u respectively. Now think of any surface: it must also have two variables, no matter how irregular it is. To discuss curvature, it is only a matter of figuring out the equation that describes those variables for whatever surface you have. Gauss takes these variables t and u that one can determine for any surface, and creates a universal description for any line on any surface:

22222222 )()(2)( ducbadudtccbbaadtcba !+!+!+!+!+!+++ This is important because now you can find the line equation for any two surfaces, and find an action, or function, to map the first surface onto the second surface. And after this mapping, now you can look at the changes that took place between the transformation of the two surfaces. As in our Sphere to plane, the great circles, longitudinal lines, became straight lines on the plane and the latitudinal circles became a set of exponentially growing circles. Or think of it inversely, you can take a plane and map it onto the sphere; what was once infinity for the plane is now the zenith pole on the sphere.

Figure 10 – Gauss’ Geodetical Measurements.

(This picture was once on Germany’s 10DM bill.)

In 1826 Gauss was tasked to geometrically survey the entire Kingdom of Hanover. This proved a good opportunity to take on a practical problem, something he could use to flesh out his “higher geodesy.” He worked strenuously on this project. He took 3,000 coordinates personally, and at the toughest times, he was only getting 2-3 hours of sleep every day. Gauss said never before had he worked so hard and received so little (Fig. 10). After the year’s worth of measurements, and diligent work, he produced his General Investigation of Curved Surfaces in

1828. With this paper, Gauss revived the method of the Greeks and Sphærics explicitly, where he begins to look at the measurement of curvature (invariance), in relation to a sphere with a fixed curvature. He takes the change of the pathways on any surface and maps that (the curvature) onto the sphere; then looks at the change in curvature itself.24 Gauss later defines the general equation of a shortest line, a line that is as a string that is tightest to the surface as the combination of several differential coefficients. It begins with the equation above,

22222222 )()(2)( dqcbadqdpccbbaadpcba !+!+!+!+!+!+++ which was the equation of the shortest lines on any surface (here, t = p and u = q), then he takes the coefficients, (a²+b²+c²) = E , (aa´ + bb´ + cc´) = F, and (a´² + b´² c´²) = G then

222 dqGdqdpFdpE ++

is your general form for lines and, later, shortest lines, in Gauss’s Curved Surfaces paper.25 These shortest lines and Gauss’s method of curvature prove to be the crucial stepping stones for Riemann and his revolutionary breakthroughs.26

2000 Year Old Shackles on Science

Bernhard Riemann (1826 - 1866), a student of Gauss, was transferring from student to teacher in 1854 when he presented his axiom-shattering paper, On the Hypotheses which Lie at the Foundations of Geometry.27 This paper would prove to be the necessary blow to formalism in science, and pave the way for a real scientific approach, one which LaRouche has used for his developments in the science of economics. Riemann had the guts to come out publicly with what Gauss had been working on, in silent, for his whole career: a purely anti-Euclidean system of mathematical physics. He writes, “It is well known that geometry presupposes not only the concept of space but also the first fundamental notions of space as given in advance…The relation of these [assumptions] is left in the dark; one sees neither whether and how far their connection is necessary nor whether it is possible… From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have labored upon it.” This is a clear trumpeting declaration, that Euclidean geometry assumes the nature of the universe a priori, with no grounds to do so and therefore one must re-discover the nature of space without any assumptions. This should be seen by the reader as the same thread of method employed by the Greeks, Cusa, Kepler and Gauss, to make their scientific breakthroughs.



Think about the universe, not the universe of science fiction movies, but as the entire universe is reflected everywhere and in everything. To determine a concept, the mind has to develop a multiplicity of actions into a one, as temperature is a concept of heat and time. Riemann discusses more generally this method, by replacing Euclidian assumptions with an n-dimensional manifold, where space is a special case of a triply extended manifold. Without previous assumptions, think of the manifold you are in as defined by physical space itself, just as you determined the sphere to be the manifold/surface you were observing, not through assumptions, but with experimentation. Riemann also defines the distinction between the beginning of the investigation itself, i.e. where is it taking place? “Notions of quantity are possible only where there exists already a general concept which allows various modes of determination. According as there is or is not found among these modes of determination a continuous transition from one to another, they form a continuous or a discrete manifold; the individual modes are called in the first case points, in the latter case elements of the manifold.” After defining the concept of the n-fold extended manifold, he poses the problem of measuring relations in the different manifolds. “…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.” In determining such measurements, Riemann uses Gauss’ “shortest lines” from his curvature and geodetic work for the measurements for any surface.28 In this paper, Riemann uses an n-manifold type of Geodesic, incorporating now, not only a surface with two variables, (p and q before) but now an infinite number of variables are possible. He says, “if the location of a point in the n-dimensional manifold be expressed by n variable quantities, x1, x2, x3, and so on to xn, then the determination of a line will reduce to this, that the quantities x be given as functions of a single variable.” This becomes the line element,

22

3

2

2

2

1 ndxdxdxdxds ++++= L and later it becomes more

generally, !=2

dxds .29

The Fraud of Globalization

LaRouche elaborates the task of mapping an economy, in his book So, You Wish to Learn All About Economics. He writes, “For us as for Riemann, experimental physics centers upon those unique experiments which prove mathematical (geometrical) hypotheses pertaining to the continuous manifold by means of experimental observations made in terms of the projected images of the discrete manifold. This possibility depends upon a geometrical principle of topology, invariance. In first approximation, invariance identifies those characteristic features of the geometry of a continuous manifold which are “preserved” through the process of projection as characteristics

of the images of the discrete manifold. In sound approximation, higher-order invariances identify those changes in the continuous manifold which are carried over into the discrete manifold as transformations in the invariance of the discrete manifold. Relativistic transformations in the metrical properties of action in the discrete manifold belong to this second, higher order class of projective invariances.”30 The pinnacle of mapping the invisible and a mathematical-physics of principles today is Lyndon LaRouche’s LaRouche-Riemann Method. To begin to grasp an approximation of this, take what LaRouche said above, and apply it first to the simple idea of the difference between a line, surface and solid. (Fig. 11) These are completely different types of geometrical concepts; they each have a different power in their creation and are entirely different in each of their extension processes. For example, the method of doubling the line is inferior to the doubling of the square, and the doubling of the cube is ontologically different than both of the former. The line exists in the square and also in the cube, such as the square exists by itself and also in the cube. What, then, is the difference? Are they the same lines? No. They can’t be, because of what was done before, in their individual process of doubling. We are “seeing invisibly” a total change in topology from one geometry to another, yet it looks the same in each case. You can see that the only thing that remains the same is the appearance. View this question once more in the context of Gauss’s 1799 paper, New Proof of the Theorem That Every Algebraic Rational Integral Function In One Variable can be Resolved into Real Factors of the First or the Second Degree.31 By the end of his paper, Gauss has developed a solution to algebraic functions, not in an algebraic solution, but using geometrical surfaces, each with unique specifications to an equation. For example, take the equation x² - 3x + 5. Gauss splits this into two equations, two complex functions, which describe a physical geometry…

1) r²cos2φ - 3rcosφ + 5 2) r²sin2φ – 3rsinφ

These become two similar surfaces which cut curves into the zero plane; these curves intersect each other and at this intersection the solution to the equation is given. Now more deeply, all surfaces of the x² “species” will have 2 humps, cutting out 2 curves, moving above and below the plane (Fig. 12). These are the boundary conditions in every equation, which ever numbers you please, (e.g. 27932

11

3522++ xx )

as long as the x is to the power of two, it will always give you two, two humped surfaces, which cut out curves on the plane, and intersect only twice. When a new power is added, e.g. if the exponent is raised to the 3rd power, x³ + Ax² + Bx + M, the entire geometry changes as shown here (Fig. 13). The boundary

Figure 11



condition has changed; you now have 3 humps per surface. Now add one more power, x4 (Fig. 14). These surfaces are each a certain manifold, characteristically defined by the “power” of the equation, not by their particulars. Now think of an economic process, if you add revolutionary ideas to the manifold that all the interaction takes place on, it changes the entire manifold and reorganizes all of the relationships in it. Or, think inversely, that the removal of these ideas in practice of the economy, results in a collapse, as we are experiencing now with the destruction of the Auto sector and the stupefying of the American population. Like FDR’s infrastructure projects, or the American Revolution, the new developments in economy change all of mankind’s relationships to each other, and more fundamentally, mankind’s relationship to the universe as a whole. The introduction of new cultural ideas or new technology, willfully, is the subject of economic science. Truly, the science of science. What was once “impossible,” has now been discovered to be a potential pathway to take. An increase in the quality of living and maximum capacity of the humans living in a region, is now a physical boundary, determined by the ideas that are being exercised in that economy. It becomes clear that the universe responds to our scientific understanding of these economic principles. “These changes in potential, in internal geometry, of the continuous manifold, are reflected (“projected”) upon the mirrored appearances of the discrete manifold, as relativistic changes in the lawful composition of action, as action is reflected as a phenomenon upon the discrete manifold.” “In between those critical points of relativistic phase-change in processes, the consistent local lawfulness of action within that interval is relevant to physics’ fundamentals only as this empirical data’s analysis describes that localized lawfulness. It is the comparison of the difference between two such sets of local laws, for successive phase-states of relativistic transformation, which is the most interesting feature of such local laws, ‘interesting’ because of the way such local laws are defined from the standpoint of reference to the critical transformation. Physics is thus viewed as the matter of willfully orchestrating desired changes in the local laws of the universe.”

“This is, broadly speaking, the map we require for attacking our problem of defining negentropic functions in economic processes.

1) The net work characteristically accomplished by the universe is an increase in the potential of the universe. This is a negentropic transformation, equivalent to an increase in the universe’s energy-flux density. The power to accomplish this work is the proper definition of “energy”.

2) It is the introduction of new singularities to an economic process, to the effect for introducing phase-changes representable as increased energy-flux density, and thus raising the potential expressed as the characteristic action within the economy as an integrated entirety, which expresses this “energy” in the economic process.

3) The most interesting, most powerful form of such transformations in potential of an economic process, are those new technologies which are themselves based on relativistic physics.”32

This question of mapping is directly related to communicating ideas of all types; what is music and art but a sensual representation of that idea, provoking your mind to hypothesize the original idea of the individual artist? And such it is with the composer of the universe - to know his art, is to know his mind and how his mind creates. Where would mankind be with out maps? Is it possible to know a change in

a process at all without a map of the changes? Could man first know anything about the different seasons without a map of the changes in the sun’s pathways? More simply, even today, when a person is lost, a map is crucial to getting home, or back on the right direction; the same is such with a nation or a culture. References 1 [email protected] 2 David Shavin, “The Courage of Gauss” (unpublished 2004 manuscript) 3 “Is Joseph Goebbels on Your Campus?” LaRouchePAC pamphlet, October 2006. www.LaRouchePAC.com.

Figure 12 – x2 Surface Figure 13 – x3 Surface Figure 14 – x4 Surface



4 Plato, Timaeus, (Rep. 529e-530b), The Collected Dialogues, (Princeton University Press, New Jersey) 5Lyndon H. LaRouche Jr., “Mathematical Physics from the Starting-Point of Both Ancient and Modern Economic Science,” EIR Special Report, January 1984, (p. 23) 6 Most of these early projections incorporated cones, where these cones mimic the circles of latitude. 7 This phenomenon is caused by the earth rotating around itself, something that was not so ‘self evident’ to anyone (unless you can feel, while completely sober, that the earth is moving). 8 Also, here are all the constellations for the zodiac, which were developed, not as pagan mystical symbols, but as something behind the “wandering stars” (planets), to be mapped against, so one can map the change. 9 In Gauss’s Copenhagen prize essay there is a mathematical proof for this - it is called later the Cauchy-Riemann Equation. 10 Richard Knox, Experiments in Astronomy for Amatures (New York : St. Martin's Press, 1976) 11 Here is shown only the top part of the sliced piece, the reader should construct the rest on his/her own. 12 NB: If you place the relative distances of your concentric circles on a linear graph, you will find the potential birth of transcendental functions. 13 These paradoxes have been carried over through the history of astronomy. This map is called the Astrolabe. 14 David Greehood, Down to Earth: Mapping for Everybody, (Holiday House: New York, 1944, 1951) 15 Gauss in 1822 discovered and named this, more specifically, a “conformal map.” 16 “However, practically all conics in use are mathematical, rather than shadow casting jobs.”- Greenhood, Mapping. See also Apollonius’ Conics 17 Although Lagrange doesn’t say the name Mercator in his paper, On the Construction of Geographical Maps, from 1779, he does say “…in order that, when the degrees of longitude are supposed constant on the map and that on the globe these are the degrees of longitude which remain constant, it is necessary that on the map, the degrees of latitude increase in the same ratio of the cosines of the latitude, or which is the same thing, in direct ratio to the secants of the latitude; by which one can conclude by integral Calculus that the distance between the equator and any parallel has to be proportional to the logarithm of the

tangent of the half-compliment of the latitude of this parallel; this is the basis for understanding scale-maps.” 18 Mercator’s friend and collaborator, Abraham Ortilius, made the first modern map book and called it an atlas, named after the Greek God with the world on his shoulders, in 1570 called, Theatrum, with more than 70 maps. (Greehood, Mapping) 19 John Noble Wilford, The Mapmakers, p 75 (Vintage 2001) 20 “You yourself in your supreme wisdom are well aware of the intimate and necessary bond that unites all sciences among themselves and with whatever pertains to the prosperity of the human society.” -Gauss to the Duke, July 1801 21 Dr. G.W. Dunnington, Carl Friedrich Gauss: Titan of Science, A Study of His Life and Works, (Exposition Press, New York, 1955). 22 Carl F. Gauss, On Conformal Representation, 1822 (Allgemeine Auflösung der Aufgabe die Theile einer gegebenen Fläche so abzubilden (the famous "Copenhagen Prize Essay") (1822), pp. 189-216. & In English: A Sourcebook in Mathematics (Dover Publications, Inc.: New York 1959) 23 e.g. take any surface, say a summer squash, and place a string taught on it, this is a shortest line. 24 See elaboration of this in Riemann For Anti-Dummies #44 – 48, located at http://www.wlym.com 25 Gauss, after this work had moved to more physical applications of his method, in electrodynamics, magnetism, etc. See Tarrajna Dorsey’s translation elsewhere in this issue. 26 Einstein said on the topic, “The best that Gauss has given us was likewise an exclusive production. If he had not created his geometry of surfaces, which served Riemann as a basis, it is scarcely conceivable that anyone else would have discovered it. I do not hesitate to confess that to a certain extent a similar pleasure may be found by absorbing ourselves in questions of pure geometry.”- Carl Friedrich Gauss: Titan of Science, p 350 27 The reference to the work of Bernhard Riemann, is to Riemann's 1854 habilitation dissertation, Über die Hypothesen, welche der Geometrie zu Grunde liegen, in Bernhard Riemanns Gesammelte Mathematische Werke, ed. by H. Weber (Stuttgart: B. G. Teubner Verlag, 1902), and reprinted by Sändig Reprint Verlag (Vaduz, Liechtenstein) pp. 272-287 (English Translation in A Sourcebook in Mathematics, op cit.) 28 These can also be called “least action pathways.” For example, on our celestial sphere, the measurement between any two stars is a part of a great circle; the shortest line or least action on that surface was this type of line.



29 “There must be some experimental proof, which demonstrates, in a measurable way, that a certain crucial experimental occurrence requires us to construct one kind of mathematical physics, rather than some other. This demonstration must have such unique significance. Riemann points to three hints, on which he has relied for elaborating the general quality of “yardstick” we require for that kind of measurement.” Lyndon H. LaRouche Jr., “Leibniz from Riemann’s” Standpoint, FIDELIO magazine, Vol. X, No. 3, Fall 1996. 30 Lyndon H. LaRouche Jr., So, You Wish To Learn All About Economics?, 1984, p. 55-59, (EIR Publications, Feb. 1984) 31This paper of C.F. Gauss’, is often referenced as the Fundamental Theorem of Algebra, Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores primi vel secundi gradus resolvi posse. (Helmstedt: C.G. Fleckeisen, 1799.) For more, one can find pedagogicals written by the LaRouche Youth Movement at http://www.wlym.com 32 LaRouche, “Mathematical Physics” p. 33 Special thanks to Peter Martinson, Nick Walsh, and my dearest

love – Shawna M. Rodarte.

On How to Do a Stereographic Projection Lopez


On How to Do a Stereographic Projection Ricky Lopez1 For most people, the necessity of “abstract” geometrical ideas are either useless or, “doesn’t concern their specialized field.” As passionate as one might attempt to tout such an idea, it can never be true. I shall attempt to show people how to create a stereographic projection of a sphere onto a plane in this essay; the importance of which can be discovered in dept simply by a study of cartography and great mathematicians such as Bernard Riemann.2 The main intention of the author is for the reader to have fun!

Looking at One Surface In And Of Itself Before one can map one surface onto another, perhaps one might investigate a few things about surfaces. Since this piece involves mapping a sphere onto a plane, one should investigate some characteristics of these two surfaces. Given the nature of the task, one must first become conscious of how a point is located on each surface. To begin with, one can investigate the plane. Now, to locate any point on the plane, or to take an action that results in some type of displacement, one can rotate and extend on a plane.

When one investigates the sphere, any point on the sphere is located with two angular displacements, as seen in the diagram with angles t and u.

The Moment of Crisis Perhaps one found that locating points on a surface wasn’t at all difficult, but rather comfortable. The challenge now is to map every single point on the sphere onto the plane; how can every

point on the plane represent every point on the sphere? Without prior knowledge on how to accomplish such a task, one should experience the frustration of being completely ignorant of what to do! What type of mathematical language must be

used to accomplish this task? How can one teach others to empirically construct this map, for without being able to teach others how can one be so sure one has even been successful in accomplishing this task? Where does one begin? Solely placing the sphere on the plane may aid in accomplishing the goal, although the problem is not resolved.

One Thing As A Function of The Other In order to tackle the problem, the author has decided to define some idea of what a function is. The author defines a function as a process in which elements in the process depend upon one another; for example, yellow teeth are a function of the amount of times the teeth have been brushed. To give another example, having a dumb political leader is a function of the amount of idiots that tolerate such a leader. In the geometrical problem in this essay, one must cause the amount extended and rotated from the origin on the plane depend upon an angle t (which in this example represents the movement down from the north pole on the sphere) and an angle u (which is the amount one might have rotated after moving down from the north pole) on the sphere. To begin with, one can look at a mapping of points with a triangle that connects one point on the sphere to a point on the plane.



To understand the importance of this triangle and to understand its properties, (which may lead to representing every point on the sphere, on the plane) investigating a cross section of this diagram shall be useful.

The amount extended on the plane shall be denoted by the letter R and the amount one has moved down from the north pole shall be denoted by the angle t. The triangle here is formed by the diameter of the sphere, the extension R, and the line connecting the North Pole to a point on the sphere (the one being mapped on the plane) and a point on the plane (the one representing the point on the sphere). Half of the task of mapping shall be completed once R has become a function of t, for the problem of making extension a function of one of the angles will only leave the problem of making rotation on the plane a function of the other angle, that is, the angle u.

The Mechanics As painful and unexciting actual mathematical calculations are to most human beings, the author has reached a point where he must unfold some mathematical calculations. First, various angles and lengths must be examined in order to get the relevant formulas and functions. The relevant lengths and angles shall be shown through diagrams and explanations.

One of the relevant angles is the angle θ, the angle nigh the north pole .

To derive another important angle one must extract the isosceles triangle in the diagram.

In Triangle B the angle θ represents the angle nigh the North Pole. The angle t in this diagram is cut in half and, therefore, Triangle A and Triangle B definitely contain within them an angle that is the same, id est half of the angle t. Another angle that both triangles contain within them that are identical, is a right angle - for they are both right triangles. Therefore,



Triangle A must also have θ as an angle for the plane-triangles contain two identical angles and, therefore, the third must also be identical. Now, one must examine two important triangles in the important geometric construction below: Examining the triangles more closely one shall be one step closer to making R a function of t!

Now some important relationships must be laid out:

!"

!"

cos

sin

2=

R,

therefore,

!

!

cos

sin

2=

R,

hence,

!

!

cos

sin2 "=R ,

and, therefore, !tan2 "=R .

One more thing must be done to make R a function of t, for one probably noticed that R is a function of the angle θ, not t, in the above formulation. This problem is easily solved by looking back at the Isosceles triangle examined above. For θ can easily be made into a function of t:

t=!" #21800

01802 !="! t#

090

2+!=t

" .

Now one can simply substitute -t/2 + 90 for θ and finally R is a

function of t: )902

tan(2 0+!"=t

R .

The Other Half of the Problem!

Now that the angle t has been made a function of extension on the plane (and vise versa), half of the task of creating a map has been accomplished. Now, one must make the other angle u a function of rotation on the plane.

The author has come up with the bright idea of making the angle u on the sphere a function of the angle φ on the plane (amount of rotation) with the formula u = φ; hopefully the reader shall appreciate the formula and conjecture the reasons why it is the most reasonable.

Conclusion The only thing left for the reader is to attempt to apply all that was written in this short essay to see if it actually works; and to continue to investigate similar subject matter to know that physical-space time wasn’t wasted actually reading this short essay. 1 [email protected] 2 Bernard Riemann (Sept. 17 1826- July 20, 1866) was a German mathematician whose discoveries paved the way for the development of general relativity. He was a student of other great mathematicians such as Carl Gauss, Jakob Steiner, Carl Jacobi and Lejeune Dirichlet. See the online biography by O’Connor, J. J. and Robertson, E. F. Georg Friedrich Bernhard Riemann, URL: http://www-history.mcs.st-andrews.ac.uk/Biographies/Riemann.html (1998)

Are You Living an Imaginary Existence? Brown


Are You Living an Imaginary Existence? Rachel Brown1

1. Approaching human relations from the standpoint of another person's potential, rather than the lower standpoint they might currently be operating from, is equivalent to bringing a new principle to a manifold.

2. Approaching any collapsing economy, except from the standpoint of bringing an added principle to the process, is clinically insane. The dynamic relations of an economy, if already in a collapse function, can potentially act like a vacuum, where one factor contributes to the hastened decay of another factor; for instance, lack of health and sanitation infrastructure contributing to the increase of disease.

3. The study of dynamic relations, as a measurable physical science, was developed by Plato and Gottfried Leibniz, and the foundations of the mathematics were established in 1799 by Carl Friedrich Gauss.2

Is it possible to know anything? How about in social processes? Maybe it’s possible to know scientific truths, like gravity, but when it comes to human beings, there are just so many factors that come into play, who’s to tell what’s the cause of anything? A murderer could have been born like that, or maybe he had a bad childhood, or maybe it was the influence of video games, who knows? A genius may have had good teachers, good genes, or maybe worked a little harder. An economic depression? It could have been a psychological reaction in the markets, an adversarial market that was started somewhere else, maybe a drought for your farms, or bad political decisions. Again, there are an infinite amount of factors, so it’s impossible to determine one cause, right? There might not necessarily be one simple cause, but that doesn’t mean the manifold of causes is unknowable. Principles can be isolated through a scientific method. Which scientific method is the question - what’s your method of measurement? Maybe if you have a problem in the constancy of the matter you’re examining, you can find constancy in what’s making the change of the matter possible. Listen to this excerpt from a report on the impact of fascist banker Felix Rohatyn’s fiscal austerity policies in New York City in the 1970’s and 80’s.

“In 1975, New York City experienced a fiscal crisis rooted in long-term political and economic changes in the city. Budget and policy decisions designed to alleviate this fiscal crisis contributed to the subsequent epidemics of tuberculosis, human immunodeficiency virus (HIV) infection, and homicide in New York City.

Because these conditions share underlying social determinants, we consider them a syndemic, i.e., all three combined to create an excess disease burden on the population. Cuts in services; the dismantling of health, public safety, and social service infrastructures; and the deterioration of living conditions for vulnerable populations contributed to the amplification of these health conditions over two decades.”3

The report describes how budget cuts, like an overall 33% spending cut for the city from 1975 to 1982, laying off 20% of the police force, eliminating the Addiction Services Agency, cutting payroll at the Health and Hospitals Corporation 17% between 1975 and 1978, and other similar measures broke down the social and health infrastructure that are in place to prevent epidemics. Also attributed as causes are deindustrialization, loss of manufacturing jobs, loss of city jobs, and suburbanization. Drug use and crime went up, police protection went down, garbage was left in the streets, families were broken up, guns and crack abounded, each factor contributing to the other, each change slightly adjusting the boundary conditions of the human potential existing in those given conditions. A multiplicity of causes creating a unified effect. Still sound impossible to determine? Thank goodness for Gauss, for establishing the basis of the mathematics by which to measure this type of process, which we call the complex domain. The development of the complex domain represented a breakthrough in epistemology, in what LaRouche identified as "hypothesizing the higher hypothesis," because Gauss was not just discovering a new idea. He was developing an infrastructure on which to re-examine all past discoveries and a foundation on which to generate new discoveries, in a drastically different way. How can you distinguish the difference between the complex domain and any previous methods? By the power of ideas they are able to generate. What is the limit of an idea contained in each system? Gauss starts by examining the notion of so-called "positive," and "negative" numbers.

“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). . . 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.”4

One example of this would be two exponential spirals, each starting at the same point, and identical in rates of rotation and extension, but going in opposite directions.

Now each "number," or magnitude of each arm of the spiral actually represents the same "length," for example, the length “x^2” appears in both spirals. This concept of number indicates not a simple representation of an object, or as a “negative”

usually connotates, the lack of an object, but is instead a change in a physical process. In this case, positive and negative are only opposite directions of action from some given starting point. This starting point can be either physically determined or arbitrarily determined.

In a simple process of extension without rotation, the two opposite directions would be represented as a line, the common conception we know as the "number line." This line is what Gauss’s student, Bernhard Riemann, would call a simply-extended manifold, with the possibility of action in only two directions, but along one basic pathway, which, as a series, would have the principle of a simple transformation from one term to the next, such as 1, 2, 3, 4, 5, …in which there is no change in the rate of change as the series progresses. Simple extension in two directions. But how would we explain another change in direction, like leaving this series to go to another one, not located on this line? We have to have some way to account for this additional change in action. Gauss found a way to solve this paradox, by solving a seemingly unrelated paradox: what is this square root of a negative number? What can this "magnitude," the square root of negative one, possibly mean, which arises consistently in algebra, but has no physical explanation? Since what we have established are not objects, but relations among magnitudes, Gauss determines the "real" existence of this magnitude as representing a change in the type of action from that which generated the original "line." Now you have a set of relations established which can describe the difference between the x and y axes of the Euclidean plane. Here is an example. If we call the unit on one side of our origin +1, and an equal unit on the opposite side is called –1, the



magnitude which is the geometric mean between these would be called the square root of –1, and the geometric mean between –1 and +1 would be the negative square root of –1. The algebraic expression for this geometric process would be 1/x=x/-1, so we find x2=-1 and x equals the square root of –1. The negative square root of –1 would represent the opposite direction from the square root of negative one, but maintain the same geometric relation.

We have started to describe a geometric domain, for the so-called imaginary numbers!

"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. "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, -i. "In forming 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.5



Thus, Gauss establishes the existence of "imaginary" numbers.

Rotational Action is Primary? Now how do we know how to move in this new domain? Do we have to come up with a whole new language and set of directions? No! Those boring, stale, linear equations have way more significance than that old TI-83 was ever able to demonstrate. Each equation represents a series of actions. Take the simple expression of x2. This means that we will begin by taking our beginning magnitude designated as x, and squaring its length while doubling its angle. For example, square the length 2. As a unit, this expresses only extension and no rotation, doubling an angle of zero is still zero. Squaring our length, however, is possible, so we end up with 2*2=4. 4 is on our original series, requires no action to deviate from it, so it is a so-called real number. If you start from negative 2, your angle here is not zero, but 180 degrees. Double that, you get 360, so you are also back at 4, but you took a different action to get there.

If our beginning magnitude is not on the original series, we start to get into complex functions. We already had action implied in our real numbers, but we could not actually “see” the continuous manifold that they were part of without the imaginary, or complex realm. Take now for our unit the term 2+i. 2+i represents a magnitude off our real number line, so now we can see the rotational nature more clearly. Let’s square this magnitude.

(2+i)(2+i)=3+4i Then do it again.

(2+i)(3+4i)=2+11i And one more time.

(2+11i)(2+i)=-7+24i. We start to see a process of spiral action, specifically, logarithmic spiral action. A method that was limited before, is now unlimited. What was unknowable as a physical process before, in the dead language of infinite straight lines, is now an expression of a universe that is “finite and unbounded,” as Einstein expressed, or in the words of Lyndon LaRouche, “finite and self-bounded.’6 Now we start to see the real power of “powers,” and maybe an idea of how the visible domain is not merely what we see, but actually the result of a knowable, lawful process.

A Change in Manifolds Now say we want to examine a change in the geometry created by adding a principle to a given manifold. Take this plane as Gauss developed, and apply the squaring function to every point on the manifold. What happens to your square-bounded surface? What happens to the straight lines?



The lines remain harmonic, or orthogonal, but their geometry is changed. They become a series of intersecting parabolas. So you have added a principle, and changed the nature of action on a surface. You’re still going to take right turns everywhere you go, but now the places you can travel are much more interesting. Each time you add a principle, you add a new type of action, and a new possibility. Does this apply then, to other subjects which we usually approach in linear terms? Let’s see now, whether this method is really universal. How about if we take population density as our expression of magnitude, and a hunter-gatherer society as our linearly, or simply-extended manifold? This manifold is not representing any power, it is simple. Human beings are using limited capabilities of mind, to wander about, picking up objects and consuming them. Are they acting any differently than animals? Now, if one of these humans makes a sovereign discovery of a principle, there is a change in every point in the manifold, every previous relation will now be changed. For example the discovery of planned agriculture, planting a food crop in a given area to be harvested, instead of simply wandering about, increases the yield and decreases the amount of time spent acquiring food, thus feeing up time for other

pursuits, like developing tools. This is a measurable change, whose unit of measurement is the amount of human beings that can be supported in a given land area. Whereas before something like 1 person for every 3 square miles could be sustained with adequate food sources, now 10 people can be supported on these same 3 square miles. So we have applied a principle, and increased the power that human beings hold. Or, you can imagine a colony being developed. What types of people would you need first to occupy the desired territory? You would definitely need skilled construction workers, or your housing situation would be pretty uncomfortable, and subject to higher rates of disease. This would be one boundary you were defining: What is the level of safety provided by your housing? Also, of course you would need to eat. Food providers define the boundary between life and death. In order to increase food supply, you would need tools, so miners and metalworkers would be very important. A transportation system would speed up the process of transporting raw ore to the place of manufacture, thus, engineers would soon be needed. How fast you are able to transport goods is another boundary



condition,another “degree of freedom.” We are changing the dynamic relations that affect each individual, or each household. What is the potential of each household to increase its ability to make discoveries, based on the infrastructure it's provided with? Since these boundaries are created by new discoveries, we can view the input-output relations of the society as being ultimately determined and bounded by the amount of discoveries being made.

Once these discoveries have been implemented, it is not the objects that are changed, but the relations between the objects. If we build the Eurasian Landbridge network of high-speed rail corridors, we are opening up new potentials. For example, in the hinterlands (vast land-locked inland region) of Eurasia, you could open up new raw material deposits, and stations of processing and manufacturing these raw materials along the way, creating the necessity for new cities or creating new economic opportunities for existing cities. New markets would be opened up, with the transportation lines directed between Russia, Western Europe, India and on to North Africa, and Asia. A good example is Central Asia, strategically located between Russia, Europe and Asia, with a population of over 53 million. It also has some of the richest raw material deposits in the world, including oil reserves, natural gas, bauxite, chrome ore, silver, gold, tungsten, and major water sources provided by the Amudar'ya and Sydar'ya river systems, along with the once large Aral Sea. Kazakhstan alone has significant concentrations of nearly every chemical element in the periodic table. After the collapse of the Soviet Union, these countries were left with independence, but little economic sovereignty, having had their economies largely dependent on monoculture cotton production, especially the agricultural economies of Uzbekistan, Turkmenistan, and Tajikistan. Despite being the source of

cotton production, the textile production took place in Moscow, thousands of kilometers away. Going back to the beginning, let's think of this in terms of adding a principle to the manifold, the Eurasian Landbridge principle of strategic infrastructure development.7 A transportation route would allow high technology machine tools to better extract the raw materials, and access to new markets. A revitalized space program would increase the demand for technology and specialized equipment, using the array of elements and resources, and the established scientific cadre (Kazakhstan was the location for the Soviet space program). Large infrastructure projects, like water canals, will help solve the problem of the shrinking Aral Sea, provide for more food production for the population, and create a demand for more machine tools, which could be produced in the area. A broader range of goods could be available at a faster rate, speeding up the process of production. Easier travel will provide for a more relaxed social process, with more access to human-to-human relations. This is crucial to have higher level discussions of important ideas regarding economics, science, music, art, education, and the nature of man. Each human being implicitly has a higher level of relationship with every other human being. A child being born, has a higher potential of making more profound discoveries. Think now, to the young person in today’s culture. In a society like this, with a future of debt, forcing yourself through strenuous but anti-cognitive classes to get good grades, thinking there are no other options, living through a politically impotent period of American history, in a culture based on momentary pleasure rather than truth, might they not be so pessimistic? This is the importance, of actually studying Gauss’ discovery of the complex domain, and not simply settling for an imaginary existence. References 1 [email protected] 2 New Proof of the Theorem That Every Algebraic Rational Integral Function in One Variable can be Resolved into Real Factors of the First or Second Degree, by Carl Friedrich Gauss, 1799, available on www.wlym.com under Classics-Gauss-Fundamental Theorem. 3 The Impact of New York City’s 1975 Fiscal Crisis on the Tuberculosis, HIV, and Homicide Syndemic; Freudenberg, Nicholas; Fahs, Marianne; Galea, Sandro; Greenberg, Andrew; American Journal of Public Health. 4 On the Metaphysics of Complex Numbers; by Carl Friedrich Gauss; translated from Gauss’s Werke, Vol. 2, pp.171-178, by

Central Asian Rail Lines



Jonathan Tennebaum, translation available in Spring 1990 edition of 21st Century Science and Technology. 5 Ibid 6 The Principle of Power, by Lyndon LaRouche, December 23, 2005 EIR, Vol. 32 No.49 7 EIR Special Report: The Eurasian Landbridge, The ‘New Silk Road’ –locomotive for worldwide economic development; January 1997

A General Theory of Earth’s Magnetism Gauss


A General Theory of Earth’s Magnetism Carl Friedrich Gauss Carl Friedrich Gauss (1777-1855), like Gottfried Leibniz and Johannes Kepler, established the foundations for more fields of study than most professionals today can even name. Though he was politically silenced after 1799 by Napoleon Bonaparte’s academic Gestapo, led by Lagrange, he still blazed the way for the first truly Anti-Euclidean geometry, which exploded from Gauss’ own student, Bernhard Riemann, in 1854. Key in this, was Gauss’ work on Magnetism and Potential. In 1828, Gauss attended the opening convention of the Berlin Geographical Society, where he was hosted by Alexander von Humboldt. Humboldt showed Gauss his own instruments for measuring the magnetic field of the Earth, and expressed his wish for a network of “magnetic observatories” around the planet. Gauss built the first of these observatories in 1833, in Göttingen, with the help of his collaborator Wilhelm Weber, and subsequently established the international Magnetischer Verein (Magnetic Union), to organize a global study of the Earth’s magnetism. In 1840, the U.S. Exploration Expedition of Lt. Charles Wilkes discovered the south magnetic pole, quite close to where Gauss predicted it would be. We present here the introduction to his 1838 Allgemeine Theorie des Erdmagnetismus, which lays the foundations for not only all future studies of planetary magnetic fields, but also for Gauss’ crucial work on Potential Theory. This translation was made by Tarrajna Dorsey, with help from Daniel Grasenack-Tente.*

Introduction

The restless fervor in recent times with which people strive to explore the direction and magnitude of the magnetic force of the earth on all parts of its surface is a delightful occurrence, insofar as the pure scientific interest is thereby visibly brought forth. Indeed, as important as the fullest possible knowledge of the divergent line is for navigation, its demand of the latter does not extend any further, and what lies beyond it [the divergent line – translator] remains almost irrelevant for navigation. However, although science may well further material interests, it is not restricted to them, but rather demands equal exertion for all elements of its investigations. It is customary that the bounty of magnetic observations is represented on the map of the earth with three systems of lines, which have been well-named the isogonal, isoclinal, and isodynamic. These lines change their form and position so considerably with the passage of time, that a depiction of them shows merely the condition of the appearances for a certain point in time. Halley’s declination map is very different from

Barlow’s 1833 representation; and Hansteen’s inclinations map for 1780 already differs very strongly from the present position of the isoclinal line; the endeavors to represent intensity are still too new to reveal similar changes, yet with the passage of time they will doubtlessly cease to remain hidden. All of these maps are still more or less incomplete, or partially unreliable: it remains to be hoped however, that, although such completeness cannot be wholly attained, due to the inaccessibility of some parts of the earth’s surface, it will draw nearer with quickening steps. Considered from the higher standpoint of science however, the most complete compilation of the phenomena by way of the observations is not itself the real goal: by this one has made only a resemblance, like the Astronomer, for example, when he has observed the apparent path of a comet on the celestial sphere. One has only building blocks, not the building, so long as one has not made the intricate phenomena submissive to a principle. And like the Astronomer, whose main task actually begins after the stars have removed themselves from his vision, relying on the laws of gravitation, calculates the elements of the true orbital path from these observations, and through these calculations is even able to present the positions of the rest of its journey with certainty; the physicist should also pose the problem to himself in this way, at least in so far as the differing and partly less favorable circumstances will permit, to investigate the form of action and the values of magnitude of those fundamental powers [german: Grundkräfte] which bring forth the phenomena of earth’s magnetism; to submit the observations, as far as they reach, to these elemental characteristics, and to thereby anticipate the phenomena for those regions in which as of yet the observations have not been able to penetrate at least to a certain degree of sure approximation. In any case it is good to have this highest goal before ones eyes, as well as to attempt to render the path toward it traversable, even if currently its incompleteness allows for no more than a distant approximation of this goal. It is not my intention to mention those earlier unsuccessful attempts, where some held the opinion that this great puzzle could be guessed without any physical foundations. One can only admit to be named physically founded, such attempts as those which look upon the earth as a real magnet, and subordinate calculations to the proven type of action which a magnet has at a distance. However, all previous attempts of this sort were similar, in that some, instead of firstly investigating how this great magnet must be structured in order to correspond adequately with the phenomena no matter how simple or how very complex a structure might be seen to emerge, they departed at the outset from a fixed simple structure, and attempted to see if the phenomena agreed well with such a hypothesis. Meanwhile, repeating itself within all of this, is solely what the

A General Theory of Earth’s Magnetism Gauss


history of astronomy and natural sciences reports from the beginnings of so many of our discoveries. The simplest hypothesis of this kind is, to assume only a single very small magnet in the middle point of the earth, or rather (as hardly anyone seriously believed in the actual presence of such a magnet), to presuppose the magnetism to be distributed within the earth, so that the total outward effect is equivalent to the effect of a fictitious, infinitely small magnet, in approximately the same way as gravitation acting upon a homogenous sphere is equal to the attraction of a mass of the same-size concentrated in the midpoint. In this presupposition, both of those points where the continuation of the magnetic axis of those central magnets cuts the earth’s surface, are the magnetic poles of the earth, whereupon the magnetic needle stands vertical, and where at the same time the intensity is greatest; the inclination will = 0 at the greatest circle midway between both poles (the magnetic equator) and the intensity will be half so great as at the poles; between the magnetic equator and a pole inclination as well as intensity depends on its distance from that equator (the magnetic width), and indeed in such a way, that the tangent of the inclination is equal to the doubled tangent of this width; lastly the direction of the horizontal needle everywhere coincides with the direction of a great circle drawn toward the north magnetic pole. However, nature only agrees with all of these necessary consequences of this hypothesis in gross approximation; in reality the line of diminishing inclination is no great circle, but rather a line of double curvature; one does not find equal intensity at equal slopes; the directions of the horizontal needle are far from all converging towards one point, etc. So even superficial contemplation is enough, to show the rejectionable nature of this hypothesis: although one does employ one of the above expressions as an approximation, to derive the position of the line of receding inclinations from such observations as have been made at a distance, with moderate inclinations. Eighty years ago, Tobias Mayer departed from a similar hypothesis, with the only modification, that he did not place the infinitely small magnet in the midpoint of the earth, but rather off-set from it by about a seventh of the earth’s radius: yet he maintains, presumably in order to avoid greater complication of the calculation, the actually completely arbitrary constraint, that the plane perpendicular to the magnet‘s axis goes through the midpoint of the earth. In this way he found, at an admittedly very small number of locations, the observed divergences and inclinations to be highly concordant with his calculations. However, an extended test would soon have shown, that one can not represent the whole of the phenomena of both these elements much better with this hypothesis, than with the first mentioned. It is well known that the determination of intensity was not known at all at that time. Hansteen had gone one step further, in that he tried to fit the hypothesis of two infinitely smaller magnets of unequal position and strength to the phenomena. The decisive test of admissibility or inadmissibility of an hypothesis will always remain the comparison of the results contained within it, with

the experiences. Hansteen had compared his own with the observations at 48 different locations, of which however only 12 are found, where the intensity is also determined, and overall only six, where all three elements are found. Here we are still met with discrepancies between the calculation and observation, which climb to nearly 13 degrees with the inclination.† If one now finds such great divergences to be inadequate of the demands which must be made of a sufficient theory, then one cannot avoid but conclude, that the magnetic form of the earth‘s body is not one, for which a concentration in one or a couple of infinitely small magnets can stand as a representative. With this it is not being denied, that with a greater number of such fictitious magnets a sufficient overlap could at last be reachable: an entirely different question is, if such a form of solution to the problem is advisable; it seems indeed, that the already extremely difficult calculations for two magnets would, with a reasonably greater number, set insurmountable difficulties against practibility. The best will be to completely leave this path, which is inarbitrarily reminiscent of the explanation of the planetary movements by a heaping of ever more epicycles. In the present treatise I will develop the general theory of earth’s magnetism, independent from all particular hypotheses of the distribution of the magnetic fluid in the earth’s body, and at the same time share the results, which I have attained from the first implementation of this method. As incomplete as these results may be, they will yet be able to give a grasp, of what one might hope to reach in the future, when a refined and repeated elaboration of the latter will first be able to have reliable and complete observations from all regions of the earth underlying it. * [email protected], [email protected] † A difference of over 20 degrees was once found with the declination: however it is reasonable to account the error of calculation not to the number of degrees of declination, but rather to the real inequality between the calculated and observed direction, where it amounts to 11½ degrees at the location being discussed.

december 2006 - larouchepac · 2018. 4. 30. · Δυναµις vol. 1, no. 2 december 2006 needed...

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