mysterium cosmographicum - larouchepac · johannes kepler for the publication of the rudolphine...

March 2007 Vol. 1 No. 3

www.seattlelym.com/dynamis

EDITORS Peter Martinson

Jason Ross Riana St. Classis

ART DIRECTOR

Chris Jadatz

LaROUCHE YOUTH

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On the Cover Urania, front piece designed by

Johannes Kepler for the publication of the Rudolphine Tables. Kepler took over this

project upon Tycho Brahe’s death, and Kepler labored almost 20 years of his life to finish these astronomical tables. Kepler

created these tables by applying his revolutionary discoveries to Tycho’s extensive observations.

In a sense, the Rudolphine Tables became the application of

Kepler’s method in his Commentaries on Mars to all the other planets. In the center of the engraving, standing in Urania’s temple, are Tycho, Copernicus,

Ptolemy and Hiparchus, deliberating in the simultaneity of eternity. If you look on the lower

left side of the temple, through one of the windows, you can see

Kepler working late into the night by candle light.

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From the Editors De Astronomia By Mike Vander Nat Never Metaphysics I Didn’t Like By Spencer Cross The Fallacy of the Equant By Jason Ross Physics, Not Statistics: Kepler’s Paradoxical Return to the Equant By Jason Ross A Preliminary Investigation of the LaRouche-Riemann Method By Michael Steger

“…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

∆υν∆υν∆υν∆υναµιαµιαµιαµιςςςς Vol. 1, No. 3 March 2007

From The Editors

Here, now, I sit in the same chair that I occupied when composing the editorial for the first issue of ∆υναµι∆υναµι∆υναµι∆υναµιςςςς, a mere seven months ago. But today, as if this chair and the person sitting in it were in a completely new universe, I write from a different place and a different time, a locus where time and place are themselves, different. Imagine: I doze in this chair one September evening, pressed by the enormity of the idea of launching this journal, relieved that words have indeed staggered onto the page. I awake to an odd sensation, as if my body’s presence in the world were somehow different. I arise from this chair. I wonder, “Is this really my body? Could I still be dreaming?” But, yes, my body does feel like its familiar self; it is not noticeably altered; my foot has the proper weight upon the floor. Perhaps this curious impression is merely the residue of sleep still clinging to my mind.

Imagine: I exit the house, stepping not onto the well-worn stairs I could descend even in my sleep, but into an unfamiliar street along a blooming park in spring, where the alien song of strange birds carries from one peculiar tree to the next.

Now imagine: While I expect a fantastic scene, as I have just described, to impinge on my senses when I step through my door, instead the street appears just as it had before. Though I am aware of a great change, this change, so profound that I expect to see it wrought everywhere, wrought into the very essence of nature itself, is not visible. Or, at least, the internal reorganization of nature is not fully visible upon its surface, yet. My body feels the same; the street looks the same, but simply trusting appearances is foolishness in the extreme.

When we launched the first issue of ∆υναµι∆υναµι∆υναµι∆υναµιςςςς a mere seven months ago, the animations team assigned the task of breaking through Kepler’s New Astronomy was just bringing the first phase of their work to a close. I and four other members of LaRouche’s Youth Movement had just arrived in Virginia to begin work on Kepler’s Harmony of the World. As we began our work, the specter of our dying economy loomed large, darkening the horizon, as the collapse of the housing bubble, the collapse of the auto industry, the collapse of the living standard of the majority of the citizens of this and other countries alike, could no longer be denied, just as they are today, though, perhaps, the acute disaster is now more generally acknowledged. We were coming upon a midterm election, an event in the political spectrum that usually pales next to the hoopla and brouhaha of presidential races. But this election was different, as we would likely never impeach the egregious Dick Cheney or contain the idiot George Bush without a sea change in congress, without a Democratic majority coming to power with the ability to fully investigate this administration’s crimes.

When we launched this journal, we knew within our organization, within LaRouche’s Youth Movement, that a powerful potential was developing, a potential to revive classical art and science, to regenerate sanity in society, to exemplify the renaissance community. We knew that we, a small but growing portion of our generation, dedicated to

recreating LaRouche’s economic discovery, were changing the reigning culture, though this was not reflected in the media, and though often, our success in influencing policy-making for the good would seem to be obscured by the great stupidity of even the better elements of Congress, and by the great fascist criminality of the Cheney administration. We knew that it was time to create a definite forum to express this changing culture, to make it more apparent. We knew that within our youth movement a developing capability for a dialogue about the principles of art and science was emerging. We wanted to make this dialogue public, to aid in inspiring and challenging the citizens of the United States and the world to become better, to become actively engaged in the future survival of civilization. And, so, we launched this journal.

Upon its inception, LaRouche wrote: The birth of this publication reflects a significant

moment in an ongoing process of the reliving of the act of original discovery in the case of the Pythagoreans, Plato, and Plato’s Academy, and, now, the foundations of modern physical science in the work of Johannes Kepler as the avowed follower of Nicholas of Cusa. So, the LYM has reached the point of actually launching what will become, hopefully, a rebirth of that specific current of science upon which the greatest achievements of European civilization had been premised, heretofore.1

That moment in late September last was pregnant with

the awesome developments that have unfolded since, while their actuality remained invisible. Shortly after we put forth the first issue of ∆υναµι∆υναµι∆υναµι∆υναµιςςςς, , , , the first animations crew returned to their homes and fully initiated the entire youth movement’s commitment to mastering Kepler’s New Astronomy. Soon it was clear that this enabled a level of discussion that had not existed before. In many ways, Jason Ross, who was a member of that first team, reflects this new capability in his articles on Kepler’s treatment of the equant. In fact, Ross comments that the question he addresses in his second paper arose because of the deep investigation of Kepler’s work generally taken up by the LYM, that the question of why Kepler reintroduces the theory of the equant for the earth when he had disproved the very idea of the equant previously, would not have been a question if people were not thoroughly engaged with Kepler’s work.

Then, in the first weeks of October, leading into the hot phase of the midterm election campaigns, an investigative team composed primarily of LYM members, uncovered the full implications of Lynne Cheney and David Horowitz’s American Council of Trustees and Alumni, ACTA, as a repressive organization targeting professors and students that they

1“Actually Relive History!”, Lyndon LaRouche, Jr., September 2006 Dunamiςςςς, Vol. 1, No. 1

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perceived as opponents to the Cheney administration agenda.2 This organization acted as a key component in the attempt to suppress the youth vote, to intimidate professors and students from engaging in political dialogue, and thus to assure yet another Republican victory on November 7th. Between Lynne Cheney’s active coercion and Howard Dean’s passive sabotage of his own party’s campaigns3, apparently, little would change politically, except for the worse, over the months before the 2008 presidential race.

After the LYM investigation exposed the nature of Lynne Cheney’s operation on college campuses, the LYM mobilized to get out thousands of pamphlets to share this fact with the students and professors who were the implicit and explicit targets of Lynne Cheney’s dirty work. They concentrated their efforts in key states, intervening into important congressional races where the Democratic candidate was often not getting the support required from the Democratic National Committee. We provided the impetus. We changed the course of history.

After this fierce political fight was waged and largely won, LaRouche conducted a webcast from Washington D.C.4 At that event, members of the LYM from the East Coast and Midwest, and some of us present from the West Coast, participated in a presentation of J. S. Bach’s Jesu, Meine

Freude motet, conducted by John Sigerson, before LaRouche’s address. That presentation, which can be viewed in its entirety at www.wlym.com, exemplified a new level of capability and understanding of classical modes of communication, of insight into the conveyance of profound ideas. Though the performance was imperfect, though the full power of Bach’s work was not yet fully developed, that performance revealed that the LYM had evolved to a new level of capability. LaRouche increased the intellectual challenge to his youth movement accordingly.

Meanwhile, I and the four other LYM members working on the second phase of the animations work, on Kepler’s Harmony of the World, continued to labor. When we launched our website5 in the beginning of February, when we revealed what we were able to do, we showed that LaRouche was right when he wrote in that introduction to the first issue of ∆υναµι∆υναµι∆υναµι∆υναµιςςςς so many months before that:

I read the reports contained within this first edition of

∆υναµι∆υναµι∆υναµι∆υναµιςςςς as marking the beginning of the true rebirth of the university from a long, downward journey into that Sophistry of a thus-self-doomed Athens which has become the characteristic contemporary mood within popular opinion and governments in Europe and the U.S.A. today. The opportunity to rejuvenate the universities has thus arrived.

2 Is Joseph Goebbels On Your Campus? John Train and the Bankers’ Secret Government, LaRouche PAC pamphlet, October 2006. Available as a PDF at www.larouchepac.com 3 “The Inside Story of Dean’s Sabotage,” Debra Hanania-Freeman, EIR January 5, 2007, vol. 34. no.1 4 “Organizing the Recovery from the Great Crash of 2007,” Nov. 16, 2006 webcast transcript in pamphlet form. Webcast archived at www.larouchepac.com 5 http://wlym.com/kepler/

Our work, like that choral presentation, also represented a new phase in the growth of the LYM, a point of departure. Now a third group is working in LaRouche’s basement on the discoveries of Carl Friedrich Gauss, pushing the work on Kepler into the realm of ideas preceding Bernhard Riemann’s development of the concept of hypergeometries. This is the key discovery leading the mind to “know what universe it is in,” as LaRouche has said, the key discovery to master, as we rediscover LaRouche’s discovery.

While this third team works on Gauss, they are also involved, along with an extended group of the LYM, in intense choral sessions with John Sigerson and Lyndon LaRouche every week. The harmonic nature of the universe, of man’s role as the embodiment of the characteristic of development in that universe, is becoming conscious, is waking within an essential group that will act as a catalyst for the rest of human society today. Soon the LaRouche Youth Movement will have the foundations to embark on making new discoveries.

Thus, while the LYM rediscovers the true nature of humanity, it is entirely lawful that we should be involved in a massive political fight that directly hinges on that concept of humanity. If the Democratic party will survive, if they will regain their potency, then they will no longer tolerate the environmentalism that they have increasingly donned as their banner; they will no longer tolerate the absurd hypocrisy and poorly hidden genocidal nature of the policies Al Gore claims must be adopted for the good of the earth. Al Gore, as a political figure, is only able to be promoted now, and to garner a following, because the truth about man’s fundamental goodness has been so debased, so abused, so forgotten.6

From this standpoint, read Spencer Cross’s article on the first astronomy with joy and good humor; read Michael Vander Nat’s engrossing dialogue, seemingly on the same subject as Cross’s, and yet, so completely different, with that sense of capturing something which has been lost. Employ Michael Steger’s pedagogical exercise to free your mind of Euclidean assumptions about the nature of space, a prerequisite for any truthful exploration of astrophysics. Along with Ross’s treatment of Kepler’s use of the equant in The New Astronomy, these articles represent the first of what will be a series of issues on Kepler’s discovery of universal gravitation. Future issues will develop the ideas presented here further and will venture into realms not yet imagined.

Suddenly, tectonic changes, changes in the essential

structure of nature as we think we know it, are occurring with great rapidity.

Peter Martinson

Jason Ross

Riana St. Classis

editors

6 “Implications of the Gore Hoax for international Policy,” Lyndon LaRouche webcast of March 7, 2007; EIR March 2, 2007, vol. 34, no.9; EIR March 9, vol. 34, no. 10; EIR March 16, vol. 34, no. 11; EIR March 23, vol. 34, no 12; EIR March 30, vol. 34., no. 13; and certainly many future issues of EIR, until the idea is fully driven home for the congress and the population at large.

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De Astronomia

Vander Nat


De Astronomia Mike Vander Nat

A dialogue between a Master and his Disciple

Disciple. What is the astronomer’s art? Master. According to the ancients, the astronomers explain the motions of the heavenly bodies which appear irregular to us and preserve the appearances so that their future positions may be predicted. D. Have not all the heavenly bodies the same motion? For I do not posses the astronomer’s art but I have often seen their risings and settings from East towards West. M. Have you not also witnessed the rising of a planet throughout the season? D. Yes, I believe Mars was clearly visible this summer last. M. Was there regularity in the time of its ascension and descent? D. I do not understand your meaning. M. In other words, did Mars enjoy the entire night, always setting on the western horizon before the Sun arose? D. If I remember correctly, Mars did reach the horizon before a new day, but now that I think of it, Mars would appear very high, beyond the eastern horizon and more toward the West. By Fall, Mars would set soon after the Sun. M. Very well. I shall tell you why Mars and the other planets all appear irregular to us when they are in fact well ordered. However, since I myself am a student of astronomy, I will borrow from the testimony of the three great astronomers: Ptolemy, Copernicus and Brahe. But this is no easy task, so I may ask you to help me from time to time. D. I would very much like to hear your explanation, although I do not think I will be of much help to you for as I said, I have not the knowledge of astronomy, save what I hear from the townsfolk. From them, the names Ptolemy and Copernicus are familiar to me but their works are not. M. Did you not say just now, you observed Mars throughout the summer? D. Yes. M. Then answer only what you have seen for yourself and when you are in doubt say what you think most reasonable. D. I shall.

M. You say, do you not, that all the heavenly bodies pass above us from East toward the West? D. I do. M. Both those we call stars and those we call planets? D. Yes. M. But are not some stars visible throughout the night, neither rising nor setting but remaining aloft? D. Yes of course, I meant that those which rise and set will move in the westward direction. The others I suppose revolve in an arc around the Northern Star which is stable. This unique star is used by navigators to arrive at far away lands and without which they might never return home again. M. This common motion then is called by astronomers the First Motion, on account of what is immediately visible to us. The ancients took more careful notice of the stars which appeared to change their positions relative to those rather fixed groups which today we call constellations. Over many years there was observed the circuit of these wanderers. D. What do you mean ‘circuit’? M. By that I mean the procession of the wandering stars through the fixed images of The Bull, The Twins, The Crab and so on, until they returned where they once began. D. Yes I recall that these are the signs of the Zodiac, and that each planet moves about the heavens on an orbit or what you called a circuit. But I have often wondered how the planets arrive at these signs, for they never seem to remove themselves from the other stars; rather, they move in common. M. Yes dear pupil, it is not apparent how they move from sign to sign, but remember this is in fact the end of astronomy, to explain the motions which carry each planet in consequence through the heavenly sphere. This then, the motion peculiar to each planet, is called proper motion. The proper motions are more difficult to observe, and this knowledge was acquired slowly from year to year and generation to generation. The astronomer follows the wanderers, recording the places they once occupied until they are no longer visible, moving too near the Sun. D. When these wanderers traverse the zodiac, keep they their distance apart?

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De Astronomia

Vander Nat


M. Do you mean to ask if they move in unison, never crossing one another? D. Yes. That is my meaning. M. The planets do cross each other and the Sun. D. How can they pass through one another? Will their courses not collide? M. Excellent question, and here I shall defer to the authority of the astronomers, for I believe their answers are most excellent. The planets then, are distant from one another and for this reason only appear to collide with the Sun. When this happens they are said to conjoin. Those that move slowest have extremely large orbits, and those that move more swiftly have smaller orbits. All the astronomers are in perfect agreement on this but they differ greatly as to the arrangement of these orbits. D. What says Ptolemy? M. Ptolemy declares the Earth to be still, at the center, or very near the midpoint of the planetary orbits. He places the Moon closest to the Earth and next Venus and Mercury enclosed by the Sun. Venus and Mercury vary in their speeds and overtake each other, sometimes passing the Sun and other times falling behind. Mars is next, placed very far from the Earth, Jupiter around Mars and finally Saturn is highest above the Earth having the slowest motion. All these bodies then are enclosed within the orb of the stars. D. Why do you say the orbits are swift or slow when they seem to move no differently than the fixed stars? M. Remember, the common motion carries all the heavenly bodies around in unison; however, planets appear differently from week to week and season to season. Since Ptolemy placed the Earth at rest, the outermost stars move quickly, the planets move somewhat slower. Copernicus and Brahe say the Earth rotates in place like dancers who turn about; if so, the stars are at rest. D. Yes I see that makes a difference. M. A great difference indeed, for when you consider the places the planets occupy, and watch over the places they move toward, an even more wonderful occurrence appears. D. Appears? M. Yes, in the astronomer’s eye. The astronomer will show that the planets move contrary to the fixed stars. D. I think I see now what you mean. If the dancer takes notice of the audience around herself, they, being at rest, will appear to her, to revolve, or rise and set as it were. M. You understand well.

D. And should someone arise and depart, slowly making his way through the crowd, the dancer’s eye perceives his motion to be contrary to the seated crowd. M. Yes exactly. D. Would not the speed of the outermost stars be greatly reduced if the Earth rotates rapidly toward the opposite direction? M. Yes. I knew you would understand, having no prior knowledge, provided you answer what seems most reasonable to you. D. I was in great confusion about the common motion of the planets and stars, but it is quite plain to me now that the planets do in fact move, as my memory can testify, but in the opposite way my eyes perceive them to move. M. This contrary motion is called the Second Motion. It is also called proper motion. The rotation of the Earth is called by astronomers the diurnal motion, which was first. Your confusion is quite common on account of the fact that the two motions of the planets are mixed together. But you have separated them remarkably well. And if the effect of the diurnal motion is removed, so to speak, you will see a most irregular motion of the planets. D. I do remember you said the motions appear irregular when in fact, they are well ordered. Please go on. I am eager to know what irregularities occur. M. Try to recall the summer last when Mars was visible. D. I do recall. M. Did you not say that Mars was more advanced beyond the East and more overhead each time you happened to see the sunset? D. I did. M. There is then a sort of chase between the planets and the Sun? D. Reason forces me to admit a chase does take place, whereby Mars gains upon Apollo, the Sun. So it is with all the gods, if they do in fact move upon orbits. There would have to be a conjunction as you said which we cannot perceive on account of the Sun’s light. M. You nearly have it, for remember that the diurnal motion may be an illusion and as we said before, the gods move contrary to the stars, that is, from West to East. D. How easily I forget. So, in the case of Mars advancing toward the Sun, as I observed, the proper motion carries Mars toward the West and it is Apollo who chases Mars.

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De Astronomia

Vander Nat


M. Exactly. Now think about the conjunction and the passing of the heavenly bodies. Does it not seem to you that the planets must also depart from the Sun after the moment of conjunction? D. It does seem so to me. M. There is a moment when a planet will at some time be opposite the Sun? D. There is. M. According to the astronomers, at this moment a most surprising act occurs. D. What occurs? M. As the planets take the place opposite the Sun, they retard, stop, and retrace their steps or like crabs, move in the opposite direction as they face us. D. This is most astonishing. I do not understand what this looks like for I have never seen it, nor have I ever heard of such a thing until now. M. I will try to explain as best I can what the astronomers call retrogression. The planets advance in the eastward direction as we discussed, but as the Sun departs the planets slow their steps until they come to rest, and then retrogress, that is, move backwards in a loop, then continue on in the eastward direction. This occurs over many days time and is seen only in the astronomer’s eye. It is a most curious pattern which occurs in various signs of the zodiac, not keeping to one sign alone. D. Tell me if I follow your meaning. You say the planets move swiftly with the approach of the Sun. M. Yes. D. The planets strive to escape the Sun. When they are passed by him they retard, as if fatigued by the chase and thankful that Apollo is in pursuit of some other god. In relief they slow to a halt and decide to reverse their path hoping to evade the Sun for good; but, they recollect halfway through their regress that they may encounter Apollo all the sooner because their courses are in the round shape, and not the straight. In this way they return to

their eastward course, quickening their steps as they anticipate the return of the Sun and strive to outrun him once again. M. Yes, you have narrated it all so poetically. I believe you were just now speaking in the dithyramb. D. I cannot say for certain if those were the words of Dionysus or my own words. Perhaps our discussion is pleasing to God, because our thoughts are upon Him. M. Very good. D. One thing more confuses me. You said the planets vary their steps, that they accelerate and also retard? M. They do.

D. I am curious how this can be seen when as you said before, the motions of the planets are mixed with the rotation of the stars. M. Remember it is the astronomer’s art to record the places each planet travels through. From them it has come down to us that each divinity moves in sequence of the zodiacal images. D. I understand that now, but I am still at a loss as to the observation of each planet’s speed; especially when they are not always visible. M. You must think hard as I answer in the best manner I can. I said this knowledge we attributed to the astronomers was acquired slowly, over many generations. When the days are counted as the planets

traverse the stars, we gather that they spend more time in one constellation than another, making more progress in one season than another. And it is a great convenience to astronomy that the signs through which the planets move are spaced evenly enough that we may observe the variance in pace. D. Very convenient indeed. If it were otherwise, I perceive a great difficulty in measuring their progress. M. Such are the wondrous designs of God. His divine reason hath provided man all that is needed to discern His works. D. Wonderful are His works, and I marvel at them. And just now something has occurred to me.

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De Astronomia

Vander Nat


M. What is it? D. When you said it is by the signs of the zodiac that we behold the movements, I remembered that it is the Sun’s traversal of the heavens which determines our year, though we depart somewhat from the pagan calendar. In light of what you said about the altering of the speeds, I wonder now if the Sun too changes its speed throughout the year. M. It certainly does. D. How astonishing indeed. Is this what you meant in saying the motions are irregular? M. Yes; and there is still more. Let us return to a point that came up earlier. We say that the planets will spend more time in one constellation than another. Know this too, that the time a planet requires to complete its circuit from a sign of origin, is quite irregular as well. D. This is all very perplexing. The astronomer’s art must be rare among men. M. So it is. But be assured that these perplexities have been overcome by the astronomers, although not removed entirely, for there is disagreement between them. D. Yes of course, how silly of me to forget. You said the astronomers differed greatly over the arrangement of the heavenly bodies. I was distracted by the mixture of the common and proper motions. Now, I would like to return to the other astronomers, for I should very much like to hear how each explains the perplexing irregularities. M. These unequal motions caused men to wonder if the heavens might somehow be defective, or if somehow they appear differently to us besides being exceedingly well ordered. D. I should like to think they are well ordered, for I cannot comprehend why God would chose to create lawless motions. M. Nor can I. D. I am very anxious to get past these irregularities, for I see therein lies the true structure of the universe which will reveal to us whether God has fashioned the world poorly or well. M. Well said. But before we go on to the other hypotheses, it is necessary to separate the two irregularities, for they are very mixed together. D. Two did you say? I thought you named several irregular motions: a reversal of the path, the increase and decrease of speed, the rising up and descending motions, and completing the signs unevenly. M. Yes, I said all these things. But answer me this. D. What?

M. Is it not so that the planets revolve through the heavenly sphere, returning once again to a place of origin? D. It is so. The planets do in fact have a proper motion forming an orbit, at least on the authority of the great astronomers, and this seems reasonable to me. M. And the Sun too, does it traverse the heavens? D. It does. M. This then, is called the First Inequality. D. I am not sure if I understand. Was this motion not called by you, the Second or Proper Motion? M. You perceive very well and to answer you, I must admit the two are very much alike. D. How then, do they differ? M. They are the same in so far as we say the planets are carried around not in the westward direction but in the eastward, which distinguishes them from the common motion. But the First Inequality signifies exactly how the planets traverse the heavens in order that they may return. I will have more to say on this later. D. I accept your answer provided you do as you say and explain more when the time comes. M. Next, did we not also say the planets are attuned to the Sun, causing them to conjoin and oppose one another, and also to rise up and make a ring across the fixed stars? D. Indeed we did. M. This then is another motion, connected to the Sun, for retrogression occurs at no other time than at opposition. D. That is clear, but I do not recognize what is unequal about this. M. What appears to be unequal is the duration of these backward motions, meaning sometimes forming a large ring, other times a smaller one. These retrogressions also cause the planets to rebound in the upright position and sometimes down-turned. For these reasons it is called an inequality. D. Why is the first one called ‘First’ and why the second, ‘Second’? M. On account of the first one being observed first I suppose, for it too belongs to the Sun and the Moon. As for the second it is absent in these two. The First Inequality being present in all bodies makes it prior to the Second, found only in some.

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De Astronomia

Vander Nat


D. Did you not say that the Sun did in fact cause these motions called the Second Inequality? Now you say it is absent in the Sun. M. Right well my pupil, I did say that and I do not contradict myself. I say it does not apply to the Sun, meaning the Sun does not reverse its direction, nor is it ever opposite to itself. Being a cause of the planet’s evasive maneuvers, it does not itself commit an evasive act. D. That is more clear. Perhaps you can clarify one more uncertainty. M. Perhaps. D. You have indicated that both inequalities affect the speed of the planets, one measured in each sign, the other measured by the entire orbit. This being so, as well as the other odd maneuvers taken by the planets, how can it be known that there are two inequalities and not three or very many inequalities? M. You have hit upon a most important question which I have asked myself; for both produce irregularities and they are always mixed together. The separation of the two was extremely difficult indeed and in order that you might distinguish them as well, be careful that you do not confuse the two. You are right in saying that they both affect the speed or velocity of the planets, but do you remember what I said a little while ago? D. What? M. That these retrogressions can occur anywhere in the heavens, not having a fixed home. D. I do. That is why I am confused now. To my thinking there may be many causes which move all the bodies unevenly through the expanse, both the stars and those we call wanderers. M. Your inquisitiveness is delightful. In order to settle this doubt, answer me this: do not all the planets have their own separate orbits, whose distance from the Earth makes some revolve more slowly than others? D. Yes they have each their own course.

M. Will the presence or absence of the Sun prevent them from arriving at their starting points, say in Aries, or in Taurus or in some other sign where they may return? D. No, it will prevent them not. M. Then if the other inequality has its own effect upon the separately moving bodies, it cannot possibly be one motion which accomplishes both effects, namely the retrogressions determined somehow by the Sun and the motion which returns the planet to a beginning place. D. It is not possible provided there be some regularity to these effects, for if not I do not see how you could say there is one or two or very many inequalities mixed together.

M. Your reasoning is excellent and I believe you would make a fine astronomer. To this I say there is regularity or near regularity between the times of opposition and conjunction. Moreover, the retrogressions occur in near order of the zodiac, moving methodically from one sign to the next. D. Then it is easier for me to see now, that the inequalities are distinct. For if the planet is passed by the Sun in a somewhat regular time, although in various places, and each planet eventually returns to his origin, the two cycles are separate. M. You have separated them admirably well. D. This cycle with the Sun and each planet seems to me less of a chase and more of a dance,

keeping nearly in time, as dancers modulate their movements to keep in time with musical rhythms. M. You are quite right. The ancients have often compared the celestial motions to creating music. D. This Second Inequality then, the dance, is very perplexing to me, and I cannot at this moment imagine what would determine such an act, even if it is rhythmic. As for the First Inequality, it seems to me it can be none other than a circle, for what other shape can it be? M. That seems reasonable, though not yet certain. This is where the great astronomers differ in their explanations. I think it would be appropriate now to distinguish the hypotheses since we have separated the awkward motions seen by us.

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De Astronomia

Vander Nat


D. I have not seen them myself, but what you relate to me on the authority of the astronomers seems reasonable and in agreement with what I have seen. M. Returning to Ptolemy then, he explains the First Inequality thus: the planets are carried around a circle, as you concluded, with the exception that the Earth is not at the center of these orbits. D. What is the reason for this? M. The reason is manifold. The planets are seen to return to their origin; but remember, this will not happen uniformly. Astronomers have discerned however that each orbit is in fact uniform in its own esteem regardless of how it is seen by us. D. What do you mean ‘uniformly’? M. I mean in equal times, from sign to self-same sign. D. Of course. Uniform in this case means ‘equal time’. M. And by so placing the Earth off center, or as he says, eccentric, the regions where a planet is more distant from the Earth appear to lapse in more time, and when it comes nearer the Earth, it will appear to pass more quickly. D. I must ask you to pause while I imagine this eccentric, it is no easy matter. M. Think carefully, while I too pause to regain my breath. D. I think I have it. If the body were carried around uniformly, seen from the center, then this would create the illusion of a slowing down and speeding up seen by the off center observer. M. That is correct. However, Ptolemy places the equalizing point not at the center but beyond the center as far removed as the Earth. D. What do you mean ‘equalizing point’? M. I mean what you said about the uniform motion; except, the place where the planet is seen to move in equal times is not at the center but beyond the midpoint. This is called the ‘equant’. D. Beyond the mid-point where? Does it matter where? M. Yes, it matters greatly. It must lie on the line dividing the orbit equally in half. The endpoints of this diameter must be the moments of slowest and swiftest motion. D. Why through these moments and not others? M. For if otherwise the two halves of the orbit would be unequal in time which is contrary to observation. D. I see that the authority of the observer must hold if we are to unfold Ptolemy’s hypothesis.

M. Upon this line, struck through the center, place the equalizing point and the Earth on opposite sides of the mid-point and equally removed. D. I do not understand why this is done. Does not the displacement of the observer explain the First Inequality? M. I see that follows but the reason is this: in order that the planet not only appear to change speed, but also to physically effect this change as the observations required. D. Do you mean a sort of compensation occurs? That, as much as the planet slows down at the distant side, it speeds up when at the nearer side. M. I suppose that is a type of compensation. D. I see that if the equant were the observer, he too would experience this illusion, if only the uniform motion was beheld from the center. And when the equalizing point is placed beyond the center opposite the Earth, the planet must change its speed in order to compensate for the longer distance in the region opposite the equant. So the inequality is both apparent and physical. M. You have envisioned it well. I detect the hand of Athena guiding you through this difficult passage. D. Indeed, she guides all those who revere wisdom. I am all the more determined now, to discover the mind of God through his creative works; therefore, I must know next what Ptolemy says of the Second Inequality. M. Ptolemy and the Alexandrians before him, supposed there to be additional circles borne upon those we call orbits. In this way, smaller circuits are completed before the planet has returned to its origin. These smaller circuits are called epicycles. D. I think I understand, but go on, for I would like to know in what manner these epicycles move. M. Imagine if you can, a circle with smaller circumference than the orbit. Next, fasten the center of this epicycle to the orbit which revolves around the Earth. As the planet is carried along the epicycle westward, the epicycle itself advances eastward thus explaining the proper motion. D. It is hard to visualize. M. First imagine the stars at rest and allow the planets to move in the eastward direction upon their eastbound epicycles. D. This seems easy enough. M. Now follow my hand as I trace the circular path of the epicycle. If I place this cycle parallel to your line of sight as Ptolemy does, do you perceive a reversal of direction?

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De Astronomia

Vander Nat


D. Yes of course. I did not think of it until now. There is indeed a sweeping back in the westward direction when the planet is coming nearer to me. I now recognize the retrogressions. M. Well done. Now remember, the planets all attune their motions to the Sun, and this we called the Second Inequality. D. I remember, and something else comes to mind when you describe the retrogressions. You stated earlier that the Sun never retrogresses, but I wonder if it can be the only body to surround the Earth which does not move on an epicycle. M. The Sun is a beating heart in the body of the world whereby all the epicycles are circulated by the Sun’s orbit, giving them each a pulse. D. That seems just, since the Sun is the source of heat. Yet something else perplexes me. M. What is it? D. You said, did you not, that the planets depart from their positions very slowly, not in one evening? M. I did say so. D. Ptolemy moved the stars ahead of all else, leaving the planets behind, did he not? M. He did so. D. How then, might the planets proceed contrary to the stars, and yet move with them as I always see them? If the stars are fastest would not the planets appear in a different sign from one night to the next? M. They would were it not for the common axis which moves the entire dome of the universe. I see you are confused, let me explain it more simply: The planets are carried eastward by the First Inequality, as well as eastward upon the epicycle which causes them sometimes to retrogress. Meanwhile the entire system is pulled in the direction of the stars, moving swiftest of all, completing its course in one day. This is accomplished by an axis through the celestial sphere which spins in the westward direction. D. Then the motions of the First Inequality are quite slow as if resisting the motion of the celestial sphere? M. I am not entirely sure if the planets are unwillingly pulled in the direction of the stars in the time of one day, but this is how I understand Ptolemy’s hypothesis. D. Ptolemy has seemed to achieve the end of astronomy: to give order to the irregular motions and preserve the appearances thereof. M. Copernicus however did not approve of this explanation.

D. What says Copernicus? M. Copernicus agreed that Heaven should be made more perfect than the Earth, but the many motions described by Ptolemy seemed to him less perfect. D. To me, Ptolemy has given excellent reasons for the irregular motions, accounting for all the subtle movements. M. Indeed he seems to have accounted for all the movements; however, Copernicus questioned whether it were wiser for the planets to change their velocity or to move evenly through the heavens; whether it were wiser for the planets to retrogress or remain perfectly circular. D. How could he know which was wiser? M. The reasoning of those who contemplate divine things is this: the universe lacks nothing, yet nothing created is superfluous; for, if unnecessarily, then created without reason. Nature creates many effects with the fewest causes. She prefers accomplishing the most with the very least, so that no cause is obsolete. D. Then Nature loves simplicity? How did Copernicus simplify the compound motions? M. I shall give you Copernicus’ explanation of the Second Inequality first, for in this did he differ most with Ptolemy, preferring a more ancient theory which moves the Earth. In this cosmography the Second Inequality is the result of an illusion caused by the Earth’s annual motion around the Sun. Those irregular retrogressions are accomplished by one simple movement of the Earth passing the others planets. D. The Sun was said to cause the Second Inequality, how can the Earth do this work instead? M. What Ptolemy saw as a motion attuned to the Sun, Copernicus believed to be attuned to the Earth. He put the Sun at rest, the Earth and all the other worlds are borne along in the easterly direction. In this schema, Mars, Jupiter and Saturn are superior to Earth, while Mercury and Venus are inferior, lying between the Earth and Sun. When Earth approaches Mars, like racing charioteers, the gallop of Mars will appear to slow down until Earth catches up with him at which time he appears still, locked at her side. D. I see it now. The Earth running the inner track must pass Mars making him appear behind, when in fact his course is straightforward and uninterrupted. M. Yes pupil. When Earth gets far enough ahead, the Sun will appear between us and Mars. When we see Mars from the opposite side of the Sun, as you observed, this is called conjunction, for those two bodies seem to meet.

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De Astronomia

Vander Nat


D. So what was said before about Apollo gaining upon Mars, is only an illusion? This too is accomplished by the departing motion of the Earth, rounding ahead of Mars. M. You understand well. D. One thing more must be made clear. M. What? D. I remember that Mars was said to quicken his steps when anticipating the Sun and to recede as if to evade the Sun. I thought that Ptolemy had given excellent reasons for this occurrence, and now I would like to hear Copernicus’ explanation. M. You shall have it. Think once more about the charioteers. Do you perceive that the Earth will pass Mars? D. Yes, according to hypothesis. M. Imagine yourself in the Earth’s chariot at the moment of conjunction of Mars with the Sun. Is not Mars so much the farther than at the moment of opposition? D. Yes of course. He is farthest of all at this point, if my geometry is correct. M. Is not the horse of Mars drawn in the opposite direction of your own at this moment? D. It certainly does seem so, as if the two horses were running away from each other. M. Yes, the effect of the Earth’s motion augments that of Mars; the two contrary motions cause Mars to appear all the faster. D. Do I understand correctly? His change in speed, his change in luminosity and the curious retrogression are all due to the Earth’s proper motion? M. Yes, the Second Inequality is entirely swallowed up by the Earth’s annual motion. D. What of the First Inequality? M. Copernicus held the First Inequality to be a circle, or near circle, keeping with the customary hypothesis, but he thought it

were less wise for the circles to vary in speed. He therefore explains the eccentricity by a concentric circle with a double epicycle. In this way, both the smaller epicycles and the concentric motion move uniformly, with a combined effect of a near circle moving unevenly. D. I am confused. Did you not say the epicycle was an illusion according to Copernicus? M. With respect to the Second Inequality, Copernicus did reject the epicycle. However to effect the First Inequality, he has carried over two epicycles, which move so as to never cause a retrogression.

D. Why are three circles necessary to accomplish the work of one? Did not Copernicus seek simplicity? M. He did. The reason I suppose was to preserve uniform movement. Together they could achieve an eccentric motion, shifting the planet ahead of and behind an even rotation, while each epicycle is perfectly uniform in its own esteem. The smallest circle revolves quickly, its rotation being double the concentric orbit. The mid-size epicycle revolves westward, that is, opposite the smaller and opposite the concentric path but at exactly the same rate as the latter.

D. In my view, the revolution of the Earth eliminates the retrogressions of other planets which is simpler, but Copernicus has introduced more epicycles than Ptolemy which is more complex. M. You have made a fine point. D. It seems he has made an inversion in the hypothesis. M. In what way? D. Both astronomers make use of epicycles: Ptolemy for the Second Inequality, Copernicus for the First. Ptolemy deems the world to be Earth- centered, Copernicus, Sun- centered. M. I agree that is a sort of inversion. D. What says Brahe?

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De Astronomia

Vander Nat


M. Brahe could not bring himself to displace the Earth from the center of God’s creation, but could not ignore the advances made by Copernicus. Brahe proposed a compromise between the two cosmographies, the geocentric and the heliocentric. He placed the Earth very near the center of the heavenly sphere and placed the Sun in orbit around us, like Ptolemy, yet the five planets have the Sun as their center, like Copernicus. D. Do the epicycles explain the Second Inequality in this case as it does for Ptolemy? M. At conjunction Mars recedes above the Earth for the Sun is his center and passes in between them. At opposition the lapping Sun draws Mars downward to the Earth. This makes the Sun’s First Inequality responsible for the Second Inequalities of the planets, very much like Ptolemy. D. How could both the Sun and Earth be center? M. The five planets keep the Sun at their center as much as possible, as if they were all epicycles of one orbital circle. This center revolves around the Earth, like our Moon, but much farther off. D. Why do you say the planets keep the Sun at center “as much as possible”? M. By this I mean they are eccentric. All the astronomers are in agreement on this point: Ptolemy places the planets eccentric to a point near the Earth, while Copernicus and Brahe place the point of attraction not in the body of the Sun, but at a point near the Sun. D. This hypothesis too seems like an inversion. Ptolemy has put epicycles on eccentrics, while Brahe placed the eccentric epicycles about the point near the Sun. M. I must admit this too seems to be an inversion. D. But why would the planets revolve around this point? How are the planets able to know where it is? M. That is a fair question and one which I cannot answer, or rather, which the astronomers in question cannot answer. For as Ptolemy put into use the equant point and Brahe has brought forth another, the end of astronomy thus far has been describe the irregular motions, not to lay down their causes. D. Is not the equant a cause of changes determined by the First Inequality? M. Truly, it is determined by the First Inequality and therefore seems geometrical in nature, not physical. For similar reasons Brahe has maintained the point near the Sun but has not improved upon the ancients by demonstrating a cause. D. What do you mean “improved upon the ancients”?

M. I mean that the ancients believed each planet moved of its own force, either by aid of a god or by its own soul desiring to move through the expanse. Ptolemy, Brahe and Copernicus as well, all satisfy the appearances with their geometrical constructions, but the cause of each motion is no more than mythical belief. D. Do you state that the hypotheses merely conform to the appearances, giving no account for the cause of these paths? M. I do indeed. D. Then it seems to me that astronomy has not yet reached its end, for unless we can lay down the motive causes once and for all, we cannot affirm which of the Worlds is correct. M. I see clearly the obstacle you have in mind, and I leave it for you to settle what to some seems fruitless, but to those who love wisdom, most productive. You are well beyond the layman in your comprehension, so by aid of the goddess you may attain the truth, and display for all mankind, those secrets of the universe. D. Would that the goddess reveal to me the beauties of divine work, that I might measure them and thereby know them. M. Let that be our prayer.

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Never Metaphysics I Didn’t Like Cross


Never Metaphysics I Didn’t Like Spencer Cross

“I know Global Warming is a fraud,

but all my constituents believe in it!”

When politically organizing the U.S. Congress, the

amoral Baby Boomer generation, or the pessimistic youth of today, the problem is not within their stated beliefs as such, but more with their approach to thinking about the universe as a whole. It is for this reason that Lyndon LaRouche initiated the curriculum for his International Youth Movement to study the mind of Johannes Kepler, specifically his approach to Astrophysics. To aid the process of our population beginning to think for themselves, this essay aims to present part of Kepler’s approach to finding the path of Mars.

The New Astronomy

An important thing to remember when beginning to master Astrophysics, is that, much like the United States from 2007-2015, Kepler had no space shuttle, or space program with which he could “measure” the stars. He had no ruler or light-year long measuring stick, only the ability to measure changes in angle, and to proceed from there. As I hope to show, ditching the ruler is the first step to overthrowing the tyrant of pseudo-science.

If you view the night sky, you’ll see the sky doesn’t change. If you look more than once, however, you’ll notice that the heavens are now arranged differently. Now either you’ve been drugged and/or are imagining things, the trees in the background have moved, or something has happened with the Earth and the heavens. A good question to ask is, what is really moving? Ptolemy thought that if the Earth moved, we would be constantly thrown against walls and the clouds would just zoom past us, so why would anyone assume we move? Regardless of who moves, we can know the relationship is changing.

The heavens appear to move East-West in a fixed formation, except for a few anomalous stars which seem to move in the same direction, but slower. Now, why does 99% of the sky move at the same speed but these few? Are these stars eating at Al Gore’s mansion, and simply too hefty to keep up? Two possibilities emerge. To think like today’s existentialist youth, we would say, “Aw sweet, those stars are totally creative, they’re just doing their own thing, and defying the authority of the rest of the world.” That is one option, but Kepler put his faith in “However, when experience is seen to teach something different to those who pay careful attention…it gives rise to a powerful sense of wonder, which at length drives men to look into causes1.”

Since we know the relationship is changing, we can turn our universe into a flip-book and take a few pictures at known astronomical intervals to try to map the change. Take

1 Kepler, Johannes New Astronomy, Ch.1

the constellation Leo, for example, and take a “picture” of it at the same astronomical time everyday. Take the “pictures” at the same fixed, geographical location, i.e. from your toilet seat2 looking out the window, and snap a picture every time Leo reaches a known point like a prominent tree or power line. You need a fixed reference point, so it will not benefit anyone to witness a full moon here, or movements of other types3. You can imagine the relationship between your fixed observation point –the view from the Crapper-, through the horizon to the constellation as a fixed line, and you may view the night sky as though the change in relationship wasn’t occurring. You are in effect stopping the change in relationship to view the night sky from a standstill. Now this probably won’t tell us much, because we are stopping the Earth-heavens rotation, and that includes everything, right?

You can use this method to begin to look at how the planet moves, in relation to the heavens around it, (i.e. a constellation). The starkest aspect is that the planet is not moving east to west slower than the heavens, but West-East, in the opposite direction from the stars! Since what appeared to our senses as a slower movement of the stars was actually a different direction, are our senses deceiving us? Kepler calls the motion of all the heavens from East to West the “first motion,” and the motion of the anomalous planets from West to East the “second, or proper motion.”

In an Arc of Crisis, You can Know an Arc

Since we know the general direction of the planet, let’s try finding its speed. Now again, how are we to know the speed of the planet? To my knowledge, there is no policeman’s speed gun capable of clocking a star, much less some galactic speedometer to reference, so then, how? It is incumbent on us to try a simple example. If Al Gore weighed 250 lbs. before he lost to a primate in 20004, then ballooned to 293 lbs. by the end of ’01, then you would know Al Gore gained weight

5.

Similarly, if you take multiple observations of the same planet, you notice that it travels different distances in the same time intervals. If between the first and second observations it travels 3° of arc in the sky (a simple measure of distance), while between the second and third observations it travels 5°, then how is the planet changing? Investigating this process more closely, you notice that the planet speeds up to a certain, fastest

2 For countries with modern sanitation, toilets are common, fixed positions. 3 See Appendix A. 4 His animal rights activism was considered a conflict of interest. 5 Recent Computer models calculate Gore’s concentration of Cake-Oreo2 (CO2) ppmv has skyrocketed in the last decades, and the cause is “likely” algorepogenic.

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speed, at which point it slows down until it reaches the slowest point, and this repeats. This cycle of the planet’s change in speed is called the first inequality.

“But the Emperor has no clothes!”

Yet again, we are confronted with the problem of

explaining a seeming miracle. Does the Star simply choose to slow down and speed up whimsically? Is the Star speeding up to get to work at the same time every year and then slowing down after a hard year on the job, or is something else going on? We can say that if the orbits are circular, and we are at the center of the circular orbit, it would appear to us that the planet travels at the same speed all the time. Since it is not uniform, we can admit that the planet changes speed, that we are not at the center, or both. Kepler had already seen several explanations for the planets’ quirkiness, so instead of bothering with that “hypothesizing” stuff, he began with the ideas of Ptolemy and Copernicus.

1. A concentric circle with epicycle: The observer is at A, with the center of the planet’s orbit (B) orbiting it. The planet travels on this smaller circle, tracing out the arc CEG, taking orders from the point B, which in turns takes orders from the point A. 2. The Second model is a circle where the observer is not at the center, called an eccentric circle. The planet γ travels around the center of the circle β, even though the observer is observing it from α, which makes the planet appear to change speed. (Today’s baby-boomer sub-species must study this model to prove they’re not the center of the universe.)

Concentric with epicycle (above), and eccentric (below)

Kepler shows geometrically how these two are equivalent if CAE = γαε. In other words, these models are two ways to describe what you observe in the heavens, and they show the same thing.

You could adopt either model to explain the positions of the planets, and both models would fit everything you see well beyond what your eyes could observe. Now, if these two very different models can both explain the phenomena you are observing, which one is right? Could they both be right at the same time? Or, could neither be correct? I won’t spoil the fun by telling you, but if you tried to explain the motions of the planets, would the largest object in the sky factor into your model? In other words, could the giant, raging ball of fusion that dominates the sky just possibly have something to do with heavenly motion?

Get a Map of the Stars!

Speaking of segues, we will now reunite with our Sun. If you are to chart the movement of Mars against the background stars, what will you see? You should construct this map yourself, but observe Diagram 2 to get a quick look. Mars slows down, changes direction, then returns. This happens about every two Earth years, but what does this tell us? Are we to believe that the planet is simply riding a roller coaster, or driving drunk, tanked on ethanol? Kepler terms these changes of the star in relation to Earth and the Sun, and the cycle of all the changes and discrepancies, the Second Inequality. The first and second inequalities combine to produce what you see in a given night, which produces these anomalous effects, but the bigger question is, why?

Diagram 2

Hold on a Second!

Now this wild motion of the star Mars seems large enough for Ptolemy and Copernicus to rethink their whole system, right? Appendices B and C are attempts to explain the motion of the planet seen in diagram 2. In order for their system to work, Ptolemy added an epicycle to his original theory, and Copernicus thought the Earth would have to move. The latter was a theory the Dark Age-scientists wouldn’t touch with a 5½ foot pole. You might notice again that they both are different,

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yet seem to explain the motions you observe in the sky. So again, who is right; or, is either?

Never Metaphysics I didn’t like

Kepler discusses that although the two models share a geometrical equivalence, their philosophical differences are as big as “Scale” Tipper Gore’s disgust with procreation6. If we are to believe that the heavens function as these models describe, then what kind of Universe do we think we live in? In current life, most humans don’t believe that our wristwatches are powered by microscopic slave factories of fairies, because to honestly think that, we would have to 1. Believe in fairies. 2. Support slavery. 3. Comprehend next to nothing about basic mechanics. 4. Believe in global warming. Now, that last one was a trick, in that you wouldn’t necessarily have to believe in it to believe the wristwatch hypothesis, but if you believed in the fairy idea, then you might also believe in Global Warming. For instance, Ptolemy believed in a concept known as “orbs.” He thought the universe was composed of a whole series of solid, glass orbs, within which the planets move. Now, if the Heavens were merely a mechanical system of solid, glass orbs that move according to a predetermined ratio, then all the creator of the universe would have to do would be to “flip the switch” or set the system of orbs in motion, and then die, because his continued actions were unnecessary. The idea is that God is perfect, (which in this line of thinking means unchanging) and once his universe is set into motion, he has done all that he would ever need to do, so he can “retire.” If we are to believe in solid orbs, then we must admit that God is dead. Nietzsche wasted his time by not reading Aristotle more thoroughly, because 2000 years before Nietzsche was born, Aristotle’s universe logically removed God, allowing the Romans to stop worrying and get back to pillaging. If we were to maintain solid orbs but have epicycles as well, we would have to devise two types of motion, one to move the center of the epicycle around the observer, another to move the planet around the center of the epicycle. This system would require two motions, each acting with respect to a different point in the universe, and in the case of the planet on the epicycle, acting with respect to a point that doesn’t even really exist! The other type of System of the Universe is one which does not have solid orbs, but one in which the planets move of their own accord in the heavenly ether. This requires that the planet knows by itself to make a circle with its path, for no reason other than its own discretion. In the case of the eccentric, the planet is traversing a circle around a point that has no physical existence! In the epicycle model, the center of the epicycle revolves around the observer, but the planet is revolving around a center, which has no physical existence! Now since the planet is changing speed in the air, you would also attribute a mind to the planet such that it knows (on its own accord) not only how to travel in an enclosed circular system by itself through the air, but how to change speed by itself. In that sense, we are attributing multiple minds to each planet, and

6 Clinically, algoraphobia, see Appendixie D.

making them into demi-gods. Is this really the consequence of scientific theory? To believe in either of them, I would either have to admit that 1. God is dead, or 2. that the universe is run by many Gods, whose motives humans are forbidden to discover?

Sun of What?

At this point you might be wondering, “how is working on Astrophysics going to do anybody any good?” The beauty and difficulty of Economics, especially as developed by Lyndon LaRouche, is that Economics is the realm where science, belief systems, and human survival are all the same thing. To use a modern example, radical ecologists and environmentalists begin with an old idea. Hitler and the Nazi movement believed that the Jews were “vermin,” who did nothing but destroy, therefore it was acceptable for them to be exploited and eventually eliminated for good. Today’s greenies have advanced this idea, and because they believe all men are created equal, they believe that mankind as a whole is vermin, which does nothing but destroy the Earth, and we need to implement policies to eliminate them. These ideologues then concoct a theory of Global Warming to fit their ideology, to give them the backing of “science” to implement their policies. It is humorously non-ironic that the global warming fanatics of today share so much with our ancient astronomical friends. In order to prove something they “really believe in,” something they are “really compassionate for,” they construct a belief that frees their conscience from a real scientific investigation into causes. To give a specific example, CO2 levels have been higher during the last 140 years, yet temperature has not been higher during those periods. Human CO2 is less than a 1% factor in the “greenhouse effect,” yet the Greenshirts of today demand we refuse Africa modern industry and sanitation, because “most Scientists might agree.” The Sun is a larger factor by magnitudes in climate theory, and most long-term cycles in global climate go back before human records, and well before the emergence of environmentalists. I would happily mention that Kepler’s astronomer-predecessors and modern-day environmentalists have both ignored (except to mystically worship) the Sun. Working through Kepler’s method of scientific thinking, is proving crucial for not only the current mis-leadership of the United States, but more importantly, for the next generation of scientific/cultural leadership, typified by the international LaRouche Youth Movement. My fellow scientific thinkers have an immense responsibility today. Because of the U.S. economic breakdown and the push to immediately end the Nation-State system by implementing green-brownshirt policies, we have to challenge the leaders of the United States to fight for the soul of the Democratic Party. We have to ask them, “If the consequences of your policy are to prevent Africa from developing for 100 years, and to sign the death warrant for the U.S. economy and constitution, are you certain about this ‘science’?”

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Appendices

Appendix A

“The Cow” Tipper, waiting to observe a full moon where she can wolf down some carbonated beverages.

Appendix B

Appendix C

Appendixie D

To date, this is Gore's best argument against population growth.

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The Fallacy of the Equant

Ross


The Fallacy of the Equant Jason Ross

Typically, actually scientific method, as teams of the LaRouche Youth Movement (LYM) have experienced this for themselves, is the crucial example, demon-strated by Johannes Kepler, of the problematic attempt to define Solar orbits in a manner congruent with the notion of an equant. All comprehensive notions of a competent modern physical science are implicitly em-bedded in the implications of the problematic nature of the assumption of the equant. It is this discovery by Kepler, which provided modern science with a rigor-ously defined notion of the ontologically efficient actu-ality of what is rightly considered a universal physical principle, such as gravitation. It was Kepler's recogni-tion of the fallacy of the equant which, according to Kepler's account, prompted Kepler's conception of the infinitesimal reflection in the very small, by a universal principle in the very large. All competent modern sci-ence is premised on an apriorism-free notion of a uni-verse defined by a process of development among a set of universal physical principles of the same, experimen-tally defined, ontological quality, in themselves, as Ke-pler's notion of universal gravitation.1


Moreover, it is no less false that the center of the world is within the earth than that it is outside the earth; nor does the earth or any other sphere even have a center. For since the center is a point equidistant from the cir-cumference and since there cannot exist a sphere or a circle so completely true that a truer one could not be posited, it is obvious that there cannot be posited a cen-ter [which is so true and precise] that a still truer and more precise center could not be posited. Precise equi-distance to different things cannot be found except in the case of God, because God alone is Infinite Equality. Therefore, He who is the center of the world, viz., the Blessed God, is also the center of the earth, of all spheres, and of all things in the world. Likewise, He is the infinite circumference of all things.2

In Part II of his The New Astronomy, Kepler takes up the motion of Mars, having identified two a priori, axiomatic assumptions which had bounded the investigations of astronomy up to that time: that the planets move in perfect circles, and that an equant point can be found for the orbit – a point from which the planet could be observed to move at a constant speed, trav-ersing equal angles in equal times – a point of uniformity.

1 Lyndon H. LaRouche, Jr., “The Dance of the Bio-Fools,” EIR, Feb 2, 2007 2 Nicolaus of Cusa, De Docta Ignorantia, trans. Jasper Hopkins

Rather than this approach to investigating the apparently irregu-lar motion of Mars, Kepler’s healthier mind posed the question: what are the characteristics of change of Mars’s apparent mo-tion?

Investigating Mars

The motion of the planets were seen to have one perio-dicity and movement through the zodiac proper to themselves, and a second movement related to their apparent proximity to the sun. Kepler’s predecessors Claudius Ptolemy, Nicolaus Co-pernicus, and Tycho Brahe offered hypotheses for these two actions.3 To investigate a planet, it is first necessary to untan-gle these two motions. Using the Copernican model, the second inequality, as it is called, does not reflect a change in the motion of Mars, but rather reflects the motion of our Earth, from which we observe it. This inequality can be removed by using specific observations: those taken at opposition. If we see the Sun and Mars at opposite sides of the horizon (one setting while the other rises), then the Sun, Earth, and Mars lie on a straight line, and the sun “sees” Mars in the same direction that we on the Earth observe it. Thus, while making observations as we stand physically on the Earth, our mind’s eye is transported to the sun, from which we can watch Mars.

Earth orbit

Mars orbit

Oppositions: when the Sun, Earth, and Mars are lined up, the

Earth sees Mars in the same direction as does the Sun.

Having eliminated the effect of our moving Earth on the observations of Mars, the first inequality still remains; Mars’s motion, freed form the second inequality, is determined to have one position on the zodiac at which its motion is fastest,

3 See Vander Nat’s “De Astronomia,” ∆υναµι∆υναµι∆υναµι∆υναµιςςςς, this issue.

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and another where it is slowest. How can this changing speed be understood?

Two geometrical hypotheses that cause an apparently changing speed are the eccentric model, and the equant model:

Eccentric motion (left) and equant motion (right)

In the eccentric model (left), the planet moves uni-

formly around the center of its orbit, but is seen from a point off-center. This causes the apparent speed of the planet to change based on how close it is to the observer. (Remember, we are still “observing” from the sun by using opposition obser-vations.) In the equant model (right), there are two centers: 1. a center of motion, the equant (top point), around which the planet describes equal angles in equal time, and 2. a center of location, from which the planet maintains a constant distance, simply called the center (middle point). In the equant model, the planet actually does change its speed, a physical change, whose appearance is further affected by its changing distance from the observer (bottom point). Ptolemy introduced the equant, rather than the simple eccentric, because he found it to better represent the motion of the planets.4 From Ptolemy through Kepler, the aprioristic assump-tion of a principle of uniformity acting efficiently in the uni-verse, was unchallenged.5 For what alternate cause of motion could be said to exist?

4 For a comparison of the two types of models, see http://wlym.com/~animations/part2/16/aside.html and http://wlym.com/~animations/part2/21/index.html. 5 Copernicus writes: “We must however confess that these movements are circular or are composed of many circular movements, in that they maintain these irregularities in accordance with a constant law and with fixed periodic returns: and that could not take place, if they were not circular. For it is only the circle which can bring back what is past and over with; and in this way, for example, the sun by a movement com-posed of circular movements brings back to us the inequality of days and nights and the four seasons of the year. Many movements are rec-ognized in that movement, since it is impossible that a simple heavenly body should be moved irregularly by a single sphere. For that would have to take place either on account of the inconstancy of the motor virtue -- whether by reason of an extrinsic cause or its intrinsic nature -- or on account of the inequality between it and the moved body. But

since the mind shudders at either of these suppositions, and since it is quite unfitting to suppose that such a state of affairs exists among things which are established in the best system, it is agreed that their regular movements appear to us as irregular…” (De Revolutionibus, I.4)

Kepler’s Model

Before directly challenging the assumptions of the

equant and of circular orbits, Kepler aims to vindicate his use of the apparent sun (the one we see in the sky) instead of the mean

sun, an imaginary point near the real sun used by Ptolemy, Co-pernicus, and Tycho to set up their planetary hypotheses. To do this, Kepler uses opposition observations made with respect to the apparent sun, rather than the mean sun oppositions used by his predecessors.6 With twelve observations of Mars at opposi-tion at his disposal, Kepler selects four with which he works out a planetary hypothesis using an equant and a circular orbit.

From the observed distance along the zodiac between one observation to the next, we know how far Mars has moved as seen by the sun (apparent longitude). From the time between the oppositions, we know how far Mars has moved as seen by the equant (mean longitude), since the equant is the hypothe-sized point around which Mars moves at a constant angular speed. With four observations, and an incredible amount of time spent on a difficult procedure, Kepler determines the best alignment of the sun, center, and equant with respect to orienta-tion and distances. He calls this model his vicarious hypothesis.

Sun

Equant

Center

The Vicarious Hypothesis (not to scale)

Kepler remarks that the solar eccentricity of this model

(the distance from the sun to the center) is 11.3% of the radius of the orbit, a value determined not by finding the center as the midpoint between Mars locations, but as the best value to use to make his model “work.” To test this model, simply determine the angles around the equant corresponding to the times of the other oppositions, draw lines from the equant to the orbit at those angles, and see where the sun would see those planetary positions.

His vicarious hypothesis is a success! Among the twelve oppositions, the largest disparity between his model and the observations is only about 2' (two minutes of arc), 7 which is the margin of observational error in the measurements them-selves. Thus, he can conclude that his model works “perfectly” all around the zodiac. This is better than the models of his predecessors, all of whom had used the mean sun rather than the apparent. Some might now rest from their labors, content at

6 See http://wlym.com/~animations/part1/meanapparent.html. 7 An arc of 1' is one-sixtieth of a degree of the nighttime sky, and is about the width of a pencil lead held eight feet away from you.

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having developed the world’s best method for determining the position of Mars, but not Kepler. He poses the question: just because any possible error in his model is too small to be ob-served, does this necessarily mean that he has found the truth? And how can we settle the debate between Ptolemy, Copernicus, and Brahe – Kepler’s vicarious hypothesis could be imple-mented equivalently in any of the three world-views, if the ap-

parent sun be used.

Another Determination of Eccentricity

Kepler then sets out to check this eccentricity, by de-

termining it directly, rather than the indirect method used in the vicarious hypothesis. Kepler will use measurements of latitude, rather than longitude, and find the physical distances of Mars from the sun at opposite points of its orbit, as a way of finding the center of the Mars path.

Mars

Mars

Earth

EarthSun

By observing the latitude of Mars north and south of the plane of the ecliptic – the plane of the Earth’s motion around the sun – and using some trigonometry, Kepler determines how far Mars was from the sun. Doing this at nearly the closest and farthest parts of the orbit means that the mean between these two gives the radius of the orbit, and the distance of the center from the sun – the solar eccentricity. With this method, Kepler determines this eccentricity to be 8.0% - 9.9% of the size of the orbit, which does not match the eccentricity determined by the vicarious hypothesis (11.3%).

A paradox emerges: how can Mars have one required eccentricity according to his longitude model, and another when investigated directly according to latitudes? What can be said of a model that gives correct results while using a parameter known to be false?8

A Crack

In an attempt to reconcile these two eccentricities, Ke-pler adjusts his vicarious hypothesis to have an eccentricity in keeping with that determined by his study of latitudes. The total

eccentricity of the vicarious hypothesis – the distance from the equant to the sun – is 18.6%, of the radius, whose half, 9.3% does fit in the range of 8.0-9.9% required by latitude observa-tions. So, if he bisects the eccentricity, moving the center to be in the middle between the equant and the sun, he can apply the eccentricity determined by latitudes to the functionally perfect vicarious hypothesis model. This also agrees with Ptolemy, who assumed a bisected eccentricity when working out his model.

8 Here, too large an eccentricity.

135°

135°

>90° <90°

Original Vicarious Hypothesis Bisected Eccentricity

In this diagram with greatly exaggerated eccentricity, the sun’s perception of Mars changes when the eccentricity is bisected.

After a time of 135° from aphelion, measured by the equant, the angle between Mars and the perihelion is greater than 90° in the vicarious hypothesis, and less than 90° in the bisected version.

Changing to the bisected eccentricity alters the vicari-

ous hypothesis, and this change is significant: the perfection of the vicarious hypothesis is lost when the bisected eccentricity determined by latitudes is introduced. When drawing lines from the equant at angles determined by the times of opposition, Ke-pler finds a gap: the bisected model is now 8’ off for the opposi-tion of 1582. This difference is not within observational margins of error.

Thus, his model cannot both give correct positions of Mars (original vicarious hypothesis) and incorporate the true eccentricity (bisected) at the same time. There is therefore no possible way of adjusting it to make it work. Kepler writes:

Therefore, something among those things we have assumed must be false. But what was assumed was: that the orbit upon which the planet moves is a perfect circle; and that there exists some unique point on the line of apsides at a fixed and constant distance from the center of the eccentric about which point Mars de-scribes equal angles in equal times. Therefore, of these, one or the other or perhaps both are false, for the obser-vations used are not false… Now, because they could not have been ignored, these eight minutes alone will have led the way to the reformation of all of astronomy.

The statistical approach to astronomy, the attempt to understand the heavens by making models of the footprints of a cause of motion, has not only failed so far to achieve perfection; Kepler has proven, conclusively, the impossibility of creating a perfect model with this approach.

What is the implication of this new category of experi-ence for the practice of Man’s mastery over nature? Kepler has demonstrated the required existence of a universal, physical (not geometrical) principle. The unavoidable, paradoxical implica-tions of the equant, force the mind to a new sort of wonder.

To attempt to present Kepler’s discovery of universal gravitation, without a thorough working-through of the paradox of the equant, were to proffer an answer to an audience incapa-ble of posing the right question.

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Kepler’s Paradoxical Return to the Equant

Ross


Physics, Not Statistics: Kepler’s Paradoxical Return to the

Equant Jason Ross

Since all competent modern physical mathematics is based on the pioneering achievements of Johannes Kepler, the argument to be made, in explanation of the intrinsic incompetence of statistical mechanics for economics, will employ the image of a planetary orbit, as defined by Kepler's uniquely original discoveries, to define a forecastable quality of true long-term cycles in an economy. That lesson, from Kepler, for economics today, is the best source of remedy for the failures intrinsic to the consistently failed methods which have been employed by economics statisticians generally

during the recent decades.1

Preface: A Dispute on the Equant

A social process, created no later than the summer and fall of 2006, when the LaRouche Youth Movement, following Lyndon LaRouche’s guidance, embarked on a mass socialized project to work through Kepler’s The New Astronomy to make the breakthroughs in economic science required to politically organize the rebuilding of the collapsing world physical-economic system, has borne fruit in a new level of discussion, allowing new questions to be developed, and breakthroughs to be made, that would never have occurred among a group of isolated individuals working on discoveries in their bedrooms, or among students listening to a lecture series. Evidence of the successful gestation of this process was borne out in one incident in early December, 2006, at a meeting of the LaRouche Youth Movement in Washington, DC. A question was raised there that was being pondered by an increasing number of members of the LYM around the world, a question that provoked a lively back-and-forth: “Why does Kepler re-introduce the equant in Part III after he has already refuted the possibility of its existence in Part II of The New Astronomy?” Follow-up discussions on the question, particularly after Lyndon LaRouche’s December 22nd paper, in which he first makes explicit reference to the equant,2 prompted the writing of this article.

These discovered, universal principles, belong to a category of experience which Kepler was the first to define, through exploring the paradoxical implications of the equant, as showing the ontologically infini-tesimal reflection of any universal physical principle.3

1 Lyndon H. LaRouche, Jr., “Re-Animating an Actual Economy,” EIR, August 4, 2006 2 LaRouche, “What the Congress Needs to Learn: The Lost Art of the Capital Budget,” EIR, Jan 12, 2007 3 Ibid.

And: That function of irony, in language, as in physical science, which distinguishes the creative mental powers typical of the specific notion of the human individual, is the same function associated with the process of discovery of a universal physical principle in physical science, as Kepler's treatment of the fallacy of the equant, in proceeding toward the discovery of a universal principle of gravitation, illustrates the existence of the apparent infinitesimal magnitude associated with the quality of action by a universal physical principle of gravitation.4

The Equivalence of Hypotheses

Claudius Ptolemy, Nicolaus Copernicus, and Tycho Brahe offered hypotheses for the motions of the planets, hypotheses which, at first glance, appear to be immensely different. Ptolemy has all planets move around the Earth with an equant and an epicycle, Copernicus has all move around the sun with a double epicycle, and Brahe combines the two, allowing the planets to move around the sun which itself moves around the Earth. Kepler, professing himself to be of the Copernican outlook, nonetheless does not begin his The New

Astronomy with a defense of the Copernican hypothesis. Rather, his first task is to demonstrate the equivalence of these three seemingly different systems.5 Kepler proves that all three world systems can have their parameters adjusted in such a way as to completely agree. If these hypotheses are capable of making exactly the same results (within observational precision), that is, if they all “work” equally well, how can one decide which among them is correct? Should “working well” even be the metric of truthfulness?6

And, what is equivalent about these hypotheses? Certainly not their outward appearances (although they can be used to create equivalent predicted appearances for the planets), but rather their equivalent assumptions. All are based on uniformity as a guiding principle of the organization of the universe: both uniform angular motion, and motion of uniform distance (circular motion). They also share a rejection of the

4 Ibid. 5 See http://wlym.com/~animations/part1/index.html for animations of this equivalence. 6 A society convinced that human beings are evil, which decides to organize itself according to that principle, will achieve results that cohere with that thought. “I know I didn’t try to do anything good, but doesn’t my failure prove that it was impossible anyway?”

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body of the sun playing any role besides celestial décor; instead of the actual sun (the apparent sun), they all use a geometrical point near the sun known as the mean sun. For Ptolemy and Brahe, this is a point moving around the Earth with the same period as the sun, but, unlike the actual sun, it moves uniformly. Copernicus considered the mean sun to be the center of the Earth’s orbit, and he set up the orbits of other planets using this point as his anchor. It is upon this use of the mean sun that Kepler targets his investigative acumen. Kepler writes that Ptolemy “chose the mean motion, thinking that the difference (if any) between taking the mean sun and the apparent sun could not be perceived in the observations, but that the form of computation and of the proofs would be easier if the sun’s mean motion were taken.” What, for Ptolemy, was a decision in favor of computational ease, became a physical absurdity for Copernicus and Brahe. Yet, they continued to use it.7 Kepler’s insistence on relating planetary orbits to the apparent sun required that he demonstrate that, unlike the equivalence of hypotheses discussed above, the difference between the mean and apparent sun does result in incompatible hypotheses. That is, this question is something susceptible of a crucial experiment.

Kepler sets out to develop his own tentative hypothesis for the first inequality of the motion of Mars, a hypothesis, which, by its use of the apparent sun, will not be equivalent to those of Ptolemy, Copernicus, and Brahe. This tentative hypothesis, based on the use of circular orbits and a point of uniform motion rather than physical considerations, Kepler calls his Vicarious Hypothesis.

The Vicarious Hypothesis

Before reading on, be sure to have read The Fallacy of

the Equant (∆∆∆∆υναµιυναµιυναµιυναµιςςςς, this issue).

Kepler’s vicarious hypothesis works better than those of his predecessors, vindicating his use of the apparent sun. However, it is not yet perfect, for its successful application requires the use of a value (the eccentricity of the orbit) known to be false. Furthermore, Kepler proves the impossibility of adjusting such a hypothesis to provide a perfect representation of the heavens. Of this constructive failure, Kepler writes in Part II of his The New Astronomy:

Therefore, something among those things we have

assumed must be false. But what was assumed was: that the orbit upon which the planet moves is a perfect circle; and that there exists some unique point on the line of apsides at a fixed and constant distance from the center of the eccentric about which point Mars de-scribes equal angles in equal times. Therefore, of these, one or the other or perhaps both are false, for the obser-vations used are not false.8

7 See http://wlym.com/~animations/part1/meanapparent.html for a development of the mean sun and the apparent sun. 8 Johannes Kepler, Astronomia Nova, trans. William Donahue, pp. 283-284.

It would seem that a totally new approach to astronomy is in order. Yet, after his demonstration of the unavoidable paradoxes inherent in approaching astronomy with the assumptions of uniform motion embodied by the equant, and of uniform distance required by a circular orbit, Kepler surprises many readers with what he does next to refute the approaches of Ptolemy, Copernicus, and Brahe. He spends the first half of Part III demonstrating that the motion of the annual orb (the Sun or Earth) is not a simple eccentric, as his predecessors insisted, but that it does conform to the equant-hypothesis! That is, it is wrong to say that the annual orb does not have an equant in its theory. But, why would Kepler add an assumption that he has seemingly just refuted?

A Different Type of Equant But, before getting into the details, let us review Kepler’s mind. In his argument for the existence of the equant, from the chapter of Mysterium Cosmographicum titled “Why a planet moves uniformly about the center of an equant,” he says of an equant with bisected eccentricity:

Therefore at the middle part of the eccentric path where it projects above the concentric circle, the planet will be slower, because it moves further away from the Sun and is moved by a weaker power; and in the remaining part it will be faster, because it is closer to the Sun and subject to a stronger power… Let A be the source of this moving spirit, namely the Sun…. Then, naturally, let the whole universe be full of a spirit which whirls along any stars or comets it reaches, and that with the speed which is required by the distance from the Sun or of their positions and the strength of its power there.9

As you can see, Kepler had always considered the equant-model a good hypothesis to explain the motion of the planets, not because of a power of uniform angular motion, but because it mimicked a physical power, a “moving spirit,” whose source is in the Sun. Here we have, over a decade before the printing of The New Astronomy, Kepler’s hypothesis of a physical cause! The failure of the vicarious hypothesis in Part II was a successful demonstration that the geometrical mimic can be differentiated from the physical truth.10 Note the difference in method: Ptolemy introduced the equant to match the observations, but Kepler, looking back at its introduction, forms a physical hypothesis for why the equant appears as a useful geometrical tool. Now, the search for what might ironically be called a physical equant can begin!

But, before moving ahead, Kepler raises an objection to this idea:

9 Johannes Kepler, Mysterium Cosmographicum, trans. A. M. Duncan, Ch. 22, pp. 217-219 10 As Kepler says in Chapter 4 of his Astronomia Nova: “The point of the equant… is nothing but a geometrical short cut for computing the equations from a hypothesis that is clearly physical.”

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[W]hat explanation will be eventually put forward for the annual motion of the Earth? For it did not need an equant either in Ptolemy’s theory or in Copernicus’s. Consequently, this is also a doubtful case awaiting the judgment of astronomy.11 How can his physical hypothesis be universal, if it is

not expressed in the motion of the annual orb, which does not have an equant?

A Barrier to Hypothesis

Kepler writes of this mental barrier to consideration of his physical reasoning in the beginning of Part III of his The

New Astronomy:

In chapter 22 of the Mysterium Cosmographicum, when I was giving the physical cause of the Ptolemaic equant or of the Copernican-Tychonic second epicycle, I raised an objection against myself at the end of the chapter: if the cause I proposed were true, it ought to hold universally for all planets. But since the earth, one of the celestial bodies (for Copernicus), or the sun (for the rest) had not hitherto required this equant, I decided to leave that speculation open, until the matter were clearer to astronomers. I nevertheless entertained a suspicion that this theory might perchance also have its equant. After I gained the recognition of Tycho, this suspicion was confirmed in me.12

Studying the Earth

The study of the annual orb is of particular interest for studying Kepler’s method of thought. Unlike their theories of the motion of Mars, which, at times, were off by degrees, the hypotheses of Ptolemy, Copernicus, and Tycho for the motion of the annual orb left nothing to be desired: the predicted locations were never found to be at variance with the actual locations of the sun. Upon what grounds might Kepler introduce an equant into the theory of the annual orb, when a simple eccentric already works “perfectly?” We seem to be facing the same problem encountered in Part I of the equivalence of hypotheses: were Kepler to show that using an equant for the annual orb gives correct values, could he claim that he is more correct than his predecessors, or only equivalently correct? Clearly, a different, active, type of reasoning is required here.

An Irony From Above

But, how can we begin to study the motion of our home the Earth, when we are moving along with it? This question is akin to the use of irony in political organizing.

11 Kepler, Mysterium Cosmographicum, p. 219 12 Kepler, Astronomia Nova, pp. 305-306

All humans have assumptions about how the world around them, and the mind within them, operate. These predispositions act upon all that we observe or consider; therefore our observations are shaped by our assumed axioms. Is it possible to examine these assumptions themselves, and, if so, how?13 How can anyone be made to see their assumptions, if all their perceptions are made according to these very assumptions? How can we see the motion of the Earth while we are moving with it?

Watching the Earth from Mars

Kepler uses an irony in the Sun-Earth-Mars relationship to allow this feat of self-reflection, by watching the Earth from the Sun and from Mars. This he does by making Mars remain motionless, an observer of our moving Earth. But, how does he accomplish this?

Mars has its year just as does our Earth. In the same way that the sun appears at the same location in the zodiac each Earth year (keeping in mind the precession of the equinox), Mars is in the same position with respect to the sun after each of its years. But, our Earth is not in the same position after one of Mars’s years. Therefore, the motionless Sun and Mars are Kepler’s two celestial eyes, watching our Earth in its course through the heavens.14

Four different positions of the Earth, with the Sun

and Mars always at the same positions.

Just as in Part II, where a dialogue involving the Sun

and the Earth was used to determine the path of Mars by using observations taken at opposition, Kepler again allows Sol, Earth, and Mars to work in harmony to determine their relationships. So, armed with this tool of creating irony, we can retrace Kepler’s footsteps as he demonstrates conclusively that the Earth (or Sun) does not move on a simple eccentric.

13 Kant believed this to be impossible; he writes of one such unquestionable assumption in his Critique of Pure Reason: “Not only in judgements, however, but even in conceptions, is an a priori origin manifest. For example, if we take away by degrees from our conceptions of a body all that can be referred to mere sensuous experience—colour, hardness or softness, weight, even impenetrability—the body will then vanish; but the space which it occupied still remains, and this it is utterly impossible to annihilate in thought.” 14 See http://wlym.com/~animations/part3/24/index.html.

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Let us create a very specific irony, to directly examine the assumption of a simple eccentric. Allow the position of Mars, to which it returns each Mars-year to be perpendicular to the Earth’s line of apsides, and, in fact, to be on a perpendicular raised from the point of uniformity of the Earth’s motion (which is also the center, according to those who came before Kepler).

F

CE D

A

Here, A is the sun, C the point of uniform motion (both

the equant and the center, assuming a simple eccentric), E and D are two positions of the Earth looking at Mars at the same position F. The motion of the planets did not provide Kepler with the good fortune of angles FCE and FCD being exactly 90°, but he did get both to be equal to 64°23'30".

By the hypothesis of the simple eccentric, this orientation should be symmetrical:

F

C

E D

A

However, observed angles CEF and CDF do not come out equal. Working from the location of Mars, the angle Earth moves around the equant, the observed positions of Mars, the orientation of the line of apsides for the Earth, and, finally, the assumption of a circular orbit, we arrive at this unsymmetrical result, indicating that C is not the center of the Earth’s orbit. Thus the mean sun is not the center of the Earth’s orbit it all, but is actually its equant. How much more absurd it now appears to set up planetary hypotheses using the mean sun!

F

C

E D

A B

B is introduced as the center of Earth’s orbit, distinct from C.

With this result, Kepler, using beautiful insight into triangles, is able to determine the distance BC to be around 1837, which is almost exactly half of the distance AC, which was taken to be 3584 (whose half is 1792); this appears to be a bisected eccentricity! Kepler is emboldened to move ahead with his physical hypothesis from his Mysterium Cosmographicum:

Such, then, was the beginning of this enquiry, timid and encumbered with such concern that the anomaly of commutation be equal on both sides [that the diagram work out to be symmetrical]. Now that we have once made a hazard of this, we are buoyed by audacity to sally forth again more freely onto the battlefield. For I shall seek out three or more observed positions of Mars with the planet always at the same eccentric position, and from these find by trigonometry the distances of that number of points on the epicycle or annual orb from the point of uniform motion [equant]. And since a circle is defined by three points, I shall use three such observations to find the position of the circle, its apsides (previously taken as a presupposition), and its eccentricity with respect to the point of uniform motion. Should a fourth observation be at hand, it will serve as a test.15

Now that Kepler has shown a paradox, a crack in the results he obtains from using Tycho’s assumptions, he is able more freely to investigate what is really happening, no longer in terms of why others are wrong, but to find what is right.

Watching the Dance

Rather than using two observations, let us use four: three to define a circle, and a fourth to test. To avoid the objections of those who suspect novelty in his introduction of the apparent sun, Kepler first does this using the hypothesis of the mean sun, finding the distance between the center of the annual orb and its equant to be around 1530, very close to the 1800 required by a bisected eccentricity.

α β

ζ

ε

η

θ

In Chapter 25, Kepler finds the distance of the center of the

annual orb (β) from the point of uniform motion (α). An animation of a similar process is at http://wlym.com/~animations/part4/41/index.html.

15 Kepler, Astronomia Nova, p. 316

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Kepler next switches his anchor to the apparent sun, and finds the distance between Sol and the center of Earth’s path. He does this in chapter 26 using the vicarious hypothesis for the position of Mars as seen from the sun, and then proceeds without this assumption in chapter 27. He finds an eccentricity of 1800 in chapter 26, and 1653 in chapter 27.

η

δ

γε

ζ αβ

Kepler uses the Sun (α) as his anchor, and finds the distance

between it and the center of the Earth’s orbit (β), again finding a bisection of the total eccentricity used by Tycho Brahe.

To confirm this bisected eccentricity, Kepler assumes it

is true, and then devises a method of determining Mars’s position based on positions of the Earth using this assumed eccentricity. His concluded positions of Mars as determined from several Earth positions agree among themselves, and with his vicarious hypothesis. Thus, he has concluded the bisection of the eccentricity in the annual orb by four different methods.

Effects on Observations Despite these demonstrations using observations of Mars, some might object to Kepler that since Brahe’s tables based on a simple eccentric never reveal an error with respect to observations of the sun, that introducing an equant could only serve to add an error to the theory. Kepler writes: “Brahe feared that if I bisected the sun's eccentricity I would vitiate his equations of the sun’s motion.” To test this, Kepler simply calculates the maximum difference between the eccentric-model and his equant-model for the annual orb, and finds this maximum difference to be on the order of seconds of arc – too small to be observed: the equant does not vitiate the equations. A remarkable consequence of this is that simply observing the sun would never have generated an irony requir-

ing the equant. Since the two models are indistinguishable by solar observations, this question could have met the same fate as the debate between the equivalence of hypotheses of Ptolemy, Copernicus, and Brahe addressed in Part I: it would have been considered undecidable by observation. The universe required Kepler’s active, experimental seeking, rather than passive observations to reveal its secret! Without observing Mars, the perceived motion of the Sun in the heavens could not have been known. In engaging in a dialogue with Mars, we come to know the Earth.

A Universal Physical Principle Now what held him back need no longer hinder him. Rather than adding an axiom of equant-motion, Kepler has removed an exception to the existence of a universal physical principle of motion; he has removed the exceptional status granted to the annual orb. With his irrefutable demonstrations that the center of the annual orb is not its center of angular motion, and that, in fact, it too possesses an equant with bisected eccentricity, Kepler can look for a universal cause of motion:

Now in my Mysterium Cosmographicum… I postponed arguing this case of the cause of the Ptolemaic equant for the sole reason that it could not be said on the basis of ordinary astronomy whether the sun or earth uses an equalizing point and has its eccentricity bisected. However, now that we have the confirmation of a sounder astronomy, it should be transparently clear that there is indeed an equant in the theory of the sun or earth… Now that this is demonstrated, it is proper to accept as true and legitimate the cause to which I assigned the Ptolemaic equant in the Mysterium

Cosmographicum, since it is universal and common to all the planets. So in this part of the work I shall make a further declaration of that cause.16

It is in “making a further declaration of that cause [emphasis mine]” that Kepler leaps across the pit into which a statistician would stumble. (See box) Kepler moves to declare this cause, with his demonstration in chapter 32, that an equant with bisected eccentricity closely imitates his physical hypothesis of the sun serving as the seat of a power moving the planets: the time for a planet to traverse equal amounts of arc is measured by its distance from the sun: “the elapsed times of a planet… on equal distances in the ethereal air are in the same ratio as the distances of those spaces from the… center of the world.” This could also be stated that the planet’s speed is in inverse relationship to its distance from the sun. But, this holds only when the equant and the sun are at equal distances from the center along the line of apsides. When they are at unequal distances along the line of

16 Kepler, Astronomia Nova, p. 372

Box: Perfectly More

Since the equant works so well in the vicarious hypothesis, perhaps a slight alteration of it would work even better. This is the temptation to make a small addition to the model, to make it “more perfect.” But, this is not Kepler’s approach. Rather, he develops our reason to be “perfectly more.” When Cusa was confronted with the paradox of the squaring of the circle, he developed a new principle, rather than seeking to fill a sieve by adding more and more sides to a polygon. Thus does Kepler, instead of making a slight change to the equant-model to better match observations, hypothesize the true, physical principle of which the equant acts as an imitation.

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apsides (as, for example, in the vicarious hypothesis) the motion created does not closely approximate speed depending on distance from the sun. Now, a solution to the mystery of the repeated appearance of bisected eccentricities can be proposed: the universal principle causing motion is imitated only by a bisected eccentricity.

Developing the Hypothesis In a state of hurried excitement, the reader sweeps through Chapters 33, 34, 35, 36, 37, and 38. We discover more about this “immaterial species… the primary agent of every motion in the universe.” But how could a physical species, which inheres in no matter, and “between the source and the movable thing is in a state of becoming, rather than of being,” be subjected to geometrical laws?

The reply is this: although the motive power is not anything material, nevertheless, since it is destined to carry matter (namely, the body of a planet), it is not free from geometrical laws, at least on account of this material action of carrying things about. Nor is there need for more, for we see that those motions are carried out in space and time, and that this power arises and is poured out from the source through the space of the world, all of which are geometrical entities. So this power should indeed be subject to other geometrical necessities.17

We discover that the sun is a magnetic body, like the Earth, and that it rotates, all before anyone had ever seen a sunspot move! The cause of motion explains why the planets all lie near a certain plane of power close to the ecliptic (hinting at a truly revolutionary insight in Part V). His new hypothesis allows him to make sense of the “variation” of the Moon’s motion: the intension of the Moon’s speed when it is along the line from the Sun to the Earth is because of the intensity of the force along this “diameter of power.” Kepler writes: “It would be preferable to attribute to the Earth a force that retains the moon, like a sort of chain, which would be there even if the moon did not circle the earth at all.”18 The cause of the planet’s ascent from and descent towards the sun are understood by a vis insita, an inherent force in the planet which expresses itself only in its relation to the species of the sun. No direct cause pushes the planet along this direction, but only an accidental one, like the shape of a riverbed causing water, which seeks to move downwards, to move out to sea in a river. What a panoply of breakthroughs his pregnant hypothesis has engendered!

A Historic Recapitulation Now, Kepler tests his hypothesis by posing his physical principle as the cause of motion. To offer a compelling conclusion, he must demonstrate that his hypothesis accurately

17 Kepler, Astronomia Nova, p. 383 18 Kepler, Astronomia Nova, p. 402

describes the motion of the annual orb. In chapter 40, he finds between the equant model – which, by its close equivalence to the eccentric model, works – and his proposed physical model, a greatest difference of 33", well within observational precision. Thus, in the case of the annual orb, no objection can be raised to the implementation of the physical principle of gravitation to understand its motion.

Circular Motion?

A dynamic universe has a different potential than does a mechanical one. Near the close of Part III, with the dynamic universe in mind, Kepler re-approaches the assumption of circular orbits. Although circles are very easy to draw with a compass or a piece of string, they are literally impossibly difficult to generate with physically driven motion.19 Analysis Situs!

20 What shape does this motion give

itself? Instead of imposing geometric concepts from outside the self-governance of the orbit, ask instead: according to the principles causing motion, which shapes are possible?21

Completing His Mission

The operation of the species of the sun is very close to the use of an equant with bisected eccentricity, and such an equant was found to be within 8' accuracy for the motion of Mars. Has Kepler yet proven that gravitation is a required principle, and not only an interesting additional surmise? Armed with his hypothesized physical principle, and calling into question any use of geometry as a possible cause, 22 will Kepler find the perfection that the geometric mimic – the equant – missed? That is the task of Part IV.

19 See http://wlym.com/~animations/part3/39/index.html. 20 G.W. Leibniz, “Analysis Situs,” in Leibniz: Philosophical Papers

and Letters, trans. Leroy E. Loemker, Kluwer Academic Publishers. 21 “This quality of experimentally premised conceptual evidence, which is associated, like the Pythagorean comma, with the notion of universals, implicitly defines the physical universe as composed not of,

but by universal principles of this quality. These do not represent a perfected set of such principles, but a set undergoing implicitly anti-

entropic developments. Any event in that universe is acting upon, and is acted upon by that universe, as Leibniz makes this point in, as referenced above, his sundry, anti-Cartesian writings on the subject of dynamics. This anti-entropic quality of the universe so defined, is echoed as the implications of Kepler's empirical demonstration of the problematic character of the implicitly anti-entropic notion of the paradox of the equant.” “Principles are not something amid, and as if connecting Cartesian-like objects in a pair-wise fashion. They are the essential, existing matter of which the universe is composed as a universe. It is a self-developing universe, in which essential action is expressed as, or in resistance to efficient action supplied by, for example, the human individual's will. This is, essentially, dynamics as its experience is traced in known history to the method of the Pythagoreans and Plato's circles.” From LaRouche: “What Congress Needs to Learn: The Lost Art of the Capital Budget,” EIR, Jan 12, 2007 22 Including the use of circular orbits

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A Preliminary Investigation of the LaRouche-Riemann Method Steger


A Preliminary Investigation of the LaRouche-Riemann

Method Michael Steger

Part 1 of 4

The advancements made in the field of physical

economic science by LaRouche over the course of the last half-century, are not separate, but are rather a continuous advancement on the questions of mathematical physics dating back to Cusa and Kepler, and onward through Gauss and Riemann, subsuming the work of Einstein and Vernadsky. But rather than describing these developments, let us rather leave the narrative to personal research and political discourse, and take up a pedagogical example to uncover certain underlying assumptions, that may reveal a more truthful method.

Keep in mind, this is only the first of a series, and the implications of our investigation into the field of physical-economy will be drawn out over the course of the entire discussion.

Consider this an experiment, which would then require a playful spirit for success, and let us attempt to communicate a necessary characteristic of the universe as it, the universe, presents itself to your field of vision. Before continuing, it is helpful to remind ourselves that we exist in the universe, which may contradict previous high-school and university doctrine.

So, might we start with this. Imagine you exist as a point in space. Your perception of sight is as from a point in space, with vision in every direction. Assume you cannot move, and are fixed in one location. To create a sense of this, just keep your head still, and look at the room around you. Make sure you are not standing too close to a wall, but look towards a part of the room or outside which has images cascading over one another, such as a tree in front of a car, or chairs standing on the floor. What do you notice, that might be different from your ‘expected’ field of vision?

How many dimensions of space are there in the area within which you see? Are there three? Forward-back, left-right, up-down? One’s friend might say, “Yes, of course there are three. Every point in the room can be described by three coordinates: x, y, and z.” Well, before assuming Euclidean space applies to our experiment, look again. Determine first what is provably known, so that we may then venture into the unknown.

For example, if you wanted, you could say there were seventeen dimensions, and then choose seventeen different axes, and give seventeen different coordinates for each location, but all seventeen would not be necessary. So, how many dimensions of space are necessary to describe the space from which you now perceive as a fixed point?

If the answer is not obvious, let us investigate further. Use your arm and finger as a linear extension of the ‘so-called’ point, and place your finger in the direction of some object in front of you. Now, pick another object in your limited field of vision, (which is only a slice of your theoretically possible field), and move you finger to the second object. Pick a third and a fourth object, and move your finger accordingly. Continue to investigate the spatial determinations necessary to describe each object’s position in relation to every other.

This may remind you of astronomical investigations, where, faced with the challenge of mapping the stars onto a flat piece of paper, you first had to determine the orientation of the stars to one another within the night time sky, before you could recreate them on the page.

For those who are unfamiliar with astronomy, describe the locus your finger traces as you place it in the various directions. (It’s important to keep your arm fully extended to maintain continuity.) Do not continue onward until you have demonstrated a working hypothesis which approximates the most necessary and sufficient locus.

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Why has the room, which, for our friend, seemed so obviously to conform to x, y, and z coordinates, become two dimensional? Our room appears to allow three directions of action. How is it possible that the positions of every object and every point in that room, can be uniquely located with two circular actions, as if each point were located on a sphere? The answer lies in the nature of the experiment, as one might guess. What were our assumptions? First, you were a point in space, which seems unrealistic, but not problematic, unless you were coincidentally convinced. Second, you were a fixed point in space. How would removing motion from your potential field of vision change the dimensions of space? Do not the dimensions of space exist independently from your perception of them? One might ask his new-age physics professor, “How many dimensions are necessary to describe the unseen monk falling in the forest?”

Return to the fundamentals of astronomy and optics and our initial problem is easily solved. You may be familiar with the idea of parallax, when the position of an object is changed due to the change of the observer’s position. Take as an example, the suggested motion of an object when perceived first with left eye closed, and then with the right. Often known as triangulation, this method provides a measurable distance, or sense of depth, for perceived objects through three given points, for example: 1. the right eye, 2. the left, and 3. the perceived object. We can then measure the object’s distance from the observer by the metric of length between the two eyes.

The circle is the perceived object, which when seen with only

one line of sight, can only be identified by its direction. But with both lines of sight, i.e., with both eyes, the object is seen as from

two positions and is perceived with both direction and depth.

This is the same method used by Kepler in his

imitation of the ancients known as the Vicarious Hypothesis, where he uses the acronychal position of the planet in the zodiac, or acronychal direction, as well as the location of the

planet as seen from the equant, to measure the assumed characteristics of Mars’s eccentric orbit. For more on this, see Kepler's New Astronomy, Ch. 16 through 19, and the relevant pedagogicals in this issue.

Equant

Earth or Sun, depending on which system.

Mars

Acronychal direction

Direction from the Equant

As above with eyesight and depth, the planet Mars is given a position of both direction and distance on a circle of some diameter when perceived from both the earth/sun and hypothetically from the equant. This is a facet of Kepler’s Vicarious hypothesis.

Let us now see how this notion of parallax applies to

the universe as it presents itself to our field of vision. Return to the experiment, while remaining as a theoretical ‘point,’ and move one step to the left or right. Has something which was previously blocked from your field of vision, now come into view? If not, continue stepping to one side until you distinctly notice something appear that was previously unseen. How would this previously unseen object map onto the first hypothesis of a sphere? If it does not fit onto the spherical surface, is there another two dimensional surface upon which it would fit? Could this be what A. Halevy meant by “mapping the invisible?”1

It should be obvious now that perceived space is not actually two dimensional, because there are objects which exist outside any two dimensional mapping. But, if by using three dimensions, such as the x, y, and z axes, we are then able to map every position within the perceived space, then must the universe as it presents itself to us necessarily be three dimensional, and must it conform to what is commonly taught in math and physics classrooms? For some, the problem is solved, and the universe coheres with commonly accepted notions found in almost every scientific institution.

But before we end our experiment, consider something about our three dimensional space: what necessary characteristic of three dimensional space, such as a Euclidean type box,

1 See ∆υναµι∆υναµι∆υναµι∆υναµιςςςς, Vol I, No. 2, at http://seattlelym.com.

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permits motion? What aspect of three spatial extensions enables our movement, so that we may then map space according to these three dimensions?

Perhaps we are assuming that motion solely depends upon our pre-conceived notion of spatial dimension, and at that, requires only three dimensions? Before these questions could be answered, we would first have to ask, “What is motion?” But, for now, let us return again to our initial inquiry, perhaps more conscious of the direction of our experiment.

Remember, in the original experiment, as a point in space, the room was physically three dimensional (as we later proved through parallax), but, as a fixed point, the room only required two dimensions (i.e. a spherical surface) to communicate its every location. So, three dimensions were mapped as only two.

Might then a mapping of space with three necessary dimensions as discovered above, require a fourth ‘dimension’ which permits motion, as the mapping of space with two necessary dimensions depended physically on three? (For those curious, how many dimensions would be necessary to map a space of only two dimensions? Try to develop a pedagogical example.)

Now, a four dimensional space may seem plausible, living after Einstein’s presentation of the concept of relativity, where the four dimensions of “space-time” have become a house-hold conception. But, what is time, other than motion? Try to consider some measurement of time that does not require motion. One may start with the obvious increments of time such as a day, a month, a season, or a year. If these seem obvious, discover how your cell phone keeps time. Is your cell phone smarter than you? (Only move forward when the answers to these questions no longer seem obvious.)

After a thorough investigation into the origins of

measured time, for one to say, then, that the dimension of time permits motion, is like saying money produces food. (For the Federal Reserve money apparently grows on food. But for them, time is only money.)

So, while attempting to determine the spatial characteristics of our local environment, we’ve encountered a necessary fourth ‘dimension’ which permits motion. To grasp this visually, imagine a holographic three dimensional room,

animated over a series of snap-shot freeze-frame images. The room was three dimensional, and the fourth dimension was that which permitted the physical changes to occur in the succession of images, conventionally referred to as time.

Let us reconsider the ground we’ve covered. A three dimensional room, perceived from a fixed point, could be perfectly replicated as a two dimensional spherical image. However, once our point of perception changed, the room transformed, and was no longer spherical or any other two

dimensional surface, but now only replicable three dimensionally, such as by a holographic image. And yet, our three dimensional Euclidean space does not necessitate change, with which we associate time, which has then forced us to confront some fourth ‘dimension.’

So, what then is responsible for the nature of change associated with this fourth dimension, which permits motion, and from which we measure time? Is it another, totally unperceived, and visually unimaginable fifth ‘dimension’? Proceeding this way, each new phenomenon encountered requires a higher dimension from which to communicate the newly perceived phenomenon, i.e. a necessary third dimension to communicate previously hidden objects unseen in two, and a fourth dimension to communicate the change in spatial relations of objects as seen in three, etc.

Following this argument, there must then be infinite ‘dimensions’ of space,

since for that which causes the motion associated with the fourth dimension, there must be the fifth, which when discovered, will then beckon the question, “What permits this fifth ‘dimension’ of action?” and hence a sixth, and so on. But, perhaps we’ve passed our boundary of skeptical reason.

If, however, this is not completely absurd, let our

investigation continue, asking: 1. What is dimension, such that it permits successively higher changes, including both extension, as well as action? 2. What is the nature of change, as perceived in higher dimensions of action? Perhaps Kepler will provide the necessary insight. (To be continued in part two.)

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mysterium cosmographicum - larouchepac · johannes kepler for the publication of the rudolphine...

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