The Circle Is Not Simply Round

by Bruce Director

Among the most interesting and provocative investigations of the thinkers of the ancient Greek speaking world, were problems concerning the construction, with straight edge and compass, of certain geometrical figures; specifically, the doubling of the cube, the trisection of the angle, the construction of the regular heptagon, and the quadrature of the circle. In most of the modern English language sources on the subject, these problems are generally portrayed as a certain type of puzzle, or brain teaser. Lacking in virtually all of this scholarship, is any conception of what these ancient Greek scientists were actually investigating. To answer the latter question, we need not hunt for some long-lost text, in which the deeper implications of these investigations are explicated. Rather, we need only to relive the discoveries ourselves, and, in the mirror of our own mind, those deeper implications will be reflected.

Instead of wasting time with today’s academics, let us take as our guide Johannes Kepler, whose new and original discoveries arose from his own re-working of these investigations of ancient Greece. In the first book of the Harmony of the World, “The Construction of Regular Figures,” Kepler presents some of the results of his re-discovery. Pertinent to this discussion, he provides the following definitions:

VII. “In geometrical matters, to know is to measure by a known measure, which known measure in our present concern, the inscription of figures in a circle, is the diameter of the circle.”

VIII. “A quantity is said to be knowable if it is either itself immediately measurable by the diameter, if it is a line; or by its [the diameter’s] square if a surface: or the quantity in question is at least formed from quantities such that by some definite geometrical connection, in some series [of operations] however long, they at last depend upon the diameter or its square. The Greek word for this is `gnorimon.’

IX. “The construction of a quantity which is either to be described or to be known is its deduction from the diameter, by permitted means, in Greek [these are called] `porima.’

“So construction generally yields either description or knowledge. But description declares mere quantity, whereas knowledge also in addition declares quality or a definite quantity. Now a line can be geometrically determined, in Greek, `takah,’ even though its quality is not yet known intellectually. On the other hand, a line or lines may be known qualitatively, but that does not yet determine them or make them determinate, that is to say if their quality is common to many other things which are different in quantity. So for such lines description is easy, knowledge very difficult. Finally, many things can be described by some Geometrical means or other; but cannot be knowable by their nature: as knowledge has been defined above.”

With these concepts in mind, take a first look at one of the classical Greek problems, the trisection of the angle. In proposition #46 of the same book Kepler restates this problem as:

“The division of any arc of a circle into three, five, seven, and so on, equal parts, and in any ratio which is not obtainable by repeated doubling from the ones which have been shown above, cannot be carried out in a Geometrical manner which produces knowledge.”

His demonstration goes like this: To bisect an arc of a circle, we first bisect the chord drawn between the two ends of the arc. A line drawn from the center of the circle, through that point, will also bisect the arc. On the other hand, if we want to trisect the arc, the situation becomes much more ambiguous. Draw an arc and its chord, and label the end points A and B. Construct points P and Q on the chord, such that A-P = P-Q = Q-B. This trisects the chord. If we now draw lines from the center of the circle through P and Q, that intersect the arc at P’ and Q’, the arcs A-P’ and Q’-B will be smaller than the middle arc P’-Q’. On the other hand, if we draw lines perpendicular to the chord, through P and Q, the result will be that arc P’-Q’ will be larger than the arcs A-P’ and Q’-B. Therefore, to trisect the arc, we have to draw lines through P and Q from a point that is somewhere between the center of the circle and infinity. Kepler shows that the position of this point gets farther from the center, as the arc gets smaller. But, this relationship is not proportional. That is, decreasing the arc by a given amount, does not change the position of the point by a proportional distance.

This is another manifestation of the phenomenon of non- linearity of circular action demonstrated three weeks ago with respect to the sine and cosine. One can illustrate that principle with the following experiment:

Draw a large circle on a black board. Draw two perpendicular diameters, one vertical and one horizontal. Get a string with a weight on it. Hold one end the string with your finger at the intersection of the horizontal diameter and the circumference, and let the weight hang down towards the floor. Now, move the end of the string with your finger along the circumference of the circle. The weight will rise, and the string will form a chord of the circle that intersects the horizontal diameter. Now watch the intersection of the string and the diameter, as you move the end of the string around the circumference of the circle. What is the relationship between the circular action of your finger, and the rectilinear movement of the point of intersection of the string with the diameter? The constant curvature of the circle, produces a non-constant motion of this point.

Back to the trisection of the angle. What Kepler’s demonstration reveals, is a kind of boundary condition with respect to the divisions of a circular arc. Dividing a circular arc in half, or into powers of two, does not produce an immediate discontinuity between the circular arc and the straight line chord defined by it. But when we try to divide by three, division of the line and the arc diverge.

Through Kepler we have now re-discovered this problem in the form confronted in 5th-4th Century B.C. One attempted solution was devised by Hippias of Elis<fn1>, who is credited with producing the first non-circular curve, today called the quadratrix. This curve was later investigated by Leibniz, Huygens, Bernoulli, et al., from the higher standpoint of that Leibniz developed out of Kepler’s discoveries.

The quadratrix of Hippias is generated as follows. Draw a square. Label the corners clockwise from the upper left hand corner A, B, C, D. Now imagine side A-D rotating clockwise around point D. As this segment rotates, point A will trace a quarter of a circle from A to C. Now imagine that as this line is rotating, side A-B, moves down the square to side D-C, at the same rate as side A-D is rotating. Side A-B remains parallel to side D-C as it moves. The quadratrix is the curve traced out by the intersection of these two lines as they move. Thus, the quadratrix is the intersection of circular rotation and linear motion.

Because of the way the quadratrix is constructed, Hippias used it to trisect the angle. That is, since both sides move at the same rate, when A-D has rotated 1/2 way, side A-B has moved down 1/2 way. Similarly, when A-D has rotated 1/3 of the way, side A-B has moved 1/3 of the way down. And so on for any other division.

To trisect an angle, mark off any angle with vertex at D, such that one side of the angle will be D-C, and the other side of the angle will intersect the quadratrix at some point K and the arc A-C at L. Draw a line parallel to the side D-C through K. That line will intersect side A-D at some point E. Now, find the point E’ that divides D-E by one third. Draw a line from E’ that is parallel to side D-C. This line will intersect the quadratrix at some point K’. Connect K’ to D and the angle formed with side D-C will be one third of the original angle.

But, has Hippias constructed a means for trisection that is a “knowable” quantity as Kepler re-stated the problem? We could construct an mechanical apparatus to draw a quadratrix, but is this curve “knowable,” that is, constructable by the circle and it’s diameter?

Ah, there’s the rub! To construct the quadratrix by “knowable” means, Hippias proposed the following:

First draw the perpendicular bi-sector of side D-C. Then draw the bisector of the angle C-D-A. The intersection of these two lines is a point on the quadratrix. Then bisect the two new segments of side D-C and the two new angles formed by bisecting C-D-A. These intersections will define two more points on the quadratrix. This process can be repeated again and again, to fill in so many points on the quadratrix, that by connecting the dots, the quadratrix can be drawn.

But, wait! All these points were determined by division by two, and so they will only precisely determine points on the quadratrix that intersect lines that divide angle C-D-A by powers of two. None of these dots will precisely determine a point on the quadratrix that trisects an angle. Those points, will always lie on the indeterminate, “filled” in parts of the curve. The “non-knowable” parts.

We seem to have hit a stumbling block. The boundary between division of a circular arc by two and division by three has re- emerged. Think about this a while. The closed door we’ve seem to run into, is, perhaps, an open hallway, through which our predecessors have strolled.

1. This Hippias is the subject of Plato’s dialogues Hippias Major and Hippias Minor. He is also mentioned in the Apology. He was apparently a traveling philosopher, with some facility in mathematics.

Double Your Mind –

by Bruce Director

This week we look at another classical Greek problem as reported by Theon of Smyrna:

In his work entitled {Platonicus} Eratosthenes says that, when the god announced to the Delians by oracle that to get rid of a plague they must construct an altar double of the existing one, their craftsmen fell into great perplexity in trying to find how a solid could be made double of another solid, and they went to ask Plato about it. He told them that the god had given this oracle, not because he wanted an altar of double the size, but because he wished, in setting this task before them, to reproach the Greeks for their neglect of mathematics and their contempt for geometry.

In future weeks we will re-live this problem. For now think about it.

The Means to Double Your Mind

by Bruce Director

According to Eratosthenes, Plato took great pleasure at the prospect, that the cognitive capacity of his fellow Greeks might be improved, when, having asked the gods for help, the gods answered with a question, that required the Greeks to think. It might improve our own cognitive functioning, and make us smile, to find out why Plato was so delighted.

Before taking up this problem directly, let’s first look at the Pythagorean investigation of the doubling of the square. Plato discussed these investigations in the famous Meno dialogue. We can reconstruct an essential feature of that discovery by the following means:

Draw a square. Double its side and draw the square on the doubled side. Repeat this process several times. If the area of the first square is considered 1, then the subsequent areas are 1, 4, 16, 32 … etc. The sides of the corresponding squares are 1, 2, 4, 8, 16, …

Now draw another square. Draw the diagonal. Draw a square on the diagonal. Draw the diagonal of the new square. Draw a third square on the diagonal of the second square. Continue this several more times.

You should see a series of squares and diagonals in a spiral formation. If the area of the first square is considered to be 1, then the subsequent squares have the areas, 1, 2, 4, 8 …, respectively. The area of each square in this series is in the same proportion to the one proceeding it, as to the one succeeding it. That is, 1:2::2:4::4:8::8:16 …

Notice that the squares produced by doubling the sides are every other one, of the series of squares produced from the diagonals. The squares of doubled area, are in between the squares of doubled sides. If we think of the curvature of the whole series, the squares of double areas are found in a smaller interval of action, than the squares of doubled sides. The proportionality between the terms of the series remain the same, but something completely new emerges in the smaller interval of action, to wit: doubled areas.

(It should be particularly thought provoking, to think of the curvature in the small, of a discontinuous series!)

Now look at an even smaller interval of action; the interval between two squares. That interval includes, the side of one of the squares, the diagonal, and the side of the succeeding square. Here again, the principle of proportionality remains the same, but a new singularity emerges — the incommensurability of the side to the diagonal of the square. (For a more complete discussion of incommensurability, see “Incommensurability and Analysis Situs” Parts 1 and 2; NF Vol. XI #22 6/9/97 97237jbt101 and NF Vol. XI #23 6/16/97 (7267jbt101).

In Plato’s Theatetus dialogue, Theatetus reports on an investigation of a still smaller interval of action of this same curvature, by Theodorus of Cyrene, a Pythagorean who was one of Plato’s teachers. Theodorus produced a series of triangles whose hypotenuses were the square roots of 2, 3, 4, 5, etc. For now, we leave to the reader the fun of re-constructing this series.

Now think back on these series of squares. All are characterized by the same principle of proportionality. Each one is a smaller interval of action than the previous one. With each smaller interval, new singularities emerge. What is invariant in every smaller interval, however, is the principle of proportionality.

Now if a square is doubled through this principle of proportionality, how is the cube doubled? A generation or two before Plato, Hippocrates of Chios, (not the same Hippocrates of medicine fame) made a crucial discovery concerning this problem. This Hippocrates supposedly was a merchant who ended up broke in Athens around 500 B.C. and started to teach thinking to earn a living. His discovery can be re-created in the following way:

If we begin with a cube whose side is one, its volume will also be one. If we double the side, the new cube will have a volume of 8. Between 1 and 8 are two means, 2 and 4. That is, 1:2::2:4::4:8. So, the cube whose volume is double, is the lesser of the two means, between the cube whose side is doubled. Since the sides of the cubes are in the same proportion as the volumes, the side of the cube whose volume is double, is the lesser of two means between 1 and 2.

Eutocius reported this discovery in his commentaries on Archimedes as, “It became a subject of inquiry among geometers in what manner one might double the given solid, while it remained the same shape, and this problem was called the duplication of the cube; for, given a cube, they sought to double it. When all were for a long time at a loss, Hippocrates of Chios first conceived that, if two mean proportionals could be found in continued proportion between two straight lines, of which the greater was double the lesser, the cube would be doubled, so that the puzzle was by him turned into no less of a puzzle.”

Plato reflected on this discovery in the Timeaus: “But it is not possible that two things alone should be conjoined without a third, for there must needs be some intermediary bond to connect the two. And the fairest of bonds is that which most perfectly unites into one both itself and the things which it binds together; and to effect this in the fairest manner is the natural property of proportion. For whenever the middle term of any three numbers, cubic or square, is such that the first term is to it, so is it to the last term, and again, conversely, as the last term is to the middle, so is the middle to the first,– then the middle term becomes in turn the first and the last, while the first and last become in turn middle terms, and the necessary consequence will be that all the terms are interchangeable, and being interchangeable they all form a unity. Now if the body of the All had had to come into existence as a plane surface, having no depth, one middle term would have sufficed to bind together both itself and its fellow-terms, but now it is otherwise, for it behoved it to be solid of shape, and what brings solids into unison is never one middle term alone but always two.”

Hippocrates’ discovery amounts to investigating the geometric principle of proportionality, in an even smaller interval, trying to find two means between 1 and 2. His successors soon discovered a new boundary had to be crossed, when investigating this smaller interval. Those familiar with Gauss’ Disquisitiones Arithmeticae and his theory of bi-quadratic residues, will recognize the seeds of those great investigations, in these ancient Greek inquiries.

In future weeks we will re-construct some of the ancient Greek studies of this problem. For now, keep thinking about it.

How Johannes Kepler Changed the Laws of the Universe, Part I

by Jonathan Tennenbaum

The following discussion begins a long journey, along a pathway of <astronomical paradoxes> leading from our discussion of “the simplest discovery,” via the revolutionary work of Johannes Kepler, to the birth of a physics characterized by non-algebraic, elliptic and hypergeometric functions.

In his “Commentaries on Mars” (also known as “Astronomia Nova”), Kepler locates the origin of astronomy itself, in a paradox going back to the most ancient times:

“The testimony of the ages confirms that the motions of the heavenly bodies are in circular orbs. It is an immediate presumption of reason, reflected in experience, that their gyrations are perfect circles. For among figures it is circles, and among bodies the heavens, that are considered the most perfect. However, when experience is seen to teach something different to those who pay careful attention, namely, that the planets deviate from a simple circular path, it gives rise to a powerful sense of wonder, which at length drives men to look into causes. It is just this from which astronomy arose among men.”

Indeed, in our previous discussion of “the simplest discovery,” the hypothetical prehistoric astronomer, observing the cycle of day and night, came upon the paradox of a growing discrepancy between the Sun’s motion and that of the constellation of stars. While the stars pursue what appear to be perfectly circular orbits, the pathway of the Sun, as recorded (for example) from week-to-week and month-to-month on the surface of a large spherical sundial, has the form of a tightly-wound coil. Each day the Sun completes one loop, making a slightly different loop the next day. In the course of a year, the spiral runs forward and then backward, doubling back on itself. More complicated still than the path of the Sun, are the motions of the Moon and planets. The latter displaying irregular, even bizarre behavior when mapped against the background of the stars. Kepler continues:

“The first adumbration of astronomy explains no causes, but consists solely of the experience of the eyes, extremely slowly acquired. It cannot be explained in figures or numbers, nor can it be extrapolated into the future, since it is always different from itself, to the extent that no spiral is equal to any other in elapsed time … Nevertheless, there are some people today who, riding roughshod over 2,000 years’ work, care, erudition and knowledge, are trying to revive this, gaining admiration of themselves from the mob … Those with more experience consider them with good reason to be incompetent….

“For it was very helpful to astronomers to understand that two simple motions, the first and the second ones, the common and the proper, are mixed together, and that from this confusion there necessarily follows the continuous series of conglomerated motions.”

Indeed, to make some sense out of the motions of the Sun and the planets, it is necessary to disentangle them from the daily apparent rotation of the heavens (“the First Motion”). This is most easily done, by recording the positions of the planets relative to the stars and their constellations, rather than relative to the horizon of the observer on the Earth. In other words, we plot the positions of the planets against a “map” of the stars (the so-called siderial positions). The resulting motions of the planets relative to the background of stars, became known as the “Second Motions.” The first and second motions combine together to give the observed motions.

In the case of the Sun, we have to overcome the difficulty, that its illumination masks the weaker light of the stars, so the Sun’s position among the stars cannot be observed directly. But there are many ways to adduce it indirectly; for example, we can observe the positions of the constellations visible in the still-dark side of the sky opposite to the Sun at the moment of sunrise or sunset, and use the relevant angular measurements to reconstruct the exact position the Sun must have on the stellar map. The result of plotting the Sun’s motion against the “dome” of the stars, is very beautiful: The Sun is found to move along a great circle in the heavens, called the ecliptic, whose circumference is traditionally divided into twelve parts named by stellar constellations (“signs of the zodiac”).

For the planets, however, the siderial motions turn out to be surprisingly complicated, and even bizarre. Kepler explains:

“Now that the first and diurnal motion had thus been set aside, and those motions that are apprehended by comparison over a period of days, and that belong to the planets individually, had been considered in themselves, there appeared in these motions a much greater confusion than before, when the diurnal and common motion was still mixed in. For although this residual confusion was there before, it was less observed, less striking to the eyes, because the diurnal motion was very swift … (In particular) it was apparent that the three superior planets, Saturn, Jupiter, and Mars, attune their motions to their proximity to the Sun. For when the Sun approaches them, they move forward and are swifter than usual, and when the Sun somes to the sign opposite the planets, they retrace with crab-like steps the road they had just covered.”

What could be the reason for this bizarre “crab-like” behavior of planetary motions, even forming doubled-back loops in the case of the planet Mars? Where is the simple circular motion, which would supposedly constitute the elementary, self-evident form of action in the Universe?

Don’t rush to supply answers from what you were taught in the past, thus cheating yourself out of the joy of reliving some earth-shaking discoveries. Let’s stop and think about this.

Remember first Plato’s parable of shadows in the cave. Are we seeing, in the bizarre motions of Mars and other planets, mere shadows of the real process? Assuming, for example, that we are seeing only a projection of the real planetary motions in space, how could we discover the “true motions” of the planets? Reflecting on this challenge, we soon find ourselves confronted with a seemingly formidable array of interconnected paradoxes.

First, given that astronomers were restricted (until recent decades) to observations made only from the Earth, how could we determine the exact location of a planet in space? In particular, how could we even determine its distance from us?

To see the elementary difficulty involved, pose the task in more general terms. Imagine an observer, located at any arbitrary point in space. In respect to distant objects, the Universe appears to that observer as if projected onto the surface of a large sphere centered at the observer — the so-called “celestial sphere.” The principle of the projection is very simple: Imagine a distant object, such as a star, emitting rays of light in all directions. The rays which reach the observer, form a very thin cone, which intersects the sphere in a tiny circle (assuming the star itself has a spherical cross-section). Now, from the standpoint of what the observer sees, the star has the same appearance as if it were a light source of appropriate size, brightness, and color and so forth, fixed to the surface of the sphere. Or, again, if we were to compare the given star to another star, at <twice the distance>, but also <twice as large>, how could the observer tell the difference? Furthermore, in the case of distant stars (and to some extent even planets, when observed by the naked eye), the ratio of the object’s diameter to distance is so small, and the cone of rays so thin, that these objects are seen as hardly more than mere points; evidently their distances could be varied over a considerable range, without the observer being able to detect the difference.

The situation becomes even more complicated, when we consider the effect of motion. First, consider the case of a distant planet moving at constant velocity in a circular orbit around the observer. As seen from the observer, the planet’s motion over any given interval of time will appear to describe a circular arc on the celestial sphere. It is easy to see, that the same apparent motion would be caused by a planet moving twice as fast, on a circular orbit of twice the radius around our observer.

Actually, the ambiguity is much greater! Construct a plane passing through the original circular orbit. That plane passes through the location of the observer, and cuts the sphere in a great circle. Now draw <any> arbitrary curve on that plane, only subject to the condition, that it encloses the observer without folding back on itself. Then it is easy to construct a hypothetical motion of a planet on that curve, which would present exactly the same appearance to the observer as original planet moving in a constant circular orbit! All we have to do is construct a ray from the observer to the location of the original planet on its circular orbit. That ray intersects the arbitrary curve in some point P. As the ray follows the motion of the original planet, rotating at constant speed, the point P moves along our arbitrary curve. If we now attach a hypothetical planet to the moving point P, its motion, as seen from the standpoint of the observer, will seem to coincide with that of the original planet. Note, that although the observed imagine will appear to move always at a constant rate around the observer, the actual speed of the hypothetical planet on the arbitrary curve will be highly variable; in fact, the planet will be accelerating or decelerating at each point where the arbitrary curve deviates from a perfect circle around the observer. Consider, for example, the case where the curve is an elongated ellipse with the observer at one focus.

The problem becomes more complicated still, if we admit the possibility, that the observer himself might be moving. The paradox already hits us with full force, when we observe the nightly motion of the stars. Are the stars orbitting around us, or are the stars fixed and the earth is rotating, in the opposite direction? Or some combination of both? Supposing the stars are fixed, and the Earth is rotating, what about the Sun? When we “clean away” the effect of the Earth’s rotation, by plotting the Sun’s apparent motion against the “sphere of the fixed stars,” the Sun is seen to move on a circle, the ecliptic. Is the center of the Earth fixed relative to the stars, and the Sun orbiting around that center? Or, is the Sun fixed, and the Earth orbitting with the same speed, but the opposite direction around the Sun, on a circle of the same radius? In each case, and in countless other imaginable combinations and variations, the observed phenomena would seem to be the same!

These arguments would appear to demonstrate the complete futility of determining the actual orbit and speed of a planet from its observed motion as seen from the Earth! We seem to be confronting Kant’s famous “Ding an sich” — the pessimistic notion, that Man can never know reality “as it really is.” Can we accept such a standpoint? Were God so cruel, as to create such a hermetic barrier to Reason’s participation in His universe?

During centuries of debate about the motion of the Earth and the celestial bodies, there were those who rejected even the concept of “true motions” as opposed to “apparent” ones, and maintained that <only observations> — i.e., sense perceptions — <are real>. From that sort of radical-positivist standpoint, it makes no difference whether we assume the Earth is fixed and the Sun is moving, or vice-versa; these are merely two among an infinity of mathematically equivalent opinions, none of which have any particular claim to truth.

One of the notable advocates of this kind of indifferentism, sharply and repeatedly denounced by Kepler, was one Petrus Ramus (1515-1572). Ramus was a leading “anti-Aristotelian” of the species of the later Paolo Sarpi. (In other words, he was more Aristotelian than Aristotle!) Ramus held a prestigious Professorship at the College de France and was known for works on philosophy, law and mathematics. In his famous book on elementary mathematics, Ramus banned incommensurables, eliminated the axiomatic approach of Euclid, and rejected the regular solids as insignificant and useless. He went over from the Catholic Church to Calvinism and found his end during the famous “St. Bartholemeus night.” Kepler put his polemic against Ramus on the very first page of the “Astronomia Nova,” quoting Ramus’ demand for an “astronomy without hypotheses,” and then giving his own, devastating reply:

Petrus Ramus, Scholae Mathematica, Book II:

“Thus, the contrivance of hypotheses is absurd; nevertheless, in Eudoxus, Aristotle, and Callippus, the contrivance is simpler, as they supposed the hypotheses to be true — indeed, they have been venerated as if they were the gods of the starless orbs. In later times, on the other hand, the tale is by far the most absurd, the demonstration of the truth of natural phenomena through false causes. For this reason, Logic above all, as well as the Mathematical elements of Arithmetic and Geometry, will provide the greatest assistance in establishing the purity and dignity of the most noble art [Astronomy – JT]. Would that Copernicus had been more inclined towards this idea of establishing an astronomy without hypotheses! For it would have been far easier for him to describe an astronomy corresponding to the truth about the stars, than to move the Earth, a task like the labor of some giant, so that in consequence of the earth’s being moved, we might observe the stars at rest … I will solemnly promise you the Regius Professorship at Paris as a prize for an astronomy constructed without hypotheses, and will fulfill this promise with the greatest pleasure, even by resigning our professorship.”

The author [Kepler – JT] to Ramus:

“Conveniently for you, Ramus, you have abandoned this surety by departing both from life and professorship. Had you still held the latter, I would, in my judgement, have won it indeed, inasmuch as, in this work, I have at length succeeded, even by the judgement of your own logic. As you ask the assistance of Logic and Mathematics for the noblest art, I would only ask you not to exclude the support of Physics, which it can by no means forego … It is a most absurd business, I admit, to demonstrate natural phenomena through false causes, but this is not what is happening in Copernicus. For he too considered his hypotheses true, no less than those whom you mentioned considered their old ones true, but he did not just consider them true, but demonstrates it; as evidence of which I offer this work…. Thus, Copernicus does not mythologize, but seriously presents paradoxes; that is, he philosophizes. Which is what you wish of the astronomer.”

What is wrong with our arguments? Provoked by Kepler’s remarks, reflect for a moment on the paradox of “unknowability” of the true planetary motions, presented above. Is the Universe really unknowable in that way? Or might it not rather be the case, that our reasoning contains some pervasive, false assumption, which is the root of the trouble?

(Note: This discussion begins a longer series, which will not run consecutively, but will nonethless constitute a coherent whole.)

From Cardan’s Paradox To The Complex Domain, Part I

by Jonathan Tennenbaum

Contrary to British-authored mythologies, the intense interest on the part of Greek geometers from Pythagoras to Eratosthenes, in so-called “unsolvable problems” of geometry, had nothing to do with an idle fascination in mathematical puzzles. At issue, in the investigation of such problems as doubling a cube, trisecting an arbitrary angle, constructing a regular 7-sided polygon etc. was nothing less than the notion of Natural Law, as a higher principle subsuming an ordered series of {sets of physical principles}, each embodying a higher per-capita power of Mankind over the Universe.

In fact, there is no absolute “unsolvability” of the above-mentioned and other problems, except relative to a given, fixed set of principles of construction such as the ruler-and-compass constructions of so-called Euclidean geometry. Archimedes, Nichomedes and Eratosthenes and others already developed a whole array of “solutions” based on introducing addition principles, embodied in higher-order curves, constructions in higher dimensionalities, and the use of various physical machanisms and instruments.

The issue was not, that a given problem were “unsolvable” in some absolute sense, but rather that: 1) it could not be solved in the terms in which it had been posed, i.e. in terms of a certain implicitly circumscribed set of principles; 2) it {could} be solved with the help of the discovery and introduction of one or more {new} principles, lying {outside} the given domain, but demonstrated to be physically valid; 3) that the arrays of principles, arising this way, are implicitly ordered by a notion of Man’s increasing {power} over the Universe.

A simple illustration is the realization that a straight line, by and of itself, could never generate a surface. Nor could a surface ever transform itself into a solid. In both cases, a process of (rotational) extension is required, acting upon the line or the plane “from the outside”, and which already embodies the principle of the higher domain. This realization was the basis for the classical differentiation of geometrical problems, between so-called linear, plane and solid species; and for the notion, that the lower-order domain is always derived from the higher one, and not vice-versa.

However, by the time of Plato the Greek geometers had already conclusively established, that the actual ordering of “powers” is {not} that of simple dimensionality in visual space. The former lies beyond the reach of visual geometry per se, but actually determines the characteristics of action as reflected in visual space.

For example, the problem of {trisecting} any given angle in a plane, only {appears} to be a “two-dimensional” or “plane” problem. In actuality, as demonstrated by Nichomedes and others, it belongs to the same domain of “power” as the doubling of a cube!

That relationship is the original focus of the following discussion. It is key to understanding, why attempts by Cardan and others in the 16th century an afterwards, to derive an algebraic formula for the solution of a arbitrary cubic equation, inevitably ran into a devastating anomaly in the emergence of so-called “impossible”, “imaginary quantities”. The second installment of this discussion, will examine Cardan’s anomaly through the eyes of Abraham Kaester, setting the stage for Gauss’ subsequent discovery of the complex domain.

Trisecting an Angle

Since the origin of all angles is rotation, we reference all constructions to a circle whose center (marked “O”) is the vertex of the given angle, and whose radius is taken as “1”. The angle itself corresponds to the circular arc of rotation between the two points P and Q on the circle, where the sides (rays) of the angle intersect the circle.

Nothing is simper in visual geometry, than to double, triple, or multiply a given angle a whole number of times. We have only to set our compass to PQ and mark off a succession of equal arcs on the corresponding circle, starting at Q. Relative to OP, the rays OR, OS, OT, joining the center O to the endpoints of those successive arcs, represent double, triple, quaduple etc. the original angle between OP and OQ.

But what about the {inversion} of that process: to {divide} a given angle into a whole number of equal parts? Bisecting an angle is easily accomplished with the aid of ruler and compass, but the problem of dividing an arbitrary angle into {3} equal parts, presents a whole different kettle of fish. Centuries of attempts to develop a general solution within the domain of the ruler-and-compass constructions of plane geometry, ended in failure. Why? Key to this is the species-relationship to the doubling of the cube, known to the Greek geometers around the time of Plato.

Evidently, dividing the angle into three or any other number of equal angles, is equivalent to dividing the corresponding {circular arc} into the same number of equal arcs.

Now, it is a relatively easy matter to divide a straight line segment (hypothetical “zero curvature”) into three or any other number of equal segments, by ruler-and-compass constructions. Someone might, accordingly, attempt the following “solution” for trisecting a circular arc: First trisect the {chord} of the given arc; then {project} the division-points from the center onto the circular arc (see Figure 1a).

This attempt fails, for reasons Cusa and Leonardo emphasized in their discussions of the {distortion} introduced by any projection between a linear and curved surface. The projected images of the three equal segments on the chord onto the circle, are no longer equal as arcs.

Figure 1

Conversely, if we project the division-points of an already trisected arc onto the chord joining the arc’s endpoints (by drawing radial lines from the circle’s center to the division-points on the circular arc) we get an {unequal} division of the chord (see Figure 1b). Furthermore, the lengths of the resulting segments on the chord, taken in and of themselves, do not manifest any simple proportionality; rather — as it turns out — the attempt to express the convoluted relationships in algebraic form, inevitably leads to what are called “equations of the third (cubic) degree”.

Turn that around in your mind. Might it not be the case, that the appearance of complicated combinatorial-algebraic relationships among ordinary “scalar” magnitudes (whole numbers, “real numbers” measuring the lengths of straight line segments etc.) reflects the fact, that we are dealing, not with self-evident realities, but rather with “shadows”, cast from a higher physical-geometrical domain? Yes indeed, as Gauss demonstrated most conclusively in his work on biquadratic residues! The same principle underlies Gauss’ “fundamental theorem of algebra”, and its pre-history in the celebrated, centuries-long controversy over “imaginary numbers” appearing in the solution of the cubic equation.

That is where we are headed right now, along a trajectory defined by the ancient problem of trisecting a given angle.

Take a closer look, first, at the {inverse process} — {tripling} an angle — and at the lawful functional relationships, which are generated among selected “shadows” cast by that process.

To triple a given angle POQ, mark off points R and S from Q on the circle, such that the arcs QR and RS are equal to PQ. The rays OP, OQ, OR and OS are equally spaced, so that the angle between OR and OP is {double} that between OQ and OP, and the angle between OS and OP is {triple} the original angle.

For purposes of illustration it is best to represent P as the right-hand endpoint on the horizontal diameter of the circle. Rather than only considering a fixed angle, imagine the point Q as moving along the circle, starting at P (angle 0) and going around in a counterclockwise direction. Call the size of the angle POQ, “alpha”. What happens to the points R and S, as the alpha grows?

Clearly, for alpha = 0 all three points Q, R, S coincide with the position P. As alpha grows, Q moves at a proportional rate along the circle, while R runs ahead at twice, S at three times that rate. When Q reaches 90 degrees, R will have reached the position opposite to P at 180 degrees, and S will be at 270 degrees (see Figure 2).

Figure 2

Next, investigate the lawful relationships among the “shadows” cast by that process under various sorts of projections.

The simplest and most characteristic case, is perpendicular projection onto the horizontal diameter (axis) of the circle (the line through O and P).

Denote by q, r, s the perpendicular projections of the points Q, R, S, respectively onto the horizontal axis (see Figure 3a).

Figure 3

How do q, r, s move, as the point Q runs at a uniform rate along the circle starting at P?

Focus first on q. Imagine one end of a string is attached to Q on the circle (the latter oriented in a vertical plane), while the other end is attached to a lead bob, so that the string hangs vertically downward. Q’s projection q is the point where that vertical crosses the horizontal axis through O and P. (This works when Q lies on the upper half of the circle; when Q is on the lower half, we have to project “upward” by extending the direction of the string until we reach the axis.) Note, that the motion of the point q is not uniform like that of Q; rather, q starts very slowly (Q near P), and then accelerates, reaching maximum speed as the angle alpha approaches 90 degrees, and then slowing down again as Q approachs the point opposite P on the left of the circle (180 degrees). At that point, q reverses direction and repeats the process in reverse, as Q runs back to P along the lower half of the circle, and so forth (see Figure 4).

Figure 4

This will be familiar to anyone who knows the so-called trigonometric functions (better termed “circular functions”); The position of the “shadow” point q relative to O, corresponds to the so-called {cosine} of the angle alpha; and q’s motion has the form of a simple “harmonic” vibration whose frequency is the number of revolutions of the moving point Q around the circle, per unit time.

More entertaining, is to watch the {simultaneous} interrelated motions of q, r and s, as the latter two oscillate along the axis with frequencies {twice} and {three times} that of q! (see Figure 5).

Figure 5

Does there exist a mutual relationship among the positions of the three “shadows” q, r, s, taken by themselves, which remains valid and invariant throughout the process?

To answer that, “freeze” the motions for a moment, and pose the question again as follows: what direct relationships among q, r, and s flow from the circumstance, that the corresponding points on the circle — Q, R and S — were generated from P by rotation through the angles {alpha}, {2 x alpha} and {3 x alpha} respectively?

(For the purpose of illustration, it is best to take the case, where Q lies in the upper right-hand quadrant of the circle with the angle POQ less than 45 degrees. The other cases are essentially equivalent.)

Note the three {right triangles} OQq, ORr, OSs, each of which have hypotenus equal to unity (the radius of the circle), and whose angles at O are {alpha}, {2 x alpha} and {3 x alpha} respectively. How are those three triangles related? In effect, the successive rotations through the angle {alpha} on the circle, imply transformations of the triangle OQq into ORr and then into OSs. But generally speaking, the triangles are neither congruent, nor similar to each other in shape (see Figure 3a).

Noting the doubly and triply self-reflexive character of the action involved — i.e. an action applied to itself, and then to the result of that — it should occur to us, that the points R and S bear the same relationship to the axis OQ, as the points Q and R do to the axis OP.

This remark suggests that we consider the perpendicular projections or “shadows” of R and S on the axis OQ — call these r’ and s’, respectively — as well as the projections r and s on original axis OP (see Figure 3b [partial construction]).

Figure 3

Note, that the right triangles ORr’ and OSs’, arising from the new projection, are congruent to OQq and ORr respectively. In fact, the latter are carried into the former by exactly the same circular rotation through {alpha}, that carries Q into R and R into S. For the same reason, Or’ = Oq and Os’ = Or.

These observations now provide the key to unravelling the relationships between the “shadows” q, r and s.

The construction required is a bit laborious, but worth going through in detail, while keeping an eye out for the underlying bounding principle. For behind the following, nested chain of similarity relationships, lurks Gauss’ complex domain.

Algebraic Equations Arise Through Projection of Rotational Action

The point of departure for unravelling these relationships is the first triangle, OQq, and in particular its base and height — the segments Oq and Qq — whose lengths we shall call X and Y, respectively (see Figure 3a). The two are linked to each other through Pythagorus’ relationship: the sum of the squares of X and Y is the square of the radius of the circle, which we took as 1.

Now proceed as follows, concentrating first on q and r.

First project the point r’, lying on the axis OQ, down onto the original axis OP, obtaining r”. This is, so to speak, “the shadow of a shadow” (see Figure 3b).

We can obtain the position of r” quite easily, because the result of projecting from a straight line (in this case OQ) onto another straight line (OP) is to transform the distances along the line by a {constant factor}, as measured from the point of intersection of the two lines.

In our present case, we can determine the factor involved by comparing the length OQ, with the projected length Oq. OQ being of length 1 (Q lies on the circle), the factor is just the value of the length Oq, namely the quantity we have called X.

Since Or” is the projection of Or’, its length is X times that of Or’. The latter, as we already noted, is equal to Oq, whose length is X. So the length of Or” is X times X, or {X squared}!

So much for r”. How do we get from there, to r? We have to compare the {direct} projection of R from the circle down to the axis OP — which gives us r — with the “double” projection, from R onto r’ on OQ, and then from r’ to r” on OP.

The difference between the two arises from the fact, that the first step in the “double” projection occurs at an angle, which is “skewed” relative to the vertical direction of the other projection. I am talking about the angle betwen the vertical line Rr, and the segment Rr’. What is that angle? With a bit of reflection, you can see it is none other than {alpha}. For, Rr’ is perpendicular to the line OQ, which itself is “rotated” by alpha relative to the horizontal line OP. (To put it a different way, the triangle ORr’ is the result of rotating the triangle OQq around O by the angle alpha. In that process, the directionality of each of the triangle’s sides is changed by the same amount.)

The result of the “skew” projection that generated r’, is that its ensuring projection onto the horizontal axis will lie a certain distance to the right of the direct projection r. By how much?

Draw the perpendicular line segment from r’ to the vertical line Rr, and let r* denote the endpoint of that segment. Then r’r* will be parallel, and equal in length, to the segment between r” and r on the horizontal axis. (see Figure 3b)

Now observe that the triangle Rr’r* is {similar} to the original triangle OQq. Indeed: by construction Rr’r* has a right triangle at r*, while the angle at the vertex R, as we just saw, is alpha.

(Note also, that Rr’r* is rotated by 90 degrees relative to OQq. This, as we shall see later, reflects the action of the complex number “i”, lurking in the background of this whole construction!)

The similarity means that the the sides of Rr’r* are proportional to the corresponding sides of OQq, by a common factor of proportionality. Comparing the hypotenuses of the two, note that the hypotenus of OQq — the segment OQ — has length 1; while the hypotenus of Rr’r* — the segment Rr’ — is equal to Qq, the length of which we designated “Y”. The ratio is thus 1:Y, i.e. the factor of proportionality is Y.

In the similarity relationship between the triangles Rr’r* and OQq, length we are looking for — namely r’r* — corresponds to the side Qq of the triangle OQq, which again has length Y. So, to get the length r’r*, apply factor of proportionality Y to length Y. The result is YY, {Y squared}!

To get from r” to r, we thus have to move to the {left} by a distance {Y squared}. On the other hand, we found the length of Or” to be {X squared}.

Our conclusion: {r is located at distance [X squared – Y squared] from O along the horizontal axis}.

Here X stands for the length Oq, Y for the length Qq. Remember, that X and Y are linked to each other, as we noted above, by Pythagorus’ relationship XX + YY = 1 (Q lies on the unit circle). From this YY = 1 – XX, so that XX – YY = XX – (1 – XX) = 2XX – 1.

The result is to express the position of r {directly} as a function of q: The distance Or is equal to 2XX – 1, i.e. twice the {square} of the distance Oq, minus one.

Don’t miss the remarkable implication: the process of {doubling} the angle, by self-reflexively applying the rotation to itself, results in a {quadratic} relationship — i.e. one involving a {square} power — among the scalar “shadows” generated by that process!

Note also certain “topological” features of the relationship of X to 2XX – 1, that reflect the different rates of motion of Q and R along the circle, as the angle alpha grows. (Remember, X measures the segment Oq, not alpha directly). For example, X = 1 corresponds to the case alpha = 0, when P, Q, and R coincide. Sure enough, for this value of X, 2XX – 1 is also equal to 1. On the other hand, X = 0 corresponds to the case where Q lies at the top of the circle (alpha is 90 degrees); in this case, 2XX-1 = -1 and, sure enough, R lies {opposite} to P, at an a angle of 2 x alpha = 180 degrees.

Those skillful in geometry, will find no great difficulty applying entirely analogous methods, to determine the position of the {third shadow}, s — first in terms of r, and then in terms of q. It turns out that the distance Os is equal to 4XXX – 3X. Thus, {tripling} an angle results in {cubic} relationship among the corresponding “shadows” — i.e. one involving the {third power} of X. Hence, by inversion, the implicit relationship between {trisecting an angle} and constructing the {cube root} of a given quantity, which is the general form of the classical problem of doubling a cube.

In the following installment we shall derive the cubic relationships for the trisection of an angle from an improved and simplified standpoint, and then turn to the celebrated paradox of “Cardan’s formula”, as seen through the eyes of Gauss’ predecessor and teacher, Abraham Kaestner.

From Cardan’s Paradox to the Complex Domain, Part II

by Jonathan Tennenbaum

“In this place it will please us to speak of the great advantages of opening up the fountain of Transcendental Magnitudes , and discovering the reasons, why certain problems are neither plane, nor solid, nor of any other degree, but surpass all algebraic equations.” (Gottfried Wilhelm Leibniz, 1686).

We began the first installment of this series, by recalling the physical-geometrical “powers” investigated by the Greeks, and the unexpected emergence of higher “powers” in connection with what appeared to be a straightforward problem of plane geometry: to divide a given angle or circular arc into two, three or a larger number of equal parts. In the following discussion we shall nail down that relationship, in particular deriving the cubic equation corresponding to the trisection of a given angle, and thereby revealing the inseparable relationship with the doubling of a cube.

The same relationship will become clear, when we later examine the implications of the self-similar spiral, itself a reflection of Gauss’s complex domain. On such a spiral, tripling a given angle of rotation, translates into taking the cube (third power) in terms of the corresponding ratio of radial distances. However, the self-similar spiral itself provides neither the means to trisect an arbitrary angle, nor to construct a cube of a given volume. However we shall see later, that both those ancient problems — and countless others — can easily be solved the using the higher principle or “power” embodied in the catenary .

The `Complex’ Composition of Angles

In the last installment we found an algebraic relationship between the scalar “shadows” generated by the doubling of an angle, in terms of the corresponding arcs on a circle of unit radius. We identified the center of the circle as “O” and called the right-hand endpoint of a chosen diameter (taken as the horizontal) “P”; also, we denoted by Q and R the points on the circle, corresponding to the angles {alpha} and {2 x alpha}, respectively, measured as rotations around O relative to the horizontal ray OP. Finally, we denoted the perpendicular projections of Q and R onto the horizontal diameter q and r respectively.

Our analysis showed, that the distance Or is equal to twice the square of the distance Oq, minus one. In other words: the position of r relative to O is given by 2XX – 1, where X measures the corresponding position of q. (See discussion on the meaning of negative values of these parameters, toward the end of this installment.)

Now I want to take on the case of tripling the angle {alpha}!

Recalling last week’s discussion, let S denote the point on the circle, corresponding to the angle {3 x alpha}, and let s be the projection of S onto the horizontal diameter. How is s related to q and r?

As a bit of reflection shows, the answer is already implied by what we did last time, to analyze the relationships for doubling an angle. Reworking the essential steps here again, if possible with an actual physical model or corresponding animated diagram as reference, should cause the principle involved to “leap out” at the reader (see Figure 6).

Figure 6

We started from the right triangle OQq, whose horizontal and vertical sides we called X and Y, respectively. We noted that the point R arises from Q, by rotating Q through the angle {alpha}. Applying that rotation to the whole triangle OQq, yields a congruent triangle ORr’, where r’ marks the perpendicular projection of R onto the axis OQ. At the same time, that rotation generates a new right triangle: ORr, whose third vertex r is the projection of R onto the original horizontal axis OP. Our analysis of the relationship of the triangles, showed that the horizontal side of the latter triangle, Or, had length equal to XX – YY. (Using Pythagorus’ relation between X and Y, we found XX – YY to be equivalent to 2XX – 1.)

Now, to get the point S corresponding to tripling the original angle, it is enough to rotate R — itself obtained by doubling the angle alpha — through the same angle once again. Apply that rotation to the whole right triangle ORr. Observe, that the resulting relationships have essentially the same form as the earlier case, when we rotated the triangle OQq through {alpha}, to obtain the result of doubling the angle. The only difference is, that the angle of the triangle ORr at O is not {alpha}, but {2 x alpha}. But our earlier analysis did not really depend on any special assumptions concerning the shape of the right triangle being rotated, but only the angle of rotation itself ({alpha} and the parameters X and Y connected with alpha).

Accordingly, suppose we have an arbitrary right triangle ORr with hypotenus 1 (i.e. R lying on the given circle) and its side Or lying along the horizontal axis OP. Call its angle at O “{beta}”, and the lengths of its horizontal and vertical sides “A” and “B” respectively (see Figure 7a).

Now rotate ORr by the angle {alpha} around O. The vertex R is carried to a point S, the which, relative to the original point P, corresponds to an angle of {alpha} + {beta}. Imagine a weighted string attached to R and hanging down vertically as the triangle ORr rotates. Observe how the angle, between that vertical and the triangle’s side Rr, grows, as the rotation progresses; in fact, that angle will be equal to the angle of rotation itself (see Figure 7b).

Figure 7

After completing the rotation of the triangle ORr through the full angle {alpha}, R is carried to the point we called S, and r to a point s’, corresponding to the projection of S onto the axis OQ. The vertical string attached to R (now at position S) hits the horizontal diameter at the point s, creating the new right triangle OSs (see Figure 8).

Figure 8

We can now unfold relationships entirely analogous to the ones we found for the doubling of the angle {alpha}, but which now apply to the generalized case of the sum {alpha} + {beta}. To wit:

Let s” and s* be the perpendicular projections of s’ onto the horizontal diameter and onto the vertical line Ss, respectively. As we noted in last week, the first projection changes lengths by a factor X; since the length of the segment Os’ is the same as that of Or, which we called “A”, the projection of Os’ — i.e. Os” — will have length X x A.

Observe, in addition, that in virtue of the process of rotation which generated an angle {alpha} between the vertical at S and the line Ss’, the right triangle Ss”s* will be similar to the original triangle OQq. At the same time, the hypotenus of that triangle, Ss’, is congruent to Rr, whose length we called “B”. Since the original triangle’s hypotenus is 1, the factor of proportionality must be B. As a result, the horizontal side s’s**, which corresponds to the vertical side Qq of OQq, has length B x {length of Qq} = B x Y. On the other hand, that distance is the same as the gap between s and s” on the horizontal axis. Since s” lies to the right of s, we must subtract that distance from Os” — whose length we just found to be X x A) — in order to obtain the length Os.

The result of this chain of relationships is: length Os = {X x A} – {Y x B} That was the horizontal side of the triangle OSs. With a little extra effort, we can also find its vertical side. The latter, Ss, is divided by s* into the two segments Ss* and s*s. The first of them, which coincides with the vertical side of the triangle Os’s*, is proportional to the horizontal side of the original triangle OQq (length X), by factor of proportionality B. So, length of Ss* = B x X The second segment s*s, is parallel to and equal in length to the vertical segment s’s”. Note, that the points O, s’, s” form a right triangle which is slightly smaller than, but similar to the original triangle OQq. The factor of proportionality is the hypothenus of the former triangle, namely Os’, which is equal in length to Or = A. As a result, the length of the side s’s”is A times the corresponding side of the original triangle, namely Qq = Y. So, length of s*s = length of s’s” = A x Y.

Putting these results together, we find: length of Ss = length Ss* + length s*s = {B x X} + {A x Y}. Summing up: the lengths of the horizontal and vertical sides of the right triangle, generated by the angle {alpha} + {beta}, are {X x A} – {Y x B} and {X x B} + {Y x A} respectively, where X,Y and A, B are the sides of the right triangles corresponding to {alpha} and {beta} (see Figure 7b).

Notice, that the horizontal and vertical sides of the triangle for the sum or composition of the two angles {alpha} and {beta}, each involve all of the four values X, Y, A and B. This “complex” interwining of parameters is merely the algebraic “shadow” of the physical process of combining two rotations .

The Cubic Equation for the Trisection of an Angle

At this point we can easily derive the relationships resulting from tripling an angle, and invert these to obtain the third degree algebraic equation corresponding to trisecting an arbitrary angle.

First, what happens when we apply our “composition formula” to doubling a given angle? In this case {beta} is the same as {alpha}, A = X, B = Y, and the horizontal and vertical sides of the triangle for {2 x alpha} come out as {X x X} – {Y x Y} and {X x Y} + {Y x X} respectively. The first one, XX – YY, we had before; and now we have 2XY as the second side.

Now take that triangle, and rotate it by alpha again. In this case {beta} is the double of {alpha}, and A = XX – YY, B = 2XY, as we just found. The result of the composition formula is now a bit more “hairy”, but lawfully so. For our present purposes we only need the horizontal component, which expresses the position of the “shadow”-point s: {X x (XX – YY)} – {Y x (2XY)} = XXX – XYY – 2YXY = XXX – 3XYY. Recalling Pythagorus’ relation XX + YY = 1, YY = 1 – XX, we can express the latter magnitude in terms of X alone: XXX – 3XYY = XXX – 3X(1 – XX) = XXX – 3X – 3XXX = 4XXX – 3X This is the result I announced at the end of last week’s discussion. The position of the “shadow”-point s, resulting from tripling the angle {alpha}, is related to that of the point q, corresponding to {alpha}, as follows: The length Os is equal (in scalar magnitude) to 4 times the cube of Oq, minus three times Oq.

Thus, tripling an angle, is reflected in an essentially cubic or third-power algebraic relationship among the corresponding “shadows”!

Recall the relationships for doubling an angle, involved a square or second-power relationship among the shadows. Thus, the linear, plane and solid geometrical “powers” of Classical geometry, seem to be subsumed within the process of successive transformations of rotation by once, twice, three times an arbitrary angle.

But, what is to prevent us from applying the composition of rotations once again, to obtain analogous relationships for 4, 5 or any other whole-number multiple of an angle? Evidently, each time we apply the rotation {alpha} we increase by one the dimensionality or degree of the corresponding algebraic relationship in the domain of the “shadows”. In this sense, the circular rotation subsumes and transcends all those algebraic dimensionalities. And by the way, didn’t Nicolaus of Cusa refer to the circle as reflecting a higher principle, bounding the linear (algebraic) world of the regular polygons? The latter constitute, of course, special cases of the whole-number division and multiplication of angles.

But returning to our cubic relationship, two remarks are in order.

First, someone who has not been utterly brainwashed by high school or college algebra courses, might rightfully object to subtracting what appears to be a simple one-dimensional magnitude — 3 times the length Oq, i.e. 3X — from the cubic or three-dimensional magnitude 4XXX. Such a subtraction would be plainly absurd; evidently sort of error or foul play has occurred!

Looking a bit closer at what we did, however, reveals, that the multiplier “3” in the above expression does not signify a simple linear magnitude. Indeed, if you check back, you will find that this “3” originated from the “1” in Pythagorus’ relationship XX + YY = 1. That 1, however, signifies the square of the hypotenus of the right triangle OQq, i.e. a two-dimensional magnitude . Thus, the magnitude “3X” is actually a magnitude of “cubic” or third order, while being at the same time proportional to X. (Much more could be said about this matter, under the rubric of the devastating fallacies arising from belief in the supposed self-evidence of “simple numbers”.)

Secondly, we implicitly assumed, in our analysis of the relationships among q, r, and s, that the points Q, R, S all lay in the same, upper right quadrant of the circle; we also spoke of “lengths” always as positive magnitudes. Whereas, as the angle {alpha} grows, the points R and especially S, race ahead of Q, and can come to lie on opposite sides and different quadrants of the circle. In these cases, the layout of the triangles and projections, upon which our derivation depended, change (see Figure 4). At the same time, note that both 2XX-1 and 4XXX-3X can take on nominally negative values, as for example when X = 1/2. What is the significance of those negative values?

Gauss himself, as well as Lazard Carnot in his famous book on the “Geometry of Position”, devoted careful attention to this question, which is closely related to the analysis situs origin, not only of the negative numbers, but also of the so-called “imaginary” numbers. The following should suffice to identify the essential point:

Real physical magnitudes — as opposed to mathematical fictions — are never “indifferent,” but are invariably associated, at least implicitly, with a notion of directionality or orientation in the Universe. “Negative numbers” arise very simply, in connection with the notion of reversal of direction or orientation. Indeed, when for example the point “r” (corresponding to the angle {2 x alpha}) crosses over to the left of the midpoint O — the which occurs at the moment {alpha} hits 45 degrees, and X becomes less than the corresponding value, namely 1/sqrt(2) — the “length” Or reverses its direction. Exactly at that point, indeed, the value of 2XX-1 hits zero and becomes negative .

Thus, the so-called “rules” of algebraic operations with negative numbers, are no mere conventions or arbitrary inventions of so-called “pure mathematics”; on the contrary, they are determined by the geometrical characteristics of rotational action . When those characteristics are taken into account, and when the differentiation of positive and negative values for the “lengths” Oq, Or, Os etc. correspond to the distinction between “right and left” relative to the chosen origin O, then the expressions 2XX – 1 and 4XXX – 3X etc. turn out to be valid for all values of the angle {alpha}.

To explore these relationships, graph the cubic function represented by 4XXX – 3X. Note, that the horizontal coordinate X of the graph corresponds to the position of the point “q”, whereas the vertical coordinate (with value 4XXX – 3X) corresponds to the position of the point “s”. (Keep this separate from the representation of motion on the circle, to avoid confusion!) Imagine the overall form that curve must have, to represent the relative motions of q and s, as {alpha} increases. Next, explore the graph numerically, by calculating the value of 4XXX – 3X for a variety of values of X between -1 and 1, noting the points of reversal of direction and their significance in terms of the interrelation of Q, S and q, s.

So far we have been focussing on tripling a given angle. What about trisecting an angle? For that case, the point S is given, and we have to find the point Q, such that S is the result of tripling the rotation from P to Q. This is evidently equivalent to determining a value of X, such that 4XXX – 3X is equal to the length Os (taking account of +/- sign), where s is the projection of the given point S onto the horizontal diameter. For, once we have the value X, we know the position of Q’s projection q on the diameter; then Q can be constructed as one of the two points of intersection of the perpendicular at q with the circle.

Thus, trisecting an arbitrary angle corresponds to solving the cubic equation 4XXX – 3X = c, where c (corresponding to the position of s on the axis) can assume any value between -1 and 1.

In terms of the graph of 4XXX – 3X, this means finding the intersection-point between that cubic curve, and a horizontal line at height “c” parallel to the X-axis. But, wait a minute! Given the “looping” form of the curve, there will be not just one, but in general three different points of intersection! What do they signify? How could there be more than one solution to trisecting an angle? And what about the doubling of a cube, which corresponds to the cubic equation XXX – 2 = 0? Could there exist three different cubes, having the same volume?

Part III takes us from the birth of “algebra” in the famous “Hisab al-jabr w’al maqalaba” of the 9th-Century Baghdad astronomer ibn Al-Khwarizmi, to Cardan’s paradox and its discussion by Kaestner.

From Cardan’s Paradox to the Complex Domain, Part III

by Jonathan Tennenbaum

As we have emphasized, the subject of Gauss’ “Fundamental Theorem of Algebra” — ostensibly the solution of algebraic equations of arbitrary degree — goes back to very long before the emergence of what came to be known as “algebra”, to the discussions emanating from the Pythagoreans and continued in the circles of Plato, on the general notion of physical magnitude . It is exactly that line of development, which culminated via Gauss’ breakthroughs in Riemann’s conception of magnitude as a self-developing multiply-extended manifold, where “extension” signifies the differential action of generation and integration of a new principle of physical action into the ongoing process.

The very nature of Riemannian physical action is such, that it generates an increasing density of discontinuities or other sorts of anomalies relative to any attempted formal representation “projection” of the action involved.

Look at algebra from this standpoint, and all fearful mysteries dissolve into pure fun. That’s what we shall pursue now, in examining the devastating anomalies which developed within algebra itself in connection with the celebrated “Cardan’s rule” for the solution of cubic equations, and which was played a central role in the disputes which culminated to Gauss’ 1799 refutation of Euler and Lagrange on the so-called “imaginary numbers” and the heredity defects of formal-algebraic method.

The origin of “algebra”

According to available accounts, the term “algebra” derives from the Arabic word “al-jabr”, signifying “completion” or “healing”. The term became current through a famous Arabic mathematical treatise, the “Hisab al-jabr w’al maqalaba”, composed by the astronomer and mathematician Abu Ja’far Muhammad ibn Al-Khwarizmi (approx. 780-850). Al-Khwarizmi worked together with Al-Kindi and others at the “House of Wisdom” in Bagdad, founded by the son of Harun al-Rashid as a center of learning. There, alongside original investigations and writings, classical Greek scientific manuscripts were collected and translated into Arabic. The “Hisab al-jabr w’al maqalaba”, later became widely used in Europe in Latin translation, transmitting the Indian (decimal) system of arithmetic and now-familiar methods for the rearrangement and solution of equations, which Al-Khwarizmi called “completion” (al-jabr) and “balancing” (al-muqabala).

Of particular historical influence was Al-Khwarizmi’s method of solving quadratic equations, such as XX + 10X = 39 . He did not express this with letters, as later became commonplace, but posed the problem instead this way: “a square plus ten of its roots is 39 units. Find such a square.”

How? Draw any square to represent, in hypothetical manner, the square we are looking for, with unknown side (“root”) X. Then “ten of those roots”, constitutes corresponds to a rectangular area with sides 10 and X. Added together the square and rectangular areas are supposed to give a total area of “39”; but there is no evident way to combine the two areas, in such a way, that the value of the square will be evident. Al-Khwarizmi proposes the following “remedy” (al-jabr!):

Divide the rectangle into 4 equal rectangles, by cutting the it parallel to the “X” side, into 4 rectangles of sides X and 10/4 (= 5/2, or 2 1/2). Now, arrange these four rectangles alongside the four sides of the square XX (you have to draw this to follow the argument!). The resulting figure is nearly a square — all that is missing, is four “corners”! “Fix” the defect by adding four squares, each 5/2 by 5/2. The result is a single, bigger square, whose sides are 5/2 + X + 5/2 = X + 10/2 = X + 5. The area of the big square will be the SQUARE of X + 5. On the other hand, the area of each of the four supplementary squares, is (5/2)(5/2) = 25/4, so the total area supplied was 4 time 25/4, i.e. 25.

Adding that same amount to the right side of the original equation, Al-Khwarizmi finds: the area of the big square, namely {X + 5 SQUARED} is 39 + the added area of the four “corners”, i.e. 39 + 25 = 64. Thus the square of X + 5 is 64, and thus X + 5 and X = 8 – 5 = 3. In fact, we can easily verfiy that the value X = 3 does indeed solve the equation XX + 10X = 39. The square Al-Khwarizmi demanded is the 3 x 3 square of area 9.

Some people may recognize in Al-Khwarizmi’s method for “completing the square” the origin of the famous formula for the solution of a general quadratic equation, which students are mechanically drilled in, without ever encountering the simple geometrical idea behind it.

(Indeed, a typical feature of the sadistic “New Math” educational reforms, pushed through in the 1960s, was to systematically suppress the geometrical underpinning of algebra. This included obfuscating the commonplace “rules” for what Abraham Kaestner called “Buchstabenrechnung” — calculation with letters representing unknown or hypothetical magnitudes –, the which became a prominent technical feature of the development of algebra in the 15th century and afterwards. Consider, for example the formula (A + B) x (C + D) = AC + AD + BC + BD.

The “New Math” typically presents this as a deduction from the so-called “associative law” of addition and multiplication. But the origin of the formula is geometrical : it simply describes the division of the rectangular area with sides A+B and C+D, by corresponding perpendiculars to those sides, drawn at the points that divide them into lengths A,B and C,D respectively. The result is four rectangles of sides A,C; A,D; B,C and B,D respectively. Similarly, the equation:

(A + B) squared = A squared + 2AB + B squared describes the special case of the division of a square of side A+B into two squares, with sides A and B respectively, and two rectangles with sides A,B. There do arise some interesting subtleties and paradoxes, when the magnitudes involved take on negative, imaginary or some other species of values, and are no longer assumed to be simple lengths, as we shall discuss below. But by depriving students of even the simplest, visual-geometrical image of these relationships, the perpetrators of the “New Math” fraud also blocked the pathway to the deeper physical principles that underlie both algebra and the geometry of visual space. In fact, Bertrand Russell’s “New Math” represents nothing but a revival of the worst features of Euler and Lagrange’s formalist method — exactly the method Gauss refuted in his first paper on the “Fundamental Theorem of Algebra”!)

Applied to a general quadratic equation of the form

XX + BX + C = 0 where B and C represent any arbitrary choice of parameters, Al-Khwarizmi’s approach yields the following result. First, we make XX + BX into a square, by adding B/2 squared = BB/4, exactly as we did for the particular case of XX + 10X above. The general case takes the form:

XX + BX + BB/4 = (X + B/2) squared

Accordingly, add BB/4 to both sides of the quadratic equation, and apply “al muqabala” — the art of “balancing” or shifting the components of an equation between the sides, in order to “box in” the value of the unknown X. To wit:

XX + BX + C = 0

XX + BX + (BB/4) + C = BB/4

(X + B/2) squared + C = BB/4

(X + B/2) squared = BB/4 – C

X + B/2 = square root of ( BB/4 – C )

X = – B/2 + SQRT( BB/4 – C )

The last in this chain of relationships is essentially nothing but the famous formula for the solution to the general quadratic equation XX + BX + C = 0 (see footnote), and the precursor to Cardan’s attempted solution to the general cubic equation.

Al-Khwarizmi’s quadratic species

But, beware the algebraicist’s sleight-of-hand! A number of subtleties and paradoxes lie buried beneath the apparently routine procedure we just went through, to “solve” the quadratic equation.

Al-Khwarizmi distinguished between at least four different species of “quadratic” problems. For example:

1) XX + 10X = 2

2) XX = 10X + 2

3) XX + 2 = 10X

4) XX + 10X + 2 = 0 each representing a distinct sort of geometrical relationship.

In the first case, we seek “a square which, when combined with the rectangle whose sides are the square’s root and 10, respectively, gives a total area of 2”. We discussed a problem of this sort already above (with 39 instead of 2).

In the second case we seek “a square whose area is 2 units more than the area of the rectangle whose sides are the square’s root and 10, respectively.” At first glance it might not be clear at all, from the geometrical picture, how to apply the method of “completing the square”. On the other hand, changing the “balance” by shifting the rectangle 10X to the other side of the equation, we can put it in a form apparently similar to the first case. In fact, from the standpoint of determining the unknown X, XX = 10X + 2 is equivalent to XX – 10X = 2.

Compared with Al-Khwarizmi’s construction for the first case, we run into a significant difference: this time we have to take away the area of the rectangle 10X from the square, rather than adding it. After cutting the rectangle into four equal subrectangles with sides 10/4 and X and trying to fit them in an analogous way into the square XX — in order to “take them away” — we find that they overlap at the corners (make the drawing, to see what I am getting at!) Thus, taking away the four areas from the square would mean to remove each of the four corners twice . But how can one “take away” an area twice, from the same place? After we have removed it once, it is gone ; and to remove the same area again would mean taking something from nothing !

Al-Kharizmi scrupulously avoided such “impossible” operations. In fact, if you add in four extra little squares of sides 10/4, one at a time, at the appropriate moments, interspersed between removing the four rectangles one at a time, you can circumvent the difficulty. The result of a suitably ordered sequence of adding the four square areas and taking away the four rectangles is a smaller square, whose side is X – 10/2 = X – 5. From this point on, the solution proceeds entirely analogously to the first case.

The third case is more tangled-up still, if we hold to a simplistic visual-geometrical conception. Indeed, when we try to rearrange the equation XX + 2 = 10X in order to be able to “complete the square”, we get XX – 10X = – 2.

The equation demands, explicitly, that we take away a bigger area from a smaller , to get a negative area — clearly an “impossible” proposition! Here again, the difficulty can be circumvented by adding the four squares (10/4 x 10/4) to both sides of the original equation, before proceeding further.

The fourth case, XX + 10X + 2 is truly “impossible” in Al- Khwarizmi’s sense. Areas are by their very nature positive magnitudes . The sum of a square (XX) and a rectangle (10X) cannot be less than zero. Hence XX + 10X + 2 cannot be less than 2, and certainly never equal to zero.

All of this indicates quite clearly, that, to the extent the famous “general formula” for the solution of a quadratic equation XX + BX + C = 0 is valid at all, it must presuppose the existence of a different domain , distinct from that of simple visual geometry, but coherent with the latter.

Kaestner on “negational magnitudes”

At first glance, that domain is distinguished by the introduction of negative numbers , which have no self-evident interpretation in terms of lengths, areas and volumes, and were long branded as “impossible” for the indicated reasons. In fact, this issue was first fully cleared up by Gauss himself, in the context of his conceptualization of the physical principle underlying the “imaginary numbers”.

In his 1758 “Anfangsgrnde der Mathematik”, which was a standard reference for the teaching of mathematics at the time Gauss began his studies at the Carolineum in Braunschweig, Abraham Kaestner introduced negative numbers in the following manner:

“Opposing magnitudes are magnitudes of the sort, that arise through consideration of such conditions, in which one magnitude reduces another — for example assets and liabilities, forward and backward motion, etc. One of the magnitudes, whichever one chooses, is called positive or affirmative; the opposite is called negative or negational.”

Kaestner proceeds to develop the arithmetic of such “opposing magnitudes” as a new domain subsuming simple “positive” magnitudes together with their “opposites”. Keastner demonstrates, among other things, why the product of two “negational magnitudes” must necessarily yield a “positive magnitude”. For example, multiplying by -1 transforms a magnitude into its opposite, so that (-1) x 2 is the opposite of 2, i.e. -2; while (-1) x (-2) is the opposite of the opposite of 2, i.e. 2 itself. As Kaestner emphasizes, these relationships apply only to such magnitudes, as admit of a unique opposite , as for example in the case of forward and backward motion along a pathway.

Relative to such a domain of “opposing magnitudes” — as typified by the so-called “real number line” — Al-Khwarizmi’s four species of equations can all be subsumed under a common form. Indeed, in the new domain the species 1) – 4) above are equivalent to:

1) XX + 10X – 2 = 0

2) XX – 10X – 2 = 0

3) XX – 10X + 2 = 0

4) XX + 10X + 2 = 0 respectively, all of which fall under the form XX + BX + C = 0, where B and C can take on both positive and negative values. At first glance, it would seem that the formula derived above, namely X = – B/2 + SQRT( BB/4 – C ) indeed provides a general solution to the quadratic equation, subsuming each of the four cases distinguished by Al-Khwarizmi. The operations he regarded as “impossible” or absurd from the standpoint of lengths, areas and volumes — such as “subtracting a larger from a smaller” — have a perfectly determinate meaning in the domain of forward and backward motion along a line. For example, subtracting a forward motion of 5 from a forward motion of 2, results in a backward motion of 3, i.e. -3, and so forth. The chain of relationships, by which we derived the general solution to the quadratic equation XX + BX + C, are valid irrespective of the values of B and C.

In particular, the general formula provides a solution to the equation XX + 10X + 2 = 0, which Al Khwarizmi regarded as “impossible”: X = -10/2 + SQRT( 100/4 – 2) = – 5 + SQRT (23) = (approx) -2.0416848, a negative magnitude.

The fallacy of closure

Have we invented a “perfect system”? A closed domain in which the algebraicist, sitting in a room with no windows, can solve all problems at the blackboard, without any need to heed the real universe outside?

On the contrary! In fact, our derivation of the “general solution” glossed over an even more devastating paradox, than the emergence of the negative numbers which Al-Khwarizmi and many of his successors treated as “impossible”. Look back at the next-to-last step in the “general solution”:

(X + B/2) squared = BB/4 – C

X + B/2 = square root of ( BB/4 – C )

Contrary to common misconceptions, the algebraic expression, “square root of,” constitutes a question , not an answer ! We are asked to find a magnitude, whose square is the given magnitude. Just because the algebraicist has invented a formal symbol in place of answering the question, does not make it any less a question! Although there exists a simple ruler-and-compass construction for determining the square root of any positive quantity, the analogous problem for cube roots leads, as we indicated in the preceeding installments, beyond the limits of ruler-and-compass geometry.

What, however, if the magnitude, whose square root is demanded, turns out to be negative ? According to Kaestner’s argument, the required square root could be neither negative nor positive. For, the square of a positive quantity is naturally positive, while result of multiplying a negative magnitude by itself, for the reasons Kaestner indicated, is likewise positive. Thus, extracting the square root of a negative magnitude is “impossible” in the domain of the so-called “real numbers”.

On the other hand, our formula for the solution of the general quadratic equation, demands that we find the square root of BB/4 – C. What if the latter magnitude were to turn out to be negative — as for example in the case, where B and C are both equal to 2, and BB/4 – C = 4/4 – 2 = -1 ? Apparently, to solve the equation XX + 2X + 2 = 0, we would have to determine the value of SQRT( -1) !

To get a notion of the problem involved, explore the values of XX + 2X + 2 for a variety of positive and negative values of X. The values turn out all to be positive, with a minimum value of 1, reached for X = -1. In fact, plotting XX + 2X + 2 (on the “Y-axis”) against the value of X (on the “X-axis”), yields a parabola lying entirely in the positive region above the X-axis; whereas XX + 2X + 2 = 0 would imply a crossing of the X-axis by the corresponding curve, which evidently never occurs.

Another “impossible” problem

A useful, related sort of case was discussed by Cardan and his associates in the context of their attempt to grapple with “imaginary numbers”.

Consider for example the following simple geometrical problem: “Given a line segment of length 2, divide the segment into two subsegments of lengths X and W, such that the product of their lengths is equal to 2.”

A bit of reflection shows, that the task is “impossible”! If we divide the segment exactly in half, then the two segments will each have length 1, and the product will be 1, not 2. If, on the other hand, if the division is not equal, then one of the two segments (X for example) will be shorter than 1, and so XW will be less than W, which in turn cannot be longer than the total length 2.

The paradox becomes even clearer, when we look at the proposed problem in terms of a division of the square of side 2, implied by any given division of the sides into subsegments of lengths X and W. Drawing perpendiculars at the points of division, the square is divided into the two squares XX, WW plus two X-by-W rectangles. The total area of the original square is 2×2 = 4. If, on the other hand, we demand that XW = 2, then the sum of the two rectangles would already be 4, and would thus completely fill up the square! But then there would be no room left inside for the smaller squares XX and WW — unless, by some extraordinary circumstance, the two areas XX and WW were somehow to cancel out , i.e. XX + WW = 0 or WW = – XX. But since the areas of squares are always positive, the latter is clearly impossible .

This “impossible” geometrical problem leads directly to a quadratic equation. In fact, we can restate the problem in algebraic form as a combination of two simultaneous equations:

X + W = 2 and XW = 2

The first equation implies W = X – 2, and consequently

XW = X x (2 – X) = 2X – XX.

In terms of X, the requirement that XW = 2 now becomes 2X – XX = 2 or, by “rebalancing” the equation: XX – 2X + 2 = 0.

Again, graphing the values of XX – 2X + 2 yields a parabolic curve lying entirely above the X-axis; the minimum value, reached at X = 1, is 1. On the other hand, if we apply our “general formula” for the solution of the quadratic equation to this case, we obtain the paradoxical value X = 1 + SQRT ( -1 ).

At this point, the formal algebraicist, like Euler and Lagrange, might seek refuge in the following comforting thought:

“After all, doesn’t the appearance of the “impossible” square roots of negative magnitudes in the formula for the solution of a quadratic equation, coincide with the case, where the equation has no real solution? Our formula gives a solution in exactly those cases, in which a solution really exists. ‘Imaginary’ magnitudes like SQRT( – 1) mean nothing, but merely signal the impossibility of solving the corresponding equation. So, our algebraic world is closed and perfect.”

If that’s what you think, you are in for a rather unpleasant surprise! Just wait for the next installment, on Cardan’s formula. Notes:

The solution to the quadratic equation XX + BX + C = 0 is commonly presented in a slightly different, but entirely equivalent form: X = ( -B + SQRT( BB – 4C ) ) / 2 .

In the case of the equation AXX + BX + C = 0, in which the square term appears multipled by an arbitrary coefficient A, the solution takes the form: X = ( -B + SQRT( BB – 4AC ) ) / 2A.

Both formulae are simply alternative manifestations of the principle of “completing the square”, and add nothing of substance to the above discussion.

Beyond Counting — A Preparatory Experiment

by Bruce Director

In previous pedagogical discussions on Higher Arithmetic, we investigated the ordering of numbers with respect to arithmetic (rectilinear) progressions, (as in the case of linear and polygonal numbers) and geometric (rotational) progressions, as in the case of geometric numbers, and prime numbers. (See Doc.#’s 97267bmd01; 97316bmd001; 97326bmd001;)

The deeper implications of these investigations, which form the basis of Gauss’ re-working of Greek classical geometry, reveal themselves, only if we rise above intuition, and investigate the nature of numbers with the mind only.

In the coming weeks, we will begin to further investigate these principles. But, it will be much more efficient, if the reader first performs the following experiments:

As discovered in an earlier pedagogical discussion, the geometric progression is constructed by beginning with a square, whose sides are a unit length, and whose area is a unit area. We then add 2, 3, 4, or more squares forming a rectangle. We then double, triple, quadruple, this rectangle forming a new square, and so forth. With each successive action, the area of the corresponding square or rectangle increases, but the type of action, doubling, tripling, quadrupling, etc. doesn’t change. A different type of number, incommensurable with rectilinear numbers, is discovered by this process. (This construction is discussed by Plato in the beginning of the Theatetus dialogue.)

In contradistinction, the rectilinear (polygonal) numbers, form a series in which each number associated with a given polygon, increases by an increasing amount, but the differences between the differences remains the same. And so, under Gauss’ concept of congruence, all polygons of the same type can be brought into a One, because the differences are all congruent, relative to a modulus which is the number of sides minus 2. The totality of all polygons, can be thought of as a series of series, ordered by successively increasing moduli.

For geometric numbers, however, there is no simple modulus, under which the individual members of any given geometric progression can be made congruent. Or, put another way, the change from rectilinear (1 dimension) to rotational (2 dimension) changes the ordering principle. We must shift tactics. The old rules, don’t apply. We must discover a new, higher type of congruence. This new higher type of congruence, opens the door to whole new domain.

To discover the nature of this domain, it is most efficient to follow in Gauss’ footsteps, and first discover these orderings experimentally, and then investigate the deeper implications, which underlie these orderings.

Each different geometric progression can by also thought of as a series of numbers associated with the underlying action, in order of increasing actions. For example, 2 for doubling. The first number in the series is a unit area, which has undergone no doubling, i.e., 2^0 or 1. The second number is the first doubling, or 2^1 or 2. The third number is the second doubling or 2^2 or 4. The third number is the third doubling, or 2^3 or 8, etc. This forms the geometric progression, 1, 2, 4, 8, 16, ….

Another example, 3 for tripling. The first number is the unit area which has undergone no tripling or, 3^0 or 1. Then the first tripling, 3^1 or 3; the second tripling 3^2 or 9; the third tripling 3^3 or 27. This forms the series 1, 3, 9, 27, ….

Now investigate the congruences of these series with respect to odd prime numbers as moduli. Begin with modulus 3. Calculate the least positive residues of the numbers of the geometric progression based on 2 with respect to 3 as a modulus. Then take 5 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, with respect to 5 as a modulus. Then take 7 as a modulus. Calculate the least positive residues of the numbers of the geometric progressions based on 2, 3, 4, 5, 6 with respect to modulus 7.

What new type of orderings emerge? What’s going on here? We will begin to investigate these questions, next week.

Beyond Counting — Part II

by Bruce Director

If you carried out the experiment in last week’s discussion, you would have discovered the reflection of an ordering principle with respect to the residues of geometric progression. The experiment should have yielded the following result.

With respect to modulus 5, the residues of the geometric progressions based on the numbers 2-5 yield the following results: (The Powers are in the first row; the residues resulting from a specific geometric progression are in the rows which follow. The base is the type of action from which the geometric progression is generated — 2 for doubling; 3 for tripling; etc.).

Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 3 1 2 4 3 1 2 4 etc.
Base 3: 1 3 4 2 1 3 4 2 1 3 4 etc.
Base 4: 1 4 1 4 1 4 1 4 1 4 1 etc.
For Modulus 7:
Powers: 0 1 2 3 4 5 6 7 8 9 10
Base 2: 1 2 4 1 2 4 1 2 4 1 2 etc. 
Base 3: 1 3 2 6 4 5 1 3 2 6 4 etc.
Base 4: 1 4 2 1 4 2 1 4 2 1 4 etc.
Base 5: 1 5 4 6 2 3 1 5 4 6 2 etc.
Base 6: 1 6 1 6 1 6 1 6 1 6 1 etc.

This is a surprising result. The unbounded, ever increasing geometric progression, is brought into a simple periodic ordering with respect a prime number modulus. No matter which type of change (base) of the geometric progression, a periodic cycle emerges with respect to a prime number modulus. Each period, begins with unity, making a sort of wave pattern. While the “wavelength” may change with the base, the “wavelength” is always either the modulus minus 1 (m-1) or a factor of m-1. No other “wavelengths” are possible. The bases whose “wavelengths” are m-1 are called “primitive roots.” (In the examples above, 2 and 3 are primitive roots of 5; 3 and 5 are primitive roots of 7.)

These orderings were investigated by Fermat and Leibniz, and, according to Gauss, Leibniz’ investigations of these orderings, were a subject of the oligarchical slave Euler’s attack on Leibniz, played out in the famous fight between Koenig and Maupertuis. In his Disquisitiones Arithmeticae and the two Treatises on Biquadratic residues, Gauss unfolds even deeper implications of these orderings, which will be discussed in future pedagogical discussions. For now, it is sufficient to reflect on the subjective questions presented by the phenomena.

In order to even begin to discover what’s going on here, you must think in an entirely different way about numbers. What accounts for these orderings? The answer will elude you, if you cannot free yourself from a conception of number associated with mere quantity of objects. Just as the discovery of valid physical principles, such as the orbit of the asteroid Ceres, will elude you, if you cannot free your mind from fixating on the mere observations. The answer lies outside the orderings themselves, and can only be reconstructed inside the mind, by reflecting on the paradoxes presented.

Instead of thinking of each number individually, think instead of a series from 1 to m-1, associated with a unique principle of generation, that contains each number. Each principle of generation is characterized by a distinct type of curvature. One principle of generation, is the principle of adding one (rectilinear). Another principle of generation is the principle of adding areas (sprial action). A third principle of generation, is the principle of congruence (circular rotation). A fourth principle of generation is the principle of prime numbers. The combination of all four characterizes a hypergeometry, the unfolding of which, generates the periodic orderings reflected in the residues of powers.

The subjective challenge, is to be able to conceive in your mind of the interconnection of these generating principles as a One, when that One cannot be expressed as a mathematical function. The functional relationship exists only in the mind. Just as the One of a musical composition exists not in the notes, or the physical characteristics of the well-tempered system, but in the Idea of the composition, which is transfinite with respect to the unfolding of the composition.

In the interest of not diverting attention from concentrating on these subjective questions, the reader is advised to continue these experiments with respect to the prime numbers 11, 13, 17, and 19. In future discussions, we will rediscover Gauss’ application of this principle in his re-working and superseding classical geometry.

The Fraud of Benchmarking

by Jonathan Tennenbaum

This week’s pedagogical discussion illuminates the issue of “non-linearity in the small” from a somewhat different angle than the preceeding ones. This week we shall take a look at the celebrated case of Mercedes’ famous “A-Class” automobile. The following documentation was researched by Rudiger Rumpf and Ralf Schauerhammer, and will be featured in a forthcoming article in the German magazine FUSION. I have translated the core of their manuscript and added some comments. —

Jonathan Tennenbaum

To remind readers who are not “car freaks” and may not have followed this affair from the beginning: Two years ago the world-famous automobile manufacturer Daimler-Benz — manufacturer of the legendary Mercedes-Benz automobile and now partner of Chrysler in the greatest mega-merger in industrial history — triumphantly introduced into the world market for low-priced compact cars its own, specially-designed model called “A-Klasse.” Not only would Class A offer people with small pocket-books the prestige and “feeling” of driving a “Mercedes,” but the car itself boasted extraordinary features. Instead of the engine being located in front of (or behind) the passenger cabin as in other cars, the A-class has its engine placed {underneath}, with the passenger cabin built on top. This so-called “sandwich” construction, never before utilized in this sort of car, offers greater flexibilities in the use of space, while at the same time the passengers sit significantly higher above the ground than in other cars.

On Sept. 23, 1997 test drivers in Denmark found that the Class A tilted onto two wheels during a swerving manouver (i.e. sharp turn of the sort needed to steer around an object) at 55 km/hour (34 mph). A month later, on Oct. 30, a Class A Mercedes flipped upside down during the so-called “Elk Test” at 60 km/hour (37 mph), slightly injuring three test drivers in the car.

The serious issue raised by these events is not so much the obvious design weaknesses of Mercedes Class A; what is significant is the kind of faulty {thinking}, embedded in the process of design and development of the car, which ultimately caused the embarassing and commercially disastrous result. In fact, the main weakness in the design and development process of Class A lay in a dependence on so-called “benchmarking” methods.

Up until recently, Mercedes has traditionally devoted much more time and investment to develop new models, than Japanese firms for example. In order to improve its “competiveness,” in the case of the Class A, Mercedes set the goal of reducing the development time from the traditional 7 years (84 months) or so, to a mere 32 months! Yet, the comparison with the development time of Japanese makes is unrealistic and misleading, because Japanese producers typically concentrate on improving already established and proved designs. In that case only about 20-40% of the components must be newly constructed, even to make a new model. But in the case of the Mercedes Class A, for example, 100% of the components had to be newly developed.

Although Mercedes is one of the leading automobile manufacturers in the world, Mercedes engineers had never before built a compact car with front wheel drive. Furthermore, the planned “sandwich” construction had never before been used in a small car. Setting these goals while at the same time cutting the development time from 84 to 32 months, placed huge pressure on the development department of Mercedes. That pressure greatly strengthened the already-existing tendency to think that computer simulations could replace actual, real-life driving tests.

In order to save development time and costs, in April 1993 Daimler-Benz engineers input the then-available, projected basic design parameters of the Class A into a computer simulation system designed to simulate the dynamic behavior of the car. This was done before even the components and parts of the automobile had been constructed. These imaginary driving tests, according to Mercedes, were supposed to be sufficient to provide “all the answers” concerning certain important design decisions as, in particular, which of three alternative construction types (“Mehrlenker,” “Verbundlenker” or “Laengslenker”) should be chosen for the rear axle assembly. On the basis of those simulations, the cheapest of the three alternatives was chosen as fully acceptable.

As a further test for the projected driving characteristics of the Class-A (which had not yet been built, even in prototype), “engineers put themselves behind the wheel of a Mercedes S 280 (a completely different model-JT) which had been programmed to simulate the dynamic behavior of the Class-A,” a Mercedes report boasts. “During a double lane change (the similar type of manouver to the now-infamous “Elk Test”- JT) which reveals the handling characteristics of an automobile as well as its safety reserves in marginal situations, the Laengslenker axle (the cheapest of the three alternatives – JT) performed convincingly…. In the context of the total concept of the Class A, the Laengslenker demonstrated itself to be the best compromise.” The “compromise” referred to, was a compromise between handling characteristics and cost. And here the fact that the Laengslenker was much less costly, clinched the decision in its favor. All other Mercedes-Benz models are equiped with the “Mehrlenker” axle, which is much costlier, including in comparison with the systems used in competing models.

In an information brochure on the Class A Mercedes also wrote: “This time there was not enough time to carry out the extensive basic investigations with different axle types, which are normal procedure for the development of a completely new automobile type.”

Other German producers calculate up to 12 months for the costly process of harmonizing and adjusting the chassis and related assemblies in connection with electronic systems such as ABS (anti-lock brake system) and ESP (electronic stability program). This is normally done only after 3 years of testing with prototypes or rebuilt older models in order to determine the proper design of the axle. This is what Mercedes itself had always done before.

For example: In 1989, driving tests in extremely demanding, mountainous areas revealed — contrary to the results of computer simulations — that the braking system of the then-newly-designed Class S auto (V-12 motor with up to 400 horsepower) was far from meeting the full stress performance requirements. The entire brake system had to be completely redesigned. But at that stage the projected beginning of mass production of the auto was still two years in the future.

In the case of the Class A, Mercedes not only did not have the necessary time, but also lacked sufficient capacities in its development department: just in the period from June 1993 to October 1997, 9 new models had their premieres. Juergen Stockmar, Director of Development at Opel, was quoted as saying that many employees in the development departments of automobile producers today are working beyond the limits of their endurance, and that the overworked condition of the employees ran like a red thread through a series of technical breakdowns and other problems.

After the disaster in October 1997 Mercedes was finally able to bring the problems of the Class A under control — although only after three separate attempts and after company head Schrempp had intervened to halt deliveries until further design and development had been carried out. But the methods used to “solve” the problem were rather dubious.

The electronic stability system (ESP), which was originally planned to be sold as an option for 1700 DM, was now included as standard equipment on all Class A autos and delivered to the buyer without additional cost. The ESP had originally been conceived as a supplementary program for safe and stable cars, to assist control of the vehicle under extreme conditions such as wet or slippery roads. But in the Class A, the ESP became indispensable even to carry out a simple avoidance maneuver on a dry surface — something which competing models had never had problems with.

The fact that Mercedes now claims it has solved the problems of Class A by supplementary installation of the ESP system, demonstrates that the fundamental problem behind the Class A disaster has not penetrated to the conciousness of the company’s board members. They are still holding to their belief in benchmarking and computer simulations.

This becomes obvious from the fact that the board of Daimler-Chrysler still refuses to withdraw its even bigger disaster, the regrettably-misnamed “Smart,” from the market. “Flop” would be a more appropriate name for this totally misconceived and technically defective product. Besides the problems shared with the Class A — shorter wheelbase and elevated center of gravity — the developers thought they could simply ignore problems that had been known for decades. These are the problems which arise when one attempts, as with the “Smart,” to realize rear-motor, rear-wheel drive in a vehicle with short wheelbase, leading to a situation where nearly two-thirds of the car’s weight falls on the rear axle.

Ignoring the long and problematic history of constructions of this type, the Mercedes engineers even installed an over-powerful engine (after all, the car was supposed to be “smart”!), with the philosophy that “electronics will fix everything.” Since the car was known to be unstable, the maximum speed was set at 130 km/h. But even below that speed the electronics cannot compensate for the fundamental fallacies in the design. Physics prevents this! The cheap electronic stabilizing program called “Trust” has revealed itself, in all tests which were not designed in advance to give a positive result, as a failure.

Readers will not have failed to recognize a recurring syndrom of today’s larger world in our story of the “A-Klasse”: Rather than correct fundamental, axiomatic fallacies in the design of policy, the reponse to each ensuing disaster is: “we’ll fix it!” The “successful” result is to carry the axiomatic fallacies forward into the next, even worse phase of disaster, whose onset has been rendered inevitable by the follies of such linear “crisis management.”

Note, also, a second point: in a multiply-connected manifold, “dimensionalities” can never be treated as Cartesian independent variables. In substituting or modifying even an apparently minor technical component within a complete functional system such as an automobile or a space vehicle, the potential nonlinear impact of that change upon the characteristic functioning of the whole system is an issue of physics, not mathematics. In a unique experiment, the components of the experimental apparatus and their characteristics, taken in and of themselves, seem to be fully “known.” But the composition of the experiment generates an irreducible anomaly, refuting exactly the sort of linear “curve fitting,” which turned Mercedes’ proud creation into a total flop.

How Aristarchus Measured the Universe

By Robert Trout

“But Aristarchus of Samos brought out a book consisting of certain hypotheses, in which the premises lead to the conclusion that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain motionless, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the center of the sun is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.”

Archimedes, in “Sand-reckoner”

In 1543, Copernicus published his book, “On the Revolutions of the Heavenly Spheres,” which located the sun and not the earth at the center of the solar system. He was severely censured for this by Aristotelian circles. In his handwritten and autographed copy, Copernicus had written “… Philolaus perceived the mobility of the earth, which also some say was the opinion of Aristarchus of Samos….” This sentence was not included in the printed version. Perhaps, he judged the publication of it simply too risky, because it would had exposed how the system of Ptolemy, which had been imposed on the West for 1300 years, had been a fraud.

Aristarchus (ca 310-230 B.C.) is credited by his younger contemporary Archimedes, and many other commentators, with establishing that the earth rotates around the sun and not the other way around. In his one still extant treatise, “On the Sizes and Distances of the Sun and Moon,” he demonstrated his method for calculating the sizes and distances of the sun and moon, which dramatically changed man’s estimate of the size of the solar system. We will examine how Aristarchus’s discoveries are a culmination of a project launched by Plato to explain the apparently erratic cycles of the universe with “uniform and ordered motions.” Aristarchus’s created a paradox, and then resolved it by a creating a Platonic idea which explained the most basic cycles of the earth’s relationship with the sun, moon and the Universe.

Aristarchus begins his treatise by demonstrating that an observer on the earth can determine when the sun, moon, and earth are situated, so that their relationship to each other is described by a right triangle, and to measure the angles of that right triangle. From this, he was able to calculate an estimate of the ratio of the distance of the earth to the sun, relative to the distance of the earth to the moon.

Aristarchus demonstrated that the phases of the moon are caused by the sun shining on the moon from different directions. Aristarchus knew the basic principles of eclipses from Anaxagoras. He also knew, probably from the previous work of Pythagoras and Anaxagoras, that the moon is a sphere. Only the side of the moon, which is facing the sun, is illuminated, and is visible to an observer on earth. When the moon appears to be near the sun in the sky, the sun is actually further away and behind the moon. The sunlight, then, falls on the back side of the moon, so an observer on earth can see only a small sliver of the moon, if anything. When the sun and the moon opposite each other in the heavens, then the sun lights up the side of the sphere which faces the observer on earth. The moon then appears full. You can demonstrate this by shining a flashlight on a small ball from different directions, and observing how the lit portion of the ball appears.

He demonstrated that when the moon appeared to be exactly half full, the angle from the sun to the moon to the earth, was then very close to a right angle. (This requires that the distance between the sun and moon is large, relative to the diameter of the sun, as it is.)

Make a drawing to demonstrate what Aristarchus was doing. (Figure 1) Near the edge of a sheet of paper, draw a small circle, representing the moon, and label its center, M. Below M, draw a second circle, representing the earth. Draw a line through the centers of the two circles. Label the point where this line intersects the top of the circle representing the earth, E, which represents the position of the observer on the earth. Finally, draw a line, which is perpendicular to the first line, through the point M. The sun will be represented by a third circle, whose center is labelled S. If the center of the sun, S, is on this perpendicular line, Angle SME will be 90 degrees, and the moon will appear, to the observer on the earth at point E, to be almost exactly half full.

Now place the sun, with it’s center, S, on the perpendicular line, at different distances from the moon, and measure the angle which an observer, standing at point E, will see between the sun and the moon (angle SEM). When the sun is close to the moon, Angle SEM will be small. As you move the sun further away from the moon, Angle SEM will become larger.

Aristarchus, in studying the relationship between the actual sun, moon and earth, calculated that when the moon was half full, indicating that Angle SME was approximately a right angle, Angle SEM was 87 degrees. He now had calculated two of the angles of the triangle SEM. From this he was able to use the knowledge of geometry, which the Greeks had discovered at that time, to calculate SE/ME, or the ratio of (the distance from the earth to the sun)/(the distance from the earth to the moon).

There are a number of ways to do this. Neither trigonometry nor pocket calculators had yet been developed. Instead, Aristarchus solved the problem by using the knowledge which the Greeks had developed of relationships between triangles, to come up with an approximation for the ratio of the two distances. It can be seen from the diagram, that when Angle SEM approaches a right angle, the ratio of the sides, (SE/ME) becomes large. He calculated that the distance from the earth to the sun was approximately 18 to 20 times the distance from the earth to the moon.

Since Aristarchus estimated that the sun was 18-20 times further away than the moon, and they appeared to be around same size in the sky, as is demonstrated by eclipses of the sun, he used the principles of similar triangles, which had been developed by Thales, to conclude that the diameter of the sun was approximately 18-20 times the diameter of the moon.

The actual value of the ratio SE/ME is around 389, which is around 20 times greater than Aristarchus’ estimate of approximately 18-20. The actual value for the angle SEM is around 89.9 degrees. The error in his measurement of this angle probably resulted from the difficulty of determining when the moon appears exactly half full, and not his inability to accurately measure angles. Try reproducing his experiment. You will see for yourself that the angle SEM is clearly near 90 degrees, although it is difficult to determine this angle with greater precision than Aristarchus did.

Aristarchus also measured the distance to the moon, using the moon’s diameter as his unit of measurement. An observer on the earth will see the moon as a small disk in the sky. Thales is reported to have measured the angular size of the moon at 1/2 degree, approximately 300 years earlier.

To demonstrate Aristarchus’ method, draw a circle approximately 2 inches in diameter representing the earth and label a point on this circle, P, representing the position of an observer on the earth. (Figure 2) Approximately 5-6 inches away from point P draw a circle of around 1/2 inch in diameter, representing the moon. (This is not a scale model.) Now draw lines from the observer’s point, P, tangent to the two sides of the circle representing the moon. Draw a line connecting the two points of tangency which are on the opposite sides of the circle representing the moon. (Aristarchus demonstrated that the length of this line is very close to the diameter of the circle representing the moon.) Since the two lines from the observer at point, P to the two points of tangency are the same length, this creates a long slender isosceles triangle.

Using principles of geometry, which were then known to the Greeks, Aristarchus was able to calculate the ratio of the length of one of the long sides of the triangle, to the length of the short side. This ratio represented the earth-moon distance, measured in moon diameters. Aristarchus calculated that the distance to the moon was approximately 26 times the diameter of the moon. Strangely, Aristarchus used an angular displacement for the sun and moon of 2 degrees in this treatise. This is 4 times larger than the 1/2 degree which was reported by Archimedes to be known to Aristarchus. By using 2 degrees as the size of the angle at P, he decreased his estimate of the distance to the moon to approximately 1/4 of what he would have calculated, had he used 1/2 degree.

Aristarchus then combined these discoveries with another piece of experimental evidence, which was known from studying eclipses of the moon, to calculate the ratio of the size of the earth to that of the sun and moon. The Greeks had estimated, by measuring the amount of time that it took the moon to travel across the earth’s shadow during an eclipse of the moon, that the shadow, which the earth made on the moon, was approximately twice the size of the moon. Knowing this, he was able to use the geometrical relationships which existed between the sun, earth, and moon, during an eclipse of the moon, to calculate the size of the earth relative to the sun and moon.

Make a drawing which is a simplified version of Aristarchus’ calculations in his treatise. (Figure 3) This drawing will not be to the correct scale, to represent the sizes and distances in either the actual solar system or Aristarchus’ estimates of them. To do that, you need to know the answers that you are seeking. (It would also require either a very long piece of paper, or else drawing the sun, moon and earth so small that you would not be able to see the geometrical relations clearly.) Near the right edge of the paper, draw a circle, with a radius about 1 1/2 inches, representing the sun. Label it’s center S. Draw a circle to represent the earth, with its center, E, 5 inches to the left of S. Make the radius of this circle about 1/2 inch. Draw a line through the points S and E, and extend it 3 to 4 inches to the left of E. Next, draw a line, tangent to the top of the two circles, and a line, tangent to the bottom of the two circles. These lines will intersect at a point which is located on the first line, if your drawing is reasonably accurate. Label that point, A. Also, label, as U, the point where the upper tangent line intersects the circle representing the sun, and label, as F, the point where the upper tangent line intersects the circle representing the earth. Draw lines connecting U to S and F to E.

The earth’s shadow forms a cone on the side away from the sun. When the moon travels through this cone, there is an eclipse of the moon. The two lines that are tangent to the sun and earth represent the boundary of this shadow cone. The Greeks had estimated that, at the distance where the moon traveled through this shadow cone, the radius of the shadow cone was twice the radius of the moon.

Finally, draw a point M on the line AS approximately 1 1/2 inches to the left of E. This will represent the center of the moon, during an eclipse. At point, M, draw a line, perpendicular to the line AS, so that it intersects the line UA. Label the point where it intersects the line UA, N. The line NM represents the radius of the shadow cone of the earth at the distance where the moon travels during an eclipse of the moon.

From these relationships, Aristarchus constructed 3 similar triangles, AMN, AFE and AUS. He goal is to find the ratio of the earth’s radius relative to the sun and moon. He has at his disposal the following. He knows that the radius of the earth’s shadow on the moon, NM is approximately twice the length of the radius of the moon. If MN is twice the length of the radius of the moon, and the radius of the sun is 18-20 times the radius of the moon, this establishes a ratio of US/MN at between 9 and 10. He has estimated the distance from the sun to the earth, ES, at 18-20 times the distance of the earth to the moon, EM. From knowing these ratios, he is able to use the relations between the three similar triangles to calculate an estimate of the ratio of the earth’s radius to the radius of the sun and moon.

I will not go through Aristarchus’ calculations, which are available in his treatise. They are made even more complicated by the relatively unadvanced state of Greek mathematics. Archimedes estimation of the value for pi, with only the mathematics available at the time, was once described as the equivalent of running the hurdles while wearing weights. Aristarchus calculated that the diameter of the sun was approximately 6.8 times that of the diameter of the earth. He also calculated that the diameter of the moon was around .36 the diameter of the earth. He now had established a value for the diameter of the sun and moon, using the diameter of the earth as his measuring stick.

Aristarchus now had values for the distances to the sun and moon using the earth’s diameter as his measuring stick. Since he had calculated the distance from the earth to the moon, EM, at 26 moon diameters, and the distance from the earth to the sun, ES, at 18 to 20 times EM, he arrived at values of approximately 9.5 earth diameters for EM and 180 earth diameters for ES.

Eratosthenes of Cyrene (ca 276-194 B.C.) was, like Archimedes, from the generation after Aristarchus. He made a remarkably accurate estimate of the diameter of the earth. With Aristarchus’s and, then, Eratosthenes’s discovery, the Greeks had demonstrated that the sun and moon could be measured with the same units that were used to measure the earth.

Aristarchus’s estimate of, especially, the distance to the sun dramatically expanded the size of the universe over previous conceptions. For example, Anaximander, a younger friend of Thales, had estimated the distance to the moon at 19 earth diameters and the distance to the sun at 28 earth diameters. Anaximander had also thought that the planets and fixed stars were closer than the sun and moon.

Although there was a large error in Aristarchus’s measurements, his discoveries were a crucial experiment which demonstrated that the sun, which appeared to sense certainty to be only a relatively small disk in the sky, was dramatically larger than the earth. Aristarchus’s had created a paradox which he could resolve, only by overthrowing the earth centered conception of the universe that was accepted at the time, as we shall see in part II.

How Aristarchus Measured the Universe, Part II

By Robert Trout

In Part I, we saw that Aristarchus discovered how to use the geometrical relationship, that exists between the sun, earth, and moon, when the moon is half full, to calculate the relative distances from the earth to the sun and moon. He then calculated the ratio of the distance to the moon relative to the diameter of the moon, using his knowledge of the relationships contained in an isosceles triangle. Finally, he used the geometrical relationships that exist between the sun, earth, and moon during an eclipse of the moon, to calculate the size of the sun and moon relative to the earth, showing that the sun was dramatically larger than the earth or moon. He was then able to measure the sizes of the sun and moon, and their distances from the earth using the earth’s diameter as his measuring stick.

Aristarchus’ methods of measurement were crude. However, his experiment demonstrated that the sun was not just larger than the earth and the moon, but very much larger. The ratio of the volume of 2 spheres is based on the cube of their diameters. He calculated that the volume of the sun was around 6860 times larger than the volume of the moon. Likewise, he calculated that the volume of the sun was around 315 times larger than the volume of the earth.

Aristarchus did not state his hypothesis that the earth orbits the sun in his treatise, “On the Sizes and Distances of the Sun and Moon.” The treatise, that Archimedes quoted, in which Aristarchus stated this hypothesis, has been lost. Its disappearance is undoubtedly a result of the suppression of true science, that occurred with the imposition of the hoax of Ptolemy. However, we can reconstruct how Aristarchus’s discovery of the relative sizes of the sun, earth and moon represented a paradox, which could only be resolved by developing a new hypothesis of the universe, with the earth orbiting the sun and not the other way around.

To sense certainty, the earth appears to be 99% of what one can see. All around, one can see the earth. The sun and moon appear to be very small discs in the sky with an angular diameter of 1/2 degree, that rise, cross the sky and set. The stars are only tiny specks in the night sky. The earth feels rather solid and unmoved, with only earthquakes a rare exception.

A good example of a cosmology, which was based simply on assertions of sense certainty is the work of Ptolemy. Approximately 400 years after Aristarchus’s work, (and after the Roman Empire had driven the region into a dark age) Ptolemy established a fraudulent cosmology, with the earth again at the center of the universe. Ptolemy stated that “the fact that the earth occupies the middle place in the universe, and that all weights move towards it, is made so patent by the observed phenomena themselves.” His “proof” that the earth was the center of the universe consisted in arguing that all objects “which have weight” fall towards the earth. It is a dramatic demonstration of the dark age into which Europe had descended, that the ideology of Ptolemy, who argued that the nature of the Universe could be determined, by what he “saw” objects doing, within a few hundred feet from the earth, was enforced on Europe as the only acceptable view of cosmology, or the study of the heavens, for 1300 years. Nicholas of Cusa, with his “On Learned Ignorance,” again overthrew the earth centered system of cosmology.

Aristarchus’ experiments has rudely overthrown sense certainty. Aristarchus had constructed, in his mind, geometrical relationships which allowed him to determine the relationship between the sun, moon, and earth which his eyes were unable to see. Aristarchus had demonstrated through reason that the sun, that small disc in the sky which appeared to circle the earth, was far larger than the earth, estimating its size to be around 315 times the size of the earth. The idea that the earth was the center of the universe, while the sun, which orbited it, dwarfed the earth in size, created a paradox. Aristarchus, whose method was based on rejecting the fetters of sense certainty, was able to construct a new hypothesis, which placed the larger body in the center. His new hypothesis also solved a host of other paradoxes, which were inherent in an earth centered conception of the Universe.

Aristarchus had as a precedent, the discovery of Heraclides of Pontus, who had demonstrated, less than 100 years earlier, that two planets orbited the sun. His hypothesis that Mercury and Venus orbited the sun explained their seemingly highly erratic movements with a “uniform and ordered movement” of circular motion. Aristarchus’ hypothesis, that the earth was also revolving around the sun, would resolve the paradox which he had created, and explain, what appeared to be otherwise erratic and arbitrary motions of the outer planets and stars, and the cycles of the earth, with “uniform and ordered movements.”

This was, of course, a tremendous leap at that time. Plutarch described Aristarchus’s hypothesis as, “Only do not, my good fellow, enter an action against me for impiety in the style of Cleanthes, who thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe, this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest, and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis.”

Make a diagram to represent Aristarchus’s hypothesis. (Figure 4) Draw 2 concentric circles. The center of the two circles represents the sun. The inner circle represents the path of the earth’s orbit. (That the orbits or the planets are actually ellipses was, of course, a discovery by Kepler not to be examined here.) According to the view of the universe, prevalent at that time, the stars were located on a celestial sphere. (Many of the leading thinkers of the time rejected the idea that the stars were located on a physical sphere, including undoubtedly Aristarchus.) The outer circle represents the celestial sphere. You can draw little constellation around it, if you like.

Now, let’s look at how Aristarchus’s hypothesis corresponds to Plato’s research project of finding “what are the uniform and ordered movements by the assumption of which the apparent movements of the planets can be accounted for.” His hypothesis of the motion of the earth combined 2 rotations. First, the rotation of earth about its axis would explain the daily cycle of the sun and the nightly rotation of the fixed stars. Second, the yearly orbit of the earth around the sun, explains the apparent shifting of the “fixed stars” from night to night. Each night, the position of the “fixed stars” appears to have rotated slightly less than one degree, making almost one complete rotation in a year. (Although, not exactly a complete rotation. Hipparchus discovered, a century later, that these two cycles were subsumed by another much longer cycle.) It should be clear from studying this diagram, that as the earth travels around its yearly orbit about the sun, the view that an observer, located on the earth, will have of the celestial sphere, will shift from night to night.

The path of the sun across the sky also varies on a yearly cycle, corresponding to the yearly cycle of the seasons. Aristarchus’s hypothesis in which the earth revolves around the sun “in an oblique circle,” or slanting or sloping circle, can, potentially, explain this.

Conceptualize Aristarchus’s hypothesis that the earth’s orbit is an “oblique circle.” The earth is travelling, yearly, around the sun in a circular orbit, while rotating, daily, on its axis. Imagine the northern direction of the earth’s axis as pointing “straight up,” while imagining that the plane of the earth’s orbit around the sun is sloped at an angle relative to the earth’s axis. (Although, globes are constructed to have the north pole pointing upward, there is no basis in the physical universe for this assumption. Since you are assuming that the north pole is pointed upward, this makes the direction of the rotation of the earth on its axis, and its orbit around the sun both counterclockwise.) During the part of the earth’s orbit around the sun, that you are imagining to be the “higher part,” the southern hemisphere is more directly exposed to the sun. During the part of the earth’s orbit, that you are imagining to be the “lower part,” the northern hemisphere is more directly exposed to the sun. This explains the yearly variation in the path, that the sun makes across the sky each day, as seen by an observer on the earth, and the resulting variations in the seasons.

Eratosthenes also studied the question of the tilt of the earth’s axis relative to the plane of the earth’s orbit around the sun, approximately 25 to 50 years later. Eratosthenes measured the angle between the earth’s axis and the plane of its orbit around the sun with a remarkable accuracy, being in error by only around 0.1 degree from the currently accepted value.

Finally, Heraclides of Pontus had demonstrated that the apparently erratic motion of the inner planets, Mercury and Venus could be explained as circular motion around the sun. Aristarchus’s hypothesis would allow one to comprehend that the apparently erratic motion of the outer planets also corresponded closely to rotation around the sun, as seen from an earth, which is also rotating around the sun.

As a result of Aristarchus’s hypothesis, numerous movements in the universe, including a number of the most important conditions regarding human existence, such as the ordering of the seasons, which according to sense certainty, just are that way, could be determined as the consequence of “the uniform and ordered movements by the assumption of which the apparent movements of the planets can be accounted for.”

In closing, let’s examine the last part of Archimedes statement that Aristarchus hypothesized, “that the sphere of the fixed stars, situated about the center of the sun is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface.” To restate this, Archimedes attributed to Aristarchus the idea that the ratio (circle of the earth’s orbit)/(distance from the earth to the fixed stars) equals the ratio (the center of the sphere)/(its surface). Archimedes criticized Aristarchus for saying this stating, “Now it is obvious that this is impossible; for since the centre of a sphere has no magnitude, it cannot be conceived to bear any ratio to the surface of the sphere.” However, Aristarchus was probably developing a metaphor to illustrate that the distance to the stars was such a large distance, that the circle of the earth’s orbit around the sun would appear to be point in comparison.

Since this work, which Archimedes quoted, has been lost, we must reconstruct how he could have estimated the distance to the fixed stars. The geometrical methods, which he used to measure the distance to the moon and sun, would work well to solve this problem.

Aristarchus has now established the hypothesis that the earth is orbiting the sun following a circular orbit, with a diameter estimated at 180 times the earth’s diameter. Aristarchus had already expanded the size of the universe tremendously over the prevailing view.

Go back to your drawing of the two concentric circles representing the earth’s orbit and the celestial sphere. As the earth travels around it’s orbit during the year, the earth will be closer to the part of the celestial sphere which is directly overhead at midnight. A star will be closer to the earth when it is directly overhead at midnight, and approximately the diameter of the earth’s orbit around the sun further away from the earth when it is very near the sun in the sky.

The angle between two adjacent stars should be larger when the earth is close to them. Aristarchus, who was very adept at this type of measurement, could have calculated the size of the celestial sphere, based on measuring the change in the size of the angle between two stars, when the earth is near them, versus when the earth is on the opposite side of its orbit from them.

Draw two dots, representing stars, on the celestial sphere, maybe 10 degrees apart. Pick a point on the circle of the earth’s orbit which is near these two dots, and draw 2 lines from this point to the two stars. The angle between these two lines represents the angle, that an observer on the earth, would see between the two stars. Next pick a point on the opposite side of the earth’s orbit, and draw 2 lines from this point to the two stars. (Don’t pick a point directly across, or the sun will block the view of the stars.) The angle between those two lines will be less than the first angle that you constructed, at the the other point on the earth’s orbit, which was nearer to the two stars.

The size of the celestial sphere’s diameter relative to the size of the diameter of the earth’s orbit, can be estimated by comparing the difference in the size of these two angles. If the celestial sphere is only a little larger than the circle of the earth’s orbit, the difference between the 2 angles will be large. If the celestial sphere is much larger than the circle of the earth’s orbit, then the 2 angles will be closer to the same size.

However, Aristarchus must have found that the angle, between two adjacent stars, did not appear to change, regardless of whether the earth was near to or far from those stars. He could only conclude that the distance to the stars must be so large, that the distance to them could not be measured using this method. There was probably no other method that could have measured the distance to the stars, with the means available at that time. This would have led him try to communicate his discovery of the immense size of the universe, by describing the distance to the stars relative to the diameter of the earth’s orbit, by comparing it to a “ratio between a sphere and a point.” Aristarchus had expanded man’s conception of the Universe beyond the wildest imagination of the majority of men who still had their minds stuck down in the mud of sense certainty!

On Archytus

By Bob Robinson

“If I were at the outside, say at the heavens of the fixed stars, could I stretch my hand or my stick outward or not? To suppose that I could not is absurd; and if I can stretch it out, that which is outside must be either body or space (it makes no difference which it is, as we shall see). We may then get to the outside of that again, and so on, asking at our arrival at each new limit the same question; and if there is always a new place to which the stick may be held out, this clearly involves extension without limit. If now what so extends is body, the proposition is proved; but even if it is space, then, since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space without limit.” Archytus, circa 400-365 B.C.

I have invented a pedagogical -that is, teachable- model of the ancient Greek Archytus’ geometric solution to the classical problem of finding two mean proportionals between two extreme magnitudes, often also called the problem of “the duplication or doubling of the cube”. (If unfamiliar with the term “mean proportionals” think, for first approximation, in terms of numbers. What are the two mean proportionals between 1 and 8, 1 and 27, and 1 and 125?) Archytus’ solution requires no numbers; instead the smaller of the two extreme magnitudes is any chord of a circle, while the larger of the two extremes is the same circle’s diameter. His solution is three dimensional, involving the intersection in a point of three “solid” surfaces: a cylinder, a torus (doughnut shape), and a cone. But it is not just three dimensional, and to call it “three dimensional” is, as we shall see, somewhat misleading for a truthful understanding of the problem, its solution, and Plato’s friend Archytus himself. To situate Archytus’ solution to the problem of the doubling of the cube, consider the following quote of Eratosthenes (circa 200 B.C.), who developed his own solution to that problem, by Theon of Smyrna. “Eratosthenes in his work entitled Platonicus relates that, when the god proclaimed to the Delians by the oracle that, if they would get rid of a plague, they should construct an altar double of the existing one, their craftsman fell into great perplexity in their efforts to discover how a solid could be made double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant , not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt for geometry.” Archytus’ intention was not merely to move from the two dimensional realm (doubling the square) to the three dimensional realm (doubling the cube), but was to move outside the realm of three abstract dimensions into the realm of physical geometry. Three abstract dimensions appear in Archytus’ construction as a sphere. The determination of two mean proportionals lies outside that sphere, on the surface of a torus and a cylinder which surround the sphere in Archytus’ construction. One must, in effect, poke a stick through the three dimensional sphere of Archytus’ construction, to see where the stick intersects the cylinder and torus. Compare this with Archytus’ astronomical notion of “stretching my hand or my stick outwards” at “the heavens of the fixed stars”. Consider the analogous case, of how it is necessary to move into three dimensions, to double the square. The diagonal of a square, which forms the side of a square with double the area of the original one, is formed by folding the original square in half, that is, by rotation outside the plane of the original square!

The Construction

Situate a transparent sphere (such as a Lenart Sphere) in a cardboard box, such that the sphere sits snugly in a hole in the top of the box and the equator of the sphere is level with the top of the box. The hole in the top of the box should be put tangent to one side (edge) of the box. Trace with a spherical compass (also included with the Lenart Sphere) a number of concentric circles of latitude on the sphere which, unlike the circles of latitude on the earth, are all made perpendicular to the equator of the sphere. That is, the compass forming the circles is pivoted on a point on the equator of the transparent sphere. Be sure to use erasable ink of a certain color, say red. Next, with the point of tangency of the hole in the top of the box and the side of the box as center, trace with an ordinary plane compass a circle on the top of the box, such that the circle has twice the diameter of the hole in the top of the box. The diameter of the sphere and hole form the radius of the new circle, so that the new circle will be just tangent to the hole at a point directly opposite where the hole is tangent to the side of the box. This new circle, which we can only represent a portion of in our model, forms the outer perimeter of the torus in Archytus’ construction. The “hole” in the middle of the torus is of null diameter, and is represented by the center around which we pivoted the new circle, that is, the point of tangency of the hole in the top of the box and and the side (edge) of the box. Obtain from an arts and crafts store some wooden hoops with about the same diameter as the sphere, and some clear acetate. Cut some of the wooden hoops in a continuously growing array of arc lengths, up to and including a couple that are semicircles. Wrap the acetate around one of the semicircular hoops, such that it forms a cylinder, and with the sphere put into its hole in the top of the box, snuggle the acetate half cylinder in between the sphere and the hole. Next, position the smaller hoop lengths in order of ascending length, starting from near the point of tangency of the larger circle traced on the top of the box and the hole in the top of the box. The base of each hoop length should be on the larger circle, and its top should rest on the acetate wall of the cylinder. The effect should be that, were the hoops to continue past the barrier of the acetate cylinder, they would all converge on the point of tangency of the hole and the side of the box, or, what is the same thing, the center of the circle on the circumference of which their bases rest. Silicone should be used to secure their positions on the top of the box and the acetate wall of the cylinder. When the construction is completed, the wooden hoops should approximate a partial (quarter) torus, which if completed would wrap around the sphere and cylinder. Trace with a dry erase marker the line of intersection, so formed, between the cylinder and the torus. When the sphere is placed in the hole in the box, maneuver it so that the center of the concentric latitude circles traced on its surface coincides with point of tangency of the hole in the top of the box and the side of the box. Obtain a laser pointer. Shine it through the sphere, from the point where the hole in the top of the box is tangent to the side of the box, so that it hits the side of the acetate cylinder. Play around with it a bit. Trace along the curved line of intersection of the torus and cylinder, so that the laser crosses in succession all the lines of latitude traced on the sphere. Next, perform the inverse operation, by tracing along each circle of latitude to see where it intersects the line of intersection of the torus and cylinder traced on the clear acetate cylinder. As you do so, note how the laser’s motion along each circle of latitude forms a distinct cone. The integral of the bases of all such possible cones is nothing but the sphere itself! That is, the sphere is the only truthful representation of all possible cones formed by rotating all possible chords of a circle emanating from a single point on that circle’s circumference.

Archytus’ Creation of Two Mean Proportionals

Archytus wishes to find the two mean proportionals between any chord of the circle, contained in the great circle of the equator of the sphere in our construction, and the diameter of that same circle, which is also the diameter of the sphere, the cylinder, and the “tube” of the torus in Archytus construction. Place the chord with its origin at the point of tangency of the hole and the side of the box. (From now on we will simply call this point, which by our construction is on the equator of the sphere and also is the center “hole of null dimension” of the torus, and is the point from which we will direct the laser pointer, the origin.) For any such chord, there is implicitly a circle of latitude on the sphere, which is everywhere equidistant from the origin, forming, as we have indicated, a distinct cone. Next, trace with the laser along one such circle of latitude on the sphere, until the laser beam crosses the line of intersection of the torus and the cylinder. There will now be three points of light associated with the laser beam, one at the origin, one on the surface of the sphere, and one on the curved line of intersection of the torus and the cylinder. Looking down from directly on top of the model, imagine a plane including the three aforesaid points of light, and perpendicular to the top of the box slicing through the sphere, the cylinder and the torus. That plane, being perpendicular to both the top of the box and to the equator of the sphere, will form a circle cut exactly in half by the equator of the sphere, just as the circles of latitude were cut in half by the plane of the equator. This cut is most efficiently represented by a semicircle drawn on a clear overlay (provided with the Lenart Sphere), and placed on the sphere so that it coincides with the origin and the point on the sphere through which the laser shines. It should be of different color than the (red) concentric circles of latitiude on the sphere, say green. Now draw a diagram of a cross section of the portion of Archytus’ model cut by the perpendicular plane. The diagram will not be precise, but simply representative of certain geometric relations that exist in the three dimensional model. Construct two tangent circles, one inside the other. At the point of tangency, mark O. The smaller circle represents the green circle in our three dimensional model, while the larger one represents a cross section of the torus. Through O, draw a straight line that forms a diameter OB of the smaller circle, and OD of the larger circle. This equatorial line represents a cross section of the plane forming the top of the cardboard box in our three dimensional model. In the diagram, through B, draw a line perpendicular to OB, that forms a tangent to the smaller (green) circle, and intersects the larger circle at C. Draw OC and CD. Where OC intersects the smaller circle, mark A. Triangle OAB, being inscribed in the smaller semicircle of the diagram, will have a right angle at A. In triangle OBC, diameter OB and tangent BC will be perpendicular to each other, so there will be a right angle at B. Triangle OCD, being inscribed in the larger semicircle of the diagram, will have a right angle at C. All three triangles share angle DOC (identical to angle BOA). So, by similar right triangles OAB, OBC, and OCD, the continued proportions OA/OB=OB/OC=OC/OD are produced. OA represents the distance from O to the surface of the sphere at A, and is equivalent in length to the original chord forming the lesser extremity of Archytus’ demonstration. OD of the diagram represents the cross sectional diameter of the torus “tube” in the three dimensional model. By the way the model was constructed, the diameter of the tube of the torus is the same as the diameter of the sphere. And diameter of the sphere is the same as the diameter of the equatorial great circle of the sphere, which formed the larger extreme in Archytus’ demonstration. Straight line BC, perpendicular to OB in the diagram, is a cross section of the cylinder, which is perpendicular to the equatorial plane in Archytus’ three dimensional model. Thus, by reference to the schematic diagram, it is easily seen that Archytus’ model really does create two mean proprtionals OB and OC between two extremes OA and OD. Well, who needs the three dimensional model if a two dimensional drawing gives us the required solution? But the diagram is only schematic, and does not give us true values for OB and OC. Only the three dimensional model does. (Indeed, even Eratosthenes’ later mechanical method for finding two mean proportionals, which is two dimensional, only minimizes the problems of approximation inherent in any two dimensional model that uses only straight lines.) The more interesting question is, is Archytus’ model simply three dimensional, or of a higher power? Consider the fact, that in Archytus demonstration, we must go outside, not only the circle, but the three dimensional sphere, to find the two mean proportionals OB and OC. On the surface of the sphere itself, this is reflected in the different orientations of the red and the green circles.

Archimedes and The Student

Bruce Director

To Archimedes came a youth desirous of knowledge.

“Tutor me,” spake he to him, “in the most godly of arts, 
Which such glorious fruit to the land of our father hath yielded 
And the walls of the town from the Sambuca preserv’d!” 
“Godly nam’st thou the art?” She is’t, “responded the wise one; 
“But she was that, my dear son, ere she the state ever serv’d. 
Wouldst thou but fruits from her, there too can the mortal engender; 
What the Goddess doth woo, seek no the woman in her.”

This poem by Friedrich Schiller (translated here by Will Wertz) was cited by Carl F. Gauss in his famous introductory lecture on astronomy. In attacking the pragmatic thinking and the “indifference and insensibility to the great and that which honors humanity,” Gauss told the audience of faculty and students at Goettingen University, the real life implications of this way of thinking. “Unfortunately one cannot conceal the fact that one finds such a mode of thinking very prevalent in our age, and it is probably quite certain that this attitude is very closely connected with the ill fortune which of late has struck so many states. Understand me correctly, I am not speaking of the very frequent lack of feeling for the sciences themselves, but of the source from which this flows, of the tendency everywhere to ask first about the advantage and to relate everything to physical well-being, of the indifference to great ideas, of the aversion to effort due merely to pure enthusiasm for the thing in itself. I mean that such characteristics if they are predominating, can have given a strong decision in the catastrophes which we have experienced.”

He then harkened back to one of his Greek predecessors, “The great happy minds who have created and extended astronomy as well as the other more beautiful parts of mathematics, were certainly not fired by the prospect of future utility; they sought truth for its own sake and found in the success of their efforts alone their reward and their good fortune. I cannot avoid reminding you here of Archimedes, who was admired most by his contemporaries only on account of his ingenious machines, on account of their apparent magic effects; he however valued all this so slightly in comparison with his glorious discoveries in the field of pure mathematics, which at that time mostly had no visible utility in themselves according to the usual sense of the word, that he wrote nothing about the former for posterity, while he lovingly developed the latter in his immortal works. You certainly all know the beautiful poem by Schiller, {Archimedes and the Student}….

Is Gauss asking you to choose between “pure mathematics” and pragmatism? If Gauss’ words seem a bit politically incorrect to you, you have succeeded in identifying a pragmatic demon in your own mind.

Take a look at Archimedes correspondence with Eratosthenes, entitled, “The Method of Treating Mechanical Problems.” In that work, Archimedes poses the problem of determining the relationship between incommensurable solids, such as a sphere, cone and cylinder, or a spheroid and a cylinder. Gauss would later identify that these solids are characterized by different curvatures. The cone and cylinder are generated by surfaces of zero curvature; the sphere by a surface of constant curvature; the spheroid by non-constant curvature.

The problem Archimedes posed to Eratosthenes was: “Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer [of mathematical inquiry], I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge….”

An example of this method is the determination that the volume of a sphere is four times the cone with base equal to a great circle of the sphere and height equal to its radius; and the volume of a cylinder with base equal to the great circle of the sphere and height equal to the diameter is 1.5 times the sphere.

(The second part of this proposition, was depicted on Archimedes tombstone.)

The difficulty in proving this proposition, lay in the differing curvatures of the volumes measured. To overcome this obstacle, Archimedes investigated the interaction of these volumes in a physical process, the pull of the Earth’s gravity.

To construct the experiment, think of the sphere cone and cylinder, all nested together in the following way. Think of a sphere, then, think of a cone whose base is formed by the equator of the sphere, and whose apex is at the north pole. Now think of that whole thing embedded in a cylinder whose bases touch the north and south pole of the sphere. The sphere will be tangent to the cylinder at its equator.

(You can draw a cross section of this arrangement, by drawing a circle with two perpendicular diameters. This represents a cross section of the sphere. Label the intersections of these diameters counter-clockwise A,B,C,D. Then draw a triangle with vertices A,B,D. This represents a cross section of the cone. Then draw a square around the circle such that A,B,C,D intersect the midpoints of the sides of the square. This represents the cross-section of the cylinder. Label the corners of the square, V,X,W,Y clock-wise from the upper left. And label the center of the sphere K.)

Now, extend the sides of the cone (A-B and A-D in the cross section drawing) and the bottom side of the square (W-Y in the drawing). Label the intersections, E and F. Now we can think of an enlarged cone, A, E, F. Also construct, an enlarged cylinder, with base E-F and height, A-C.

Then, Archimedes imagines that this entire complex of solids is resting on a balance, whose bar is twice the length of A-C, with A at the midpoint. (To depict this in our cross-section drawing, extend double line A-C and label the new endpoint H.)

Finally, draw a line perpendicular to C-A-H and parallel to B-D. This line will intersect the cross sections of all the figures previously imagined. Archimedes, using the Pythagorean theorem, and the principles of Euclidean geometry, determines the proportional relationships existing among these cross-sections.

To determine the relationship of the volumes of the sphere, cone and cylinder, Archimedes investigates under what conditions the various cross-sections of the cone, cylinder and sphere are balanced. Using the proportions he just calculated, he is then able to determine that the volume of the sphere is 4 times the cone and the volume of the cylinder is 1.5 times the sphere.

(The actual calculation is not difficult, but it would be too cumbersome to describe in this format. If you don’t have a copy of Archimedes piece, send an e-mail to BMD, and I will supply a copy of this proposition.)

Is Archimedes procedure a physical demonstration, or a mathematical one. Isn’t he investigating geometrical objects, which have no physical existence, with respect to a physical process, the pull of Earth’s gravity? As he said to Eratosthenes, this procedure makes the proposition clear, but he still requires a geometrical proof, or, as Kepler would later state with respect to the divisions of the circle, “knowability”?

Forty years after Gauss identified the consequences of pragmatism on the political condition of Europe, then former U.S. President John Quincy Adams gave a speech in Cincinnati, Ohio on the occasion of the laying of the cornerstone of the U.S.’s first astronomical observatory. After an extensive discussion of the history of astronomy, Adams ended his speech:

“But when our fathers abjured the name of Britons, and `assumed among the powers of the earth, the separate and equal station, to which the laws of Nature, and of Natures, God entitles them,’ they tacitly contracted the engagement for themselves, and above all, for their posterity, to contribute, in their corporate and national capacity, their full share; aye, and more than their full share, of the virtues, that elevate, and of the graces that adorn the character of civilized man….

“… We have been sensible of our obligation to maintain the character of a civilized, intellectual, and spirited nation. We have been, perhaps, over boastful of our freedom, and over sensitive to the censure of our neighbors. The arts and sciences, which we have pursued with most intense interest, and persevering energy, have been those most adapted to our own condition. We have explored the seas, and fathomed the depths of the ocean, and we have fertilized the face of the land. We–you- -you, have converted the wilderness into a garden, and opened a paradise upon the wild. But have not the labors of our hands, and the aspirations of our hearts, been so absorbed in toils upon this terraqueous globe, as to overlook its indissoluble connection even physical, with the firmament above? Have we been of that family of the wise man, who, when asked where his country lies, points like Anaxagoras, with his finger to the heavens.

“Suffer me to leave these questions unanswered. For, however chargeable we may have been, with inattention or indifference, to the science of Astronomy, heretofore — you, fellow citizens, of Cincinnati — you, members of the Astronomical Society, of this spontaneous city of the West, will wipe that reproach upon us, away….”

Gauss vs. Empiricism

by Jonathan Tennenbaum

Lyn has emphasized, how Carl Friedrich Gauss’ 1799 dissertation on the so-called “Fundamental Theorem of Algebra”, constituted a devastating refutation of the leading scientific authorities of his day, including Jean-Louis Lagrange and Leonard Euler.

Gauss first points out fundamental flaws in purported proofs of the “Fundamental Theorem”, put forward by D’Alembert, Euler and Lagrange in succession, showing that they were based on arbitrary assumptions and fell far short of actually establishing the proposition, they claimed to demonstrate. Then Gauss presents his own, rigorous proof, based on different principles.

So what’s the big deal? Certainly, Gauss’s ruthless exposure of gaping “holes” in D’Alembert’s, Lagrange’s, and Euler’s proofs was a scandal in and of itself, suggesting — as Gauss himself clearly intimates — a shocking degree of conceptual sloppiness on the part of men who were considered to be standards of scientific rigor. Also, Gauss’ concise proof, going to the heart of the matter in a few pages, contrasted with the voluminous and prolix treatises of Lagrange and Euler, that Gauss tore apart in the first half of his dissertation.

But the real issue is not one mathematics per se, but of physical principle. A PRELIMINARY access to that issue opens up, when we turn from the particularities of the Fundamental Theorem itself (although they are important and indispensible) and first ask the question: WHY did D’Alembert, Lagrange and Euler FAIL? What was wrong with their THINKING?

Someone might say: well, the 20-year-old Gauss was a towering genius. But in what did that genius really consist? Was Gauss’ proof itself so fantastically clever, complicated and ingenious? No, not at all! It is quite simple, natural and direct, once one masters the basic principles involved. From a purely {formal-technical} standpoint, there is nothing in Gauss’ proof, which was not well within the range of what Lagrange, Euler and many others could easily have done.

So, what prevented them from doing so? Ah! Here we come to the exact same “mechanism”, which causes the collapse of seemingly all-powerful empires — empires possessing vast, apparently overwhelming material and intellectual resources, but which collapsed nonetheless. Why is it, that the ruling elites of such empires — and their army of sometimes highly talented, skillful and experienced advisors, analysts and other “lackeys”, selected from the “cream of the cream” of the population — why did they invariably FAIL, at crucial junctures of history, to take actions that might have prevented, or at least greatly delayed, the collapse of their systems? Why is it, that we invariably discover, as the crucial element that finally dooms such empires to destruction, an obsessive insistence on expending every last ounce of resources, skill and cunning, in the attempt to make an intrinsically FAILED system “work” — even against the laws of the Universe –, rather than to accept a CHANGE in its the underlying, flawed axioms of that system.

So, we have Lagrange and Euler, both highly skillful, knowledgeable and experienced mathematicians, much more so than most leading academic authorities today, but who FAILED most decisively, where Gauss SUCCEEDED. The case of Euler is particularly instructive, because he was, to all accounts, extremely industrious and in some ways quite shrewd and perceptive, as well as a virtuouso master of algebraic methods. Euler also made some not-insignificant experimental discoveries, in number theory and other fields, as Gauss himself pointed out. But, at the same time, Euler was a crude phillistine in philosophical terms, and above all a fanatical EMPIRICIST, in the precise sense that Lyndon LaRouche has identified, with great precision, in a recent paper. Lyn writes, in part:

“Empiricism has the form of a synthesis of three, ostensibly mutually exclusive, categorical elements, as follows:

“1. First, the empiricist assumes that no experimentally verifiable knowledge exists outside the bounds of simple sense-certainty.

“2. Secondly, therefore, every cause-effect relationship which can not be located explicitly in an sense-observed agency, is related to a domain of such forms of attributed bias in statistical behavior of observable events, or to some anonymous agency to which neither sense-certainty nor cognitive reason provides access.

“3. Thirdly, the second element leaves available a niche for the creating the illusion of the existence of purely magical spiritual powers, operating entirely outside the reach of access by sense-certainty, but able to make arbitrary interventions, even capriciously, into the domain of sense-certainty.”

Let us compare this characterization by Lyn, with the clinical evidence Leonard Euler supplies for his own case. Take, for this purpose, the same paper, that is the immediate target of criticism in Gauss’ dissertation of 1799. This is Euler’s “Recherches sur les racines imaginaires des equations”, published in the “Memoires de l’Academie des sciences de Berlin” in the year 1749.

Euler begins by writing down the general form of an algebraic equation involving a single unknown, “X”. By “algebraic equation” Euler meant any formula, built up from the “unknown” X and specific integers, fractions or so-called irrational numbers, by means of the algebraic operations of addition, subtraction, multiplication and division, and then set equal to zero. So, for example: 2XX – 5X + 10 = 0, or XXX + 3 XX – 5X + 21 = 0 and so forth. (XX means X times X or “X squared”, XXX means “X cubed”, and so forth). (Apparently more complicated cases can occur, as for example (XXXX – 4)/X = 0 or others, that involve divisions. It turns out, that divisions can be eliminated by manipulations of the algebraic equation. But these technical aspects are not important for the point we want to make here. )

The problem (as Euler understands it) is to find a specific number or magnitude, which, when put in place of “X” in the formula, yields the value zero when the additions, subtractions, multiplications and divisions are all carried out. Such values became known as “solutions” or “roots of the equation”. Thus, the equation XX – 4 = 0 has two roots, namely 2 and -2. Referring implicitly to the work of Cardan and others in the 16th century, Euler notes (my emphasis):

“It happens (in general) that not all the roots are REAL quantities, and that some of them, or perhaps all, are IMAGINARY quantities.”

Thus, for example, the equation XX + 1 = 0, appears to have no solutions or roots among the magnitudes, that Euler regarded as “real”, namely positive or negative quantities corresponding to positions to the right or left of zero on the “number line” of standard textbook mathematics. For, whether X is positive or negative, XX (X squared) is always positive, so XX + 1 will always be at least 1, or larger, for any X on the “number line”. Hence, a solution or root of XX + 1 = 0, if one could speak of such a thing at all, could only be an “imaginary” entity, as when a formal algebraicist, merely playing with symbols, might reason:

XX + 1 = 0 implies

XX = -1, which implies

X = “the square root of -1”.

But such a value of “X” could have no real existence, because it corresponds to no point on the “number line”. Euler writes:

“One calls a magnitude IMAGINARY, when it is neither greater than zero, nor less than zero, nor equal to zero. This would therefore be something IMPOSSIBLE, as for example sqrt(-1), or in general a + b sqrt(-1), because such a quantity is neither positive, nor negative, nor zero.”

Thus, for Euler a magnitude such as sqrt(-1), which is neither more, nor less than, nor equal to zero, lies outside the domain of sense-certainty, and is therefore “impossible” or “imaginary”. On the other hand, a few paragraphs further down in his paper, Euler insists, that mathematicians must study and utilize these very same “impossible” quantities! Euler writes:

“Although it seems that the knowledge of the imaginary roots of an equation would be devoid of any use, since they furnish no (real) solutions to any problem, nevertheless it is very important in analysis to become familiar with the imaginary quantities. Because we thereby not only obtain a more perfect knowledge of the nature of equations; but the analysis of the infinite can enjoy considerable benefits.”

Euler goes on to remark, that various methods for the calculation of integrals and other mathematical problems, require the use of “imaginary quantities”, even though he himself has denounced those same quantities as “impossible”! Here Euler displays the second and third characteristics of empiricism, detailed by Lyn above, and which are curiously inconsistent with the first point: A mathematician must learn to communicate with GHOSTS, “imaginary quantities”, which are unreal and yet lend the mathematician MAGICAL POWERS to manipulate the visible universe!

Here Euler reveals exactly his empiricist problem. A true physical principle, which can only be generated as an IDEA in a single, sovereign human mind, cannot be known, he thinks. Instead, there are only sense perceptions, on the one hand, and “other-worldly” entities with magical powers, on the other. Above all, formalism itself — like a cult ritual — is supposed to convey magical powers, as many modern physicists for example, ascribe awesome powers to the so-called “quantum mechanical formalism” today. A revealing example is Euler’s contorted attempt to formally justify the “rules” for multiplication with simple NEGATIVE NUMBERS, in his algebra text from 1770:

“It remains still to solve the case where – is multiplied by – or, for example – a by – b. It is obvious initially that as for the letters, the product will be ab; but it is dubious still if it is the sign + or well the sign – that it is necessary to put in front of the product; all that one knows, it is that it will be one or the other of these signs. However I say that it cannot be the sign -; because – a by + b gives – ab and – a by – b cannot produce the same result that – a by + b; but it must result the opposite from it, i.e. + ab; consequently we have this rule: + multiplied by + made +, just as – multiplied by -.”

This, explicitly cultish, gobble-dee-gook formalism, became a paradigm for the teaching of mathematics, leading to generation after generation of crippled minds.

Only a bit less openly occult is Jean-Louis Lagrange, whose famous 1788 “Méchanique analytique” became a prototype for modern “systems analysis”. In his preface Lagrange writes:

“You will find no diagrams in this work. The methods I present, require neither constructions, nor geometrical or mechanical reasoning, but only algebraic operations, proceeding in regular and uniform manner. Those who love analysis will note with pleasure, how Mechanics thereby becomes one of its new branches, and will be grateful to me for having extended the domain of analysis in this way. “

Lagrange thus pretended, to MAKE PHYSICS INTO A BRANCH OF FORMAL MATHEMATICS — exactly the opposite of what Leibniz stood for, and what Nicolaus of Cusa and Plato before him had stood for. Common to Lagrange and Euler, is the demand, that no physical principles should be permitted to intrude upon the domain of mathematics! Yet, it is easy to demonstrate, in the typical contemporary physicist and physics student, a fanatical quality of belief in the supposed magical powers of the so-called “Lagrangian”.

As Gauss points out, that while pretending to “correct” the defects of Euler’s purported “proof” of the fundamental theorem, Lagrange maintains Euler’s implicit, though baseless assumption, that all roots of an algebraic equation, whether “real” or “imaginary”, must be capable of a formal algebraic representation, in terms of addition, subtraction, multiplication, division and the so-called extraction of roots (square roots, cube roots, fourth and fifth roots and so forth). But exactly the impossibility of such a universal formula for the roots of an equation, was key to Gauss’ s understanding of the significance of the complex domain.

Gauss himself repeatedly refers to the fundamental difference in method, between his approach and the stubborn empiricism of Euler, in his early writings in particular.

So, he writes in the introduction to his Disquisitiones arithmeticae, “When I, in the beginning of the year 1795 first took up this sort of (number theoretic) investigations, I knew nothing about the work of the moderns (Euler, Legendre et al) in this field … While I was engaged in other work, I chanced upon a remarkable arithmetic truth — when I am not mistaken, it is the theorem in D.A. Article 108 (**) –; and since I found it not only very beautiful in and of itself, but suspected that it must be connected with other remarkable properties, I devoted my entire energies to comprehending the PRINCIPLE upon which those properties are based, and obtaining a rigorous proof. When at last I succeeded in my wishes, the beauty of these investigations had taken such a hold over me, that I could not tear myself away from them; so it came about, that, as each thing led to another in turn, I had accomplished most of what is contained in the first four sections of this book, before I had seen anything of the similar work of other Geometers.”

In a letter to his former teacher at the Carolineum, E.A.W. Zimmermann in October 1795, soon after he had entered Göttingen University, Gauss wrote:

“I have seen the library and hope to derive from it a considerable contribution to a happy existence in Göttingen. I already have several volumes of the Commentaries to the Petersburg Academy (by Euler) at home, and I have looked through many more. I cannot deny, that it is very unpleasant for me, to see that the largest part of my discoveries in indeterminate Analysis were only discovered for the second time. What comforts me is this: All the discoveries of Euler, that I have found so far, I had also made by myself, plus some more, too. I have found a more general, and, I believe, more natural standpoint, and an immeasurable field (for further discoveries) lies in front of me; Euler made his discoveries over a period of many years, and only after many successive tentaminibus (attempts). “

Euler’s attitude toward the so-called “imaginary” or “impossible” numbers, reflected exactly his own crippling intellectual problem: for him, IDEAS — physical principles grasped by the mind — couldn’t really exist, as objects of concious deliberation. So, he was reduced to sniffing around, with his nose to the ground, for some sort of magic formulae by which he might manipulate the world. By concentrating on issues of PRINCIPLE, Gauss had overtaken a lifetime of trial-and-error-style number-theoretic investigations by Euler, within less than a year.

FOOTNOTE

(**) It is worth giving here, at least briefly, some idea about the subject of Article 108 of Disquisitions Arithmeticae, as this is closely connected with the genesis of Gauss’ Fundamental Theorem of Algebra.

Gauss calls a given whole number A a “quadratic residue” relative to a prime number p, if there exists a “square number” — i.e. the square of a whole number: 1, 4, 9, 16, 25, 36, 49 etc — such that p divides the difference between that square and N. In Gauss’ language of congruences, the latter condition is expressed by saying, that “N is congruent to some square modulo p”. Of course, the square numbers themselves always fullfill that condition, but the more interesting case is when N is not the square of a whole number. For example, we can easily see that 2 is a quadratic residue relative to the prime number 7 — 7 divides the difference between 9 (a square!) and 2. Also 2 is a quadratic residue of 17 (the square 36 is congruent to 2 modulo 17), and of a whole series of other prime numbers. On the other hand, it turns out that 2 is NOT a quadratic residue relative to 5, nor is 2 a quadratic residue relative to 11, 13 and a whole series of other primes. Thus, the prime numbers fall into two series or species: those for which 2 is congruent to some square, and those for which 2 is not congruent to any square. Taking as N instead of 2 any other non-square number, we get a different division of the primes into species. (The harmonic interrelations of those species are the subject of an extraordinary discovery, which is the centerpiece of the Disquisitiones Arithmeticae — namely the so-called “Law of Quadratic Reciprocity”).

Another way to look at this, implied by Gauss, is to consider the realm of congruences relative to a given prime number, as defining a geometrical domain of a special type. In that domain, congruent numbers are considered to have the same “shape” and to be otherwise equivalent. So, for example, instead of saying, for example, that “2 is congruent to a square modulo 7”, we would actually regard 2 itself as a “square number” in the congruence domain defined by 7. Within that domain, for example, 2 and 9 ( = 3 squared) would be considered equivalent and indistinguishable. Thus, 2 will be “square-shaped” or “of the second power” in some prime domains, but not in others. The prime number p (the “modulus”) are thus not simple numbers, but topological characteristics.

Now, what about the number -1? This special case — which turns out to be of crucial importance in all of Gauss’ “higher arithmetic” — is the subject of the cited Article 108. As we noted above, the criterion for -1 to be a quadratic residue relative to a given prime p, is that p divides the difference between some square number and -1. But subtracting -1 from a number, means the same thing as adding 1 to it; so the condition is equivalent to saying, that there is a square number, such that when 1 is added to it, the result is divisible by p. Examples are easy to find. For example, 4 + 1 is divisible by 5, so -1 is a quadratic residue of 5. Similarly, -1 is a quadratic residue modulo 17. Also, 25 + 1 = 26 is divisible by 13, so -1 is a quadratic residue modulo 13, too, and so on for a certain series of prime numbers. But it turns out, for example, that -1 is NOT a quadratic residue relative to 7, nor relative to 23, nor 29, nor for a whole other series of prime numbers.

As a bit of reflection shows, one can characterize the two species of prime numbers also as follows: Take the series of squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 etc. and add one to each of them: 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122 etc. The primes of the first species — those for which -1 is a quadratic residue — are the ones which divide at least one of the numbers in the latter series. Those primes which do not divide any of the numbers 2, 5, 17, 26, 37, 50 etc, form the second species. (How many terms of the series do you need to check, to tell whether a given prime divides one of them?).

For the first species of prime numbers, “-1” is a square number in the corresponding congruence domain, i.e. sqrt(-1) corresponds to a specific value in the domain (for example, 5 is equivalent to sqrt (-1) in the domain of congruences modulo 13), while for the second species, the introduction of sqrt(-1) requires going OUTSIDE the domain.

Two years before his discovery of his first proof of the “Fundamental Theorem of Algebra” , Gauss uncovered the harmonic law governing the distribution of the primes into these two species: the first, as it turns out, are the primes which leave a remainder of one when divided by 4 (such as 5, 13, 17, 29…) and the second species are the primes leaving a remainder of 3 when divided by 4 (such as 3, 7, 1, 19, 23 …).

Most significant is the circumstance, that BOUNDING PRINCIPLE involved is physical in nature, and has nothing to do with any formula or formal proceedure. So. for example, in the case of primes of the first class, the existence of a value for sqrt(-1), is “forced”, as a singularity, by the overall geometry of the domain; while the specific value of sqrt(-1) assumes in that domain, must be discovered, a posteriori, by direct observation.

Motion Is Not Simple!

By Jonathan Tennenbaum

Things are not what they appear, nor does the world function as the naive varieties of “common sense” (horse sense) would have it do. Those who subscribe to Rene Descartes’ doctrine of “clear and distinct” truths, and pride themselves on not listening to anything that smells of “theory,” or cannot be explained in five words or less, are liable to be ripped off by the nearest used car dealer (or stock broker?). For a pedagogical exercise, consider the following sales pitch, invented by wily old Descartes himself.

As every simpleton thinks he knows, the universe consists of “matter and motion.” (In fact, the famed J.C. Maxwell marketed his famous textbook on physics under that title.) To measure the performance of your used car engine, Descartes tells you, just ask “how much car (weight in pounds or tons) it moves how fast (miles per hour).” You just multiply the pounds together with the miles per hour, to get the handy performance rating at Rene’s Used Car Lot. For example, how would you choose between:

Car A: a two-ton “super classic,” with wall-to-wall marble ashtrays and other extras. Flooring the accelerator, it reaches 40 mph in 30 seconds. Rene urges us to buy this “hell of a car.”

And:

Car B: a lower-class model, weighs half as much as the “super-classic,” but reaches much less than twice the speed, namely 60 mph in the same 30 seconds.

A glance at both cars tells you, that their bodies are essentially junk. If anything, the only items of significant value are the engines. Now, Rene will let you have Car A for the same price as Car B, which (“as a friend”) he points out, is a “fantastic deal.” Car A is a “bit slower” but, as you can easily calculate yourself, with two-thirds the speed but twice the mass, its engine performance rating is more than 30% larger than Car B’s.

Rene adds another generous offer: If you prefer the smaller, faster car, he will switch the engines for you, and install Car A’s motor into Car B, free of charge. Could you turn down such a deal? After all, with Car A’s engine and half the weight, other things being equal, Car B should zip up to 80 mph in the same time it took to bring the heavier car up to 40.

Rene’s enthusiasm makes you a bit suspicious, on several points. Simplest of all: does the product of mass and attained velocity really represent the work performed in accelerating a car or other massive object up to a given state of motion? “Clear as day!” Descartes explains, appealing to our “horse sense” with the following argument:

Suppose we have a two-ton object. For it to have a speed of 40 miles per hour means, that in any given hour, those two tons move a distance of 40 miles. Dividing the object into two parts, each of 1 ton mass, we see that each of those has been moved 40 miles by that same motion. Obviously, it would be the equivalent amount of motion to move the two halves one at a time, instead of simultaneously, over the same 40 miles. In other words, in the first half hour we move the first half 40 miles, and then during the second half hour we move the other half 40 miles, the result being to move the whole mass 40 miles in the course of that hour. Or, again, since the two halves are identical in terms of mass, it represent the same effort to take only one of them, and move it 40 miles in the first half hour, and then just continue to move it another 40 miles in the second half hour. Thus, with an equivalent process we have moved one ton, 40 plus 40 = 80 miles in the given hour. We repeat this for every succeeding hour. Thus, two tons moving at 40 mph is equivalent to one ton moving at 80 mph. QED.

Corollary: Car A’s motor is a better buy than Car B’s.

An admirable specimen of deductive-type reasoning. But, if you swallow the axiomatics of this argument, you are going to be cheated! Can you prove them wrong? Such a demonstration will be given in next Tuesday’s briefing. [jbt with ap ]

MOTION IS NOT SIMPLE! -PART 2

In refuting Descartes on the measure of “quantity of motion” and related points, Leibniz pointed out three interrelated fallacies. First is the implicit assumption, that physics can be subsumed within a deductive form of mathematics. Second is the implicit assumption of “linearity in the small,” that physical action has the form of singularity-free continuous motion or extension in a three-dimensional Euclidean-like space. Third is the assumption that matter is characterized by nothing but such passive qualities as space-filling (extension), inertia, and resistance to deformation. In fact, in his 1686 piece on “A memorable error of Descartes,” and in other locations, Leibniz gave a simple demonstration of physical principle, showing that the process of change of velocity (acceleration) of material bodies involves something which is absolutely incompatible with Descartes’ assumptions. Leibniz demonstrated, for example, that the work of acceleration is NOT proportional to the mere product of the mass with the velocity attained, but (to a first degree of approximation) increases as the SQUARE of the velocity! To accelerate a mass to twice a given velocity, we need, not twice, but FOUR times the work.

If you have never stopped to consider, how utterly incomprehensible such a result is from the standpoint of naive sense-certainty and “horse sense,” do yourself that favor now.

Review the Huygens-Bernouilli-Leibniz discussion on the cycloid-brachistochrone for a richer development of the same point. Also Leibniz’s discussion of Descartes’ error and of the required notion of “anti-entropic” substance, in his “Treatise on Metaphysics” (Article 18 and preceding and following articles). Consider the relevance of Nicolaus of Cusa’s treatment of the Archimedes problem, and review the whole matter again from the higher standpoint of Lyn’s writings, including on the issue of “time-reversal.”

Here is Leibniz’s paper of 1686, referred to above: “Seeing that velocity and mass compensate for each other in the five common machines, a number of mathematicians have estimated the force of motion by the quantity of motion, or by the product of the body and its velocity. Or, to speak rather in geometrical terms, the forces of two bodies (of the same kind) set in motion, and acting by their mass as well as by their motion, are said to be proportional jointly to their bodies or masses and their velocities. Now, since it is reasonable that the same sum of {motive force} should be conserved in nature, and not be diminished–since we never see force lost by one body without being transferred to another–or augmented, a perpetual motion machine can never be successful, because no machine, not even the world as a whole, can increase its force without a new impulse from without. This led Descartes, who held motive force and quantity of motion to be equivalent, to assert that God conserves the same quantity of motion in the world.

“In order to show what a great difference there is between these two concepts, I begin by assuming, on the other hand, that a body falling from a certain altitude, acquires the same force which is necessary to lift it back to its original altitude, if its direction were to carry it back and if nothing external interfered with it. For example, a pendulum would return to exactly the height from which it falls, except for the air resistance and other similar obstacles, which absorb something of its force, and which we shall now refrain from considering. I assume also, in the second place, that the same force is necessary to raise a body of 1 pound to the height of 4 yards, as is necessary to raise a body of 4 pounds to the height of 1 yard. Cartesians, as well as other philosophers and mathematicians of our times, admit both of these assumptions. Hence it follows, that the body of 1 pound, in falling from a height of 4 yards, should acquire precisely the same amount of force as the body of 4 pounds, falling from a height of 1 yard. For, in falling 4 yards, the body of 1 pound will have there, in its new position, the force required to rise again to its starting point, by the first assumption; that is, it will have the force needed to raise a body of 1 pound (namely, itself) to the height of 4 yards. Similarly, the body of 4 pounds, after falling 1 yard, will have there, in its new position, the force required to rise again to its own starting point, by the first assumption; that is, it will have the force sufficient to raise a body of 4 pounds (itself, namely) to a height of 1 yard. Therefore, by the second assumption, the force of the body of 1 pound, when it has fallen 4 yards, and that of the body of 4 pounds, when it has fallen 1 yard, are equal.

“Now let us see whether the quantities of motion are the same in both cases. Contrary to expections, there appears a very great difference here. I shall explain it in this way. Galileo has proven that the velocity acquired in a fall of four yards, is twice the velocity acquired in a fall of one yard. So, if we multiply the mass of of the 1-pound body, by its velocity at the end of its 4-yard fall (which is 2), the product, or the quantity of motion, is 2; on the other hand, if we multiply the mass of the 4-pound body, by its velocity (which is 1), the product, or quantity of motion, is 4. Therefore the quantity of motion of the 1-pound body after falling four yards, is half the quantity of motion of the 4-pound body after falling 1 yard, yet their forces are equal, as we have just seen. There is thus a big difference between motive force and quantity of motion, and the one cannot be calculated by the other, as we undertook to show. It seems from that that {force} is rather to be estimated from the quantity of the {effect} which it can produce; for example, from the height to which it can elevate a heavy body of a given magnitude and kind, but not from the velocity which it can impress upon the body. For not merely a double force, but one greater than this, is necessary to double the given velocity of the same body. We need not wonder that in common machines, the lever, windlass, pulley, wedge, screw, and the like, there exists an equilibrium, since the mass of one body is compensated for by the velocity of the other; the nature of the machine here makes the magnitudes of the bodies–assuming that they are of the same kind–reciprocally proportional to their velocities, so that the same quantity of motion is produced on either side. For in this special case, the {quantity of the effect}, or the height risen or fallen, will be the same on both sides, no matter to which side of the balance of the motion is applied. It is therefore merely accidental here, that the force can be estimated from the quantity of motion. There are other cases, such as the one given earlier, in which they do not coincide.

“Since nothing is simpler than our proof, it is surprising that it did not occur to Descartes or to the Cartesians, who are most learned men. But the former was led astray by too great a faith in his own genius; the latter, in the genius of others. For, by a vice common to great men, Descartes finally became a little too confident, and I fear that the Cartesians are gradually beginning to imitate many of the Peripatetics at whom they have laughed; they are forming the habit, that is, of consulting the books of their master, instead of right reason and the nature of things.

“It must be said, therefore, that forces are proportional, jointly, to bodies (of the same specific gravity or solidity) and to the heights which produce their velocity or from which their velocities can be acquired. More generally, since no velocities may actually be produced, the forces are proportional to the heights which might be produced by these velocities. They are not generally proportional to their own velocities, though this may seem plausible at first view, and has in fact usually been held. Many errors have arisen from this latter view, such as can be found in the mathematico-mechanical works of Honoratius Fabri, Claude Dechales, John Alfonso Borelli, and other men who have otherwise distinguished themselves in these fields. In fact, I believe this error is also the reason why a number of scholars have recently questioned Huygens’ law for the center of oscillation of a pendulum, which is completely true.” [Adapted from {Gottfried Wilhelm von Leibniz: Philosophical Papers and Letters}, LeRoy E. Loemker, ed. (Chicago: University of Chicago Press, 1956); vol. I, pp. 455-458)]

The astronomical origins of number theory, Part 1

by Jonathan Tennenbaum

Once our prehistoric predecessors created the concept of a day, year, and other astronomical cycles, a new fundamental paradox arose: By its very nature, a cycle is a “One” which subsumes and orders a “Many” of astronomical or other events into a single whole. But what about the multitude of astronomical cycles? Must there not also exist a higher-order “One” which subsumes the astronomical cycles into a single whole?

We can follow the traces of Man’s hypothesizing on this issue, back to the most ancient of recorded times, and beyond. The oldest sections of the Vedic hymns — astronomical songs passed down by oral tradition for thousands of years before being written down — are pervaded with a sense of the implicitly paradoxical relationship among various astronomical cycles, as an underlying “motiv.” That motiv, in turn, shaped the long historical struggle to develop and perfect astronomically-based calenders, as a means to organize the activities of society in accordance with Natural Law.

A familiar example of the problem involved, is the relationship of the day (as the cycle of rotation of the entire array of the “fixed star”) and the solar year. Egyptian astronomers made rather precise measurements of the solar year, including the slight, but measurable discrepancy between a solar year and 365 full days. Four solar years constitute nearly exactly 1461 days (4 x 365, plus 1, the additional “1” appearing in the present-day calender as the extra day of a “leap year”). The use of a 4-year cycle was taken as the basis of the so-called Julianic calender. In reality, however, the apparent coincidence of 4 years and 1461 days is not a perfect one; a small, measurable discrepancy exists, amounting to an average of about <11 minutes per year>. This tiny “error” eventually led to the downfall of the Julianic calender, around 1582, by which time the discrepancy had accumulated to a gross value about 10 days!

Another classical example is the cycle of Meton, invented in ancient Greek times in the attempt to reconcile the cycle of the synodic month (defined by the phases of the Moon) with the solar year. Observation shows, that a solar year is about 10.9 days longer than 12 synodic months. Assuming the first day of a year and the first day of a synodic month coincide at some given point in time, the same event will be seen to occur once again after 19 years or 235 synodic months. That defines the 19-year cycle of Meton, which was relatively successful as the basis for astronomical tables constructed in Greek times. But again, more careful observation shows that this apparent cycle of coincidence is not a precise one. A slight discrepancy exists, between 19 years and 235 synodic months, which would cause any attempted solar-lunar calender based on rigid adherence to the Metonic “great cycle,” to diverge more and more from reality in the course of time.

The same paradox emerges, with even greater intensity, as soon as we try to include the motions of the planets into a kind of generalized calender of astronomical events. In fact, after centuries of effort, no one has been able to devise a method of calculating the relationship of the astronomical cycles, which will not eventually (i.e., after a sufficiently long period of time) give wildly erroneous values, when compared to the actual motions of the Sun, stars, and planets! No matter how sophisticated a mathematical scheme we might set up, and no matter how well it appears to approximate the real phenomena within a certain domain, that domain of approximate validity is strictly finite. Outside that finite region, the scheme becomes useless — its validity has “died.”

What is the reason for this persistent phenomenon, which we might call “the mortality of calenders?” Should we shrug our shoulders amd take this as a mere negative “fact of life?” Or is there a positive <physical existence> waiting to be discovered — a new, relatively transcendent <physical principle>, accounting for the seeming impossibility of uniting two or more astronomical cycles into a single whole by any sort of fixed mathematical construction?

According to the available evidence, the Pythagorean school of ancient times attacked this problem with the help of certain geometrical metaphors, perhaps along something like the following lines.

The simplest notion of an astronomical cycle embodies two elementary paradoxes: First, a cycle would appear to constitute an <unchanging process of change>! Indeed, the astronomical motions, subsumed by a given cycle, constitute <change>; whereas the cycle itself seems to persist <unchanged>, as if to constitute an existence “above time.” Secondly, we know that the <real> Universe progresses and develops, whereas the very concept of a cycle would seem to presume exact repetition.

Reacting to these paradoxes, construct the following simple-minded, geometrical-metaphorical representation of astronomical cycles:

Represent the unity of any astronomical cycle by a circle A, of fixed radius. Roll the circle along a straight line (or on an extremely large circle). Choose a point P, fixed on the circumference of the rolling circle, to signify the beginning (and also the end!) of each repetition of the cycle. As the circle rolls forward, the point P will move on a <cycloidal path>, reaching the lowest point, where it touches the line, at regular intervals. This is the location where the cycloid, traced by p in the course of its motion, generates a singular event known as a <cusp>. Denote the series of evenly-spaced cusps, by P, P’, P” etc. The interval between each cusp and its immediate successor in the series, corresponds to a single completed cycle of rotation of the circle A.

(For some purposes, we might represent the length of an astronomical cycle simply by the linear segment PP’, and the unfolding of subsequent cycles by a sequence of congruent segments PP’, P’P”, P”P”’ etc., situated end-on-end along a line. In so doing, however, it were important to keep in mind, that this were a mere projection of the image of the rolling circle, the latter being relatively more truthful.)

The fun starts, when we introduce a <second> astronomical cycle! Represent this cycle by a circle B, rolling simultaneously with the first one on the same line and at the same forward rate. Let Q denote a point on circle B, chosen to mark the beginning of each new cycle of B. A second array of points is generated long the line, corresponding to the beginning/endpoints of the second cycle: Q, Q’, Q” etc.

Now, examine the relationship between these two arrays of singularities P, P’, P” … and Q, Q’, Q” …. Depending on the relationship between the cycles A and B (as reflected in the relationship of their radii and circumferenes), we can observe some significant geometrical phenomena.

(At this point, it is obligatory for the readers to explore this domain themselves, by doing the obvious sorts of experiments, before reading further!)

Consider the case, where we start the circles rolling at a common point, and with P and Q touching the line at that beginning point. In other words, P = Q. If the radii of A and B are <exactly equal>, then obviously P’ = Q’, P” = Q” and so on. If, on the other hand, the radius (or circumference) of A is shorter than that of B, then a variety of outcomes are possible.

For example, the end of A’s first cycle (P’) might fall exactly in the middle of B’s cycle, in which case A’s second cycle will end exactly at the same point as B’s first cycle (P” = Q’). The same phenomenon would then repeat itself in subsequent cycles.

More generally, we could have a situation, where one cycle of B is equivalent in length to three, four, or any other whole number of cycles of A. It is common to refer to this case by saying, that A divides B evenly, or that B is an integral multiple of A.

The next, more complex species of phenomena, is exemplified by the case, where the endpoint of 3 cycles of A coincides with the endpoint of 2 cycles of B. Note, that in this case Q’ (the endpoint of B’s first cycle) falls exactly between the endpoint of A’s first cycle (P’) and the end of A’s second cycle (P”), while P”’ = Q”.

The defining characteristic of this type of behavior is, that after starting together, A and B seem to diverge for a while, but eventually “come back together” at some later time. Insofar as the lengths of A and B remain invariant, that same process of divergence and coming-together of the two processes must necessarily repeat itself at regular intervals. (Indeed: from the standpoint of the cycles A and B, the process unfolding from any <given> point of common coincidence, taken as a new starting-point, must be congruent to that ensuing from any <other> point of coincidence.) Aha! Have we not just witnessed the emergence of a third, “great cycle,” C, subsuming both A and B?

The length of this third cycle, would be the interval from the original, common starting-point of A and B, to the <first> point afterwards, at which A and B come together again (i.e., where the rotating points P and Q touch the line simultaneously at the same point). This event intrinsically involves two coefficients (or, in a sense, “coordinates”), namely the number of cycles completed by A and B, respectively, between any two successive events of coincidence.

Seen from the standpoint of mere scalar length per se, the relationship of C to A and B would seem to be, that A and B both divide C evenly; or in other words, C is a multiple of both A and B. More precisely, we have specified that C be the <least common multiple> of both A and B. In our present example, C would be equivalent (in length) to 3 times A, as well as to 2 times B.

Those skilled in geometry will be able to construct any number of hypothetical cases of this type. The simplest method, from the standpoint of construction, is to work <backwards> from a fixed line segment representing “C”, to generate A and B by dividing that segment in various ways into congruent intervals. For example: construct a line segment representing C, and divide that line segment into 5 equal parts, each of which represents the length of a cycle A. Then, take a congruent copy of C, and divide it (by the methods of Euclidean geometry, for example) into 7 equal parts, each of which represents the length of B. Next, superimpose the two constructions, and observe how the set of division-points corresponding to cycles of A, fall between various division-points of B. Try other combinations, such as dividing C by 15 and 12, or by 15 and 13, for example.

Carrying out these exploratory constructions with sufficient precision, we are struck with an anomaly: the “near misses” or “least gaps” between cycles of A and B.

In the case of division by 7 and 5, for example, observe that before coming together <exactly> after 7 cycles of A and 5 cycles of B, the two processes have a “near miss” at the point where B has completed two cycles and A is just about to complete its third cycle. In terms of scalar length, three times A is only very slightly larger than two times B. For different pairs of cycles A and B, dividing the same common cycle C, we find that the position and gap size of the “near misses” can vary greatly. For example, in the case of division by 15 and 12, the “least gap” already occurs near the beginning of the process, between the moment of completion of A’s first cycle and that of B’s first cycle. But for division by 15 and 13, the “least gap” occurs near the middle, between the end of B’s 6th cycle and A’s 7th cycle.

Resist the temptation to apply algebra to these intrinsically geometrical phenomena. Don’t fall into the trap of collapsing geometry into arithmetic! Although we can use algebra and arithmetic to calculate the division-points and the lengths of the gaps generated by the division-points, there is no algebraic formula which can <predict> the location of the “least gap”!

The Astronomical Origins of Number Theory, Part II

by Jonathan Tennenbaum

In the previous article, we began to investigate the relationship between two astronomical cycles A and B, representing these by circles of different radii rolling on a common line. We were investigating especially the case, where the cycles A and B can be brought together under a “great cycle” C, whose length is a common multiple of the lengths of A and B. Our attention was drawn to the anomalous phenomenon of “near misses” — i.e., points where the two cycles nearly end together, but not exactly. The irregularity of this phenomenon suggests, that we have not yet arrived at an adequate representation of the “great cycle” C and its relationships to A and B.

Take a new look at the circles A and B, rolling down the line. In our chosen representation, the rate of forward motion of the circles is the same, and they make a common point of contact with the line at each moment. But what is the relationship of rotation between A and B? Would it not be essentially equivalent, to conceive of A as rolling on the inner circumference of B, at the same time B is rolling on the line? It suddenly dawns upon us, that the geometrical events occurring between A and B in the course of any “great cycle” C (including the phenomenon of “near misses”), are governed by the indicated, <epicycloid> relationship of A and B alone!

Accordingly, leave the base-line aside for the moment; instead, generate an epicycloid curve by rolling the smaller circle A on the inside of the larger circle B, the curve being traced by the motion of the point P on A. Observe, that an equivalent array of cusps is generated, in a somewhat more convenient way, if we roll A on the <outside> of B instead of on the inside. Experimenting with our first example of a “great cycle,” observe that the epicycloidal curve in this case wraps around B twice, before closing back on itself, while A completes 3 complete rotations. Also observe, that the points where P touches the circumference of B — i.e., the 3 cusps of the epicycloid — divide B’s circumference into 3 equal arcs. Observe, finally, that the points of contact of A, while it is rolling, with the locations of the cusp-points of the epicycloid, include not only P, but also the opposite point to P on A’s circumference. In fact, each of the 3 equal arcs on B’s circumference correspond, by rolling, to one-half of A’s circumference.

Aha! That arc-length (i.e., one-third of B, equivalent to one-half of A) constitutes a <common divisor> of A and B. Comparing the epicycloidal process of rolling A against B, with the earlier process of A and B rolling on a common straight line, what is the relationship between the <common divisor>, just identified, and the <least gap> generated by the two cycles?

To investigate this further, carry out the same experiment with the pair of cycles A and B, obtained by dividing a given cycle-length C by 7 and 5, respectively. Rolling A on the outside of B, we find that the epicycloid must go around B 5 times, before it closes on itself. That corresponds to the “great cycle” C. In the course of that process of encircling B five times, the rolling circle A will complete exactly 7 rotations, generating 7 cusps in the process; these 7 cusps divide the circumference of B into 7 equal arcs, each of which is equivalent to one-fifth of the circumference of A. Those equivalent arcs all represent a <common divisor> of A and B.

Accordingly, construct a smaller circle D, whose radius is one-fifth that of A (or, equivalently, one-seventh that of B). In the course of a “great cycle” C, D makes 35 rotations. One cycle of A is equivalent in length to 5 cycles of D, and one cycle of B is equivalent in length to 7 cycles of D.

Compare this with the “least gap” constructed in Figure 5 of last week’s article. Evidently, the “least gap” generated by A and B, is equivalent to the <common divisor> of A and B, generated by the epicycloidal construction described above. Those skillful in mathematical matters will easily convince themselves, that if C corresponds to the <least common multiple> of A and B in terms of length, then D corresponds to their <greatest common divisor>.

Evidently, C and D constitute a “maximum” and “minimum” relative to the cycles A and B — C containing both and D being contained in both. Out of this investigation, we learn, that <if A and B have a common “great cycle,” then they also have a common divisor>; or in other words, they are <commensurable>. Also evidently, the converse is true: if A and B have a common divisor D, then we can easily construct a “great cycle” subsuming A and B. If fact, if A corresponds to N times D, and B corresponds to M times B, then A and B will fit exactly into a “great cycle” of length NM. (The length of the minimum “great cycle” is defined by the least common multiple of N and M, which is often smaller than the product NM; for example, if N = 6 and M = 4, the least common multiple is 12, not 24.)

Return now to our original query about the possibility of uniting a “Many” of different astronomical cycles into a single “One.” The result of our investigation up to now is, that there will always exist a “great cycle” subsuming integral multiples of cycles A and B into a single whole, as along as A and B are commensurable — i.e., as long as there exists some sufficiently small common unit of measurement, which fits a whole number of times into A and a whole number of times into B. Does such a unit always exist?

Remember the result of an earlier pedagogical discussion, in which we reconstructed the discovery of the Pythagoreans, of the <incommensurability of the side and diagonal of a square>! A pair of hypothetical astronomical cycles A and B, whose lengths (or radii) are proportional to the side and diagonal of a square, respectively, could never be subsumed exactly into a common “great cycle,” no matter how long! If we start A and B at a common point, they will <never> come together exactly again, although they will generate “near misses” of arbitrarily small (but nonzero) size!

This situation presents us with a new set of paradoxes: First, although A and B have no simple common “great cycle,” the relationship of diagonal to side of a rectangle is nevertheless a very precise, <lawful relationship>. This suggests, that the difficulty of combining A and B into a single “whole” does not lie in the nature of A and B per se, but in the conceptual limitations we have imposed upon ourselves, by demanding that the relationships of astronomical cycles be representable in terms of a “calender” based on whole numbers and fixed arithmetic calculations. Secondly, what is the new physical principle, which reflects itself in the existence (at least theoretically) of linearly incommensurable cycles? In fact, the work of Johannes Kepler completely redefined both these questions, by overturning the assumption of simple circular motion, and introducing the entirely new domain of elliptical functions. The bounding of elementary arithmetic by <geometry>, and the bounding of geometry (including so-called hypergeometries) by <physics>, is one of the secrets guarding the gates of what Carl Gauss called “higher arithmetic.”

How Johannes Kepler Changed the Laws of the Universe

Part II of an Extended Pedagogical Discussion

by Jonathan Tennenbaum

In Part I of this series (which readers should review before proceeding further here), I presented a series of arguments, purporting to demonstrate that there is no way to determine the actual movement of a planet in space, from observations made on the Earth. To this effect, I showed how, for any given pattern of observed motions, to construct an infinity of hypothetical motions in space, each of which would present exactly the same apparent motions as seen from the Earth. Short of leaving the Earth’s surface — an option not available to Kepler and his contemporaries — the effort to determine the actual orbits of the planets would appear to be nothing but useless speculation. Actually, similar sorts of arguments could be used to “prove” the futility of Man’s gaining any solid knowledge at all about the outside world, beyond the mere data of sense perception per se!

But wait! Man’s history of sustained, orders-of-magnitude increases in per-capita power over Nature since the Pleistocene, demonstrates exactly the opposite: The human mind <is> able, by the method of hypothesis, to overleap the bounds of empiricism, and attain increasingly efficient knowledge of the ordering of the Universe. Kepler’s own, brilliantly successful pathway of discovery, in unravelling the form and ordering of the planetary orbits, provides a most instructive case in point. The conclusion is unavoidable: the arguments I presented earlier in favor of a supposed “unknowability” of the planetary motions, must contain some fundamental error!

Kepler’s emphasis on <physics> and the <method of hypothesis>, as opposed to the impotence of mere “mathematics and logic,” should help us to sniff out the sophistry embedded in those arguments.

Did we not, in constructing a multiplicity of hypothetical orbits consistent with given observations, implicitly assume that those motions took place in a non-existent, empty mathematical space of the Sarpi-Galileo-Descartes-Newton type, rather than the real Universe? Did we not implicity collapse the “observer” to an inert mathematical point, ignoring the crucial factor of <curvature in the infinitesimally small>? Didn’t we overlook the inseparable connection between <human knowledge>, <hypothesis>, and <change>?

Human knowledge is not a contemplative matter of fitting plausible interpretations to an array of sense perceptions. On the contrary, knowledge develops through human intervention <to change the Universe> — a process which involves not only generating scientific hypotheses, but above all <acting> on them. The “infinitesimal” is no mathematical point; it possesses an internal curvature which is in demonstrable correspondence with the curvature of the Universe as a whole. That relationship centers on the role of the sovereign, creative human individual as God’s helper in the ongoing process of Creation.

For example, even the most banal application of “triangulation” in elementary geometry, reflects the principle of change. Rather than impotently staring at a distant object X (for example, a distant mountain peak, or an enemy position in war) from a fixed location A, “triangulation” relies on <change of position> from A to a second vantage-point B and so on, measuring the corresponding angular shifts in X’s apparent position relative to other landmarks and the “baseline” A-B.

Notice, that when we shift from A to B, we not only change the apparent angle to X, but we change the entire array of relationships to every other visible object in the field of view of A and B. Taken at face value, the two spherical-projected images of the world, as seen from A and as seen from B, are <formally contradictory>. They define a <paradox> which can only be solved by <hypothesis>. So, we conceptualize an additional dimension, a “depth” which is not represented in any single projection per se. The same metaphorical principle is already built into the binocular organization of our own visual apparatus. Compare this with the more advanced principle of Eratosthenes’ measurement of the Earth’s curvature, and the methods developed by Aristarchus and others to estimate the Earth-Moon and Earth-Sun distances.

The circumstance, that even our sensory apparatus (including the relevant cortical functions) is organized in this way, once again underlines the fallacies embedded in Kant’s claimed unknowability of the “Ding an sich.”

It was Kepler himself, who first used the combination of Mars, the Earth and the Sun — without leaving the Earth’s surface! — to unfold a “nested” series of triangulations which definitively established the elliptical functions of the planetary orbits and their overall organization within the solar system. But, as we shall see, the key to Kepler’s method was not simple triangulation in the sense of elementary geometry, but rather his shift away from naive Euclidean geometry, toward a revolutionary conception of <physical geometry>.

Turn back now to the paradoxes of planetary motion as seen from the Earth, particularly Mars, Jupiter, and Saturn. Mapping the motion of these planets against the stars, we find that they travel around the ecliptic circle (or more precisely, in a band-like region around the ecliptic), but <not at a uniform rate>. Although the predominant motion is forward in the same direction as the Sun, at periodic intervals these planets are seen to slow down and reverse their motion, making a rather flat “loop” in the sky, and then reverting to forward motion once again. This process of retrograde motion and “looping”, invariably occurs around the time of the so-called opposition with the Sun, i.e., when the positions of the Sun and the given planet, as mapped on the “sphere of the stars,” are approaching opposite poles relative to each other. Curiously, around that same time, the planet appears the brightest and largest, while in the opposite relative position — near the so-called conjunction with the Sun — the planet appears smaller and weaker, while at the same time displaying its most rapid apparent motion!

Although the “looping” of Mars (for example) recurs at roughly equal intervals of time, and is evidently closely correlated with the motion of the Sun, the period of recurrence is <not> equal to a year, <nor> is it the same for Jupiter and Saturn, as for Mars! The so-called synodic period of Mars — the period between the successive oppositions of Mars to the Sun, which coincides with the period between successive “loops” in Mars’ orbit — is observed to be approximately 780 days. In the case of Jupiter, on the other hand, the opposition to the Sun and formation of a loop, occur at intervals of about 399 days, or roughly once every 13 months.

But there is an additional complication. The planet does not come back to its original position in the stars (its siderial position) after a synodic period! The locus of the “loop”, relative to the stars, <changes> with each cycle of recurrence. After ending its retrograde motion and completing a loop, Mars proceeds to travel something more than a full circuit forward along the ecliptic, before the looping process begins again. Long observation, shows that the locus of each loop is shifted an average of about 49 degrees forward along the ecliptic, relative to the preceeding one.

Our experiments on the behavior of epicycloids, strongly suggest, that what we are looking at is some sort of epicyloid-like combination of two (or more) astronomical cycles! If so, then one of them would be the one producing the “looping,” and having a cycle length equal to 780 days, the synodic period of Mars. The other cycle — which <cannot be observed directly>, because it is strongly disturbed and distorted by the looping — would be the one determining the overall, net “forward” motion of Mars along the ecliptic. The fact, that Mars travels 360 plus 49 degrees along the ecliptic, before “looping” recurs, suggests that the cycle determing the looping has a somewhat <longer> period, than the cycle responsible for the net forward motion. In fact, the synodic cycle would have to be about 13.5% longer than the other cycle, to give the shift of 49 degrees forward from loop to loop. Or, to put it differently: if the hypothetical cycle of forward motion along the ecliptic, generates an angle of 360 plus 49 degrees in the time between successive loops — i.e., 780 days — then the time needed by that same forward motion to complete a full cycle of exactly 360 degrees, would be 687 days, or about 1.88 years. Of course, this whole reasoning assumes that each cycle progresses at a uniform, constant rate.

Let’s stop to reflect for a moment. On the basis of assumptions which, admittedly, require further examination, we have just adduced the existence of a 1.88-year cycle of Mars, a cycle which is <not directly observable>. Firstly, as Kepler remarked, Mars’ apparent trajectory <never closes>! Evidently we have a phenomenon of “incommensurability” of cycles. Moreover, the Mars trajectory itself does not lie exactly on the ecliptic circle, but winds around it in a band-like region like a coil. When Mars returns after going around the eliptic, it does not return to the same precise positions. So, where is the cycle? If we leave aside the deviations from the ecliptic, and just count the number of days Mars needs to make a single circuit within the ecliptic “band,” we get many different answers, depending on when and where in the “looping” cycle, we begin to count. Again, the observed motion of Mars is not strictly periodic. The 1.88-year cycle is born of hypothesis, not of direct empirical observation.

Historically (and as per the discussion in Kepler’s “Astronomia Nova,” the adduced 1.88-year cycle was referred to as the “first inequality” of Mars, while the cycle governing the “looping” phenomenon, was called the “second inequality.”

Now, our analysis up to now has been based on the assumption, that the underlying motion of a “cycle,” is uniform circular motion. That assumption dominated astronomical thinking up to the time of Kepler, and not without good reasons. After all, didn’t the approach of combining circular motions prove rather successful, earlier, in unravelling the motion of the Sun? We found that the Sun’s apparent motion can be understood as a combination of two circular motions: a daily rotation of the entire sphere of the stars, and a yearly motion of the Sun along a great-circle path (the ecliptic) on that stellar sphere. In the case of Mars (and the other outer planets), we evidently are dealing with a combination of <three> degrees of rotation: the daily stellar rotation; the “first inequality” with a period of 690 days; and the “second inequality” with a period of 780 days.

We are not finished, however. As Kepler would have emphasized, “the devil is in the detail.” To undercover a new set of anomalies, we must drive the fundamental hypothesis which has been the basis of our reasoning up to now — the hypothesis of uniform circular motion as elementary — to its limits. This is exactly what Kepler does in his {Astronomia Nova}. As his point of departure, he reviews the three main methods, developed up to that time, to construct the observed motions from a combination of simple circular motions. These were: 1) the method of epicycles associated with Ptolemeus, but actually developed by Greek astronomers centuries earlier; 2) the method of concentric circles, associated with Copernicus, but which had been put forward 14 centuries earlier by Aristarchos, and probably even by the original Pythagoreans; and finally 3) the method favored by Kepler’s elder collaborator, Tycho Brahe, which combines elements of both.

The differences between the constructions of Ptolemy, Copernicus and Tycho Brahe do not concern their common assumption of simple circular motion as elementary; at first glance, they merely differ in the way they combine circular motions to produce the observed trajectories.

(Readers should construct models to illustrate the following constructions!)

In the simplest form of Ptolemy’s construction, the Earth is the center of motion of the Sun and the primary center of motion of all the planets. The “first inequality” (of Mars, Jupiter or Saturn) is represented by motion on a large circle, C1 (called the “eccentric”), centered at the Earth, while the planet itself is carried along on the circumference of a second, smaller circle C2 (called the “epicycle”), whose center moves along C1. That motion of the planet on the second circle, corresponds to the “second inequality.” In the case of Mars, for example, the planet makes one circuit of the second circle in 780 days, while at the same time the center of the second circle moves along the first circle at a rate corresponding to one revolution in 690 days. It is easy to see how the phenomenon of retrograde motion is produced: At the time when the planet is located on the portion of its epicycle closest to the Earth, its motion on the epicycle is opposite to the motion on the first circle, and somewhat faster, yielding a net retrograde motion. From the angle described by the retrograde motions we can conclude the ratio of the radii of the two circles. To account for the transverse component of motion in a loop according ot this hypothesis, we must assume that the plane of the second circle is slightly skewed to that of the first circle. Ptolemy used an somewhat different, but analogous construction to account for the apparent motions of the “inner” planets Mercury and Venus.

In the simplest form of the so-called Copernican construction, the circular motions are assumed to be essentially concentric, centered at the Sun, although in slightly different planes. The apparent yearly motion of the Sun is assumed to result from a yearly motion of the Earth around the Sun. As for Mars, we represent its “first inequality” by a circle around the Sun, upon which Mars is assumed to move directly. The “second inequality,” on the other hand, now appears as a mere artifact, arising from the combined effect of the supposed, concentric-circular motions of the Earth and Mars. Since the Earth’s period is shorter than that of Mars, the Earth periodically catches up with and passes Mars on its “inside track.” At that moment of passing, Mars will appear from the Earth as if it were moving backwards relative to the stars. On the other hand, as the Earth approaches the position opposite to Mars on the other side of the Sun, Mars will attain its fastest apparent forward motion relative to the stars, the latter being exaggerated by the effect of the Earth’s motion in the opposite direction.

In Tycho Brahe’s construction, the planets (except the Earth) are supposed to move on circular orbits around the Sun, while the Sun itself (together with its swarm of planets, some closer, some farther away than the Earth) is carried around the Earth in an annual orbit.

Now, in his discussion in {Astronomia Nova}, Kepler emphasized that the three constructions, when carried out in detail, produce <exactly the same apparent motions>. From a purely formal standpoint, it would seem there could be no basis for deciding in favor of the one or the other. Yet, from a conceptual standpoint, the three are entirely different. And since Man does not merely contemplate his hypotheses, but <acts> on them, every conceptual difference — insofar as it bears on axiomatics — is eminently <physical> at the same time, even if the effect appears first only as an “infinitesimal shift” in the mind of a single human being.

Next week, by pushing the theories of Ptolemy, Copernicus, and Brahe to their limit, Kepler will evoke from the Universe a most remarkable response: All three approaches are false!

How Johannes Kepler Changed the Laws of the Universe

Part III of an extended pedagogical discussion

by Jonathan Tennenbaum

“It is true that a divine voice, which enjoins humans to study astronomy, is expressed in the world itself, not in words or syllables, but in things themselves and in the conformity of the human intellect and senses with the ordering of the celestial bodies and their motions. Nevertheless, there is also a kind of fate, by whose invisible agency various individuals are driven to take up various arts, which makes them certain that, just as they are a part of the work of creation, they likewise also partake to some extent in divine providence….

“I therefore once again think it to have happened by divine arrangement, that I arrived at the same time in which he (Tycho Brahe) was concentrated on Mars, whose motions provide the only possible access to the hidden secrets of astronomy, without which we should forever remain ignorant of those secrets.” (Kepler, Astronomia nova, Chapter 7).

Last week we briefly reviewed the three main competing approaches to understanding the apparent planetary motions, examined by Kepler: those of Ptolemy, Copernicus, and Tycho Brahe. Kepler emphasized the purely formal equivalence of the three approaches, at least in their simplest versions, but he pointed out crucial differences in their physical (i.e., ontological-axiomatic) character, while also noting some deeper, common assumptions of all three. Kepler first of all attacked Ptolemy’s method, on the grounds of its arbitrary assumptions, which reject the principle of reason:

“Ptolemy made his opinions correspond to the data and to geometry, and <has failed to sustain our admiration>. For the question still remains, what <cause> it is that connects all the epicycles of the planets to the Sun…” (My emphasis – JT).

“Copernicus, with the most ancient Pythagoreans and Aristarchus, and I along with them, say that this second inequality does not belong to the planet’s own motion, but only appears to do so, and is really a byproduct of the Earth’s annual motion around the motionless Sun.”

In his Mysterium Cosmograpium, Kepler had pointed out:

“For, to turn from astronomy to physics or cosmography, these hypotheses of Copernicus not only do not offend against Nature, but assist her all the more. She loves simplicity, she loves unity. Nothing ever exists in her which is superfluous, but more often she uses one cause for many effects. Now under the customary hypothesis there is no end to the invention of circles; but under Copernicus a great many motions follow from a few circles.”

In the Ptolemaic construction, each planet has at least two cycles, and not only the “first inequality,” but also the “second inequality” is different for each one. Not only does the hypothesis of Aristarchus eliminate the need for many “second equalities” — deriving them all, as effects, from the single cycle of the Earth — but countless other specifics of the apparent planetary motions begin to become intelligible.

Truth, however, does not lie in the simplicity of an explanation per se. Indeed, very often the “simplest” explanation, one in which everything appears to fit together effortlessly, and all irritating singularities disappear, is the farthest from the truth! When things become too easy, too banal, watch out! To get at the truth, we must always generate a new level of paradox, by pushing our hypotheses to their breaking-points. This Kepler does, by focussing on the implications of certain irregularities in the planetary motions — overlooked in our discussion up to now — which would be virtually incomprehensible, if the cycles of the “first and second inequalities” were based only on simple circular action.

Indeed, on closer examination, we find that the “loops” of the planet Mars (for example) are not identical in shape, but vary somewhat from one synodic cycle to the next! Nor is the displacement of each loop, relative to the preceeding one, exactly equal from cycle to cycle. Furthermore, even the motion of the Sun itself along the ecliptic circle, upon close study, reveals itself to be alternately speed up and slow down significantly in the course of a year, contrary to our tacit assumption up to now.

Indeed, already in ancient times astronomers wondered at the paradoxical “inequality” of the Sun’s yearly motion. In fact, when we carefully map the Sun’s motion relative to the “sphere of the fixed stars,” we find, that although the Sun progresses along the ecliptic at an average rate of 360 degrees per year, the angular motion is actually about 7% faster in early January (about 0.95 degrees per day) than in July (about 1.02 degrees per day). This variation causes quite noticeable differences in the lengths of the seasons, as these are defined in terms of a solar calender. Indeed, the four seasons correspond to a division of the ecliptic circle into four congruent arcs, the division-points being the two equinoxes (the intersection-points of the ecliptic with the celestial equator) and the two solstices (the points on the ecliptic midway between the equinox points, marking the extremes of displacement from the celestial equator and thereby also the positions of the Sun on the longest and shortest days of the year). Due to the changes in the Sun’s angular velocity along the ecliptic, those four arcs are traversed in different times. In fact, the lengths of the seasons, so determined, are as follows (we refer to the seasons in northern hemisphere, which are reversed in the southern hemisphere):

Spring: 92 days and 22 hours; Summer: 93 days and 14 hours; Fall: 89 days and 17 hours; Winter: 89 days and 1 hour.

This unevenness in the solar motion confronts us with a striking paradox: How could we have a “perfect” circular trajectory, as the Sun’s path (the ecliptic) appears to be, and yet the motion on that trajectory not be uniform? That would seem to violate the very nature of the circle. Or shall we assume, that some “outside” force could alternately accelerate or decelerate the Sun (or Earth, if we take Copernicus’ standpoint), without leaving any trace in the shape of the trajectory itself? Furthermore, how are we to comprehend this variation, if we hold to the hypothesis, that the elementary form of action in astronomy is uniform circular motion? On the other hand, if we give up uniform circular motion as the basis for constructing all forms of motion, then we seem to open up a Pandora’s box of a unlimited array of conceivable motions, with no criterion or principle to guide us.

One “way out” — which only shifts the paradox to another place, however –, would be to keep the assumption, that the Earth’s motion (and that of the other planets) is uniform circular motion, but to suppose that the center of the orbit is not located exactly at the Sun’s position. This notion of a displaced circular orbit was known as an “eccentric”; both Ptolemy and Copernicus employed it in the detailed elaboration of their theories, to account for the mentioned irregularities in planetary motions. Assuming such orbits really exist, it is not hard to interpret the speeding-up and slowing-down of the Sun’s apparent motion as a kind of illusion due to projection, in the following way: Taking Copernicus’ approach for example, the “true” motion of the Earth would be a uniform circular one; but the Sun, being located off of the center of the Earth’s orbit, would appear from the Earth to be moving faster when the Earth is located on the portion of its eccentric closest to the Sun, and slower at the opposite end. On this asssumption, it is not hard to calculate, by geometry, how far the center of the eccentric would have to be displaced from the Sun, in order to account for the 7% difference in observed angular speeds between the perihelion (closest distance) and aphelion (farthest distance) of the eccentric.

From the standpoint of this construction, the “true” motion of the Sun (or the Earth, in Copernicus’ theory) would be that corresponding exactly to the mean or average motion of 360 degrees per year, while the apparent motion would vary according to the varying distance between Earth and Sun. Accordingly, Tycho Brahe and Copernicus elaborated their analyses of the apparent planetary motions on the basis of the assumed “true” circular motion of the Sun (or Earth).

This exact point becamce a focus of debate between Kepler and Tycho Brahe. Kepler writes:

“The occasion of … the whole first part (of Astronomia nova) is this. When I first came to Brahe, I became aware that in company with Ptolemy and Copernicus, he reckoned the second inequality of a planet in relation to the mean motion of the Sun … So, when this point came up in discussion between us, Brahe said in opposition to me, that when he used the mean Sun he accounted for all the appearances of the first inequality. I replied that this would not prevent my accounting for the same observations of the first inequality using the Sun’s apparent motion, and thus it would be in the second inequality that we would see which was more nearly correct.”

This challenge eventually led to the breakthroughs which Kepler announced in the title of Part II of his Astronomia Nova: “Investigation of the second inequality, that is, of the motions of the sun or earth, or the key to a deeper astronomy, wherein there is much on the physical causes of the motions.”

Kepler had reason to be suspicious about the assumption of perfect circular orbits as “elementary.” On the one hand, Kepler was a follower of Nicolaus of Cusa, who had written, in the famous Section 11 of Docta Ignorantia,

“What do I say? In the course of their motion, neither the Sun, nor the Moon nor the Earth nor any sphere — although the opposite appears true to us — can describe a true circle … It is impossible to give a circle for which one could not give one even more perfect; and a heavenly body never moves at a given moment exactly the same way as at some other moment, and never describes a truly perfect circle, regardless of appearances.”

On the other hand, already Ptolemy knew that the tactic of uniform motion on displaced, “eccentric” circles, fails to fully account for irregularities turning up in the “first inequality” of the planets Venus, Mars, Jupiter, and Saturn (particularly Mars). To explain the accelerations and decelerations of the planets, which still remain after the effect of the “second inequality” is removed, and to reconcile those with other features of the apparent motions, it was not sufficient to merely displace the circle of the “first inequality” from the observer on the Earth. Ptolemy (or whoever actually did the work) accordingly introduced a new artifice, called the “equant”: On this modifed hypothesis, the motion along the circumference of the eccentric circle, instead of being itself uniform and constant, would be driven forward by a uniform angular rotation around a fixed point called the “equant,” located at some distance from the center of the circle. In the case of Mars, for example, the Earth and the equant would be located on opposite sides of the circle’s center. This would result in a real acceleration of the planet going toward its nearest point to the Earth (and deceleration moving toward the opposite end), adding to the effect of viewing this from the Earth. Actually, on the basis of the “equant” construction, Ptolemy and his followers, were able to make relatively precise calculations for all the planets (except Mercury). It was first using the more precise observations of Tycho Brahe, that Kepler could finally give Ptolemy the “coup de grace.”

Copernicus rejected the “equant,” essentially on the grounds that it de facto instituted “irregular” motions (i.e., non-circular motion) into astronomy. To avoid this, Copernicus and Brahe invented still another circular cycle (in addition to the “second inequality”) to modify the supposed uniform motion on the eccentric circle. We seem to be headed into a monstrous “bad infinity.”

But, isn’t there something absurd and wholly artificial about the idea of a planet orbiting in a circle around a mere abstract mathematical point as center? And being propelled by an abstract ray pivotting on another mathematical point? Kepler writes:

“A mathematical point, whether or not it is the center of the world, can neither effect the motion of heavy bodies nor act as an object towards which they tend … Let the physicists prove that natural things have a sympathy for that which is nothing.”

The same objection applies also, of course, to the device of the epicycle, whose center is supposed to be a mere mathematical point. Later Kepler adds:

“It is incredible in itself that an immaterial power reside in a non-body, move in space and time, but have no subject … And I am making these absurd assumptions in order to establish in the end the impossibility that every cause of the planet’s motions inhere in its body or somewhere else in its orb … I have presented these models hypothetically, the hypothesis being astronomy’s testimony, that the planet’s path is a perfect eccentric circle such as was described. If astronomy should discover something different, the physical theories will also change.”

Aha! While seeking means to accurately determine the real spatial trajectory, Kepler explores the notion, that something like the effect of the “equant” might actually exist, as <a new mode of physical action>:

“About center B let an eccentric DE be described, with eccentricity BA, A being the place of the observer. The line drawn through AB will indicate the apogee at D and the perigee at F. Upon this line, above B, let another segment be extended, equal to BA. C will be the point of the equant, that is, the point about which the planet completes equal angles in equal times, even though the circle is set up around B rather than C …” Copernicus notes this hypothesis among other things in this respect, that it offends against physical principles by instituting “irregular celestial motions … the entire solid orb is now fast, now slow.” This Copernicus rejects as absurd.

“Now I, too, for good reasons, would reject as absurd the notion that the moving power should preside over a solid orb, everywhere uniform, rather than over the unadorned planet. But because there are no solid orbs, consider now the physical evidence of this hypothesis when very slight changes are made, as described below. This hypothesis, it should be added, requires two motive powers to move the planet (Ptolemy was unaware of this). It places one of these in the body A (which, in the reformed astronomy will be the very Sun itself), and says that this power endeavors to drive the planet around itself, but possesses an infinite number of degrees corresponding to the infinite number of points of the ray from A. Thus, as AD is the longest, and AF the shortest, the planet is slowest at D and fastest at F… The hypothesis attributes another motive power to the planet itself, by which it works to adjust its approach to and recession from the Sun, either by strength of the angles or by intuition of the increase or decrease of the solar diameter, and to make the difference between the mean distance and the longest and shortest equal to AB. Therefore, the point of the equant is nothing but a geometrical short cut for computing the equations from an hypothesis that is clearly physical. But if, in addition, the planet’s path is a perfect circle, as Ptolemy certainly thought, the planet also has to have some perception of the swiftness and slowness by which it is carried along by the other external power, in order to adjust its own approach and recession in such accord with the power’s prescriptions, that the path DE itself is made to be a circle. It therefore requires both an intellectual comprehension of the circle and a desire to realize it…

“However, if the demonstrations of astronomy, founded upon observations, should testify that the path of the planet is <not quite circular>, contrary to what this hypothesis asserts, then this physical account too will be constructed differently, and the planet’s power will be freed from these rather troublesome requirements.”

Kepler’s hypothesis (which undergoes rapid evolution across the pages of “Astronomia Nova”) means throwing away the notion, that the action underlying the solar system has the form of “gear-box”-like mechanical-kinematic generation of motions. Instead, Kepler references a notion of “power” and a constant activity which generates dense singularities in every interval. While for the moment, the circle remains a circle in outward form, we have radically transformed the concept of the underlying process of generation. In a sense, that shift in conception amounts to an infinitesmal deformation of the hypothetical circular orbit, which implicitly changes the entire universe. The successful measurement of deviation of a planet’s path from a circular orbit, would constitute a unique experiment for the hypothesis of a new, non-kinematic principle of action. That is the “deeper astronomy” of Kepler!

So we come back to the problem: How to determine the precise trajectory of a planet in space, given observations made only from the Earth, and taking into account the fact, that the Earth itself is moving? Having identified the “second inequality” as the crux of the problem of apparent planetary motions, Kepler turns the tables on the whole preceeding discussion, and uses Mars and the Sun as “observation posts” to determine orbit of the planet whose motion is the most difficult of all to “see” — the Earth itself!

But, how can we use Mars as an observation-post? Mars is moving. No matter! Let us assume that <part> the hypothesis of Aristarchus remains true, namely that the planets have closed orbits, and that motion along those orbits is what produces the so-called “first inequality” determined by the ancients. In that case, Mars — <regardless of whether or not its orbit is circular!> — periodically returns to any given locus in its orbit. Furthermore, we already know the period-length of that recurrence: it is the 1.88-year cycle which we adduced last week, by <indirect means>, from the study of Mars’ bizarre apparent motions.

So, make a series of observations of the apparent positions of Mars and the Sun, relative to the stars, at successive intervals 1.88 years apart! If our reasoning is sound, Mars will occupy (at least roughly) the same actual position in space, relative to the assumed “fixed” Sun and stars, at each of those times. On the other hand, at intervals corresponding to integral multiples of 1.88 — 0, 1.88, 3.76, 5.64, 7.54 years etc, — the Earth will occupy <unequal> positions, distributed more and more densely around its orbit, the longer the series is continued (the phenomenon of relative incommensurability).

Now make two “nested” types of triangulations. Assuming first that the orbit of the Earth is very roughly circular, use the observations of Mars’s apparent position, as seen from two or more of those positions of the Earth, to “triangulate” Mars’ location in space. Next, use that adduced location of Mars, plus the angles defined by the apparent positions of Mars <and the Sun> relative to the stars, to triangulate the position of the <Earth> in space at each of the times 0, 1.88, 3.76 years etc. Then use these adduced positions of the earth to develop an improved {hypothesis} of the earth orbit. Apply the improved knowledge of earth’s orbit to correct the triangulation of Mars’ position. Use the improved localization of Mars to revise and correct the values for the Earth’s positions. Finally, use the adduced knowledge of the Earth’s orbital motion to “triangulate” a series of positions of Mars, and other planets!

The experiment was successful. Ramus, Aristotle, and Kant were demolished. The door was kicked open for a revolution in physics, and a new mathematics of non-algebraic, non-kinematic functions.

Predictions Are Always Wrong

by Phil Rubinstein

Of late in dealing with the outlook of the population, we often have to face the impact of linearity most directly with respect to the sense of time. This occurs in the form of “can you predict…?”, “can you tell us when…?”, or “your prediction was wrong, it didn’t happen”, etc. All of this reflects a view of space-time that is one of a linear extension, with space as a filled-up box and in effect no concept of time, since time can exist only as change, action, becoming. It is precisely this linearity that simplifies language to dumbness, reduces music to noise and makes all science and geometry of the post-Kepler period incomprehensible.

It is no accident that one can find a nearly completely modern expression of this in Aristotle’s “On Interpretation” — he says first in section III “… verbs by themselves, then, are, nouns and they stand for or signify something…. [T]hey indicate nothing themselves but imply a copulation or synthesis, which we can hardly conceive of apart from the things thus combined.” And then, “we call proposition those only that have truth or falsity in them.” Were this only the ancient outlook of a discredited Aristotle no problem would ensue, but in fact this is the root of the thoroughly modren outlook of Russell, Frege, Carnap, etc. In fact, On Interpretation could be a handbook for information theory. While Aristotle like his modern followers recognized that the thoroughly deterministic outlook that follows from this contradicts the actual choices made by human beings, his resolution is to introduce mere contingency, a kind of randomness, which is allowed to the empty future.

The reality is best grasped by taking an approach rooted in physical economic planning. Begin with a moment in history defined by a resource level determined by an existing science and technology. A horizon can be hypothesized at which the social cost of resources usable at that level of technology would lead to a critical degeneration or inability to maintain capital or labor. That crisis defines the necessary present deployment of advanced technologies to create new scientific breakthroughs. This, however, requires greater density of use of resources, labor and soon, thus, the horizon is changed. Take the example of fossil fuel, nuclear fission, then nuclear fusion. Our present resources may be either stretched to extend the horizon, but that merely worsens the crisis. If we choose to accelerate the use of fission energy, the demand on existing resources USES UP those resources more rapidly. If we plan to achieve fusion, the rate of usage increases.

Thus, the future is changed for present action at each step. The problem then becomes to determine the actual activity required in the present. As this occurs, the relationship between now and the future is constantly altering: that also alters all other activity, allocation of resources, labor, and so on. In this way, the present is itself an incommensurable. It is a perfect example of non-constantly changing action. It is this subjectivity that lies at the root of understanding physical space-time as something both of constantly changed activity of a multiply connected type in the sense of Leibniz.

From this standpoint, one can see that not only is the future causing the present, but that implicit in any hypothesis of this type is an inversion that is assymetric. As the forecast is made it immediately brings us to a new concept of the path of action itself. The relationship of past, present, and future is altered.

This also has implications for language, such as the fundamental role of the subjunctive, and in physics, such as non relativistic relativity and non-statistical quantum theory. That could be raised for future discussion, but at least never let us be caught in Aristotelean conceptions of the future.

Prime Numbers

By Bruce Director

“Can we deny that a warrior should have a knowledge of arithmetic?…

“…. It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being…

“…. For, if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul, perplexed and wanting to arrive at a decision, asks, `Where is absolute unity?’ This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

“And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?

“Certainly.

“And all arithmetic and calculation have to do with number?

“Yes.

“And they appear to lead the mind towards truth?

“Yes, in a very remarkable manner.

“Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician….

“… Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study {until they see the nature of numbers with the mind only;} nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being….”

–Plato’s {Republic} Book VII

Elementary considerations concerning prime numbers, directly present us with the fundamental questions to which Socrates refers in the above passage. To grasp this, however, one must confront, and overcome, any influence of Euler and the Enlightenment, in one’s own thinking.

What are Prime Numbers?

Prime Numbers are integers, which are indivisible, by any other number, except one and itself. Composite numbers, are integers, which can be divided by another number. In the {Elements}, Euclid presents a proof that any composite number is divisible by some prime number. Since, any composite number by definition can be divided by another number, that other number is either another composite number or a prime number. If it is a prime number, the case is proved. If it is a composite number, that new number can be divided by another number, which is either a prime number or a composite number. By this method, you will eventually get to a prime number.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C., known as the Sieve of Eratosthenes. This is a method for eliminating the composite numbers, from a group of integers, leaving only the primes.

List the integers from 1 to any arbitrary other integer A. Now, beginning with 2 (the first prime number after 1), strike from the list, all numbers divisible by 2, for they are composite numbers. Do the same with 3 (the next prime number), then 5, etc., until you come to the first or prime is first.

This raises the question, what happens when you try to construct all integers from the primes alone? First, you’d make all the integers composed only of 2. Then you’d make all the integers composed only of 3 and combinations of 2 and 3, and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a non-linear way. Compare that process with constructing the integers by addition? Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference, is between the mental act of addition, and the mental act of division. Don’t try resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the “in-betweeness.” It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly, is a reflection of the truth, that there exists a higher hypothesis which underlies the foundations of integers. An hypothesis, which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, “to see the nature of number in our minds only.” Our minds ascend, as Socrates indicates, to contemplate the nature of true being. We ask, “If the domain of primes is that from which the integers are made, what is the nature of the domain, from which the primes are made?”

View the above result from the standpoint of Leibniz’s {Monadology}:

“29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit}.

“30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of being, substance, the simple and the composite, the immaterial, and even of God, conceiving what is limited in us as without limit in him. These acts of reflection furnish the principal objects of our reasoning.” [bmd]

Prime Numbers, Part II

“The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it….

“The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

–Nicolaus of Cusa, “On Conjectures”

Last week, as Socrates says, we, began “to see the nature of number with the mind only.” This week, we further unfold the essence of number, requiring our mind, to lift itself into a new higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such, an annoying distraction in itself. To avoid these distractions, follow the proscription of Plato, Cusa and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic, as developed by C.F. Gauss, which considers only the relations between whole numbers. Gauss, elaborated a visualization of the complex domain, translated by Jonathan Tennenbaum, in the Spring 1990 issue of Twenty First Century, under the title “Metaphysics of Complex Numbers,” to which the reader is referred.

In the previous discussion, we discovered an hypothesis underlying the essence of number, through reflection on the prime numbers. By its extension, we will be lead to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number, from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers, maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas, positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting one, or, adding -1. Positive composite numbers are formed by muliplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite one another. If positive is right, negative is left. If positive is up, negative is down. One dimension–two directions.

In short we have extended our concept of number, by conceiving of a two-fold unity: 1 and -1.

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2 X 2 =4; 3 x 3 = 9; -2 x -2 = 4; -3 x -3 = 9. The nmber being squared is called, the square root, of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A pradoxical question arises: “Can one form a negative quadratic number?” Or, conversely, “what is the square root of a negative number?”

The simplest case, which subsumes all others, is the case of the quadratic unity, 1. 1 x 1 = 1 -1 x -1 = 1. The square root of 1 = 1 or -1, as both numbers squared equal 1. What, then is the square root of -1?

Within the concept of number as one-dimensional, (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the “devil’s” interval.) Euler and others, sought to limit progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss, who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain–the complex domain. Instead of avoiding the paradox presented, by thinking of it as imaginary, or impossible, Gauss asked, in what higher domain, must such a magnitude exist? A shift in hypothesis, which his student, Riemann, would later designate as from n to n + 1 dimensions.

Gauss elaborates his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis, as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He, says he was only waiting (over 30 years) for a suitable place, in which to announce his new hypothesis to the public.

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter i. (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

In the complex domain of two dimensions, the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative, the other dimension, is associated with +i and -i. The complex domain is ONE domain, indivisible, of two dimensions. A new hypothesis, under which, the positive and negative, i and -i are made congruent (harmonic) with each other. In this new domain, all numbers are of the form a + bi, where a (short for 1 x a) designates the positive-negative dimension, and bi designates the +i -i dimension, (1 + i dimensions). This is not a combination of two different numbers, but one number, with two parts. One dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where b=0. Complex numbers where neither a nor b is 0 are called mixed complex numbers.

Reflect on the difference between the domain of one dimensional numbers and the complex domain.

In the domain of one dimension, unity is two-fold, 1 and -1. In the complex domain, unity is four-fold, 1 and -1, i and -i. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its four-fold unity, each number has four associates, found by multiplying that number by -1, i, -i. For example, a + bi is associated with -b + ai, -a – bi, b – ai. (The reader can confirm this for himself, by multiplying a + bi by i, -1, -i respectively.)

There is a special, unique, relationship in the complex domain– the relationship between a number a + bi, and its conjugate, a – bi, that is, when the sign of i is reversed. The product of a number and its conjugate is a^2 + b^2 and is called its Norm. (Notice the similarity to the Pythagorean)

Gauss then investigates the nature of prime numbers in the complex domain. Just as in one dimension, all whole numbers are either prime or composite. However, not all one dimensional prime numbers, remain prime in the complex domain. For example, in the complex domain 2 = (1 + i)(1 – i); 5 = (1 + 2i)(1 – 2i), 13 = (3 + 2i)(3 -2i). In fact, Gauss showed, that all one dimensional prime numbers of the form 4n+1 are no longer prime in the complex domain, but all one- dimensional prime numbers of the form 4n+3 remain prime. (All one-dimensional prime numbers are either of the form 4n+1 or 4n+3. Not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain–mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1 + 2i, is a mixed complex prime number, because its Norm, 1^2 + 2^2 = 5, which is a one dimensional prime number. So is 1 – 2i.

Reflect further on the difference between the one dimensional domain of positive and negative numbers, and the complex domain. The complex domain, is not simply the one-dimensional domain, in two directions, as in the fraud perpetrated by Cauchy. IT IS AN ENTIRELY DIFFERENT DOMAIN, lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain, which determines the singularities. Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you’re in? Through the creative powers of your mind. Analysis situs. If 5 is a prime number, you’re in one dimension, if not, you may be in the complex domain, but, maybe not. Gauss speculates about the possibility of numbers of higher dimensions than two. What happens to prime numbers in domains of more than two dimensions?

Isn’t this the key to improving performance in sales and intelligence?

Mind Over Mathematics–Prime Numbers, Part III

CAN YOU SOLVE THIS PARADOX

Over the previous two weeks, we’ve demonstrated, by reliving a discovery made by the young Carl Friedrich Gauss when he was 10 years old, how numbers are creations of the mind. Once this basic principle is understood, the mind is no longer a slave to formal rules concerning numbers. Problems such as adding all the numbers from 1 to 100, which at first appear tedious and perhaps even difficult, are easily solved, once the mind breaks the formal rules, and re-orders the numbers according to a new, higher principle.

This week, we’ll look more deeply into the nature of numbers, and in doing so, we’ll gain an increased mastery over our minds’ creative process.

What Are Prime Numbers?

Among the whole numbers, there exist unique integers known as Prime Numbers, which are distinguished by the property that they are indivisible by any other number except themselves and 1. Thus, 2, 3, 5, 7, and 11 are all examples of prime numbers. Composite Numbers are integers which can be divided, not only by themselves and 1, but by some other number.

It had already been discovered by the ancient Greeks, and written down by Euclid (flourished c. 300 B.C.) in his “Elements,” that all numbers are either prime or composite, and that any composite number is divisible by some prime number.

You can prove this for yourself, in the following way. Any composite number can by definition be divided by some other number, and that other number is either another composite number or a prime number. If it is a prime number, we need go no further. If it is a composite number, then that new composite number can be divided by another number, which is either a prime number or a composite number, and so on. By this method, you will eventually get to a prime number divisor.

For example, 30 is a composite number, and can be divided into 2, a prime number, and 15, a composite number. In turn, 15, can be divided into 3, a prime number, and 5, also a prime number. So the composite number 30 is made up of, and can be divided by, prime numbers 2, 3, and 5.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C. This approach, known as the Sieve of Eratosthenes. is a method for eliminating the composite numbers from a group of integers, leaving only the primes.

List the integers from 1 to any other arbitrary integer A. Now, beginning with 2 (the first prime number after 1), strike from the list all numbers divisible by 2, for they are composite numbers. Do the same with all numbers divisible by 3, the next prime number; then those divisible by 5, etc., until you come to the first prime number whose square is greater than A. (If A = 100, then you only need do this procedure with primes less than 11.) See Figure 1.

This method allows us to find the prime numbers which are smaller than any arbitrary number–but is there some large number after which there are no prime numbers? In other words, are there an infinite number of prime numbers?

That the answer is yes, is easily proved by showing that, no matter how many prime numbers are found, there can always be found one more. First, find all the prime numbers less than any arbitrary number, a, b, c, …, z. Now multiply all these prime numbers together, and add 1 to the product [(a x b x c x d x … x z) + 1]. Call this new number A. If this new number A is a prime number, then you’ve found another prime, which was not known before. If it is a composite number, then it must be divisible by some prime number. But, that prime number cannot be one of those already known, as the known prime numbers (a, b, c, …, z) will always leave a remainder of 1, when divided into A.

Primes Are Nonlinear

After we have found a large number of primes, it is obvious that, in the small, there is no regular, or linear, pattern of distribution of the prime numbers. Gauss showed, however, that over a large interval, the distribution of the primes is approximated by a logarithmic curve; that is, the larger the prime numbers become, the more spread apart they tend to be. This approximation breaks down increasingly, the smaller the interval. A nightmare for Leonhard Euler–nonlinearity in the small. (See Figure 2.)

{Again, what are Primes?} It is easy to see that any composite number can be decomposed into prime numbers, by division. For example, 12 can be decomposed into 2 x 2 x 3, or 2^2 x 3. The number 504 can be decomposed into 2 x 2 x 2 x 3 x 3 x 7, or 2^3 x 3^2 x 7.

Gauss was the first to prove Disquisitiones Arithmeticae, Article 16) that a composite number can be decomposed into only one combination of prime numbers. In the above examples, no combination of prime numbers other than 2 x 2 x 3 will equal 12. Likewise for 504, or any other composite number.

This remarkable result, which Gauss says was “tacitly supposed but had never been proved,” provokes a fundamental question concerning the nature of the universe. The fact that Gauss was the first to consider this result important enough to prove, is another indication of his genius.

With Gauss’s proof, and the preceding discussion, it is shown that prime numbers are that from which all other numbers are composed. The primes are primary. The word the ancient Greeks used for “prime,” was the same word they used for “first” or “foremost.”

This raises the question, what happens when you try to construct all integers from the primes alone? First, you’d make all the integers composed only of 2, such as 4, 8, 16, …. Then you’d make all the integers composed only of 3, and of combinations of 2 and 3, such as 6, 9, 12, …, and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a nonlinear way.

Compare that process with constructing the integers by addition. Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference is between the mental act of addition, and the mental act of division. Don’t try to resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the “in-betweenness.” It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly is a reflection of the truth that there exists a higher hypothesis which underlies the foundations of integers–a hypothesis which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, “to see the nature of number in our minds only” (from Plato’s “Republic”). Our minds ascend, as Socrates indicates, to contemplate the nature of true Being. We ask, “If the domain of primes is that from which the integers are made, what is the nature of the domain from which the primes are made?”

View the above result from the standpoint of Leibniz’s “Monadology”:

“29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit.}

“30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of Being, Substance, the Simple and the Composite, the Immaterial, and even of God, conceiving what is limited in us as without limit in Him. These acts of reflection furnish the principal objects of our reasoning.”

Next week: When is 5 not a prime number.

Prime Numbers, Part IV

 CAN YOU SOLVE THIS PARADOX?

“The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it….

“The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.”

–Nicolaus of Cusa “On Conjectures”

Last week, we investigated the nature of prime numbers. This week, we further unfold the essence of number, requiring our minds to lift themselves into a new, higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such to be an annoying distraction, in itself. To avoid these distractions, follow the prescription of Plato, Cusa, and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic as developed by Carl Friedrich Gauss, which considers only the relations among whole numbers.

In the previous discussion, we discovered, through reflection on the Prime Numbers, a hypothesis underlying the essence of number. By its extension, we will be led to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting 1, or adding -1. Positive composite numbers are formed by multiplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite to one another. If positive is right, negative is left. If positive is up, negative is down. One dimension–two directions.

In short, we have extended our concept of number, by conceiving of a twofold unity: 1 and -1.

An Anomaly Appears

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2×2=4; 3×3=9; -2x-2=4; -3x-3=9. The number being squared is called the square root of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A paradoxical question arises: “Can one form a negative quadratic number?” Or, conversely, “What is the square root of a negative number?”

The simplest case, which subsumes all others, is the case of the quadratic unity, 1: 1×1=1, and -1x-1=1. The square root of 1=1 or -1, as both numbers squared equal 1. What, then, is the square root of -1?

Within the concept of number as one-dimensional (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the “devil’s interval.”) Euler and others sought to limit the progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain–the complex domain. Instead of avoiding the paradox presented, by thinking of the square root of -1 as imaginary, or impossible, Gauss asked, in what higher domain must such a number exist? In other words, a shift in hypothesis, which his student, Bernhard Riemann, would later designate as changing from {n} to {n}+1 dimensions.

Gauss elaborated his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He says he was only waiting (over 30 years) for a suitable place in which to announce his new hypothesis to the public.

Extending The Hypothesis

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter {i.} (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

Gauss conceived of the complex domain as a domain of two dimensions, in which the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. If the domain of positive and negative numbers is thought of as a one-dimensional series, the complex domain can be thought of as a series of one-dimensional series. (See Figure 1.) Movement within a series is associated with the concept of positive and negative. Movement from one series to the next, is associated with +{i} and -{i}.

In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative; the other dimension is associated with +{i} and -{i}. The complex domain is {one} domain, indivisible, of two dimensions. A new hypothesis, under which the positive and negative, {i} and -{i}, are made congruent (harmonic) with each other. In this new domain, all numbers are of the form {a}+{bi}, where {a} (short for 1x{a}) designates the positive-negative dimension, and {bi} designates the +{i}-{i} dimension, (1+{i} dimensions). This is not a combination of two different numbers, but one number, with two parts. One-dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where {b}=0. Complex numbers where neither {a} nor {b} is 0, are called mixed complex numbers.

One Dimension, and Complex Domain

Reflect on the difference between the domain of one-dimensional numbers and the complex domain.

In the domain of one dimension, unity is twofold, 1 and -1. In the complex domain, unity is fourfold, 1, -1, {i,} and -{i}. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its fourfold unity, each number has four associates, found by multiplying that number by -1, {i,} -{i}. For example, {a}+{bi} is associated with -{b}+{ai}, -{a}-{bi}, {b}-{ai}. (The reader can confirm this for himself, by multiplying {a}+{bi} by {i}, -1, -{i}, respectively.)

There is a special, unique relationship in the complex domain–the relationship between a number {a}+{bi}, and its conjugate, {a}-{bi}, that is, when the sign of {i} is reversed. The product of a number and its conjugate is {a}2+{b}2, and is called its Norm.

Gauss then investigated the nature of prime numbers in the complex domain. Just as is the case in one dimension, all whole numbers are either prime or composite, in the complex domain. However, not all one-dimensional prime numbers remain prime in the complex domain. For example, in the complex domain 2=(1+{i})(1-{i}); 5=(1+2{i})(1-2{i}), 13=(3+2{i})(3-2{i}). In fact, Gauss showed that all one-dimensional prime numbers of the form 4{n}+1 are no longer prime in the complex domain, but all one-dimensional prime numbers of the form 4{n}+3 remain prime. (All one-dimensional prime numbers are either of the form 4{n}+1 or 4{n}+3. But, not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain–mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1+2{i} is a mixed complex prime number because its Norm, 12+22=5, is a one-dimensional prime number. So is 1-2{i}.

Reflect further on the difference between the one-dimensional domain of positive and negative numbers, and the complex domain. The complex domain is not simply the one-dimensional domain in two directions, as in the fraud perpetrated by Cauchy. {It is an entirely different domain,} lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain which determines the singularities.

Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you’re in? Through the creative powers of your mind. If 5 is a prime number, you’re in one dimension; if not, you may be in the complex domain. But, maybe not, as Gauss speculates about the possibility of numbers of dimensions higher than two.

For further reading see, in English translation, Gauss’s “The Metaphysics of Complex Numbers,” 21st Century Science and Technology magazine, spring 1990.

From Nicolaus Of Cusa To Leonardo Da Vinci: The “Divine Proportion” As A Principle Of Machine-Tool Design, Part I

Can You Solve This Paradox?

by Jonathan Tennenbaum

The following two-part discussion is intended to prompt a richer reflection on what was presented earlier, concerning Analysis Situs, the paradox of “incommensurability” in Euclidean geometry, and Nicolaus of Cusa’s discovery of a higher geometry based on “circular action.” At the same time, I will set the stage for a new series of pedagogical demonstrations, to be developed in coming weeks.

When you have encountered a new physical principle, you cannot just put it in your pocket and walk away. The new principle, if validated, implies a more or less revolutionary change in the entirety of existing knowledge. We have the task of integrating the new principle (“new dimensionality”) into a new, comprehensive hypothesis-system, incorporating the results of all pre-existing valid demonstrations of principle (i.e., the valid side of existing knowledge), as well as the new demonstration, as a new manifold of “dimensionality N|+|1.” What is the measure of the change in the per capita productive power of society, associated with the “impulse ratio” (N|+|1)/(N)? And how can we push the new manifold “to its limits,” uncovering new experimental anomalies which will provide us the stepping-stones on the way to future manifolds N|+|2, N|+|3|,|…?

From Cusa to Leonardo and Beyond

Would this sort of process be a fair way to characterize what happened during the 50-year period from Nicolaus of Cusa’s <cf2>“De docta ignorantia,”<cf1> to the collaboration of Leonardo da Vinci and Luca Pacioli on the “Divine Proportion”? Is it valid to conceptualize the scientific developments of the European Renaissance, from the Council of Florence through Leonardo and beyond, as a process of “integrating” Nicolaus of Cusa’s crucial discovery, with the best previous accomplishments of Classical Greek, Arab, and other European civilization?

Before entertaining the possible merits of such a working hypothesis, we should first make sure to reject any temptation to impose “linearized” misinterpretations on what Nicolaus of Cusa actually discovered. Here, as always, there is no substitute for “re-experiencing” the {process} of discovery, which at the same time constitutes its real {content.}

Among the most “tempting” and commonplace misinterpretations, for present-day readers, is to substitute naive visual imagination’s image of circular motion in empty space, in place of the radically different ontological conception of “circular action,” which Nicolaus actually adduced in his discovery. The promotion of this error by the Venetian agent Paolo Sarpi and his successors, as a willful fallacy, was key to the Enlightenment assault on the European Renaissance. Among other things, it provided the basis, via Galileo, Newton, D’Alembert, Lagrange, Euler, et al., for the elaboration of a so-called “analytical mechanics” as the model for an “Establishment science,” thoroughly “sterilized” against the seeds of discovery.

Circular Motion and Circular Action

Yes, there is a connection between the visible phenomena of rotation or circular motion, and Cusa’s principle of circular action. But the connection is that of a shadow to the real object, whose existence it lawfully reflects.

Two brief quotes from Nicolaus of Cusa himself might be helpful in this context. Both are taken from his mathematical essays on the quadrature of the circle and related topics. The first emphasizes the <cf2>Analysis Situs<cf1> principle of “relationship of species” as crucial to his discovery:

“Since polygons are not magnitudes of the same species as the circle, it is still the case, even though we can always find a polygon which comes closer to the circle than any given polygon, that among things, which can be made smaller or greater, the absolutely largest can never be attained in existence or possibility. In fact, the area of the circle is the absolute maximum relative to the areas of the [inscribed] polygons, which are capable of being more or less and therefore cannot reach the circular area, just as no number can ever attain the encompassing power of the Unity, nor the Composite the power of the Simple.”

Full Scope of Circular Action

Another essay ends with a magnificent stretto, in which Nicolaus reveals the full scope of his conception of “circular action,” encompassing the relationship between hypothesis, higher hypothesis, the hypothesis of the higher hypothesis, and “the Good”:

“We assert, therefore, that there exist beings of the nature of the circle, which could not be their own origin, since they are not like the absolutely greatest circle which alone is eternity. The other circles, which, indeed, seem not to have a beginning and an end, since they are conceived through abstraction from the visible circle, nevertheless, since they are not infinite Eternity itself, are circles whose being derives from the first, infinite and eternal circle. And these circles are, in a certain way, Eternity and complete Unity relative to the polygons inscribed in them. They possess a surface which incommensurably exceeds the surfaces of all the polygons, and they are the first images of the first, infinite circle, even though they cannot be compared with the latter on account of its infinity. And there are beings having an unending circular motion around the being of the Infinite Circle. These contain within themselves the power of all the other species, and from their enveloping power they develop, in imitation, all the other species; and, beholding everything within themselves, and beholding themselves as the image of the Infinite Circle, and through beholding this image–themselves–they raise themselves up to the eternal Truth or to the very Origin. These are the beings endowed with Reason, who comprehend everything by the power of their minds.”

Machine-Tool Design Prototype

By what mode of action do we expand the “enveloping power” of the human race, exercising increasing dominion over the Universe, and knowing Reason in the mirror of its own active participation in developing the Universe? What could be more fruitful, to deepen our understanding at this point, than to follow the track of Nicolaus’s discovery into the busy workshops and “design bureaus” of Leonardo da Vinci and his Renaissance friends! Here is the prototype of the “strategic machine-tool design sector,” which has been key to the emergence and survival of the modern nation-state up to the present.

Much oligarchical effort has been expended, over the centuries, to mystify and conceal the “machine-tool principle” underlying Leonardo’s work in all fields. For example, Leonardo is often portrayed as a “speculative genius” whose designs were wildly impractical in his day. As a matter of fact, much of Leonardo’s time was spent in direct collaboration with machine-building workshops and factories, as well as with construction teams involved in infrastructure and other projects, developing solutions to problems as they came up. Thus many, if not most, of Leonardo’s actual designs were implemented in his day.

Another malicious piece of gossip, spread by Joseph Needham among others, was that Leonardo made “no fundamental breakthrough” in the principles of machine-design. That assertion is commonly coupled with the assertion, that Leonardo was not a scientist, and that the real breakthrough, leading to the Industrial Revolution, came with the formal mathematical physics of Galileo, Newton, et al. For example, a book on Leonardo’s engineering work, published by one L. Olschki in 1949, claims: “The technical principles employed by Leonardo were hardly different from those handed down from antiquity and the Middle Ages…. He never attempted to frame new theoretical approaches or theories of mechanics.”

Leonardo’s Breakthrough

Leaving aside such malicious nonsense, get out a good collection of Leonardo’s sketches. Concentrate particularly on his designs for machines and mechanical devices of machines. Looking over those sketches, ask yourself: What was Leonardo’s crucial breakthrough in these matters? What is stunning, revolutionary, about Leonardo’s approach to the design of machines, and related matters, which went decisively beyond what had existed before? I am not talking about individual “inventions,” so often played up as isolated entities; I am asking for a “One.”

Whoever tends to read Nicolaus of Cusa’s principle of circular action as merely a form of “motion,” in the manner indicated above, will be plunged into a rather profound paradox at this point.

Looking at Leonardo’s designs, what do you see except mere mechanical linkages–assemblies of gears, pulleys, and levers, which transmit motion from one place and direction to another, without “adding” any new motion? Didn’t Archimedes already describe the basic mechanical principles involved, as typified by the action of the lever or pulley? Or is there something more than just “mechanics” in Leonardo’s machine-designs, something absolutely banned from the textbooks of “analytical mechanics,” but which is a key to the unprecedented rate of increase in the productive powers of labor, unleashed by the Renaissance?

To be continued.

From Nicolaus of Cusa to Leonardo da Vinci:

The “Divine Proportion” as a Principle of Machine-Tool Design Part II

CAN YOU SOLVE THIS PARADOX?

by Jonathan Tennenbaum

Lyndon LaRouche’s discoveries in physical economy provide the key to unlocking the secrets of Leonardo da Vinci and the Italian Golden Renaissance, to a degree which would have been impossible at any earlier time, before LaRouche’s work.

Observe that the leading features of Leonardo’s designs for machine tools and other machines–most emphatically including the method of “non-linear perspective” employed in his drawings–all cohere with one central conception:

The emergence of {nation-state physical economy} as a {living process} based on development of the cognitive powers of individual members of society, imposes a unique “curvature of space-time” upon the Universe, such that each and every particular must be conceptualized and measured by reference to the “horizon” defined by that curvature.

That central conception subsumes the following features and consequences@s1:

1. A physical economy is a special type of living process, whose maintenance and growth depends on development of the cognitive powers of the individual members of society.

2. The action of human Reason upon the Universe, occurs {solely} through the instrumentality of living processes. That is, through the activity of sovereign human individuals, working in and through society, upon the expanding domain of Man-altered Nature which constitutes the “substrate” of physical economy as a living process.@s2

Reason’s Dominion Over the Universe

3. Hence, Leonardo da Vinci’s conception of non-linear-perspective curvature is based on a relationship of Nicolaus of Cusa’s “species”: Living processes exercise increasing dominion over inorganic processes, and human Reason exercises increasing dominion over the entire Universe via its dominion over living processes (i.e., human individuals, the physical economy, and an expanding biosphere).

4. In particular, the required notion of “technology,” appropriate to the maintenance and development of physical economy, {cannot} be derived from inorganic physics. No mere physical laws, of the sort suitable to “inorganic physics,” could ever account for the impact of a new machine or other invention on increasing the productive powers of labor.@s3 Although it is possible to design a machine on the basis of a simple hypothesis, we cannot measure its economic {effect} that way. The survival of human society, therefore, depends on shifting attention from the mere “engineering approach” of simple hypothesis, to encompass the “horizon” defined by higher hypotheses. Leonardo’s drawings have the included purpose, to communicate exactly that conception.

5. For these and related reasons, Leonardo’s studies of anatomy, and his collaboration with Luca Pacioli on the “Divine Proportion,” were decisive inputs to his approach to machine-tool design. Leonardo sought to apply to the design of machines, a reflection on the principles and means by which living organisms exercise dominion over the inorganic domain.

6. When a living organism incorporates non-living material into its active domain, it {imposes} its own characteristic {ordering} upon that material. (One day soon, the environmentalists might turn against plants and trees, denouncing them for imposing their “authoritarian values” upon poor, defenseless dirt!)

Harmonic Proportions

7. Leonardo, Pacioli, and others demonstrated how the peculiar space-time ordering of living processes finds its lawful {visible} expression in self-similar elaborations of the harmonic {proportions} derived from the division of the circle and sphere. The latter all belong to the dominion of the circle’s “Golden Section.”

8. This sort of approach points to a principle of {harmonic composition of motion} for the evolution of machine-tool designs integrating an increasing number and density of degrees of freedom. The harmonic principle of the “Golden Mean” will be reflected, not necessarily in the individual machine per se, but rather in the context of the evolutionary series of species of technology. The latter constitutes, on the one side, a central functional feature of the growth of physical economy as a living process, while at the same time embodying an ordering of mutually inconsistent theorem-lattices of increasing “power” under the principle of “higher hypothesis.”

9. In the continuation of this process, with the increase in energy-flux density and precision of machine-tool design, discoveries in microphysics oblige us to replace the concept of “motion” by a generalized notion of “harmonically ordered physical action.” The approach of Leonardo (and later Kepler) received preliminary, but brilliant confirmation in the domain of atomic and nuclear physics.

Beauty of Leonardo’s Drawings

10. Hence, the stunning beauty of Leonardo’s drawings! He communicates not merely a set of “specifications” for a machine, but a {conception}–a conception of that invention as seen in the perspective defined by the creative principle of the Universe as a whole. By this method of “non-linear perspective,” Leonardo is able to communicate the creative process itself, and not merely a particular product. Thus, Leonardo’s designs and machines are vehicles for the communication of higher ideas, for the generation of higher qualities of labor power. Like great Classical music, they embody Reason’s ironic reflection on the principle of life.

Notes:

1. Besides Lyndon LaRouche’s writings, I would especially recommend juxtaposing to our discussion, the relevant articles by Dino de Paoli on Leonardo and related matters.

2. Have you stopped to consider the significance of the fact, that we need a brain in order to think? Actually, we need more than that: To develop, individual creative reason must continually expand and intensify its “active domain.” By the term “active domain,” I mean, roughly, the region of the Universe which is directly subject to the deliberate actions of a given individual. The growth of the active domains of members of society, is obviously correlated to increase in per capita and per hectare consumption of energy and other components of the market baskets, as it is to increase of the productive powers of labor. To the extent that the creative contributions of individuals are communicated and realized by society, their active domains may encompass the entire physical economy, and more. Would it be justified to conside, that growth of the “active domains,” in some respects represents an enlargement of the physiological processes of the brain, as an instrument for the development and realization of valid ideas?

3. Some might reject such a categorical proposition as preposterous. Don’t we know countless examples of inventions, whose labor-saving effects can easily be explained by any physics student? For example:

@sb|Levers, pulleys, and similar devices permit a single man to lift a weight which would otherwise require the muscle power of many men.

@sb|Ball-bearings and similar devices improve the performance of an existing machine by reducing friction and wear (a major focus of Leonardo’s work, by the way).

@sb|Steam engines and other power-generating devices multiply the amount of useful power at the disposal of an industrial operative, etc.

What could be more obvious, than the increase in productivity, caused by the above-mentioned inventions? Yet, such a casual affirmation overlooks at least one decisive point: What about the direct and indirect {costs} (in real terms) of developing, producing, and maintaining a given machine or technical improvement? How can we be {sure,} in any given case, that that additional cost will not actually exceed the saving in labor, or other benefits provided?

Observe, for example, the vastly greater complexity and intensity of motion of Leonardo’s machines, compared to the rudimentary gadgets of pre-Renaissance Europe. Even the simple act of introducing ball-bearings and related devices into machine design, adds new degrees of freedom to the system as a whole, raising the demands on the {quality of labor} required for the manufacture and maintenance of the machine. Actually, the purpose of the machines themselves, as means for urban-centered development of the nation-state, is not to “economize labor” per se, but to rather to {uplift its cognitive quality}. (Thus, while industrialization subsumes as a necessary aspect the reduction and final elimination of manual labor, a healthy industrial society actually increases the “work load” which must and can be borne by the average member of society.)

Reflecting upon such matters, we realize that the increase in the productive powers of labor, associated with the introduction of a new machine into the productive process, can hardly be determined from a mere analysis of the machine itself. It requires that we carry out a measurement of the entire economic process within which a proposed new machine design is to be “inserted.” Since the insertion of a new technology changes the characteristics of the economic process, that measurement must take into account, not only the present, but also its projected development in the future. In the last analysis, there is no adequate answer which does not center on the rate of improvement of the cognitive powers of labor, associated with any given “pathway” of economic development. Herein lies the cause of the essential “incommensurability” of real economic growth, relative to any linear sort of engineering or “systems analysis” standards of measurement. The significance of the Golden Section (“Divine Proportion”) comes once more to the fore.

How To Purge Your Mind of “Artificial Intelligence”: Introduction to a new pedagogical series

by Jonathan Tennenbaum

One of the reasons why you don’t really understand the significance of Plato’s five regular polyhedra, is because you have never questioned your own, completely unfounded assumption, that the sphere is a figure in 3-dimensional space.

We all remember the type of horror movie, where the Earth has been invaded by alien beings with the capability of taking over the minds and wills of human victims. The victims look the same as before, but their brains have been hollowed out or short-circuited by some sort of implanted devices, so that they effectively are no longer human any more.

However frightening the experience of such a horror movie, it hardly compares with the real horror story, of what standard school mathematics education has done to the minds of nearly everyone. As a result of what was done to you, the creative, cognitive processes of your mind are “turned off” most of the time, even when you are engaged in what you consider to be intellectual effort. Instead, a form of “artificial intelligence” operates, that was installed through school education and related, mostly early experiences. Under adversive conditions of intense cultural pessimism, even those who have known the joy of real thinking, will tend to revert back to those previously implanted, school-room (i.e. “career-oriented”) habits of “artificial intelligence”.

This “artificial intelligence” (otherwise known as Aristotelianism) excludes from consideration exactly that, which is the object of human cognition. The tactic is to divide the Universe into “sufficiently small” domains of experience, in which the cognitive considerations, thus ignored, are assumed “not to matter”. Afterwards, the flattened, linearized pieces are fitted together again to construct a parody of human knowledge. The typical symptom of artificial intelligence is an obsessive fixation on the presumed existence of “objective, hard facts” –a fixation whose most revealing manifestation, perhaps, is the inability to conceptualize the fundamental significance of the sphere and the five Platonic solids.

A remedy is at hand, however. This being the best of all possible worlds — and not the vicious, artificial world of a horror film– the condition just described is more than reversible. By rooting out the problem at its deepest origins, we may be enabled, not only to restore our own creative powers to fullest blossom, but also to discover a means to render future generations forever immune to the disease of oligarchism.

That is the issue we are committed to fighting through, in the following series of pedagogical discussions on the sphere and the Platonic solids. Here, on this battlefield of choice, we are resolved to smoke out and defeat the internalized enemy of the human mind. In the process, all the most “advanced” topics we met with in our previous work — including Gauss’ biquadratic residues, modular functions and the Gauss-Riemann domain of multiply-connected action — will reveal themselves as the most elementary sorts of notions, already implicit in the original discovery of the regular solids’ uniqueness, more than 2500 years ago. All has been buried under the myth, that a non-existent “plane geometry” was the starting-point for Greek mathematics.

Start, therefore, with the following task: Given a clear night in which the stars are visible throughout the sky, how can we make a preliminary, but conclusive measurement of the curvature of the Universe?

The First Measurement of the Universe –

Part II of a series

by Jonathan Tennenbaum

The spread of mythologies in the name of “history of science,” began very early.

The Greek historian Herodotus reported, that geometry was invented in Egypt and transmitted from there to the Ionians. He also claimed, however, that geometry arose in connection with the practical problem, of measuring and reconstructing the division-boundaries of agricultural fields after each periodic flooding of the Nile (geo-metry = earth-measurement). If Herodotus intended the term, “geometry,” to signify some specialized knowledge relevant to surveying, there may be an element of truth to the latter assertion; but if he meant the geometry of Thales, Anaximander, Pythagorus and Plato, then the account is certainly wrong and highly misleading. This story of geometry’s alleged practical origin (whether Herodotus is to blame for it or not), found its way into the subsequent histories of science, up to this day. It reminds us of the theory of the “opposable thumb” and other absurdities of Friedrich Engels’ “dialectical materialism.” Contrary to this, the overwhelming evidence — including that contained in Plato’s Timaeus, in the Vedic and other ancient calendars, as well as the implied navigational skills of the “peoples of the sea” –, demonstrates that {all physical science originated in astronomy}. Astronomy, in turn, was cultivated in some form already tens, probably hundreds of thousands of years before the classically recorded Egyptian civilization, by maritime cultures spread across the globe. Geometry begins with nothing less, than Man’s attempt to measure the Universe as a whole.

This should indicate that the practice of basing school mathematics education on so-called “plane and solid geometry” — a practice that has dominated European education, despite the Renaissance, for over two millennia — is profoundly in error. Henceforth, the teaching of geometry should begin with the {failure} of plane and solid geometry, to account for the most elementary features of visual astronomy. That failure has a precise, knowable structure; to characterize that singularity, is to carry out the first scientific measurement of the Universe.

Bearing in mind that we are dealing with matters of fundamental importance, we need not apologize for the elementary nature of the following account. It should help refresh the mind on familiar matters, while opening some new flanks at the same time.

Constructing a Star Chart

Imagine you are a prehistoric astronomer, attempting to produce a star chart on a clay tablet or papyrus sheet. You require that the chart should accurately represent the shapes of the familiar constellations of stars, and also the mutual orientations of the various constellations relative to each other, so that the chart can be used for navigational purposes.

As far as individual constellations are concerned, you find no difficulty drawing any one of them separately. You just naively transfer the image of what you see, {as if unchanged}, to the tablet. No problem? But, as you begin to map {larger} portions of the sky, adding more and more constellations to the chart, difficulties arise. The constellations don’t fit together. You begin again, with another constellation as starting-point. Once again, things don’t fit. Why? Although in each case you can specify the point at which the mapping process begins to break down, the underlying cause clearly lies {outside} the specifics of each attempt.

This problem embraces paradoxes, of the sort any curious child will have observed. I stand up and look straight ahead at some point on the horizon. Now I look to the right of that, and more to the right, and so on, until, by continuing my action of “looking to the right,” I turn all the way around and come back to the original point…from the {left}! Or instead, if I start by looking straight ahead as before, and now look {up}, and keep turning my head in that “upward” direction further and further, I end up bending backward until I am moving my head {downward} toward the ground and seeing everything upside down!

(Let no one laugh off these simple paradoxes of linearity, who is not prepared, for example, to explain to any child or adult, how it can happen that the Earth can be in two different days, depending on the position on the Earth’s surface, at one and the same moment in time.)

These sorts of paradoxes give rise to unavoidable, interwoven {periodicities} in our attempt to construct a star chart — as for example when I attempt to represent the observer’s looking “to the right” and “upward” by motion “across” and “up” on the chart.

(At a more apparently “advanced” level, the same problems plague the cartesian-like coordinate systems still used by astronomers to record the positions of the stars. To describe one such system in a perfunctory manner: Given any star, let “y” be its angular “height” above the horizon (i.e., the magnitude of angle from the position of the star “downward” to the point “directly below it” on the horizon), and “x” the angle along the horizon from that point to some chosen fixed point on the horizon. We might thus represent the position of any star by a point in the cartesian plane, whose rectilinear coordinates are proportional to x and y, repectively. The resulting mapping, however, grossly distorts the shapes and angular relationships of the constellations, especially those in the vicinity of the overhead or zenith-point, where the mapping “explodes.”)

This mere descriptive approach, however falls short of identifying the underlying cause of the problem. In particular, it does not answer a crucial question which ought to pose itself to us: Does the difficulty arise only when we want to map {large portions} of the sky; or is it already present, albeit so far unnoticed, in the attempt to represent any {arbitrarily small} portion of the sky?

The Spherical Bounding of the Universe

To progress further, we need to examine the internal characteristics of that action by which we, as ancient astronomers and navigators, are attempting to measure the Universe. The ancient astronomer makes a series of {star sightings}, measuring, in effect, the {rotation} from one direction in the sky to another. Imagine that a movable “pointing-rod” of fixed length is fixed at one end to a universal joint at our point of observation. Observe that the tip of that rod moves on a {spherical surface} whose center is the fixed pivot point, and whose radius is the rod’s length. Imagine we were to construct a transparent spherical shell of that dimension around the center, and mark the shell at each position where the end of the rod points to a star. The result would be a spherical star-chart, whose markings would coincide {exactly} to the observed star positions when viewed from the center of the sphere (and only then).

We have demonstrated a {spherical bounding} of our action to measure the Universe! The sphere is not an object in the sky, but a determinate feature of our act of measurement: a representation of its underlying {ordering-principle}. Does that make it arbitrary or “purely subjective”? By no means! This phase of astronomy is a necessary step in the self-development of the Universe, and thus an imbedded characteristic of the Universe itself.

It now appears, that the ancient astronomer’s problem of drawing a star chart on a clay tablet or papyrus is equivalent to the problem of mapping the inner surface of a sphere onto a plane surface. (Note: “inner surface of a sphere” signifies — paradoxically enough — a {completely different} geometrical ordering-principle, than the “outer surface.” “Inner surface” signifies the ordering of the surface with respect to the spherical center only.)

There exist innumerable possible methods to attempt such a projection, each of which fails in a different way. The simplest is the method of central projection onto a plane outside the sphere, defined as follows: For any locus on the inner spherical surface — corresponding to a pointing-direction from the center — prolong that direction outward until it intersects the plane. Readers should thoroughly investigate this species of projection with the help of a transparent plastic sphere and a suitable light source, noting several important characteristics.

For example: the action of simple rotation (e.g. of the pointing-rod) generates a {great circle} on the inner surface of the sphere; the projected image of a great circle, so constructed, produces the effect of a {straight line} on the plane surface. Encouraged by that result, examine the effect of the projection on various arrays of great circles. At the same time, observe that the projection maps only a {half} of the spherical surface, a hemisphere, onto the plane. The boundary of that hemisphere — a great circle whose location we can determine by cutting the sphere by a plane surface parallel to the projection-plane — defines a {singularity}: the mapping “blows up” when we approach that boundary circle. In the vicinity of the boundary, the projection introduces wild distortions relative to the relationships on the inner spherical surface. The least distortion apparent occurs farthest away from the boundary, in the “polar region” of the hemisphere.

The “catastrophic” distortions near the boundary, and the circumstance, that only half of the sphere is mapped (or actually much less, if we want to avoid the worst distortions), suggests to our ancient astronomer the following tactic: Instead of trying to map the entire spherical surface (or night sky) at once, divide the surface into regular, congruent regions, and construct the “truest possible” mapping for each one. The combination of such sectoral charts would hopefully fit together to replace a single one. Note, that a complete set of central projections, of the sort we now envisage, corresponds to a {regular array of great circles} on the sphere, each constituting the singular boundary of the corresponding mapping.

Out of the corner of our mind’s eye we might already have anticipated a new source of failure: The attempt to “fit” the mappings together at the edges of the chosen regions, will result in {discontinuities}!

We have entered into the domain governed by the five regular solids. We propose to explore that domain, from a new standpoint, in next week’s pedagogical discussion. To finish this one, consider the following:

We saw, that in order to reduce the effect of distortion in each spherical mapping to a minimum, the portion of the spherical surface mapped, should be made as small as possible. But, how finely can the surface of the sphere be subdivided?

The characteristic of linear, planar, solid or cartesian geometry in general — a characteristics which distinguishes such hypothetical, “virtual” geometries from the real Universe — is the purported possibility of unlimited, self-similar subdivision or “tiling” of space. Take a square in the plane, for example; by connecting the midpoints of the opposite sides, we can divide the square into four congruent subsquares, and so on ad infinitum. An analogous construction applies to any triangle. Similarly, a cube in so-called “solid geometry” can be divided into 8 (or any cubed number) of congruent cubes.

What about the inner surface of the sphere? Take the division of the spherical surface into six congruent, curvilinear-square regions — i.e. a regular spherical cube. What happens when we try to subdivide those regions into smaller, congruent curvilinear squares? What happens for the division of the spherical surface, defined by the regular octahedron, and the other regular solids? What is the {common source} of the barrier to further subdivision?

The First Measurement of the Universe –

by Jonathan Tennenbaum

Part III: Anti-Deductive Ordering Principles

How does the One subsume the Many? The key to the Enlightenment’s “coup d’etat” against the Renaissance, was to remove the Platonic conception of higher hypothesis/change from its newly reestablished, leading role in scientific work, and replace it by the principle of {logical-deductive consistency}. Britain’s Hollywood-style promotion of Newton and his famous Law of Universal Gravitation — a “discovery” actually lifted out of the pages of Kepler’s “New Astronomy” — marked a late turning-point in this neo-Aristotelian coup. Generations of gullible minds were seduced by the promise of a “world formula”: a single mathematical law, or set of laws, from which the entirety of physical phenomena could supposedly be derived by logical deduction and calculation. The British-Venetian propaganda machine succeeded in installing this cultish idea, which Leibniz had denounced and torn to shreds in his correspondence with Clarke and elsewhere, as the academically-accepted “norm” and “ideal” of the natural sciences to this very day.

Do you think you have been immune to this operation? How many times a day, in organizing, do you try to explain to contacts the relationship between two events X and Y, by attempting to prove to them, in a deductive fashion, one of the following three propositions:

X implies Y;

Y implies X;

some Z exists, that implies both X and Y.

What if the most crucial events occuring in the Uuiverse — including those most intimately related to each other — cannot be reduced to deductive consistency? In other words: what if the actual ordering of cause and effect in the Universe is anti-deductive?

Take, for example, the question:

“Why does the well-tempered system permit only a discrete set of musical tones (12 in number) within each octave? Why does it reject an unbroken continuum of pitches, including the pitches {in-between} the 12 pitch-levels of the well-tempered scale?”

Consider, as a response, the proposition:

The necessity of a specific, discrete series of musical pitches for well-tempered polyphony, flows from the same underlying cause, which determines the {impossibility} of a singularity-free mapping of the celestial sphere onto a plane surface.

Does that mean to say, that we can {deduce} the well-tempered system of music from the geometrical properties of the sphere? Wouldn’t we thereby be falling into a species of irrational, cultish belief: “sphere-worship,” or (in an earlier phase of our discussions on these issues), “spiral-worship,” or “the cult of the Golden Mean”?

Reflecting on the difficulty experienced by many in grasping the significance of the regular solids, my attention was called to a crucial step in those solids’ derivation, which few people have even noticed, and even fewer have thought through in a rigorous way. The point in question touches upon the much-misunderstood concept of the “celestial sphere.” Omitting or glossing over the relevant step, opens the door to serious confusions and misinterpretations of a sort which appear to be rampant among us, and can derail the whole effort. It is therefore urgent to clarify this matter now, before proceeding further along our orbit. The habit of focussing unblocked attention on just such matters, as the professionally-educated trend to dismiss as “too trivial to be worth thinking about,” which most often yields flashes of insight into the most advanced issues in science.

These are some of the reasons, why I deliberately began last week, not with the sphere per se, but with astronomy and the problems posed by the attempted construction of a flat star map. Note: I made no assumptions about the shape of the heavens or anything like that, but set out instead to {measure} — first in rough way, by attempting to draw the sky directly onto a flat surface, and then using a pivoted pointing-rod as an instrument. That {action} of attempted measurement, called forth an ordering principle, the which (for reasons indicated last week) I qualified as “spherical.”

We must, however, not gloss over a very crucial point here: The ordering principle in question is {not directly visible to the eye}; the immediate result of my measurement effort was a pattern of distortions and {discontinuities} — singularities of “failed” mappings!

So, forget the sphere as a visible form. Get it out of your head entirely. It tends to drag your thinking into a downward, aristotelian direction. Don’t say “sphere,” until you have generated the concept.

(An aside: Remember, Baby Boomers tend to throw words at things, as a substitute for working problems through. This is called “verbal skills”: the magic powers by which Baby-Boomers were typically raised and taught to manipulate their liberal parents, and to succeed in school, university and career … reinforced, naturally, by an occasional temper-tantrum. For this and related reasons, it is {mandatory} that the reader actually carry out the experiments indicated in last week’s, this and the following pedagogical discussions. The worst mistake of all, is to think you don’t need to actually carry out a construction or related experiment, because you presume you already know what the result will be, or can discern it from the text. That is pure information theory, pure post-industrial ideology. DO the experiments described. Don’t just do them in your head, don’t just try to imagine them, don’t watch somebody else do them, don’t read a description of them… DO THEM! Otherwise, you may read the words and make interpretations, but you won’t know what I am really talking about. Afterwards you may devise more elegant and powerful ways to evoke the relevant concepts, for which I and others will be most grateful.)

Make sure you have really worked through the main experiment from last week — the attempt to represent the visible arrangement of stars in the sky, on a flat surface. If a full, clear sky is not available, you can do a roughly equivalent experiment in the middle of a room. Take a large piece of paper, and try to draw your whole surroundings, as they appear to you, on that flat surface.

Leaving aside for a moment the spherical projections described last time, let’s examine the problem anew from the standpoint of multiply-connected circular action. Taking the cue from Leonardo da Vinci and later Johannes Kepler in his “Snowflake” paper, examine how the most elementary, multiply-connected features of circular action are expressed in the harmonic motions of the human body.

Standing straight from the vantage-point of your drawing experiment, point your arm straight ahead. Now, rotate the arm to point to the right (or left). That defines a first interval of rotation. From that position pointing right, now rotate into the straight upward direction. That defines a second interval of rotation. Finally, rotate down from the upward direction to the straight-ahead direction. With these three rotations you have generated a triangle: not a visible triangle, but a {triangle of rotational action}.

Now compare the two intervals: {forward==>up}, and {right==>up}. Observe, that if and rotate our whole body around to face to the right, then the interval {forward==>up} now becomes the interval {right==>up}. Observe also, that if we bend forward at the waist, so the trunk of our body is pointing straight forward, then the motion of our arm, which produced the interval {right==>up}, now does {right==>forward}. In this way, by {rotating the rotations}, each of the three rotations forming our “triangle” can be rotated into any of the others. We have an equilateral rotational triangle! The rotations which carry each of those three rotations into any of the others, constitute the {angles} of the triangle. (Note, that the rotations in question are all of the type described in ordinary geometry as “right angles.” Anticipate the paradox: a triangle whose angles are all right angles!)

Explore, in the same way, the interrelations of the total of 12 mutually-similar rotational intervals and 8 equilateral rotational triangles arising from what Kepler identified as the astronomically-derived, “three distinctions” embedded in the construction of any animal: forward-backward, up-down, left-right. Don’t confuse them with coordinate axes in cartesian space; we make no assumption of scalar, linear extension here, but only angular, rotational action, implicit in our astronomical measurements.

Indeed: What is the crucial distinction of the manifold of rotational action, we have begun to explore, as compared to a flat, cartesian manifold?

Note the following: In a flat plane, for example, the linear displacement “to the left-and-right” vis-a-vis motion “up-and-down” are apparently {independent} degrees of action. If we, for example, move one unit distance to the right in a plane, and then one unit upward, the result will be the {same}, as if we would first go up, then to the right. Compare the composition of motions, that constituted our “rotational triangle.” Are rotation “up,” and “to the right,” for example, strictly {independent} dimensions of action? Or is not the very existence of the equilateral right-angled triangle, just generated, characteristic of the multiple-connectedness of the rotational manifold?

— —————–

Note: Our ongoing pedagogical exploration should provide guide-posts for sorting out the real history of ancient geometry, and demolishing encrusted mythologies. The following note from my own, preliminary readings, will hopefully encourage comments and contributions by others.

An 1870 German treatise on the development of geometry before Euclid, refers to an ancient Egyptian treatise on geometrical constructions, the so-called Rhind papyrus from 1100-1000 B.C. According to the German author, that papyrus documents familiarity with the regular solids, as well as the elements of spherical (i.e. rotational) geometry. The papyrus contains a note to the effect, that it is a copy of a treatise dating from much earlier, probably to 3400-3200 B.C. Much later, around 600 B.C., the Ionian Thales devoted much of a lengthy visit to Egypt, to studying the methods and results of Egyptian astronomy. Back in Miletus, Thales and his school, including most notably Anaximander, reworked the Egyptian results and launched a revolution in Ionian-centered scientific development. The next phase appears to eminate from the philosophical-political movement of Pythagorus, who (among other things) is credited by later Greek writers with discoveries concerning the construction of what were referred to as the “cosmic figures” (kosmica skema). Of course, the Greek sense of “cosmic” has nothing to do with the present-day connotation of the mystical or other-worldly. Quite the opposite: the Greek expression connotes “ordering” in the sense of “ordering of the Universe” or “the Universe as ordering principle.” This is the platonic conception Wilhelm von Humboldt’s brother Alexander intended as the title of his many-volume summary of the natural science of his day: “Kosmos.”

It would also be worthwhile to investigate the obvious astronomical origin of the Chinese “Book of Changes,” which (among other things) contains unmistakable references to the characteristic, octahedral singularity of visual astronomy, explored above in a preliminary fashion. The Chinese and Egyptian developments are evidently coherent.

Part V: The Curvature of Visual Space

by Jonathan Tennenbaum

When we attempt to relive Kepler’s discovery of the efficient ordering-principle of the solar system and its crucial empirical feature — the exploded planet between Mars and Jupiter — a chief obstacle we encounter is our own, deeply-ingrained assumptions concerning the nature of space. However much some people might scream and hurl epithets at Newton, when you scratch the surface, you often find their idea of space essentially coincides with Newton’s; indeed, it seems virtually impossible to them to imagine anything essentially different, than an infinitely-extended, featureless void in which straight-line motion (or something equivalent to it) is the elementary form of action. This typically goes together with an awful sense of smallness, the existentialist’s squatting in the middle of an endless parking lot.

Fortunately, remedies are at hand. The Universe is much, much smaller than you think. What at first glance might appear to be “merely subjective” paradoxes of visual geometry, can help us free ourselves from the prison of Cartesian space, and provide a preliminary insight into a notion of anti-Newtonian curvature of the Universe, in which no isolated events are possible. The following, experimental exploration paves the way.

For this purpose, let’s go back once again to our starting-point — mapping the stars — from a somewhat different angle. Rather than attempting to draw the heavens (or other features of your surroundings) onto a piece of paper, proceed as follows. Take a flat surface of transparent material (such as plexiglass) and fasten it somehow in a fixed position in front of you, so that you can see a chosen constellation of stars (or arrangement of objects in a room) through that transparent window. Using a marker pen, and being careful not to change your eye’s position in space (better use one eye and keep the other closed) mark onto the window the positions of the stars or other objects that you see. By construction, the positions and configuration of the resulting marks will coincide {exactly} with those of the corresponding objects — at least, as seen from the chosen vantage-point of your eye.

What is the problem with this procedure? For one thing, we evidently cannot map more than the one-half of the visible world which lies in front of us through the window, at one time. We might of course set up the window on the opposite side of our vantage point, turn around at map the other “half-world”. But the two maps do not fit together smoothly. Separating the two half-worlds is a singularity, where our ray-of-sight becomes more and more skew to the surface, and finally does not touch it at all.

This is not the only difficulty. Taking the window down from its fixed position and examining directly various parts of the image drawn on it, we that they generally {do not} match what we see at all, but become more and more distorted as we move outward from the center (i.e. the region of the window directly in front of the vantage-point). Distorted how? Take a constellation of stars, for example. As the ancients did, we assist our memory by imagining the stars of the constellation joined by imaginary lines in the sky; the resulting {shape}, as reflected in specific angles between those lines, helps us to recognize the constellation. Now look at the image of constellations which are far from the center of our projection. If we measure the {angles} made by the corresponding lines on the surface of the window, we find they are generally very different from the angles we see in the sky. Try it!

The paradoxical nature of the difficulty involved, will become clearer, it you do the following very simple additional experiment: Stand in the middle of an approximately cubical room, facing one of the walls. That wall, bounded by the edges where it meets the ceiling, floor and adjacent two walls, is clearly {square} or at anyway {rectangular} in shape; and that is exactly what you see when you look toward the middle of the wall. In particular, the angles at the four corners are obviously right angles, right? But now look directly into one of those corners. You see three lines representing the edges of the cube coming together… at equal angles of each 120 degrees! What happened to the right angle at the corner of the wall? It now appears as a 120 degree angle! How is that possible? How can an angle change just because I look at it differently? Note, that the change is not explainable as the result of a shift in the position of the vantage-point in space. That point remains the same; all that changes is the {direction} we are looking in.

You can use the device of our transparent window to verify this bizarre phenomenon. If I set up the surface parallel to the wall, and trace out the outlines of the boundaries of the wall as they appear from my vantage-point, I get a square. The images of the boundaries are straight line segments intersecting at right angles. If I now look straight at the upper right-hand corner, and hold the transparent surface at a perpendicular to my line-of-sight to the corner, what I mark on the surface is three line segments intersecting symmetrically at a common point. The 90 degre angle has now become 120 degrees!

Do you think the cubical shape of the room itself is the cause of this problem? I say no. For, imagine we would install a large, transparent sphere (for example) around our position at the mid-point of the cubical room. Taking the common midpoint of the cube and sphere as our vantage-point, we could project the image of the 12 edges of the cube onto the spherical surface. If done with great care, in fact, an observer situated at our vantage point, and looking only at the pattern of lines traced on the spherical surface, would have the impression of standing in the middle of a cube! Again, looking first toward the middle of what appears to be one of the walls, and then looking toward one of the apparent corners, the observer would experience {exactly} the same change of corner angle, from 90 to 120 degrees, as before.

(Some industrious persons should prepare “pedagogical museum” demonstrations for each local, along the lines just sketched. The key item to be procured, is a set of large (preferably at least 20 cm-diameter), transparent plastic hemispheres, which fit together to form a full sphere. Use water-soluble markers to trace the spherical equivalent of the cube, and later the octahedron and other regular solids in succession, on the surface of the sphere. Now have people look with one eye into any of the hemispheres, from a location close to the midpoint of the corresponding sphere. Note, that the great circles, corresponding to the edges of the solids, at appear as straight lines when viewed from the spherical center. In the case of the octahedron, when we look toward the middle of any face, we see what appears to be an rectilinear equilateral triangle, whose angles are 60 degrees. But when we look at any vertex, we see edges intersecting at right angles! Demonstrate an equivalent phenomenon, by placing a small, bright light source (e.g. the bulb of a small halogen lamp) at the center of the sphere, and examining the projected images of the curvilinear solids onto flat screens (e.g. heavy white cardboard) placed in different positions outside the sphere.) With a bit more care, we can demonstrate the same phenomenon of shifting angles in the observation of any constellation of stars that sweeps across a sizeable section of the sky. As we shift the center-point of our vision from one star to another, the apparent, overall shape of the constellation {changes}. This can be verified using projection on a flat transparent window.

Evidently, the cause of these phenomena is not located “out there” somewhere, not in some specific feature of the objects we are observing, but rather in the “infinitesimally small” of visual space itself.

Investigate this further with the help of the following experimental device. Take a small ball of polystyrene or a clump of putty to represent any given observation-point. (It is best to mount the ball of clump at the top of a slender rod or stick, whose lower end is fixed to a flat base). Taking something like slender bamboo skewers (thin shashlik sticks or equivalent), we can represent any given {direction} from the given observation-point, by sticking the tapered end of a skewer, pointed in the given direction, into the center of the ball or clump. So, for example, let two such sticks, stuck into the ball, point in the directions of any two stars. Note the {angle} formed by the two directions. What is the value or measure of that angle? It would appear to be nothing but the magnitude of the {rotation} necessary to rotate the one direction into the other. Note we have made an implicit assumption or hypothesis here: the notion, that for any two directions taken from our vantage-point, it is actually possible to transform the one into the other by a simple rotation.

Now consider {three} directions, represented by three sticks pointing out from the common center. These might, for example, represent the directions of three stars, as seen from the given vantage-point. What are the {true angles} formed by that triangular constellation of stars? I don’t mean here the angles we might imagine are formed “out there” between the stars themselves, as objects supposedly existing somewhere in some sort of space, hundreds or thousands or millions of light-years away; let’s avoid making any assumptions about that. Rather, I mean the angles formed directly inside a hypothetical “monad” located at the given vantage-point.

Looking at the configuration of the three sticks, perhaps you might suddenly realize something you never noticed before: Any two pairs of directions define an {angle} — a unique rotation carrying one to the other. That defines 3 angles of rotation. But this is not all! Name the three given directions A, B, C. Compare the two rotations from A to B and A to C, respectively, and recall our earlier discussion of the notion of “rotation of rotation”: From the standpoint of the direction A, the two rotations, A->B and A->C are characterized, beside a definite magnitude or angle of rotation, by two different {directions of rotation}. Between those two directions of rotation there is an {angle}, namely the angle of a rotation carrying the one direction of rotation to the other. Thus, any three directions determine not 3, but a total of {6} angles of rotation!

(Some may be accustomed to a different approach to the same relationships in terms of standard solid geometry, as follows: two directions from the common center define a common {plane}. Two such planes, defined for example by pairs A, B and B,C, intersect to form a “plane angle”. The three “additional” angles are the angles formed by the pairwise intersections of the three pairs of planes through A,B and A,C and B,C. Fundamentally, however, the planes in question represent nothing but directions of constant rotational action, and the concept of “plane angle” is just a disguise for multiply-connected rotational action. We require no assumption of self-evident linear extension, of the sort which pervades so-called “standard classroom mathematics”.)

Now examine the relationship between the array of six angles, just defined, and the {changing} shapes which an observer, located at the given vantage-point, will observe when looking in different directions at one and the same constellation of stars. Just look at the effect of projection of the three directions onto a variable plane.

To close this week’s work, try a final experiment: What is the effect of two rotations, carried out in succession? Hold a book in front of you, for example. First rotate it 90 degrees in the clockwise direction. After that, rotate it 90 degrees around the horizontal axis (the upward part rotating downward away from you). Note the resulting orientation of the book. Now, do the rotation on the vertical axis first, and then the clockwise rotation. Why is the result different? Compare this with the case of combining relatively linear displacements, as when we slide the book on a table a certain distance, parallel to itself, in each of two different directions. Is the multiply-connectedness of the rotational manifold, just demonstrated, responsible for the paradoxes of vision?

The First Measurement of the Universe

Part VI: What Is a Singularity?

by Jonathan Tennenbaum

We now enter a crucial phase of our journey, which begins by discovering the axiomatic implications of what at first appears as a mere optical illusion, takes us to Kepler’s discussion of “the curved and the straight” in his Mysterium Cosmographicum, and on from there to a fresh view of the regular solids.

First, an experiment.

Take a large transparent hemisphere, held or fixed with its border-circle in the vertical plane, so you can look into the inside with your eye in the spherical center and the pole of the hemisphere straight in front of you.

Trace a great circle (or actually half-circle) approximately at mid-height on the hemisphere, so that it has the appearance of a horizontal line when viewed from the center. Now trace a “vertical” great circle, which cuts the horizontal one at right angles at a point X. Viewed from the hemisphere’s center, this second line appears as a straight line running perpendicular to the first one. Note, that relative to the horizontal line (circle) running right-left, the up-down line is perfectly {symmetrical}: it does not “lean” in either direction. Now chose a point Y about 30 degrees to the right (or left) on the horizontal circle from the position of X, and draw a third great circle through Y, at right angles to the original circle. What do you see when you look from the sphere’s center? You see a straight, horizontal line with perpendiculars drawn to it at two points X and Y, don’t you? And as perpendiculars intersecting the common base-line, they must be parallel to each other, must they not? Indeed, focussing attention on a point mid-way between X and Y, the perpendiculars appear as perfectly parallel, vertical straight lines coming off the horizontal.

But as you look upward from the horizontal line, you notice that the “parallels” come closer and close together, as if leaning toward each other! How is that possible, if they remain straight? You recheck the angles at the horizontal line. No question, they are right angles, which means {complete symmetry}: the perpendiculars cannot lean in either direction, left or right. And yet, you just found them converging toward each other! Did they somehow get bent? You follow each of the perpendiculars carefully, and find no divergence anywhere from what appears to be {perfect straightness}! How could it happen, that perfectly straight lines, making right angles to a common line, stop being parallel?

Compare this paradox with that of our earlier investigation of the great-circle triangle with 90 degree angles, traced on the surface of a sphere. Looking from the center the sides appear as perfectly straight lines, forming an equilateral triangle. Looking at the angles, you see that each one is a right angle. Try to draw what you have seen on a piece of paper. Impossible? Why? A triangle is a triangle isn’t it? If so, why can’t you draw it?

Evidently, something anomalous is going on here, which is much simpler and more fundamental than most ordinary sorts of optical illusions. I suggest, that the problem is not located in your visual apparatus per se, but in your mode of {interpretation} of visual perceptions, so that you experience the paradox as “unheimlich”, as bursting forth inside your own mind. Let’s try to trap this critter for closer examination:

1. An arbitrary configuration of great circles, when viewed from the center of the sphere, appears to the viewer as a configuration of {perfectly straight lines}.

2. No {single} view of that configuration contains {anything} which were incompatible with the assumption, that what we are looking at, is an array of straight lines drawn in a plane.

3. A difficulty arises only, when we compare {more than one view} of the same apparent configuration. Indeed, when we try to {correlate} our various perceptions of that array, as we look from the center in various directions, we encounter phenomena (i.e. the equilateral triangle with right angles, or the converging perpendiculars) which are absolutely incompatible with the assumption just articulated.

4. Why does this surprise and baffle us? Evidently, one and the same perception, and one and the same array of predicates (the straight-line images) can be interpreted in more than one way, from the standpoint of more than one set of assumptions concerning the geometry in which those predicates are embedded. It would seem as if the very appearance of an array of straight lines tends to evoke, in our minds, the assumption of a linear, plane geometry. Whereas, that same appearance is not only consistent with a curved geometry, but the {changing} array of appearances, arising when we change our direction of viewing, is compatible {only} with a non-zero curvature of a certain type.

5. That, then, is where the implicit flaw is located; not so much, I submit, in the formal assumption of a plane geometry per se, but rather the deep-seated tendency to regard the characteristics of a geometry as something emanating from, or self-evidently determined by, the predicates (appearances, objects, isolated “facts”) in and of themselves. Whereas, what {distinguishes} the curved from the flat geometry, in this case, is not the predicates per se, but the characteristic of {change} in the adducible relationships within the array of predicates, or more precisely, in our cognition of that change and its implications.

Kepler’s Argument

From this standpoint, let us turn to the kernel of Kepler’s argument in his Mysterium Cosmographicum, the section entitled “The sketch of my main proof”. I hope the pedagogical devices of this and the preceeding pedagogicals will throw some new light on Kepler’s notion of “the curved versus the straight” (or, for reference to our present discussion, “curved versus linear, or flat”), not as an “objective” contraposition of types of forms in space, but rather in terms of the {mental processes} we have just begun to explore, and particular the process of {shift of basic assumptions}, from one type of geometry to another. Much more could be said about this, and we shall come back to it again, but let us go ahead and read what Kepler writes:

“God wanted, that Quantity should be created before all other things, in order that a comparison of {curved} and {straight} might occur. Exactly for this reason I find the Cusaner (Nicolaus of Cues) and others possessed of divine greatness, namely because they attached such high importance to the behavior of the straight and the curved toward each other, and dared to attribute the curved to God, the straight to the created things…What the Cusaner ascribed to the circle, and others to the space enclosed by the sphere, I attribute only to the surface of the sphere alone. I am firmly convinced, that no curved thing is more noble and more perfect than the spherical surface. For, the (solid) ball is more than the spherical surface, and is mixed with the linear, by means of which alone the interior is filled. A circle arises only in a plane, i.e. only when the ball or sphere is cut by a plane….

“But why did God chose the difference between curved and straight, and the noble nature of the curved, when he wanted to form the world? Why, indeed? Only because the most perfect architect must necessarily construct a work of the highest beauty…. In order that the world might be the best and most beautiful, in order that it might be able to receive this idea, the All-wise Creator produced Magnitude and brought forth the Quantities, whose entire nature is comprehended, in a sense, in the differentiation of the two concepts, the straight and the curved… It is probable, that God from the first moment selected the curved and the straight, in order to engrave in the world the divine nature of the Creator; to make possible the existence of these two concepts, quantity was created; and in order that quantity might be conceived, He created before everything else (spatial) body.

“As we before chose the sphere, because it is the most perfect quantity, so we now make a {single} jump to the bodies, which are the most perfect among the straight quantities and consist of three dimensions.”

In this light, let’s review the ground we have traversed, once more. We had two geometries, a linear geometry represented on a plane surface, and a spherical geometry. The two geometries are fundamentally incompatible, hence Kepler’s expression: a “jump”. We cannot construct a single, consistent, “literal” representation of the curved surface of a sphere (i.e. the manifold of rotations), within the bounds of linear plane geometry. Might there exist some {other} form of representation? We already have the answer on the tip of our tongues: Given the impossibility of a consistent representation of spherical geometry within the linear, plane domain, the spherical domain’s {existence} could have no other lawful manifestation within that linear domain, than through the generation of characteristic patterns of {anomalies}! What, then, is the {minimum} set of anomalies, sufficient to characterize what Kepler describes as the “single jump” from the “straight” to the spherical geometry?

Aha! We already encountered a relevant sort of anomalies, which arose in the attempt to {correlate} different views of one and the same configuration of great circles on the spherical surface, as seen by looking in different directions from the center of the sphere.

Juxtaposing an Array of Projections

Investigate this phenomenon, by placing a light source at the center of the transparent (hemi)sphere, and projecting images of various great circles traced on the spherical surface, onto a large, flat screen which we have mounted in any chosen, variable position relative to the sphere. We see that the images projected on the screen are indeed straight lines. Indeed, if we keep the screen in its given position, and replace the light source again by our eye, then the positions of the straight lines on the screen will be seen to coincide {exactly} with the appearances of the lines (great circles) on the sphere, as seen from that center.

Note, however, that the actual array of lines projected on the screen, including the magnitude of the angles between those lines, change greatly when you move the screen from position to position around the sphere. What is the significance of those changes? Evidently, the various projections correspond to what we referred to above as “different views of one and the same array of great circles” when viewed in {different directions} from the center.

The relationship becomes clear, when we determine the line running from the center of the sphere to the nearest point P on the plane of the screen; in other words, the perpendicular from the sphere’s center to the plane of the screen. Let Q mark the position where that line passes through the spherical surface. If we trace the projected images on the screen, and hold the screen in front of us so we are looking at P, then what we see is a “photograph” of how the sphere appears to us, when we look from the center into the direction of the point Q.

For any given position of Q, the corresponding positions of the screen are defined by the perpendicular planes to the corresponding direction. Clearly, moving the screen closer or further away from the screen, while keeping it perpendicular to that line, only blows up or contracts the dimensions of the image, while keeping the angles the same. If we choose to regard such changes as non-essential — which indeed seems justified in view of our search for a “minimum” representation of the anomalies — it makes sense to choose only one plane for each Q. Which one? The only unique choice, at this present stage, is to slide the screen up to the sphere until it touches it, at Q; in other words, project onto the {tangent plane} to the sphere at Q.

If, now, the anomalies which we observed earlier, are connected with the discrepancies or changes between such projections, when made in different directions, then two tasks confront us: First, to determine a minimal set or sets of projections, needed to display the type of anomaly in question; and second, to characterize the anomaly itself.

Bearing in mind what was said in the next-to-last paragraph, each projection involves the choice of a point Q on the sphere, such that the line from the sphere’s center through Q defines the direction of the projection; the screen being located at the tangent to the sphere at Q, perpendicular to that line. Accordingly, choosing an array of projections of the indicated type, amounts to choosing a {set of points “Q” arranged on the spherical surface}. Each projection is equivalent to a “snapshot” of the spherical surface, taken from the sphere’s center with our “camera” pointed at the corresponding point Q. The interesting phenomena will obviously be located in the regions where any two projections, say corresponding to points Q and Q’ from the set, intersect or overlap with each other.

With a bit of thought, we conclude that the character of the transformation or change between two such projections, depends only on the change of relative directions, i.e. of the relative positions Q and Q’ on the sphere; or in still other words, on the {arc} between them. As a result, to obtain the simplest, minimum characterization of the anomalies we are looking for, we must choose the array of points Q in such a way, that the arcs between adjacent points of the set, are all {equal}. In other words, they must form a {regular} array.

Isn’t it now obvious, where our journey is taking us? Look at the array of tangent planes (our “projection screens”) corresponding to the various points of the regular array of points Q. They form a kind of “envelop” within which the sphere is inscribed. Observe the edges formed where adjacent planes intersect, and cut off the portions of those planes which protrude on the outside of the intersections, in the obvious manner. What do you get?

Finally, do the following experiment. Trace a single great circle on your transparent sphere, and install a small, but bright light source in the middle of the sphere. Next, using some appropriate translucent material (i.e. plastic sheet), build a regular solid around your transparent sphere. The points of tangency with the sphere, defines the regular array of points “Q” in our previous discussion. Now observe the image of the great circle, projected onto the faces of your regular solid. It appears as a closed chain of straight line segments, whose “links” are at the edges of the solid. Observe the {discontinuous change of angle of inclination}, when the image crosses an edge, from one face of the solid to an adjacent one. Does this not remind you of the refraction of light? Finally study how te image behaves, when the sphere is rotated inside the solid.

The First Measurement of the Universe

Part VII– Prelude to the Pentagramma Mirificum

By Jonathan Tennenbaum

Recapitulation

Pursuing Kepler’s juxtaposition of the “curved and the straight” in terms of the attempted mapping of a spherical surface onto a plane, I last week suggested the following:

Given the manifest impossibility of a simple, consistent representation of spherical geometry within the linear, plane domain, the spherical domain’s {existence} could have no other lawful manifestation within that linear domain, than through the generation of characteristic patterns of {anomalies}! What, then, is the {minimum} set of anomalies, sufficient to characterize what Kepler describes as the “single jump” from the “straight” to the spherical geometry?

I sought to answer that question, by studying the anomalies, which occur when I project the sphere from the center onto a tangent plane. The most obvious anomaly is the apparent {incompatibility} between any two such projections: they don’t fit together, at least not in any way that can be accounted for by “plane geometry.”

I ended up with the idea, that a minimum representation of spherical curvature would be achieved by an array of projections, whose “incompatibilities” all have the same form. This led to the requirement, that the {directions} or midpoints of the projections, should form a {regular array} on the surface of the sphere. I argued, that this amounts to {projecting the sphere onto the faces of a regular solid}.

Some readers surely recognized at least one major inadequacy in my approach, namely, that I posed the choice of a regular array as a {requirement}, but I didn’t account for {where} such regular arrays come from and {why} they must exist.

The effort to fill this lacuna will takes us into new territory, a territory inhabited and ruled by a wonderous creature, called the “pentagramma mirificum.”

Now, the territory in question has a fearful reputation: Many of those who venture in, either never return again, or if they do, they tend to emerge in a scrambled-up state, suffering from dizziness and giddyness and unable to report what they saw in a coherent way. To avoid falling victim ourselves, it is necessary to proceed step-by-step, and above all to fix our conceptual bearings from the start.

The Singularities of Rotation

For most of us today, the idea of simple rotation of a body around a well-defined axis seems self-evident. That idea is deeply embedded in human culture, it would seem, since no later than the proverbial invention of the wheel. Have you ever stopped to think, that an {hypothesis} is required? Indeed, if we put aside astronomy, and observe the motions of “natural” Earth-bound objects, then, apart from man-made objects, constructed or selected on the basis of that very hypothesis, we hardly find any case of rotation around a well-defined, precise axis. Pick up an irregular rock, throw it into the air or try to spin it on a hard surface. You never get a simple rotation, but rather a very complicated, wobbly motion. Imagine someone challenges you to go into a forest, without any modern tools or other products of our technical culture, and construct a wheel from the natural materials you find there. How would you do it? How would you, for example, starting from “nothing”, build a rotating table for producing pottery, an elementary form of machine-tool? If you make the attempt, you might develop a healthy respect for the level of technology embodied in the simplest household artifacts of ancient cultures.

So the idea of simple rotation as a fundamental principle of physical action, does not arise from mere sense-perception of objects around us. Nor does the notion of {axis of rotation} as a subsumed singularity of such action arise so. Might it not be the case, that the notions of rotational action and axis of rotation, at least insofar as they became a concious principle of ancient machine-tool design, developed from astronomy? Remember how, millenia later, Gauss and Wilhelm Weber initiated a revolution in electrodynamics, by carrying over principles of astronomical measurement, into the microscopic domain. But let’s be careful not to jump over crucial steps here. Bringing machine-tool principles from heaven down to the Earth, is no self-evident linear process.

Observe the heavens on a clear night. Do you see the rotational motion of the stars? Not directly, anyway. But suppose, we as very prehistoric astronomers, have once established, with the help of our memory, the existence of the daily cycle of star motion and finally the circumstance, that the individual stars move in what appears to be a system of concentric circles in the sky. (Having filled in, in our minds, those motions unseen during the interruption of daylight and the periodic disappearance of stars below the horizon.) Now someone will probably jump up and say: What’s the big deal? You already have it: rotation!

But wait a moment. {Where} is the {axis} of that rotation? Through what points in space does it pass? Does it go through your body? Does it pass through somebody a mile away, who also observes the circular motions in the heavens? Or does the rotation have any axis at all in the sense of a line going through There is still a big topological distinction between the cyclical motion we adduce in the heavens, and that of the wheel we are about to invent.

To make my point a bit clearer, take out the measuring apparatus introduced earlier, consisting of a thin pointer rod or stick whose end pivots around a fixed locus, the latter corresponding to the view-point of the observer. (Many variants of this sort of instrument are possible; what is important is the functional result, namely to determine and record the {directions} in which stars are sighted, when seen from the given locus). Now examine the {manifold} of positions of the pointer rod, as it follows a given star in the course of an evening. Supplement those positions according to the presumed motion of the star when it is not directly visible. Do the same thing for a variety of stars. What do you find?

In the course of a daily cycle, the pointer rod describes the surface of a {cone}! [Show this with the bamboo skewers (shashlik sticks) stuck into a small ball of putty or styrofoam, or equivalent means.] Different stars determine different cones. Some are narrow, some wide, in correspondence to the apparent size of the circular path of the star in the heavens. But the entire array of cones is ordered in a very beautiful way, as a {nested} series. We find there are stars which barely change position in the course of the night (and day); such a star generates a very thin cone. A star a bit further away from that region of the heavens generates a larger cone, which contains the first one, and so on. The cones open out more and more, until they become virtually flat (for stars near the so-called celestial equator, which I shall define rigorously in a moment); after that they begin to close away from the direction of original narrow cones, becoming narrower again.

Now pay attention to the really interesting features of this family of cones, its {singularities}! On the one side, we have the narrow cones, which, as they become narrower and narrower in a nested manner, appear to converge toward a certain definite direction in the heavens, common to all. How shall we characterize that singular direction? It is the direction of {least motion}.

On the other hand, as we examine stars located progressively further and further away from the locus of the heavens corresponding to least motion (known as the “celestial pole”), the cones open outward, and we encounter another singularity: an ambiguity separating two subfamilies of cones, the ones opening toward the pole, and the ones opening in the opposite direction. That ambiguity corresponds to a hypothetical, “perfectly flattened” cone, which makes what we today call a {right angle} to the direction of least motion. The corresponding “ring” around the heavens, correponding to all possible directions of the pointing rod moving in the flattened cone, is known as the celestial equator. Stars in this region have the {greatest motion}, compared to everywhere else. At the same time, we conceive the existence of a {second} pole of {least motion} opposite to the other pole, although the earth under our feet blocks it from view.

Now, how do the characteristics of motion, which we have adduced from our observation of the stellar motions as a whole, project from the astronomical scale, down to our own, earthbound scale?

Take a putty or styrofoam ball, and 4-5 shashlik sticks. Insert one stick into the ball so that it points in the direction of the celestial pole, and insert the remaining sticks so that they point in the direction of as many stars, including one on or near the celestial equator. With time, of course, those stars will change position. Is there a single continuous motion of the ball, such that each of the pointers remains pointed at its assigned star? Now we have it: the {rotation of a solid body around a (relatively) fixed axis} is the form of action, on our earthly scale, which corresponds to the adduced motion of the heavens. The axis is the direction of the stick which points to the celestial pole. This also suggests a possible astronomical determination of a “primordial straight line”: motion along the constant direction of a pole. (There is more to be said on this, but not now).

Now imagine our pointing device placed in the middle of a transparent sphere. If we mark the locations on the surface of the sphere, corresponding to the positions of various stars as seen from the center, then the two celestial poles correspond to two points on the sphere, located opposite each other from the center, and the celestial equator corresponds to a great circle, located on the plane through the center perpendicular to the direction from the center to the poles (that plane is the “flattened cone” we spoke about earlier). We can represent the relationship of the pointing device to the heavens, by the relationship of the center of the sphere to the spherical surface. The daily motion of the stars corresponds to a rotation of the entire sphere, around the axis through the center and the two poles. In that rotation process of the sphere, the poles constitute the {regions of least motion}, the equator the {region of maximum motion}. That, in turn gives us the principle of the wheel, which combines maximum motion on the perimeter with minumum motion of the axel.

The Singularities of Multiple, Self-reflexive Rotation

Now obviously, the preceeding astronomical derivation of rotation and its singularities, does not exhaust the Universe. Although the Sun does have a daily rising and setting, if we plot its position on the transparent sphere, now made to rotate so that it follows the motion of the “fixed stars”, the relative locus of the Sun {changes}. In fact, it traces out what looks like a circle on the sphere (corresponding to the so-called ecliptic in the heavens). That circle intersects the equator (the circle corresponding to the celestial equator) in two points, apparently exactly opposite to each other (these are the Spring and Fall equinox points, the two days when night and day have equal length.) In fact, if we apply rotation to the sphere, taking as the axis of that rotation the direction through the center and those two points, we find that we can rotate the equator {exactly} onto the ecliptic. At the same time, the points corresponding to the celestial poles are carried into new positions, which have the same relationship to the ecliptic as the original poles had to the original equator. Evidently those new positions constitute the poles of a {third rotation}; a rotation whose equator is the ecliptic. You will probably see these relationships a bit more clearly, when we generate them in a slightly different way, in a moment.

Still another degree of rotation reveals itself, when we move our observation-point to a different location on the Earth’s surface. As we go toward the north, the axis of rotation of our sphere must be {rotated} into an increasing angle relative to our apparent horizon increases. This fourth degree of rotation Thales and probably Eqyptians and others long before, interpreted correctly, as a manifestation of the curvature of the Earth.

These circumstances, among others, suggest, that action in our Universe involves nothing less than a combination of many degrees of rotational action. What is the interrelation or interconnection among those various rotations? Our comparison of the celestial equator with the ecliptic suggests the idea: rotational action applied to rotational action.

To explore this notion further, as follows, using the surface of a medium-sized plastic or styrofoam ball, and marking singularities with a non-permanent marker). Start with a first rotation R, which generates as singularities two poles N, S and an equator E (mark them). Now imagine any arbitrary second rotation R’. The second rotation generates a second pair of poles N’, S’ and a second equatorial (or “great”, i.e. maximum) circle E’. The two equators intersect in a pair of points X, Y lying on opposite sides of the spherical center. Examining the two great circles and their corresponding pairs of poles marked on the sphere, it is manifest how to rotate one onto the other: the required rotational axis, is the axis passing through the intersection-points X,Y of the two equators. In other words, X and Y are the {poles} of the rotational action which carries E to E’. That same action carries the poles of R into the poles of R’. Note, that the equator corresponding to that third rotation, passes through {all four poles} of the original two rotations.

If you look carefully now, you will find at least two fish flapping around in your net.

The first fish is the remarkable suggestion: all rotations can be generated from any single one, by rotating it in the manner just described! Actually, that is not exactly true; we only demonstrated we could rotate the chief {singularities} of the two rotations into coincidence with each other, but didn’t take into account the different modes and quantities of rotation — fast or slow, continuing (indefinite) rotation or terminated (definite) rotation, and in the latter case through what magnitude of angle, etc. So, the more accurate conclusion so far would be: any rotation can be obtained from any given one, by applying a definite rotation to that one, and in addition possibly changing its mode and quantity.

The second fish, caught in the corner of our eye, so-to-speak, is the suggestion of a self-reflexive sort of “connectivity” among rotations. Let’s try to catch this one. To do that, avoid the element of arbitrariness in the angle between E and E’ in the previous discussion, by considering the effect of a {continuing} rotation of E, i.e. one that does not stop at E’. Call that continuing rotation R1. The poles of R1 are still the points we called X and Y. As I noted before, the equator E1 of R1 contains the poles N, S of the original rotation R; in fact, each of those poles traces out E1 in the course of rotation by R1.

Interesting. That means E1 is, in a sense, covered {twice} in the course of a single cycle of R1. Look at the process a bit more closely. As we proceed to apply the rotation R1, the equator E rotates into a {variable} circle E’ which intersects E in X and Y, making ever larger angles to E. Suddenly, however, we run into a singularity: when the angle is what we now call 180 degrees, E’ coincides with E, although with an {opposite direction} of rotation! At that same moment, the poles N and S have {reversed position}. As we continue to rotate further, E’ separates again from E, only to coincide with it again when the total angle of rotation is 360 degrees, i.e. a full cycles. To sum up the result: rotation applied to rotation, results in the division of a full cycle into {half-cycles}, divided by a singularity corresponding to “reversal of direction.”

Choose a victim, and ask him to prove to you, why “1/2” and “-1” are equivalent as geometrical numbers!

So far, we have rotations R and R1, poles N,S and X,Y and equators E and E1 respectively. Observe, that E and E1 intersect at two additional points, Z and W, which lie opposite each other across the spherical center, and divide both equators in half. At the same time, notice that {E is carried into E1} by a rotation whose poles at Z and W. Call the corresponding {continuous} rotation R2, and its equator E2. Note that all four points N, S, X and Y lie on E2, and they divide a full cycle of rotation according to R2, into {four} congruent sections.

If we start with E and begin to apply R2, we again get a variable circle E”, intersecting E at the points Z and W. After a rotation of 90 degrees, E” coincides with E1. Continuing past that, we get to the singularity at 180 degrees, when E” coincides with E, except for a reversal of direction. Next, at 270 degrees, E” coincides with E1 but with reversed direction, before finally coming back to E. What shall we call the rotation from E to E1, the which, when repeated, takes us to the reversed-direction version of E? Call it “i”, otherwise famous as the first imaginary or complex number.

Now go back to your victim, and demand that he immediately explain to you the equivalence of “1/4” and “i”. Also the equivalence of both of these with “3”, since we required a uniquely-determined series of {three} rotations to divide a full rotation into four congruent subcycles and to generate “i”.

With the addition of the rotation R2, its poles W, Z and its equator E”, a new phenomenon occurs: closure! The attempt to continue the process of generating new rotations and poles from the configuration just created, in the manner pursued so far, yields nothing new. If, for example, we take the intersection of E1 and E2, we get the poles N and S of the original rotation; and that one-fourth of a full cycle of that rotation carries E1 into E2. Thus, the construction process has an intrinsic {periodicity}, returning to the starting-point after three steps.

Any {two} of the rotations R, R1, R2 are carried into each other by the third, through the same quarter-cycle of rotation. The equator of each of the 3 rotations contains the poles of the other two, which in turn divide that equator into 4 congruent segments. The combination of all 3 equators E, E1, E2 divides the surface of the sphere into 8 congruent regions, bounded by 12 arcs and 6 vertices (the poles). Each of the regions is bounded by an equilateral curvilinear triangle whose angles are all 90 degrees.

Note: There is nothing {arbitrary} about that configuration. If you begin with {any} continuous rotation (as R), and rotate it around {any} axis that lies on R’s equator (as R1), then you end up with the {same} — i.e. precisely congruent — configuration of three rotations R, R1, R2 and the same array of singularities (poles, equators, division of the spherical surface).

The reader has surely recognized the curvilinear octahedron, discussed in Part 3 of this series, and may also be familiar with the way the octahedron produces — with hardly any outside assistance! — the cube, and the cube the tetrahedron. But here we seem to encounter a natural boundary. To proceed further we must add a singularity. That will bring us face to face with the legendary pentagramma mirificum.

The First Measurement of the Universe

Part VIII: Pentagramma mirificum

By Jonathan Tennenbaum

“It is as if one were travelling, alternately, in two worlds. In one world, there is action-at-a-distance along straight-line pathways, a linear, empiricist or Cartesian world. In the adjoining world, a circular action is produced by {rotation}, not by action-at-a-distance along straight line pathways… These two worlds are two Types, of which the rotation-world is the superior, the bounding, the limiting, the determining, the higher one.” (Lyndon LaRouche, in “Cold Fusion: Challenge to U.S. Science Policy”, Chapter III)

We have now come to the construction, that Kepler’s enthusiastic contemporary Napier dubbed, “the wonder of the pentagram.” My description in words will be a bit awkward, probably unavoidably so, nor could static diagrams by themselves convey the required sense of self-reflexive, multiply-connected rotation. There is no substitute for the reader’s active exploration and replication of the following constructions.

The locus of the pentagrammum is the rotational manifold, that arose as a product of our attempt to map the heavens (see Parts 3 and 4 of this series). We have represented such rotations in two ways: {first} in terms of changes of direction, as when we observe the sky from a single viewpoint, i.e. the celestial sphere as seen “from the inside”; and {second}, in terms of the rotations of a spherical surface as seen “from the outside.” I shall start with the second representation, which is easy to experiment with, using plastic spheres and erasable markers of various colors to mark the singularities (great circles, poles etc.)

Start with an arbitrary rotation of the sphere. Call the equator for that rotation E1, its pole R. Choose any position P on the great circle E1. Think of P and R at first as reference-positions, relative to which we now juxtapose a third, arbitrary (variable) position. Let that third location, Q, be given anywhere on the sphere outside the equator E1. (To avoid certain difficulties, which I shall discuss below in part, it were best to begin with the case, were Q does not lie too far away from either P or R, i.e. forming an arc of less than 90 degrees from either of those two locations.)

Now, by “unfolding” what is implied in the relationship between the arbitrary locus Q and the two loci P and R, we obtain the following, seemingly endless {chain} of singularities:

First, there will be a {unique} great circle passing through P and Q, corresponding to the least rotation which carries P to Q (*1). Construct that great circle. Call the pole of that rotation S, and call the circle itself (i.e. the equator of the rotation) E2.

Next, there will be a unique great circle E3 passing through Q and R, corresponding to the least rotation carrying Q to R. Call the pole of that rotation T.

Again, there will be a unique great circle E4 passing through R and S. Call its pole U.

Still once more, there will be a unique great circle E5 passing through S and T. Call its pole V.

And so forth. At first glance, this chain of relationships might seem to go on ad infinitum: a “bad infinity.”

But do the experiment. You will find, to your initial surprise, probably, that the process {closes} by itself, after generating the {fifth} point! Indeed, the pole U appears to {coincide} with the starting-point of the chain, P, while V coincides with Q and so forth. The whole process repeats, generating {exactly} the same sequence of five great circles and poles once again! The points P, Q, R, S, T form the vertices of a (generally) non-regular, 5-sided spherical polygon.

This periodicity was first studied in detail, as far as we know, by Johannes Kepler’s contemporary Napier. What strikes us as so extraordinary (mirificum!), is the circumstance, that the character of periodicity does not depend on the choices of the initial points P, R and Q. More precisely: all that is assumed in the construction, is three arbitrary points P, Q and R, subject only to the condition, that P lies on the equator of a rotation whose pole is R. (As the reader can easily ascertain, the latter condition signifies that P and R are separated by an arc of 90 degrees as seen from the center of the sphere).

Evidently, the self-closure of the chain into the form of a non-regular spherical pentagon, reflects a {universal} characteristic of spherical geometry, having no obvious equivalent in simple plane geometry. That characteristic determines the outcome of the construction, as it were, “from outside”; standing above and beyond the seemingly arbitrary choice of starting-points for the construction.

But let’s try to see more clearly, {why} the pentagrammum {must} close. For this purpose, let’s review the chain of relationships once again, this time from “inside” the spherical geometry. We shall find that the pentagrammum is already implicit in the simplest astronomical observations.

Under a clear night sky, stand facing due north, looking toward the corresponding northermost point on the horizon. Take that point as your “P”. At the same time, note that the zenith point (directly overhead), corresponds to the pole of the horizon. In other words, if we point our arm toward the horizon and rotate our arm left-right so that it follows the horizon, then the axis of that rotation will be vertical, and the poles are the zenith and the point opposite to the zenith, “directly down” under our feet. Call the zenith-point “R”.

Now choose a star anywhere in the sky. Designate its position “Q”. Observe the relationships between Q, the horizon-point P and the zenith-point R. Note two imaginary arcs formed in the sky: from P to Q, and from Q to R. These are the first two sides of our pentagram. Note also the arc from the zenith R down to P, which makes right angles to the horizon — a celestial right triangle! That same arc will be a {diagonal} of the pentagram.

Now trace out the arc PQ, by pointing first at P, and then applying the relevant rotation to your arm until you are pointing at Q. Point with your other arm in the direction of the axis of that rotation, i.e. toward its pole. Call that pole “S”.

It might be helpful, in grasping these relationships, to tilt yourself in such a way, that the arc PQ appears as your new “horizon”, and your new “above” (zenith) is S. Similarly for the arc QR and its pole T, the arc RS and its pole U, etc.

In this way, we trace the pentagram as an imaginary “constellation” in the sky, unfolded from the relationship of any given star Q to the reference-points P and R. Note, that if we change the position of Q, the shape of the pentagram will also change.

Now, what makes the chain of arcs and poles close after exactly 5 steps? Perhaps the reason will emerge, if we draw up a list of the chain of relationships in the construction:

1. P is a point on the horizon. 2. Q is any arbitrary point off the horizon. 3. R is the pole of the horizon (i.e. the zenith). 4. S is the pole of the rotation P->Q. 5. T is the pole of the rotation Q->R. 6. U is the pole of the rotation R->S. 7. V is the pole of the rotation S->T. 8. W is the pole of the rotation T->U. 9. X is he pole of the rotation U->V. etc. etc.

From the list itself, we don’t see any reason why U should coincide with P, V with Q and so on. Have we failed to take account of something? Recall last week’s discussion of the multiple-connectedness of spherical rotation. Aha! We didn’t pay attention yet to the various {angles} in the pentagramma. For example: the very first angle in the construction, which is the angle formed at P between the horizon and the arc PQ. This is the angle an observer would have to tilt himself by, in order to make the great circle containing PQ into his new “horizon.” Or in other words, in the language of our earlier constructions on the sphere: it is the (lesser) angle formed between the great circles E1 and E2, which in turn is the amount by which we would have to rotate the sphere itself, to carry E1 into E2. Evidently, the point P, which is the intersection of E1 and E2, represents one of the {poles} of that rotation.

Now what happens to the pole of E1 (i.e. R), when we carry out that rotation of E1 into E2? Evidently, E1’s pole is carried to E2’s pole, i.e. R moves to S. Our conclusion: the rotation from E1 to E2 — a rotation whose pole is P — {coincides} with the rotation R->S. The latter rotation, however, appears as the 6th step in our list above, where the point “U” is defined as its pole.

So P and U are poles of one and the same rotation! Now we begin see why the chain of relationships closes.

“The Theory of Ambiguity”

But here a difficulty arises: The circumstance, that P and U are poles of the same rotation, does not necessarily mean they {coincide}. They might instead be {antipodes}. Indeed, a rotation always has {two} poles, at diametrically opposite positions on the sphere.

By failing to consider the ambiguity in our expressions, such as “S is the pole of the rotation P->Q” or “S is the pole of the great circle E2”, we left open two possible choices. If at each step of our construction, we permit either of the two choices, we evidently end up with many more possible constellations. The chain is no longer uniquely defined, and in some cases will not close after 5 steps. The chain only becomes well-defined, if we introduce some “external” criterion for choosing between the two poles at each step: as, for example, by requiring that P, Q. R, S, T etc all lie on the same hemisphere. Alternatively, we could require that all the rotations (arcs) are all less than 90 degrees — which can always be accomplished by the proper choice of poles –, or that at each step the pole chosen should be “upward” with respect to an observer who has tilted himself through the angle between the successive great circles (or more precisely, the lesser of the two pairs of complementary angles, the one less than 90 degrees) to get from one great circle as his “horizon” to the next one. In the course of the construction, that “upward” direction pivots around in a closed cycle, pointing always toward the “interior” of the pentagram. Examining the entire configuration of five great circles, we see that not one, but {two} identical pentagons are formed on the sphere. Their vertices — 10 in all — are antipodes of each other.

At first glance, the ambiguities might seem a bothersome complication. Yet, as Gauss and Riemann developed the point in great richness: it is the ambiguities which determine, to a great degree, the whole character of a process. We began to study a similar, related case in Gauss’ approach to the Pothenot problem. Evariste Galois, a disciple of Gauss, referred to this elementary part of analysis situs as “the theory of ambiguity” (*2).

If you think the problem of ambiguity can be avoided, just try to define the vertices of the pentagram as a {continuous function} of the variable Q. Watch what happens when Q approaches, and crosses, the great circle through P and R, or when Q runs around the back and underside of the sphere as seen from P and R. The ambiguities associated with the double nature of the poles, come out as discontinuities in any attempt to impose a single, simple continuous function on the pentagram relationships.

Higher Self-similarity

Now behold the array of self-reflexive relationships subsumed by the pentagram, which the reader should be able to verify without much trouble:

The {diagonals} PR, QS, RT, SP and TQ are all equal in magnitude, corresponding to quarter-circle arcs (90-degree arcs).

Each vertex P, Q, R, S, T is a pole of the rotation defined by the opposite side (arc) of the pentagram (i.e. P is the pole of the arc SR etc.). The exterior angle formed at each vertex by the great circle-prolongations of the adjacent sides, is equal to the angle spanned by the arc on the opposing side as seen from the center of the sphere.

Of the total of 20 intersection-points of the five great circles, 10 are vertices of the two, antipodal pentagons formed by those circles (namely the intersection-points of E1-E2, E2-E3, E3-E4, E4-E5 and E5-E1). At the other 10 intersection-points of the great circles (those of E1-E3, E2-E4, E3-E5 and E4-E1), {right angles} are formed.

Most important is the self-reflexive characteristic, that the pentagram can be “regrown” from any three consecutive vertices. In other words: if for example I take R, S and T as starting-points, instead of P, Q and R, and construct a pentagram from {them} in the same way as before, then I end up with {exactly the same figure}.

Thus, although Napier’s pentagram — unlike a regular pentagon — can take on a continuum of different visible shapes, including very irregular ones, the periodic character of the construction-process points to a higher form of symmetry and self-similarity. Instead of the five equal angles and sides of the visible regular pentagon, the pentagram embodies five equal {transformations}. The reader who has worked through the above constructions, should already have a sense of this (*3).

Needless to say, the existence of such a five-fold, self-similar periodicity embedded in the rotational manifold, points toward the existence of the duodecahedron whose sides are regular pentagons. Indeed, as we shall see next week, the pentagramma mirificum is the crucial singularity leading us beyond the domain of the spherical octahedron and its “children” — the spherical cube and tetrahedron, as well as the corresponding straight-line polyhedra which they bound — to the duodecahedron/icosahedron and the notion of a unique, universal characteristic of the “rotational world”, subsuming, and reflected in, the whole simultaneous array of the five regular solids.

——————————————————–

(1) By “least rotation” I have in mind the following. Given two loci X, Y on the sphere, there are {many} rotations of the sphere which carry X into Y. The total angle through which the whole sphere must be rotated, in order to carry X to Y, will differ depending on which axis we choose. For example, if we choose the axis passing through the midpoint between X and Y, then a rotation of 180 degrees is required to carry X to Y. If, on the other hand, we take the rotation which carries X to Y along the arc of the great circle through those two points, and whose axis passes through the corresponding poles of the sphere, then in general a much smaller angle will be required. In fact, the latter choice of axis provides the least rotation carrying X to Y.

(2) LaPlace and Cauchy bear direct, personal responsibility for Galois’ early death at the age of 21, as they do for the tragic, early death of another brilliant Gauss disciple, the Norwegian Niels Abel.

(3) One way to express that periodicity, albeit in a somewhat formal way, is as follows: Given any three loci A,B,C on the sphere, such that A and C are separated by an arc of 90 degrees (i.e. A,B,C satisfy the requirements to be consecutive vertices on a Napier pentagram), construct a locus D as the nearest pole of the rotation A->B, and constitute the {new} triple of loci “B,C,D”. Now conceptualize a transformation T, which carries the triple “A,B,C” to “B,C,D” (so defined), as a kind of geometrical function. T has the effect, when applied to three consecutive vertices of a Napier pentagram, of “shifting ahead by one” in the order of vertices. Thus, T(“P,R,Q”) = “R,Q,S”, T(“Q,R,S”) = “R,S,T” and so on. Evidently, the effect of applying T {five times}, is to come back to the original triple. Although T is not at all like a simple rotation, T’s self-similar periodicity makes it the higher analog of rotation by 360/5 = 72 degrees, which is the characteristic transformation of a regular pentagon.

The Pentagramma Mirificum and Cardinality

by Bruce Director

Before starting this pedagogical discussion, make sure you’ve worked through the report by Jonathan from two weeks ago (99056jbt001). In that discussion, you will have constructed the pentagramma mirificum, and begun to discover why Napier and Gauss referred to it as “mirificum”, i.e. miraculous. This week, we’ll take a further look.

First, from the construction itself a very surprising property emerges. Each side of the spherical pentagon, is the equator of the opposite spherical vertex, and, that vertex is the pole of that equator! Make sure you have grasped that property before proceeding.

In recent discussions, both written and oral, Lyn has emphasized the importance of knowing the difference in cardinality between a spherical surface and a flat one. These words will be only that, mere utterances, unless you work through a crucial paradox that brings this concept fully alive into your mind. For that, we have Gauss’ fragmentary investigations of the pentagramma mirificum, into which we will take a preliminary look today.

To begin, think first of the property mentioned above. Each side of the spherical pentagon, is perpendicular (when extended) to the other sides of the pentagon, that are not adjacent to it.

Compare that to a pentagon drawn on a flat piece of paper. Extend the sides of that pentagon. The non-adjacent sides will intersect at points outside the pentagon, forming a pentagram. (Like on the sphere, the pentagon and pentagram are not regular ones). The angles at which these non-adjacent sides intersect cannot all be perpendicular, but in the pentagram we constructed on the sphere, they all were.

Try a little experiment. On a flat piece of paper, draw a line labeled a. Now draw another line perpendicular to a called b. Now draw a third line perpendicular to b called c. Now, draw a fourth line perpendicular to c called d. Can you draw a fifth line perpendicular to d that is also perpendicular to a? But, when we constructed the pentagramma mirificum on the sphere, this is precisely what we did. In fact, on the sphere, our construction “automatically” closed after five perpendicular transformations. On the plane, the construction closes after only four.

Okay, this doesn’t surprise you. Of course, you say, the plane and sphere are of two different curvatures and so, as Lyn says, the geometry of each will be different in every small interval. So, it is to be expected that things that occur on the spherical surface, do not occur on the flat one. These are mere words, unless you can form in your mind a concept of the difference in cardinality between the two surfaces. I am NOT speaking of WHAT is different between the two surfaces, but the nature of the difference. (Think of the Socratic concept of change, embodied in Kepler’s concept of congruence, later adopted by Gauss in his geometrical study, Disquistiones Arithmeticae.)

The nature of that difference, is precisely the direction of Gauss’ fragmentary investigation into the pentagramma mirificum, and can be discovered by looking at fragment 2. (Before proceeding you should review part 6 of pedagogical discussion on spherical geometry by Jonathan Tennenbaum filed in the Alpha computer as 99036bmd001).

Go back to the spherical pentagon and take another look at the “self-polar” property. On the spherical pentagon, a “line” (great circle) connecting any vertex to its opposite side, will intersect that side in a right angle, since each vertex is a pole and the opposite side is its equator. If you were to connect each vertex to the opposite side, the five “lines” (great circles) might or might not all intersect each other in the same point. Taking the inversion, you could pick a point inside the spherical pentagon, and be able to draw perpendiculars from each vertex to its opposite side, so that they all intersect at the chosen point.

Now, perform a variation on the experiment discussed in part 6. Draw the pentagramma mirificum on a clear plastic hemisphere, and project that pentagramma on a flat surface by placing a light source at the center of the hemisphere. The flat surface should touch the hemisphere at one point. The spherical pentagramma will project, on the flat surface, a straight line pentagram. Keeping the hemisphere still, move the flat surface. First pivot it around the same point. Then move it from point to point. (To be most effective, make your flat surface out of stiff plexiglass covered with tracing paper. Trace the projected pentragam on the tracing paper. Use a different piece of paper each time you change the projection by moving or pivoting the plexiglass. That way you can draw a series of snapshots of the different projections, corresponding to the different angles and places the flat surface makes with the hemisphere. When tracing the projections, be sure to mark the point at which the flat surface touches the hemisphere.)

Now take the array of projected flat pentagrams drawn on the pieces of tracing paper, and draw lines from each vertex that intersect the opposite side at right angles. These lines will all intersect at one point, and that point will be the one at which the plexiglass touched the hemisphere! The self-polar property of the spherical pentagramma remains embedded, in the projected plane one. In fact, as Gauss notes in his fragmentary investigation, {every} plane pentagon is nothing more than the projection of a spherical one.

To help make this point sink in, take its inversion. Start with an arbitrarily drawn pentagon on a plane. Draw the perpendicular lines from each side to the opposite vertex. These lines will all intersect in one point. But, this pentagon was just drawn on a flat paper. No sphere was used in its construction, yet the spherical property of the pentagramma mirificum is still in there. Spherical action is present, even without the sphere.

(On this point, I refer the reader to the very important, but far too little read, Science Memo by Lyn on Cold Fusion, written in prison and released during the 1992 Presidential campaign.)

We leave you this week, by introducing for future contemplation, another piece of Gauss’ fragmentary investigation. Go back to the spherical pentagramma drawn on the hemisphere with a flat surface touching the hemisphere at one point. The light source at the center of the hemisphere will project a cone, whose apex is the center of the hemisphere, and whose base will extend through the points of the spherical pentagon. The flat surface will cut that cone obliquely, defining an ellipse, that circumscribes the projected pentagon. What does this have to do with Kepler’s determination of elliptical orbits, Gauss determination of the orbit of Ceres, and Gauss’ later investigations into the perturbations of planetary orbits? In future weeks, we will delve into these questions.

An Exercise in The Division of the Sphere

by Bruce Director

To begin with, the hastily written end of last week’s pedagogical might have caused some confusion for those who carried out the construction. And, as Kepler said, “A hasty dog produces blind pups.” In the next to the last paragraph, the reader was asked to draw an arbitrary pentagon, such that the altitude lines all intersected in one point. The intersection at one point, of the altitude lines of the plane pentagon, will only occur on those pentagons which are central projections of a spherical one. An arbitrary plane pentagon, may not necessarily be such a projection. In those cases, the altitude lines will not necessarily meet at one point. However, those pentagons, can be transformed into projections of spherical ones. We will take that up at a later date.

That said, this week we will look at the difference in cardinality between a flat and spherical surface from another standpoint; the principle that Kepler and Gauss called congruence, or in the Greek harmonia. Unfortunately, due to the political mobilization of the past weeks, this week’s discussion is also written hastily, so I beg your indulgence in advance for what may seem to be a rushed presentation. However, the issues are crucial, and I would not want to postpone your enjoyment of working through them, by delaying it’s appearance in the briefing.

In the second book of the Harmonies of the World, Kepler re-introduces the Greek concept of harmonia, as equivalent to the Latin term congruentia, or in English, “to fit together.” Kepler, demonstrates that on a surface of zero curvature, or a plane, certain polygons, i.e. squares, triangles, and hexagons, will fit together perfectly. He called this a perfect congruence. However, in three dimensions, (i.e. solid angles) the formation of perfect congruences is entirely different. Perfect congruences can be formed by three, four, or five triangles, three squares and three pentagons. In this way, the uniqueness of the five Platonic solids is demonstrated.

But, there is still an underlying assumption not completely revealed in the above demonstration. The difference in which perfect congruences can be formed, is a function of something not stated explicitly, an underlying curvature of space. On the other hand, if we look at this principle of congruence from the standpoint of the surface of the sphere, as the Greeks and Kepler undoubtedly did, we see that this difference in congruence between two and three dimensions, is a reflection of the difference in cardinality between a surface of zero curvature– a plane, and a surface of constant curvature — a sphere.

To create this concept in your mind, think of a crucial difference between a plane and sphere. On a plane, the sum of the angles of any triangle are equal to two right angles, or 180 degrees. On a plane, triangles can change their size and relative shape, but the sum of the angles are always 180 degrees. Additionally, there is no maximum triangle. A triangle can be as big as can be imagined.

Think of a triangle on a sphere. In the constructions in the earlier pedagogical discussions on this subject, we constructed triangles with three 90 degree angles, e.g. the triangle between the zenith, a point straight ahead on the horizon, and a point directly to the left or right on the horizon. The great circles which form the sides of this triangle intersect each other at 90 degree angles, and the area enclosed by them is 1/8 the entire area of the sphere. Now, in your mind, move one of the horizon points towards the other, keeping the zenith and second horizon point fixed. What happens to the angles of the triangle and the area enclosed? The great circles intersecting on the horizon will remain at 90 degrees each, but the angle between the great circles meeting at the zenith will decrease from 90 degrees to 0. Simultaneously, the area enclosed by the triangle will also decrease. When the two horizon points meet, the resulting triangle, will look the same as a great circle from the horizon to the zenith. This “triangle” will have two 90 degree angles at the horizon, and a 0 degree angle, for a total of 180 degrees. In other words, when the sum of the angles of a spherical triangle are 180 degrees, the triangle ceases to be!

Next, do the reverse. Rotate one of the horizon points away from the other, keeping the zenith and the second horizon point fixed. The great circles intersecting the horizon will remain at 90 degres, but the angle at the zenith will increase, and, so will the area enclosed by the triangle. When the two great circles intersecting the horizon come together, the resulting “triangle” will have a zenith angle of 360 degrees, and two base angles of 90 degrees, for a total of 540 degrees. The area enclosed by this “triangle” will be 1/2 the surface of the sphere. But this “triangle” will appear to be the same as the triangle of 180 degrees, but constructed in exactly the opposite manner.

From this we can begin to arrive at a concept of a maximum and minimum triangle on the sphere. To get this idea more firmly in the mind, think of an arbitrary triangle on the sphere. If we increase the lengths of the sides of this triangle, the area enclosed will increase, as well as the sum of the angles. But, as the triangle grows, the angles between the sides will get greater and greater, until the angles are all 180 degrees, for at total of 540 degrees. Like in our previous example, this maximum triangle, encloses an area equal to 1/2 the surface of the sphere. On the other hand, if we shrink this triangle, the angles will get smaller and smaller, and so will the area enclosed.

From this demonstration, you should now be able to grasp the concept, that, unlike on a surface of zero curvature, a triangle on the surface of a sphere, has a maximum and minimum area, and the sum of the angles has a maximum and minimum boundary. But, there is a crucial distinction between the nature of the minimum boundary and the maximum. The minimum sum of the angles is 180 degrees, but that is the sum of the angles of a plane triangle. Since the sphere is nowhere flat, even in the smallest interval, a 180 degree triangle does not exist on the sphere. On the other hand, the maximum boundary, is a great circle, which, when considered as the maximum triangle, contains three 180 degree angles and encloses an area equal to 1/2 the sphere. Consequently, the sum of the angles of a spherical triangle, is always greater than 180 degrees, but never greater than 540 degrees. And, the area of a spherical triangle is always greater than zero, but never greater than 1/2 the area of the sphere. Since the area of the triangle is proportional to the sum of the angles, and since a triangle whose angles equal 180 degrees has zero area, the area of a triangle is proportional to the amount by which the sum of its angles are greater than 180 degrees. This quantity is called “spherical excess.”

With this principle established in our minds, lets look at the formation of perfect congruences on the surface of the sphere. As Kepler did, we want to find what spherical polygons will make such perfect congruences. However, since the angles of a spherical polygon change with size, we must consider both shape and size when forming perfect congruences.

We begin with discovering which perfect congruences can be formed with triangles. Because of the crucial difference in the nature between the minimum and maximum triangle, we must start with the maximum triangle, i.e. a triangle whose angles sum up to 540 degrees and that encloses 1/2 the area of the sphere. We can then shrink this triangle, until we find one whose size is such that it can make a perfect congruence with at least three other triangles. Because, unlike the plane, the sphere is bounded, this process has two boundary criteria. First, since the triangles must form a perfect congruence, that is “fit together,” the angles of the triangles must add up to 360 degrees when they come together at a common vertex. And, since the sphere is bounded, these congruences must divide the total area of the sphere evenly.

Since the area of the sphere is 4 x Pi x the cube of the radius, the area of the maximum triangle, on a sphere whose radius is 1, will equal 2 Pi. This same area can be thought of as an angular change from the center of the sphere, or 360 degrees. The area of the entire sphere will thus be 4 Pi, or as measured from the center of the sphere, 720 degrees.

To make a perfect congruence with three spherical triangles, we shrink the maximum triangle until it has three angles of 120 degrees, or 1/3 of 360. The total sum of the angles of such triangles will be 3 x 120 or 360 degrees making a spherical excess of 180 degrees. Since the total area of the sphere is 720 degrees, 4 such triangles will fit exactly onto the sphere, forming a spherical tetrahedron.

If four spherical triangles are to be fitted together, we must continue to shrink the triangles until the internal angle are 1/4 360 degrees or 90 degrees. The angles of these triangles will have a sum of 270 degrees, or a spherical excess of 90 degrees, or 1/8th the entire surface of the sphere, forming a spherical octahedron.

For five spherical triangles to be fitted together, the internal angle must be 1/5 of 360 degrees or 72. The total sum for these triangles will be 216, making a spherical excess of 36 degrees, or 1/20 the total area of the sphere. This forms the spherical icosahedron.

If we make our triangle still smaller, so that six triangles fit together, the internal angles will be 60 degrees, for a sum of 180 degrees. But we already discovered that such a triangle can’t exist on a sphere, and so we’ve reached the boundary of dividing the sphere into equal regions with triangles.

Now, try dividing the sphere with spherical squares. Like with the triangle, a great circle is the maximum square, comprised of 4 180 degree angles, for a total of 720 degrees. The sum of the angles of a square on a surface of zero curvature, is 360 degrees. So the maximum spherical excess of a spherical square is 720 degrees – 360 degrees = 360 degrees. If we make the square smaller so that 3 can be fitted together, the internal angles must be 120 degrees, with the angles of each square having a total sum of 480 degrees. This makes a spherical excess of 480 degrees – 360 degrees = 120 degrees, or 1/6 the total area of the sphere, forming the spherical hexahedron, or cube.

If we make the square smaller, so that 4 fit together, then the internal angles must be 90 degrees, for a total sum of 360 degrees. But this is equal to the maximum spherical excess, and so such a square cannot exist on the sphere.

We can similarly show that the spherical pentagon will divide the sphere into the spherical dodecahedron, and that is the limit of equal divisions of the sphere. We leave this demonstration to the reader.

From this standpoint the nature of the difference in cardinality between the sphere and the plane can be seen anew. When we begin with the maximum polygon, a great circle, we form the simplest perfect congruence, division in half. Then as we descend from the maximum polygon, there are certain sizes which form perfect congruences, or harmonies. The polygons in between the maximum and the harmonic ones, form imperfect congruences or even dissonances. The spherical divisions, corresponding to the five Platonic solids, are the only perfect congruences, or perfect harmonies of the sphere surface. Work through this construction yourself, so we can discuss it further in future weeks.

Some Wisdom from Friends

by Bruce Director

Next week we will bring to a conclusion, this series of pedagogicals on spherical action, that began in the Dec. 18, 1998 briefing, and continued through last week’s discussion on the the spherical development of the five Platonic solids. You are strongly urged to review this series as a whole. In the meantime, this week we offer you some words of wisdom from our predecessors, Cusa, Kepler and Gauss.

Nicholas of Cusa

Leonard Ignorance Book 1, Chapter 23:

“… Hence, Parmenides, reflecting most subtly, said that God is He for whom to be anything which is is to be everything which is. Therefore, just as a sphere is the ultimate perfection of figures and is that than which there is no more perfect figure, so the Maximum is the most perfect perfection of all things. It is perfection to such an extent that in it everything imperfect is more perfect — just as an infinite line is an infinite sphere, and in this sphere curvature is straightness, composition is simplicity, difference is identity, otherness is oneness, and so on. For how could there be any imperfection in that in which imperfection is infinite perfection, possibility is infinite actuality, and so on?

“Since the Maximum is like a maximum sphere, we now see clearly that it is the one most simple and most congruent measure of the whole universe and of all existing things in the universe, for in it the whole is not greater than the part, just as an infinite sphere is not greater than an infinite line. Therefore, God is the one most simple Essence (ratio) of the whole world, or universe. And just as after an infinite number of circular motions an infinite sphere arises, so God (like a maximum sphere) is the most simple m easure of all circular motions….

“… Therefore, all beings tend toward Him. And because they are finite and cannot participate equally in this End relatively to one another, some participate in it through the medium of others. Analogously, a line, through the medium of a triangle and of a circle, is transformed into a sphere; and a triangle is transformed into a sphere through the medium of a circle; and through itself a circle is transformed into a sphere.”

Johannes Kepler

Epitome of Copernican Astronomy; Book 4

[written in Q and A format in original]

“What is the cause of the planetary intervals upon which the times of the periods follow?

“The archetypal cause of the intervals is the same as that of the number of the primary planets, being six.

“I implore you, you do not hope to be able to give the reasons for the number of the planets, do you?

“This worry has been resolved, with the help of God, not badly. Geometrical reasons are co-eternal with God — and in them there is first the difference between the curved and the straight line. Above (in Book 1) it was said that the curved somehow bears a likeness to God; the straight line represents creatures. And first in the adornment of the world, the farthest region of the fixed stars has been made spherical, in that geometrical likeness of God, because as a corporeal God — worshipped by the gentiles under the name of Jupiter — it had to contain all the remaining things in itself. Accordingly, rectilinear magnitudes pertained to the inmost contents of the farthest sphere; and the first and the most beautiful magnitudes to the primary contents. But among rectilinear magnitudes the first, the most perfect, the most beautiful, and most simple are those which are called the five regular solids. More than 2,000 years ago, Pythagoreans said that these five were the figures of the world, as they believed that the four elements and the heavens — the fifth essence — were conformed to the archetype for these five figures.

“But the true reason for these figures including one another mutually is in order that these five figures may conform to the intervals of the spheres. Therefore, if there are five spherical intervals, it is necessary that there be six spheres; just as with four linear intervals, there must necessarily be five digits.

“Why do yo call them the most simple figures?

“Because each of them is bounded by planes of one species alone, viz., triangles of quadrilaterals or pentagons, and by solid angles of one species alone — the three primary figures by the trilinear angle, the octahedron by the quadrilinear angle, and the icosahedron by the quinquelinear angle. The other figures vary either with respect to the angle or with respect to the plane….

“Why do you call these figures the most beautiful and the most perfect?

“Because they imitate the sphere — which is an image of God — as much as rectilinear figure possibly can, arranging all their angles in the same sphere. And they can all be inscribed in a sphere. And as the sphere is everywhere similar to itself, so in this case the planes of any one figure are all similar to one another, and can be inscribed in one and the same circle; and the angles are equal.”

Carl F. Gauss

Letter to Gerling; April 11, 1816

“It is easy to prove, that if Euclid’s geometry is not true, there are no similar figures. The angles of an equal-sided triangle, vary according to the magnitude of the sides, which I do not at all find absurd. It is thus, that angles are a function of the sides and the sides are functions of the angles, and at the same time, a constant line occurs naturally in such a function. It appears something of a paradox, that a constant line could possibly exist, so to speak, a priori; but, I find in it nothing contradictory. It were even desirable, that Euclid’s Geometry were not true, because then we would have, a priori, a universal measurement, for example, one could use for a unit length, the side of a triangle, whose angle is 59 degrees, 59 minutes, 59.99999 seconds.”

Kick the Newton Habit

by Jonathan Tennenbaum

(In partial celebration of the second anniversary of the pedagogical discussions.)

Whoever has worked through the previous installments of this series in a thoughtful manner, should now have a fairly solid grasp of 1) how the rotational manifold and spherical curvature arise in the most elementary astronomical measurement of the Universe; 2) the characteristic sorts of anomalies, that result from any attempt to map a spherically curved surface onto a flat surface; 3) the origin of the regular solids in this context, as a single interconnected unity with the dodecahedron/pentagrammum as the centerpiece; and out of this, 4) why the five regular solids constitute the necessary and sufficient (least action) expression of the singularity, that separates spherical curvature, as a {type}, from flat, linear geometries typified by classroom plane geometry.

While much more could be said on these geometrical matters, and the pedagogy should be further refined, the preceeding installments provide at least a first approximation to what is needed. But one big issue has still been left hanging. Many readers, I am sure, have a nagging thought in the back of their minds, concerning the meaning of the whole exercise. To put it crudely, but otherwise accurately, I read the thought as follows:

“Your spherical geometry is lots of fun, and now I understand the regular solids much better. But I just gotta ask you: Do you really want me to believe, that the {solids} determine the planetary orbits? They are just abstract ideas, aren’t they? How could they have any {physical} effects? I mean, don’t get me wrong, I know Newton was a bad guy and all that, but … what should I say?… I really can FEEL that gravitational force. It’s really there. You know what I mean?”

Here we have a clear case, where no decisive progress can be made, until certain entrenched, false ideas and habits of thinking are fully demolished and the rubble cleared out of the way. People should study Lyn’s most recent memo (reproduced in Friday’s briefing), which deals with exactly this topic. In honor of the second anniversary of the pedagogical discussions, I would like to add a few additional observations.

Firstly, observe that the form of the indicated, nagging doubt corresponds {exactly} to the what many people react to, in Lyn’s “triple curve” characterization of the curvature that is governing the collapse of the present global financial-economic system. They cannot accept the idea, that the reason for the collapse — and the emergence of the Russia-India-China-Iran “survivor’s club” — lies entirely {outside} the domain of Newtonian-like mechanical causality. They see events as being caused by the interaction of a huge number of “forces”: market forces, political forces, sociological forces, “lone assassins” etc. They reject the idea, that the entire manifold of current history might be shaped {as a whole} in such a way, that the possible courses of events at this juncture are restricted to a very few alternative pathways, and no others. Ignoring this higher bounding of history, they entertain all kinds of scenarios and “solutions,” which do not exist in reality.

Just so, the Newtonian imagines arbitrary planetary orbits at arbitrary distances from the Sun, while the real solar system permits only a discrete array of harmonically-determined orbital bands. (Offending objects, it appears, are ejected from the system, or end up in the “garbage can” of the asteroid belt).

What is the problem? Project a curved surface on a flat surface, and observe the distortions produced. If you stubbornly insist on regarding the linearity of the flat surface as an inherent feature of reality, then you will be obliged to invent a complicated system of “forces” to explain the distortions in the image.

This is exactly what Sarpi, Galileo, Newton, Descartes etc. did.

Look at the Universe. Look at the impossibility of constructing a “flat” projection of the heavens, and look at the spherical geometry (often refered to as the celestial sphere) we demonstrated to underly all astronomical measurement. Look at the hierachy of {periodicities}, {cycles} which the ancients found to govern all apparent motions of the stars and planets. Look at the spherical (or spheroid) curvature of the Earth, measured by Erathosthenes, and the spheroidal curvatures of all other celestial bodies. Look at the harmonic system of the planetary orbits, whose unique coherence with the regular-solid spherical harmonics was demonstrated by Kepler. Look down toward the microscopic scales. Look at Kepler’s founding of crystallography (in the snowflake paper), and Mendeleyev’s ensuing discovery of the periodic system of the elements. Look at the Huygens-Fresnel demonstration of the spheriodal geometry underlying the process of light propagation, and its organization in cycles of wavelength and frequency — work that demolished Newton’s linear fallacy of “light corpuscles” travelling in straight lines. Look at the Ampre’s preliminary discovery of the non-linear (angular) nature of electromagnetic action. Look at Wilhelm Weber’s derivation, from his own experimental confirmation of Ampre’s principle, of the necessary existence of an essential singularity of electromagnetic action at a “critical length” corresponding to subatomic scales. Look at the implicit (if somewhat flawed) extension of the Huygens-Fresnel-Gaus-Weber work to atomic physics, by Planck, De Broglie, Schrdinger and others. Finally, look at Dr. Robert Moon’s preliminary demonstration of the Keplerian ordering of the subatomic domain. Compare this with the harmonic characteristics of living processes, and with the harmonic characteristics of human Reason, as reflected for example in the well-tempered system of bel canto polyphony. And so forth.

Review, thus, the panorama of the Universe, as the actual process of discovery has thus far revealed the Universe to be. Do you find, anywhere in this, any trace of a supposed primacy, or even mere existence of simple, straight-line action in the Universe? No, not the slightest trace! Rather, we discover everywhere reflections of a universal curvature, coherent (to a first approximation) with the characteristics of spherical bounding as understood by Nicolaus of Cusa and Kepler.

But now arbitrarily stipulate, that all events in the Universe are taking place in an “empty”, featureless, euclidean three-dimensional space, extended indefinitely in all directions. Stipulate straight-line motion at constant velocity as the “natural” form of action inhering in that notion of space-time. Build that into your physics as a basic assumption. You have now transformed the entirety of the actual physical evidence into a gigantic anomaly!

Any motion, for example, which departs from constant, straight-line motion — i.e. all real motions! — is anomalous. So, postulate the existence of “forces” that are “bending” the motions into the observed curved trajectories. Elaborate that curve-fitting into a sophisticated mathematical structure. Congratulations! You have just received a Nobel Prize for virtual reality! The main anomaly left to be explained, is how Galileo, Sarpi, Newton etc. were able to get away with it.

“But can’t you understand, I really FEEL those gravitational forces.” We can hear Descartes swearing, pathetically: “I feel it, therefore it exists”! But sense perceptions are mere phenomena, they have no meaning in and of themselves. Some action, some change has occurred. So what?

Consider the following experiment: we suspend a magnet by its midpoint on a thread. A meter or so away, we set up a coil. When we pass an electric current through the oil, the magnet on the other side of our table rotates. What is the significance of that correlation of events? Does it mean that some physical entity (Leibniz called Newton’s forces “occult qualities”) emanates from the coil, reaches out through space across the table to the magnet, and turns it? Or were it not more reasonable, in place of such extravagant and arbitrary speculations, to report, that the magnet {responded} to a {change in the Universe}, which we generated with our actions, and that the Universe is manifestly bounded in such a way, that the correlation of events in the Universe takes a certain form, and not another. The phenomena remain the same, including the weight-lifter’s conviction, that he is working against “gravity”.

But the nagging starts in again and somebody asks, “Well, if you don’t believe in forces, then please {explain} to me, {why} the planets go around like that, why the Earth is spherical, and so forth.”

“You want an explanation? Forget it. That’s the way it is, buddy. Our Universe is (approximately) spherically-bounded, and you’re going to have to live with it!”

Sometimes, in science as in organizing, blunt answers are appropriate. Sometimes you make a bad mistake by trying to “explain” things. (Explain in terms of what?) Why? Because a certain mode of demanding explanations is really just a ruse for refusal to except reality. Because, what the person is really saying is, “I will refuse to except that X is happening, if the existence of X contradicts my deepest beliefs.” What people commonly mean by “explanation,” is to demonstrate the {deductive consistency} of an event, with their own underlying assumptions and beliefs. But, what if their beliefs are wrong? If the entire coherence of the evidence contradicts a firmly-held belief or habit of thought, then as scientists and truth-seekers, we must part with those beliefs and habits.

Riemann put forward exactly this, {opposite} sense of “explanation” in a posthumous fragment on scientific method:

“If an event occurs, which is necessary or probable according to the given system of concepts, then that system is thereby confirmed; and it is on the basis of this confirmation through experience, that we base our confidence in those concepts.”But if something unexpected occurs, being impossible or improbable according to the given system of concepts, then the task arises, to enlarge the system, or, where necessary, to transform it, in such a way that the observed event ceases to be impossible or improbable according to the enlarged or improved system of concepts. The extension or improvement of the conceptual system constitutes the {`explanation’} of the unexpected event. Through this process, our understanding of Nature gradually becomes more comprehensive and more true, while at the same time reaching ever deeper beneath the surface of the phenomena.”

Thus, “explanation” in Riemann’s sense means a successful {change} in fundamental concepts and assumptions, which have been overthrown by the generation of an event in the Universe, which is incompatible with the previously prevailing beliefs and assumptions. The question, which Riemann does not fully answer, but LaRouche does, is the nature of the {bounding principle} of that process of change.

These remarks bears crucially on the deeper side of the fallacy of Newtonianism. The epistemological equivalent of straight-line action and Cartesian-Newtonian space-time, is deductive reasoning. What we encounter is a strong resistence to the notion of an efficient {bounding} of events, which does not have the form of logical-deductive implication. The existence and form of such bounding principles is an {experimental} question; they cannot be derived from mathematics. Their existence is demonstrated historically, however, by the manner in which the Universe reacts to creative human Reason, by such changes as lead to harmonically-ordered increases in the relative potential population density of the human species. Thus, Nicolaus of Cusa and Kepler understood the ontological significance of spherical (and higher) curvature, as a lawful expression of the principle of perfection of human Reason.

The Twelve Star Egyptian Sphere That Generated The Great Pyramid And The Platonic Solids

by Pierre Beaudry

“The history of astronomy is an essential part of the history of the human mind.” Jean Sylvain Bailly

THE SHADOW OF A DOUBT

Over 50 centuries ago, it was the Egyptians, not the Greeks, who invented and built the Five Platonic Solids. This can now be proven with no more than a shadow of a doubt. In the present pedagogical, we shall demonstrate that not only did the ancient Egyptians construct the five Platonic Solids, at about 3,000 BC, but that those Platonic Solids were in fact, derivative parts of the same Twelve Star Egyptian Sphere, (Figure 1) otherwise known as the Pythagorean Sphere, which served primarily as an astronomical normalizing instrument, and as a blueprint for the construction of the Great Pyramid of Gizeh. In other words, we shall finally resolve the enigma of the Great Pyramid by showing that it could not have been built without an explicit knowledge of the astronomical spherics that generated the five regular solids.

Figure 1

THE INTENTION OF NORMALIZING

The first thing that Egyptian pyramid builders did, in attempting to replicate the canopy of the heavens, was to normalize angular measurements, with respect to the changing position of the stars in the canopy of the heavenly sphere. As Lyndon LaRouche put it, recently: “: what the Greek conception of the spherics was from the Egyptians. You’re looking at the universe as a sphere. You don’t know what its diameter is: you just know it’s very large, and you’re trying to interpret things, not by measuring intervals, but measuring angles. And you’re looking at angular changes, and your looking at trying to normalize your relationship, as an observer on a rotating earth, to a planet.” Such was the intention of the pyramid builders in their founding moment of astronomy. The Egyptians knew they could not accomplish that task by simply establishing an equal division of the sphere. They had to determine a norm by which constant change could be measured. From that intention, the construction of equal partitioning of a sphere gave interesting results, but it did not give them a sense of normalizing and closure over what is constantly changing. So, they partitioned their sphere in such a way that the yearly cycles of the heavens were made to correspond to angular rotations of a circle based on 360 degrees. The divisions of the circle became the angular reflexions of yearly cycle of 360 days. That provided closure between the calendar and physical geometry of astronomy. This inadequacy function, however, between geometry and reality was expressed in such a way that, 360 Egyptian days were considered human days, while the additional five and 1/4 days, for a total of 365 and 1/4 days a year, represented additional “holy days” as gifts from God.

INSIDE AND OUTSIDE

Although the equal partitioning of the sphere was absolutely necessary, not only for astronomical purposes, but also as a means of building the sphere itself and the pyramid observatory, the Egyptians were attempting to mix equal partioning with constant change. This was very difficult to do, since all of the movements of the heavens are irregular, including the slow movement of the fixed stars. So, the very physical construction of a sphere required both equal and unequal partitioning. They derived angular proportionality by what was later to become known as the Thales Theorem (Figure 2), that is,{when a line crosses a triangle, and its direction is parallel to one of the three sides, the other two sides are divided proportionately}. From this theorem of proportionality, two geometric constructive rules were established. The first stated that when two lines cross each other, the opposite angles are equal. The other stated that when three lines cross each other, at the same point, their three angles are equal to two 90 degree angles, or form an angle of 180 degrees.

Figure 2

However, there was an even greater difficulty to be surmounted. The construction of the required sphere could not be done except from the epistemological phase space of a Riemannian type of complex domain. That is to say, the geometer-architect had to locate himself both inside and outside of the sphere he was building. That is not a comfortable position for any person to be in. Like God, he had to be both inside and outside of the universe. He had to be self-conscious of being outside of the experiment at the same time that he was at the center of the scientific experiment he was designing. In fact, that thought-object was the most important component of that Egyptian discovery.

RE-CREATING THE HEAVENS

Although the Egyptians had constructed different spheres of three, four, and six great circles, the sphere they required was made up of ten great circles, which divided each other into golden sections of the divine proportion, and projected all of their angular sections from its center, outwardly, to form a completely close-packed surface of twelve pentagonal stars. As we shall see, in a moment, the only way to project such a spherical surface onto the sphere of the heavens was from the angular measurement of the Great Pyramid of Gizeh. This represented the act of man re-creating the heavens, in imitation of God, the creator of the universe. The canopy of the heavens was, thus, divided into a twelve-part zodiac, whose spherical blazonry inscribed a great celestial dodecahedron, as Plato later acknowledged in his {Timaeus}. This was a most beautiful sight to see. The close-packing of stars displayed on the ceiling of the Pyramid Text Chamber, located at Sakkara, confirms that this was a projection of the sphere of the heavens reflecting the principle of proportionality between God’s work and man’s work.

PARTITIONING OF THE CIRCLE

The ten circles divided each other into six angular partitions, each of which was assigned 60 degrees, and each 60-degree angle was, in turn, divided into three parts, two parts of 22 degrees and one part of 16 degrees. This partitioning provided the angular measurements for the spherical golden section of divine proportion generating twelve starred pentagons projected onto the surface on the entire sphere. The complete partitioning of circles was done by alternating 2 partitions of 22 degrees, and one of 16 degrees, all of which formed the unequal divisions of each of the ten circles; that is, [6 x 44] + [6 x 16] = 360 degrees.

DECLINATION OF A PLANET

Being descendents of the people of the seas, the Egyptians knew that spherical measurements, which are non-linear measurements, had to apply to their celestial calculations of the spherical surface. We know, from Herodotus and others, that very early on, going back at least 40 thousand years, before the pyramids were built, the people of the sea were able to navigate by angular measurements, and that the distances of the sea routes were given in spherical degree units, from a Zenith function. In fact, Astronavigaters were able to translate 60 nautical miles into one degree of a great circle.

The 10-circle sphere had another advantage. Angular triangulation, taken from the pole and from the zenith, was essential in order to determine the declination of a planet, and the Thales Theorem of proportionality provided that capability. Though the declination, [or right ascension], of a planet such as Venus or Mercury cannot be obtained by meridian observations, and the declination [or right ascension] of an outer planet, such as Mars, Jupiter, and Saturn, could only be established at night, another means of calculation was required. The people of the seas had devised a celestial triangulation, which was based and modeled on navigation; that is, they devised a triangulation of great circles intersecting coastal ports with the North Pole on the spherical surface of the seas. If a sailor was lost at see, he could easily discover his position by an angular triangulation of two stars. Today, such triangulation can be readily reconstituted from a list of latitudes and longitudes of ports that can be found at the end of any good Atlas.

DIVINE PROPORTIONALITY

What the pyramid builders had discovered, in the process of constructing spherics, was not only the physical principle that would provide them with an instrument that established astronomical proportionality between God and man, but that such a {proportionality principle}, which should properly be called the {Imhotep principle}, after the architect-geometer of the first Egyptian pyramid of Sakkara, could only be expressed by means of angular measurements upon which both science and religion would be based. This represented the ancient principle of balance and reciprocity, also known as the principle of Ma’at.

According to a report given to Herodotus by an Egyptian priest, in {The History, 2.124.}, Pharaoh Khufu, and his architect-geometer, had shut down the idol worshiping centers of Egypt altogether, banned the satanic practice of sacrifice, and eliminated slavery. In fact, Khufu’s building of the Great Pyramid was accompanied with waging a real political revolution against the cult of the Magi. I will show, in a future pedagogical, that the pyramids were not built by slave labor, but by lever machines based on the princple of Ma’at, otherwise known as the {Shadoof principle}. The Pharaoh waged an all out war against the sophistry of the Satan worshiping cults that were thriving on a false difference between belief and knowledge. This evil was very similar to the Mithra cult practices that freemasons have perpetrated throughout history, and that the Martinist Synarchists are practicing today. However, the geometric evidence developed in the spherical pedagogy, which was necessary for the construction of the Great Pyramid, shows that a true belief in God and true creative human cognition could not be separated.

SQUARING THE CIRCLE

In the process of working out their sundry calculations, the Egyptian pyramid builders discovered that their spherical composition had received from God, a very special attribution. They were given the gift of seeing through a glass darkly, the ability to prove by shadow. The divine proportionality principle, which had been implanted into their minds, like a compass, had been also inscribed within the very architecture plan of the Great Pyramid apex angle, whose shadow was cast from the crucial division of this 10-circle sphere. The attribution of this incommensurable proportionality was understood as a paradox between non-linearity and linearity. The paradox was formulated in the following manner: {The height of the pyramid must be to the perimeter of its square base as a radius equal to the same height is to the circumference of its circle.} Historically speaking, this is how the principle of divine proportion was first established as the paradox of Squaring the Circle. From that historical moment, no less than 50 centuries ago, this paradox of the Great Pyramid became the most important gift of cognitive discovery in all of human civilization, and became known as the cognitive function of shadow projection onto the irregular and dimly lit wall of Plato’s Cave. [Figure 3. Squaring the Circle]

Figure 3

THE PROOF BY SHADOW

The shadow of the Great Pyramid’s apex, N, established at 76 degrees, comes from the angle of projection onto the plane, which cuts through the sphere at the equator. This is what Figure 4 shows with the projection of angle N + 1 between side A of the spherical pentagon and the side B of the star extention. In other words, if you were to project a light source from outside of the ten-circle sphere onto the curved edges of one of its spherical starred-pentagons, the shadow angle formed by the side of the pentagon and and its starred extension would project, onto the center plane of the sphere, not a curved angle, but the apex angle of the Great Pyramid at 76 degrees, a change of manifold, from N to N + 1. Thus, the principle of proportionality bridges the incommensurable gap between the sphere and the polyhedron, between God and man, as through a glass darkly.

Figure 4

From that elementary projection, the pyramid builders then established the meridian cross-section of the Great Pyramid with the angular apex measurement of 76 degrees, and located it at 30 degrees latitude north, in Egypt. The meridian great circle of the sphere was divided into four angular segments forming two Great Pyramid apex angles of 76 degrees each, and two neighboring-angles of 104 degrees each. The apex angle of 76 degrees and the pyramid base angle of 52 degrees, which is half of 104 degrees, show that the cross-section of the great Pyramid is rotated all around the circle. The multiple mapping of this cross-section onto the circle is for the purpose of pedagogical illustration only. [Figure 5. The Great Pyramid partitioning of the Great Circle.]

Figure 5

So, by fitting three great circles together into angles of 76 and 104 degrees, that is, a base equatorial circle and two other great circles of similar divisions, each of the ten circles will reflect the cross-section of the Great Pyramid, throughout its angular measurements. The different illustrations of Figure 5 should be used as a guide for the reader to construct his own Twelve-Star Egyptian Sphere and polyhedron. Thus, by partitioning the circles into all of the marked angles, which must all be drawn with great care and precision on both sides of each circle, the surface angles of the starred pentagons are set into position, and the entire 10-circle sphere can be constructed into a twelve-star spherical matrix, a Platonic Solid Planisphere. [Figure 6. Four different views of the Twelve Star Egyptian Sphere and Twelve Star Egyptian Polyhedron.] It is recommended to construct half of a sphere at a time.

Figure 6a

Figure 6b

Figure 6c

Figure 6d

THE BIRTH OF THE PLATONIC SOLIDS

Since the partitioning of a 4-circle sphere into six equal parts is known to generate the Cube, the Octahedron, and the Tetrahedron, the characteristic shape of the Cuboctasphere can be traced on the surface of that 10-circle Egyptian Sphere. However, the Dodecahedron and the Icosahedron, which are generated by the partitioning of a 6-circle sphere into 10 equal parts, are nowhere to be found in the Egyptian Sphere. But, if the 6-circle sphere is added to the 10-circle sphere, the two last Platonic Solids can be integrated. When a set of six great circles partitioning each other into ten equal parts is inserted within the angles of the hexagonal intervals of the twelve stars of the 10-circle sphere, then, the new sphere has a total of sixteen great circles, and generates, simultaneously, the Great Pyramid, the Twelve-Star Egyptian Sphere, the Twelve Star Egyptian Solid, the Cuboctasphere, and the Icosidodecasphere, all integrated into a single sphere. Lucas Pacioli and Leonardo Da Vinci demonstrated that the golden section was a projection of the spherical hexagon and decagon onto the plane. Thus, the spherical mixture of ten-sidedness and of six-sidedness reflected the spherical origin of the golden section in Ancient Egypt.

This integral 16-circle sphere also pertains to the musical system based on C-256. Since the solar system itself was based on the same natural tuning system, as Kepler demonstrated from his own reference to the spherics of Pythagoras, it was only fitting that the Egyptians, who were the teachers of the Greeks, incorporated the same idea of the harmony of the spheres, within the construction of the Great Pyramid. It is not difficult to imagine that the astronomical data, which had been gathered from the Meridian Great Gallery of the Great Pyramid, 5,000 years ago, could easily have been monitored with a chiming water-clock device that corresponded to the twelve-tone series of our musical system, since this Egyptian Integral Chora Sphere, in fact, can be shown to be geometrically apportioned in accordance with the passing tones of the six human voices. [Figure 7. Division of the great circle according to the six human voice register shifts.]

Figure 7

[Figure 8. The 16-circle Integral Chora Sphere.]

Figure 8

PERPLEXITY OF THE RIEMANNIAN PHASE SPACE

All in all, this is very perplexing. But, what was most perplexing of all, in this whole process of discovery, was the fact that the solution to the paradox of the Great Pyramid could not be found inside of the pyramid itself. The irony is that it could only be discovered in the angular measurements of the sphere, that is, from the Riemannian domain. This is like the principle of the non-living, which can only be discovered from the higher manifold of the living. Similarly, when you look for the source of this paradox, inside of the sphere, you discover the shadows of the regular solids, and when you attempt to explain the presence of those Platonic Solids, in that ambiguous shadow form, you discover that their appearance can only be explained from the principle that produced the Great Pyramid, which is what established the paradox in the first place. In other words, the very unfolding elaboration of the ten-circle sphere makes it impossible for the Egyptians not to have known, and built, the five regular solids, because that sphere was their birth receptacle, their generative phase space, that Plato called {Chora}. And, without that sphere, the Great Pyramid itself could not have been built at all. Thus, the perplexity dissipates when this paradox is resolved, not before. This means that the Great Pyramid of Egypt, and the Five Platonic Solids, are all historically bounded together and can never be separated from their common generative principle, which resides outside of them; and, the cement that bonds them together is the paradox of Squaring the Circle. That should make everyone laugh and be happy.

THROUGH A GLASS DARKLY

This proves, within the shadow of a doubt, that the Egyptian builders of the Great Pyramid were the first geometer-astronomers to have conceived and constructed, by angular measurements alone, the Five Platonic Solids, and in this capacity, have served as midwives to the Greeks in matters of pedagogy and science. Thus, the Great pyramid of Gizeh has projected, during more than 50 centuries of history, its universal shadow over Greek civilization, and over all human beings, past, present, and future. Could there be any greater gift to mankind than to reproduce this discovery, as if through a glass darkly? This is your heritage. Are you going to pass it on to the next generations?

The curvature of “rectangular numbers” Part I

by Jonathan Tennenbaum

Our pedagogical discussions concerning the problem “incommensurability” in Euclidean geometry demonstrated, among other things, that the shift from linear to plane, or from plane to solid geometry cannot be made without introducing new principles of measure, not reducible to those of the lower domain. Thus, the relationship of the diagonal to the side of a square can only be constructed in plane geometry, and is inaccessible — except in the sense of mere approximations — to the mode of measurement characteristic of the simple linear domain (i.e., that embodied in “Euclid’s algorithm”). In the following discussion, we propose to explore that change from a somewhat different standpoint.

I choose, as a point of departure for this exploration, the issues posed by any attempt to compare the areas of various plane figures. The famous problem of “squaring the circle” falls under this domain. But I propose, before looking at that, to start with something much simpler. For example: How can we compare the areas of arbitrary polygons, by geometrical construction? Or, to start with, take the seemingly very simple case of rectangles. Let’s forget what we were taught — but do not know! — namely the proposition that the area of a rectangle is equal to the product of the sides. (Actually, even if the assumptions of Euclidean geometry were perfectly true, the proposition in that form is either false or highly misleading: an AREA is a different species of magnitude, distinct from all linear magnitudes.) In the interest of making discoveries of principle, let us resolve to use nothing but geometrical construction.

Experimenting and reflecting on this problem, the insightful reader might come to the conviction, that the problem of the relationships of area among rectangles of different shapes and sizes, pivots on the following special case: Given an arbitrary rectangle, how to construct “many” other rectangles having the equivalent area. Or perhaps even to characterize the entire manifold of rectangles of area equivalent to the given one.

The first line of attack, which might occur to us, were to find a way to cut up the given rectangle into parts, and rearrange them somehow to form other rectangles. Should we admit any limitation to the shapes and numbers of the parts? To avoid a bewildering bad infinity of options, let us focus first on what would appear to be the “minimum” hypothesis, namely to divide the given rectangle into congruent squares (i.e., squares of equal size). A bit of reflection shows us, that such a division is only possible for the special case, that the sides of the given rectangle are linearly commensurable (i.e., are multiples of a common unit of length). So, for example, if the sides of the given rectangle are 3 and 4 units long, respectively, then by cutting the rectangle lengthwise and crosswise in accordance with divisions of the sides into 3 and 4 congruent lengths, respectively, we obtain a neatly packed array of 12 congruent squares. We discover, that it is possible to rearrange those squares to obtain five other rectangles: 4 by 3 (instead of 3 by 4), 2 by 6, 6 by 2, 1 by 12, and 12 by 1 (i.e., six in all counting the original one, or three if we ignore the order of the sides).

Experiment further. If we start, for example, with a square, and divide the sides into five congruent segments, we obtain 25 congruent squares. The “harvest” of rectangular rearrangements is disappointingly small: all we find is the long, skinny 1 by 25!

Carrying out such simple experiments, the attentive reader might detect a number of potential pathways of further inquiry. One of these would be to ask, for a given total number of congruent squares, how many different rectangles can be formed as arrangements of exactly that number of squares? We can then organize the number into species or classes, according to the resulting number of rectangular arrangements (or “rectangular numbers” as the Greek geometers called them). The class of numbers for which only one rectangular arrangement is possible (disregarding the order of the sides) are known as “prime numbers.” After these, we have a class of numbers with exactly two rectangular arrangements, such as 6, 10, 14, 15, 21, etc. (The otherwise mind-destroying game of “Scrabble” might be put to good use, by employing the wood squares for experiments.)

For the present purposes, however, we would like to construct as many different rectangles as possible out of the original one. We note, that the number of rectangles generated from any given division of the rectangle is very narrowly bounded, and certainly does not include all geometrically constructible ones. How to obtain more? If we stick to the method of division into squares, the only option is to increase the number of divisions. So, for example, we can bisect the unit length in our 3 by 4 rectangle, obtaining a division into 6 times 8, or 48 squares. This raises the total number of rectangles obtained by rearrangement to 10 (5 not counting the order of the sides). By repeated such subdivisions, we might hope to increase the density of population of rectangles so generated, whose areas are all equivalent to the area of the original rectangle. It might be interesting to see how the population grows, as we add new divisions.

But, should we be satisfied with this approach? Aren’t we plunging into a “bad infinity” of particulars? Is there no way to obtain an overview of the whole domain? And remember, our geometrical domain is not limited to linear commensurability of sides. Indeed, a bit of reflections suggests, that for EVERY given segment, there must exist a rectangle, whose area is equivalent to the given one, and one of whose sides is that length. How might we construct such a rectangle?

For a glimpse at a higher bounding of our problem, try the following construction: Take the rectangles constructed from any given rectangle by divisions into squares and rearrangement, as above, and superimpose them by bringing the lower left-hand corners into coincidence and aligning the sides along the vertical and horizontal directions. What do you see?

The Curvature of Rectangular Numbers, Part II

The general task, posed in last week’s discussion, was to generate the manifold of rectangles whose areas are equivalent to a given rectangle. The initial tactic chosen, was to divide the given rectangle into an array of congruent squares, and rearrange them into rectangles of different dimensions, but equivalent area. It became clear, that this tactic only yields a discrete “population” of rectangles (“rectangular numbers”), whose number depends on some characteristic of the number of divisions chosen. On the other hand, if we arrange the resulting rectangles in such a way, that their lower left-hand corners coincide, and their sides are lined up along the horizontal and vertical axes, then a hidden harmony springs into view: the upper right- hand corners of the rectangles, so arranged, appear to describe a HYPERBOLA, or at least a hyperbola-like curve. The idea suggests itself, that the discreteness of dividing and rearranging parts to form individual rectangles, is bounded from the outside by a higher continuity (ordering), whose presence reveals itself in the hyperbolic “envelope” of the rectangles.

To proceed further, let us change our tactic, concentrating on the idea, that there must exist a PROCESS of TRANSFORMATION which generates the entire manifold of equivalent-area rectangles and hyperbolic “envelope” at the same time. We might adopt the attitude, that any pair of rectangles of equivalent area expresses a kind of INTERVAL within the implied “hyperbolic” ordering of the whole.

With this in mind, start with any given rectangle, and consider the following approach. If we triple the length of the rectangle, keeping the width the same, then we obtain a rectangle whose area is clearly equivalent to three times that of the original one. If we then reduce the width of the new rectangle to one-third of its original value, while keeping the length unchanged, then the area of the resulting rectangle (with three times, the length, but one-third the width of the original) will clearly be equivalent to the original rectangle’s area. In fact, we might verify that equivalence in the former, discrete manner, namely by dividing the original rectangle lengthwise into three congruent rectangles, and then rearranging them to obtain the new one. In the same way, we could quadruple the length of the original rectangle and reduce its width to one-fourth, and so on. Obviously, nothing prevents us from applying the same procedure with ANY factor (i.e. not only 3 or 4), or from reversing the roles of “length” and “width” in this procedure.

At this point, something might occur to us, which allows us to “jump” the gap between the discreteness of our former procedure, and the underlying ordering of the problem. Up to now, we have considered as primary a process of multiplying or dividing lengths or widths by some integral number. But now we realize, that the crux of the matter, lies not in this duplicating or dividing up, but rather in the relationship of “INVERSION” between the transformation applied to the length and the transformation applied to the width. This suggests a new approach, which does not depend upon whole-number relationships at all.

Thus, take any rectangle with length A and width B. Now imagine A prolonged to ANY ARBITRARY LENGTH X. Those two lengths, A and X, define an interval. Evidently, what we must do, is to “invert” that interval with respect to B! In other words, construct a length Y, for which the interval (proportion) “Y to B” is (in relative terms) congruent to the interval “A to X”.

The required construction can be approached in many different ways. For example, generate a horizontal line, and erect a perpendicular line at some point P. Starting from P, lay off a vertical line segment PQ, whose length is equivalent to X, and determine a point R between P and Q, such that PR is equivalent to the length A. Next, chose an arbitrary point S, lying to the left of P on the horizontal line, and construct a vertical line segment ST whose length is equivalent to B. Now, generate a straight line through the points T and Q. Leaving aside the case, where that line happens to be parallel to the horizontal axis, the line through TQ will intersect the horizontal axis at some point O. Finally, generate a straight line through O and R. That straight line will intersect the vertical line ST at some point U. Reflect on the relationship formed, relative to “projection” from O, between the line segments on the two vertical lines from P and S. Evidently, the interval of PR to PQ (i.e. A to X) is congruent to the interval of SU to ST, the latter being equivalent to B. Thus, SU gives us the value Y for the required “inversion” of the transformation from A to X. In other words, the transformation of A to X, and the transformation from B to Y are inversions of each other, and the rectangle with sides X, Y will have the equivalent area to the rectangle with sides X and Y.

Consider the case, in which the value of X is changing, and observe the manner in which the positions of O and U vary in relation to X. The hyperbolic envelope is already implicit.

Those skillful in geometry will be able to devise essentially equivalent constructions, which make it possible to generate the hyperbolic envelope and the entire array of equi-area rectangles at the same time. Just to give a brief indication: Start with a rectangle, whose sides A and B lie on vertical and horizontal axes. Let O and M denote the lower left-hand and lower right-hand corner-points of the rectangle. Generate any ray from O, with variable angle, which intersects the upper horizontal side of the rectangle, at a point P. Prolonging the right vertical side of the rectangle upward, the same ray will intersect that vertical line at some point, Q. Now draw the vertical line at P and the horizontal line at Q. Those two lines intersect at a point R. Now examine the relationship of the rectangle with upper right-hand corner R and lower left-hand corner O, to the original rectangle. Examine the motion of R as a function of the angle of the ray from O.

For those who feel the compulsion to scribble algebraic equations, now is the time to kick the habit! The whole point here is to think GEOMETRICALLY. The notion of “geometrical interval” supercedes that of discrete arithmetic relationship…

The Refraction Of Light And The Circle

By Larry Hecht

The law for the reflection of a ray of light, was known since ancient times. Imagine a plane mirror, resting on a table-top. A beam of light, directed at the mirror, forms an angle with the mirror’s surface called the “angle of incidence.” About 2,000 years ago, scientists knew that the beam, after striking the mirror, would reflect off in the opposite direction, the reflected ray making the same angle with the mirror’s surface, as the incident ray.

A related phenomenon is the refraction of light: A ray of light, passing from one medium, such as air, to another, such as glass or water, is bent (refracted) as it crosses the interface between the two media. Imagine the smooth surface of water contained in a home aquarium tank. A ray of light strikes the surface, where we measure the angle of incidence. The light ray continues on, below the surface of the water, but its path has changed direction! It is bent, or refracted, such that the angle it makes with the surface of the water, measured downward from that surface — called {the angle of refraction} — is greater than the angle of incidence.

As we increase or decrease the angle of incidence, the angle of refraction also increases or decreases. But in what proportion? The most skilled investigators of the laws of optics from the Hellenic age, to the Islamic Renaissance, on to the early European Renaissance, could not discover the lawful relationship of angle of incidence, to angle of refraction. The answer was found by the Dutch republican scientist Willebrord Snell, a student of the famous Simon Stevin, in 1620. Perhaps the reason no one had found it earlier, is that the proportion is a transcendental one; that is, it expresses the relationship of a circular arc to a straight line, or chord of the circle. In geometry, this relationship is called the “sine.” It is the same proportionality discussed by Nicholas of Cusa, in the {De Docta Ignorantia}, where he demonstrates the incommensurability of straightness and curvature.

Precisely this incommensurable proportion, defines the lawful relationship between the angle of incidence and angle of refraction of a ray of light. Snell’s beautiful discovery, was to show, that no matter how the angle of incidence may vary, the ratio of the sine of this angle, to the sine of the refracted angle, remains the same. This is Snell’s Law of Refraction, also called the Law of Sines. The beauty and simplicity of it, and its relationship to Cusa’s crucial breakthrough, are unfortunately disguised by the poor teaching of geometry today, in which the trigonometric functions (sine, cosine, and tangent), are usually seen only as linear ratios; that is, as ratios of sides of a right triangle.

To see clearly, what a sine actually is, and also to better understand Snell’s Law, let us look at the description of the law given by Snell’s countryman, Christiaan Huyghens, in the closing chapter of his {Treatise on Light}, written in 1673. [The reader will have to draw this simple diagram. — ed.] In a circle whose center is O, draw a horizontal diameter CD. Let the circle represent the cross-sectional view of the air-water interface, such that the area above the diameter CD is air, and the area below, is water. Now designate a point, A, at about the two o’clock position on the circumference, from which a ray of light originates, and proceeds to the circle’s center, O. Here it encounters the surface of the water, where it is bent downward, so that its direction is toward a point B, at about the seven o’clock position on the circle. Angle AOD is the angle of incidence. Angle BOC is the (larger) angle of refraction. But also notice, that what we call angle AOD, is a measure of circular rotation: the arc AD. And, similarly, angle BOC is the arc BC.

The problem, to repeat, was to find the lawful relationship between these two angles, or arcs. From A, drop a perpendicular to the diameter CD. Do the same upwards from B. The lengths of these perpendiculars, are the sines of the angles AOD and BOC. Snell discovered, that whatever the incident angle AOD, the refracted angle, BOC, will adjust itself, such that the ratio of their sines, will remain constant. How does the ray of light, know how to do that?

An even greater “willfulness” on the part of the insensible light ray, was discovered during the remainder of the Seventeenth Century. First, Pierre de Fermat showed that the path which the light ray “chooses” from A to B, is the shortest possible in time — that is, takes the least time. This is true, anywhere along the extended line OB, not just where it intersects the circumference of the circle. Next, Jean Bernouilli investigated the refraction of a light ray, in a medium of continuously varying density, such as the atmospheric air, as it rises and thins above the earth’s surface. Being continuously refracted at each interface of the denser, with the less dense, air, the path of the light ray is a curve. Bernouilli discovered, with great excitement and delight, that the curve which the refracted light follows under such conditions, is the cycloid — the same curve which, he had just discovered, was the path of least time, for a falling body under the influence of gravitation.

We leave to future discussion, the investigation of this “higher willfulness” of inanimate objects.

Science and Life: The Importance of Keeping People in a Healthy, Unbalanced State

By Jonathan Tennenbaum

The following three-part series is ostensibly devoted to some crucial paradoxes raised by the discovery of the so-called “mitogenetic” or “biophoton” radiation of living organisms, by the great Russian biologist Alexander Gurwitsch. At the same time, I hope to provoke reflection on one of the unique and so far irreplaceable functions, which Lyndon LaRouche has performed in the life of our organization. Attention to this point may be the most efficient pathway towards understanding what is really at stake, in the issue of “nonlinearity in the small.”

Part 1: Parmenides revisited

Two leading biologists, Dr. Lebensfroh and Professor Todtkopf, were recently overheard arguing about the nature of living processes. Although the two have opposite opinions, they share a common, underlying error of axiomatic assumption, which is pervasive among even the best scientific professionals today. What is the fundamental error? Here is the dialogue.

TODTKOPF: So, you keep up with this “vitalist” obsession of yours, that there is something unique about living processes. How can you reject the fundamental accomplishment of modern biology?

LEBENSFROH: What you call “biology” has long since degenerated into blatant reductionism and mechanicism, losing sight of the real objective, which is “life.” To me, biology should be defined as the study of exactly {those} aspects of living processes, which distinguish them {absolutely} from non-living processes.

TODTKOPF: I say there are no such differences. A living organism is nothing but a very complex aggregate of molecules, interacting and combining with each other according to the known laws of chemistry and physics. Everyone knows that biology today is just a specialized branch of physical chemistry. The triumph of molecular biology is a great victory of science over naive superstition and metaphysics. For centuries unscientific people clung to the romantic idea, that some sort of “life force” or “living fluid” inhabits the tissue of animals and plants and lends them their “living” quality. But nobody ever found this living force. So it was a great breakthrough, when chemists demonstrated that living organisms are composed of exactly the same atomic elements and particles that we find in the inanimate world, in the atmosphere, in rocks and so forth. Looking for anything more is like grasping for ghosts in thin air. But fanatics continue to defend the notion of a “life force” up to this very day. I remember the uproar which was created when Justus Liebig published his book on “Chemistry and its Applications to Agriculture and Physiology” in 1840, showing that living tissue is composed nearly entirely of the simple elements hydrogen, oxygen, carbon and nitrogen, and that plants can grow on inorganic material alone. Liebig’s proposal to introduce mineral and chemical fertilizers into agriculture met fanatical resistance, even among scientists, who insisted that the nutrition of plants must involve organic material in some essential way. Even today, there is a big market for food grown with “organic fertilizers”, and many people believe that plants grown with mineral fertilizers are somehow different and even poisonous to the health. But these ideas have been refuted long since. There is no special material in living organisms, the atoms are exactly the same there as in this dirty piece of rock.

LEBENSFROH: But in living tissue the atoms are organized and transformed into complex organic molecules, like proteins and DNA or example, which are not found in the inorganic world. Only living processes do that.

TODTKOPF: People like you didn’t want to believe it, when the great chemist Friedrich Woehler succeeded in {artificially} synthesizing the organic substance {urea} from oxygen, hydrogen, carbon and nitrogen in the laboratory. That was 1828. Until then, many biologists and chemists believed that living organisms had their own, fundamentally different chemistry, and that the most important molecules composing living tissue could never be produced outside living tissue. The famous chemist J.J. Berzelius even put forward the “vis vitalis” hypothesis, according to which the characteristic difference between living and nonliving systems lay exactly in former’s supposedly unique powers of chemical synthesis. This idea was the original basis for the division between “organic” and “inorganic” chemistry, which turned out to be just conventional and not fundamental. After Whler, countless other organic molecules were synthesized, and today, we can make amino acids, small proteins (peptides), and pieces of DNA in the laboratory with no trouble. So, there is no special chemistry and no magical synthetic powers of living organisms.

LEBENSFROH: Aren’t you cheating with that argument? You left out the fact, that living human beings — chemists — carried out those laboratory syntheses. So they are still products of living processes, even if the reactions that produce them occur in a test tube. The organic molecules would never arise by themselves, without human intervention.

TODTKOPF: Not true. Researchers have demonstrated in laboratory experiments, that amino acids — the building-blocks of proteins — can be generated by electric discharges in a gas similar to the Earth’s original atmosphere. The Nobel Prize-winning chemist Manfred Eigen has shown, that in a “soup” of chemicals, more and more complex molecules can evolve from simpler ones in a purely spontaneous manner, through a kind of natural selection process among competing chemical reaction-cycles. Given enough time, I am sure all the complex biomolecules would eventually arise in such a self-organizing “chemical soup”. Eigen proposes that the first primitive living organisms actually evolved in this way, and I believe him. It was a gradual process, and there was never a definite point when you suddenly had “life,” and before just a lot of reactions.

LEBENSFROH: You mean to say, that if your mother had only been 5% pregnant and if you were 95% dead, you would still be speaking to me now?

TODTKOPF: Sometimes I feel that way.

LEBENSFROH: But, seriously, you cannot deny that living organisms behave completely differently from non-living matter?!

TODTKOPF: This is just a matter of degree of complexity. Naturally, the more complex a system becomes, the more circus tricks it can perform. But in principle, every chemical process going on in a living organism could be carried out just as well in a test tube. We are already doing DNA synthesis and other sorts of enzymatic reactions that way. It’s just when you put all those molecules and reaction processes together, that you get the effect of life.

LEBENSFROH: What about {growth}? Only living processes grow in a self-similar, exponential way. Whatever you say about the origin of living processes, the {power of growth} distinguishes them absolutely from non-living matter.

TODTKOPF: Really? Crystals can grow too, can’t they? Haven’t you watched how sugar or salt crystals grow in a water solution? Would you say those growing crystals are alive?

LEBENSFROH: No,no, wait a minute. Uh, crystals don’t grow in an exponential way, but actually more like an arithmetic or rather cubic series, as additional layers are added on, surface by surface.

TODTKOPF: And what do you say about a {chemical chain reaction}, as we find in the detonation processes in various explosives? Furthermore, in the 1920s the Russian chemist Semionov discovered the phenomenon of “branched chain reactions”, in which a population of enzymatic molecules grows exponentially, by catalysizing the synthesis of identical molecules in a mixture of reactants. These “autocatalytic” processes display exactly the same growth-curve characteristics, as cultures of bacteria and other living organisms.

LEBENSFROH: But this only works until as the mixture of reactants is used up. After that the process stops, doesn’t it?

TODTKOPF: Don’t living organisms also stop, when their source of nutrition is exhausted? After all, living organisms, like bacteria, never actually grow exponentially. Their growth curve is always an “S-curve”, as growth slows when the bacterial population has reached a maximum density, where the available sources of nutrition become marginalized and the culture reaches an equilibrium or stationary state. And such S-curves are typical of thousands of autocatalytic chemical reactions, which we can make in a laboratory. So in terms of the growth curve, you can’t tell the different between the growth of various chemical species in an auto-catalytic, branched chain reaction, and a population of bacteria which grows in the same way.

LEBENSFROH: But what about the population of the human species? The human population has grown exponentially over history.

TODTKOPF: I don’t think that can continue indefinitely. After all, resources are limited. But even if human multiplication could continue without limit, you mean to say that only human beings are living organisms, and animals and plants are not?

LEBENSFROH: No. The growth of the human population, and its impact on the biosphere in terms of a multiplication of domesticated plant and animal species, demonstrates that the {totality} of living material on the Earth, taken as a whole — what Vernadsky called the biosphere, including human beings — the biosphere has the potential for {unlimited growth} in the Universe. Actually, this was the directionality of evolution even before human culture emerged. So we can say, that living organisms are uniquely characterized by the {potential} for exponential growth, as part of the growing biophere.

TODTKOPF: Well then, from your rather involuted argument you will have to reocognize countless {inorganic} processes on the Earth as “living”, if they are connected with the growth of the biosphere in any way, won’t you? After all, the combustion of oil, or the production of steel, has increased exponentially with the expansion of human population and its economy. So would you include combustion or steel production as living processes?

LEBENSFROH: Of course not. You are just twisting my argument into nonsensical shape.

TODTKOPF: Then where do the inorganic processes leave off, and the “living” process begin? You claim there is a categorical, absolute distinction between the two. Would you say that the oxidation of glucose in cells is a living or nonliving process? It’s really just a form of combustion isn’t it, burning sugar for energy.

LEBENSFROH: This is just a trick of yours, to rip an individual chemical processes out of the organic context of the living process of which it is a part. In fact, the unique characteristic of living organisms is their indivisible unity or “wholeness”, which means that all processes going on in an organism are interconnected and subordinated to a single overall principle, and that all react together as a whole — rather than an assembly of parts — to every change in the organism’s environment. No mere mechanical or other non-living physical system has such characteristics.

TODTKOPF: Wrong again. Modern quantum physics has gone far ahead of you, and identified what are called “macroscopic coherent states” in {non-living matter} — states you would be forced to admit have every bit of that quality of “one-ness” you ascribe to living organisms. Even the wave-front of a light wave displays this quality, as Fresnel already demonstrated in his analysis of the diffraction of a light-beam at a sharp edge: When part of a light-wave encounters an obstacle, the entire wave-front “reacts”, and the direction of propagation is changed. Within scale-lengths of the order of a single wavelength of light, the light wave behaves as an indivisible whole. Beginning the early 1920s, entirely analogous characteristics were demonstrated for beams of electrons. Modern quantum physics teaches us, that even a single electron involves a process distributed over a large region of space, and which “feels” all the event events occurring within that space. Furthermore, we today have countless experimental proofs, that there is no such thing as a truly isolated, independent particle, atom or molecule. Rather, in a certain sense each particle in the Universe “knows” and reacts to what is happening with every other one, without having to be informed by any sort of signal! Our lasers, superconductors and even the semiconductor devices which are the basis of today’s computers and communications systems, are all based on that principle. In such devices, huge numbers of atoms behave as if they constituted a single coherent entity. The fact that we can demonstrate this sort of “holistic” behavior in so-called nonliving systems, has been a major breakthrough, demystifying the characteristics of living organisms and demonstrating, once again, that there is no categorical distinction between living and non-living processes.

LEBENSFROH: You are bluffing. You are ignoring the crucial property of living organisms, which is their ability to {reproduce themselves}, based in the unique process of mitosis or cell division. No reductionist or mechanicist theory could possibly describe such a self-reproducing process.

TODTKOPF: Evidently you are not familiar with the work of John von Neumann on {self-reproducing machines}. Although such machines have not actually been built yet, von Neumann proved their feasibility in principle long ago, and he even worked out how such machines would have to be programmed. Essentially, a self-reproducing machine would consist of a complex of computer-controlled, automated industrial processing-units and robots, all directed by a central computer. The robots gather raw materials from the surrounding area and feed them into the industrial process-units, which in turn produce materials and parts to match those from which the central computer, robots and industrial process-units themselves were constructed. As the final step, the central computer directs the assembly of those parts into a second copy of itself and its robots and industrial processing units. Obviously, such a machine would have to be extremely complex, and indeed, this is the fundamental point that John von Neumann and others have stressed — that there is a lower limit to the necessary, minimum complexity of a self-reproducing machine. This explains why qualitatively new types of phenomena occur when systems become as complex as cells. So a living cell is just an extremely complex kind of self-reproducing machine, with a bit of holism thrown in, if you want.

LEBENSFROH: You mean to say, you are a von Neumann clone.

TODTKOPF: No doubt about it. That’s where all of us modern biologists come from.

What is the fundamental error in this whole discussion? What do Dr. Lebensfroh and his unfortunate colleague have to learn from Gauss’ Determination of the Orbit of Ceres?

The Importance of Keeping People in a Healthy, Unbalanced State

by Jonathan Tennenbaum

Part II

Lebensfroh felt frustrated and a bit depressed after his encounter with Prof. Todtkopf last week. He was sure he had been right, and Todtkopf wrong, when he insisted that living processes could not be reduced to the same physics as nonliving processes. But in spite of this, Todtkopf seemed to have come out ahead in the debate. Todtkopf’s arguments reminded him of prosecutors who can “prove” or “disprove” anything, by a selective arrangement of supposedly unassailable, “hard facts.” Lebensfroh had tried to defend life, and lost his case. It wasn’t any particular argument, but the <whole debate> that had somehow missed the point. Lebensfroh felt embarrassed, like someone who had lost his wallet to a pickpocket.

Returning home, Lebensfroh sank deep into his armchair. He went through the discussion with Todtkopf again in his mind. Where was the mistake? Lebensfroh had presented a series of properties A, B, C, D …, each one of which he considered to be a unique and exclusive property of living processes: the synthesis of complex organic molecules, exponential growth, self-replication, “wholeness,” and so forth. One after the other, Todtkopf returned the argument, by presenting examples of nonliving processes which seemed to have the same properties, and maybe even all of the properties Lebensfroh had come up with. Lebensfroh was dismayed. What he thought he had understood very well before the argument started — namely the unique nature of life — now seemed to have evaporated into something intangible and elusive, even in his own mind.

Suddenly he had a new thought. He recognized it came from something he had read long ago by Cardinal Nicolaus of Cusa, concerning the nature of the circle. The thought was: If someone would specify any set of points A, B, C, D … on the circumference of a circle, would that determine the circle as the curve passing through those points? Well, obviously not, someone else could just connect the points by <straight lines>, getting a polygon, which is not the same as the circle. No number of points, so supplied, could ever suffice to distinguish the circle from a mere polygon. What, then, is the characteristic distinction of the circle?

Lebensfroh’s gloom and frustration disappeared, like the popping of a bubble. In his mind’s eye, Lebensfroh caught a glance of the old cardinal’s face, smiling at him. Lebensfroh smiled, too. “Thanks, Nick,” he heard himself say.

The next day Lebensfroh met Prof. Todtkopf again.

Todtkopf: Well, I hope you have given up your silly idea after our last conversation.

Lebensfroh: Indeed. I will never again lose sight of the false, lying nature of so-called “scientific facts.”

Todtkopf (shocked): What do you mean!? Facts never lie. Facts are the very foundation and essence of truth.

Lebensfroh: Wrong. I say, truth lies entirely outside, above and apart from mere “facts”; and no single fact, nor any collection of facts, however comprehensive, could ever represent truth. Only ideas, not facts, can represent truth.

Todtkopf: Are you crazy?

Lebensfroh: I will show you. See how I draw this circle, and now I mark points A, B, C, D, etc. on it, which represent what you call “facts”….

Todtkopf: Don’t talk to me about geometry. I am a biologist. I don’t go there!

Lebensfroh: The problem is, the conception I want you to understand, cannot be communicated without a certain type of metaphor …

Todtkopf: I am a scientist, not a poet.

Lebensfroh: Well I tell you it is <absolutely impossible> to grasp what a living process is, without metaphor. Because there is an ordering of ideas in science, and the conception of “living process” is a strictly <higher type> than any conception which can be communicated in a linear way. This would be obvious to you if you had worked through Gauss’ determination of the orbit of Ceres, for example. The nature of living processes, and the absolute, “strong” gap separating them from all non-living processes, lies in the characteristics of <change> manifested in the virtually infinitesimally small.

Todtkopf: I have no idea what you mean.

Lebensfroh: Well, I see we’ll have to approach this through an example. I have it! Let’s look at a unique case, which poses the relevant paradoxes in the strongest form: a physical economy, which is a very special sort of living process.

Todtkopf: What do you mean by “physical economy”? I remember reading something about that.

Lebensfroh: I mean the physical process by which a human population reproduces the material conditions for its continuing existence, at ever higher levels of potential population density.

Todtkopf: So it’s more than just the living population and its immediately activity, but also the physical processes in mining and industry, which deal with inorganic materials, as well as things like farming?

Lebensfroh: Of course. Physical economy subsumes the processes of agricultural, mining and industrial production; distribution and consumption of goods; housing, education, and health services; cultural activities, scientific research, administrative and related activity and so on — everything necessary for the maintainance and development of human society from one generation to the next. In a sense, all these things form the tissue and organs of the physical economy as a coherent living entity.

Todtkopf: Ha! Now I have caught you in a contradiction! Just a moment ago you restated your old thesis, that there is a categorical distinction between living and nonliving processes, true?

Lebensfroh: True.

Todtkopf: And according to that you would distinguish between living and nonliving matter, wouldn’t you?

Lebensfroh: Yes.

Todtkopf: Then tell me this. A piece of rock sitting somewhere in a mountain, is that living or nonliving material?

Lebensfroh: Nonliving, of course.

Todtkopf: And when that same rock is mined, and the ore is transported to a factory, and metal is produced, and that metal is worked up into parts, and the parts assembled into a machine, and that machine is integrated into the production process — would you not say, that the material of the rock has become part of the physical economy?

Lebensfroh: Yes.

Todtkopf: So then, if the physical economy is a living process, then the material which constitutes it must be living, must it not?

Lebensfroh (hesitating): Well, I guess so.

Todtkopf: Then one and the same material is both living and nonliving, or else you will have to tell me at what point the rock, or ore, or metal, or machine, became “living,” in your sense!

[Lebensfroh realized he was about to fall into the same trap, as he had done in his earlier debate with Todtkopf. Focussing on his happy idea about the circle, he recovered quickly and continued.]

Lebensfroh: Exactly. That is just the point. We are dealing with a multiply-connected manifold.

Todtkopf: There you go again with your mathematics! Tell me plainly now: do you or do you not regard the machines in a factory as being <living>, in virtue of their being integrated as parts of the “tissue” of the physical economy, which you call a living process?

Lebensfroh: In a sense, absolutely, yes. But the “living” aspect of these things does not lie in the things themselves as isolated entities, but in the characteristics of the process of <change> in which they actively participate. And the chief characteristic of change, which defines a physical economy as <living> (as opposed to pathological, dying states of an economy), is <scientific and technological progress>. That progress takes the form of an incessant series of “pulses” or “shocks” of <change in the organization of production> — shocks which originate in fundamental scientific discoveries of principle, and propagate, like waves, throughout the tissue of the economy. Those pulses or shocks reflect the action of a higher geometry — one characterized by human creative reason — upon the ensemble of lower geometries conposing the tissue of the physical economy.

Todtkopf: You mean to say, without those pulses, the tissue of the economy would degenerate and the economy would “die”?

Lebensfroh: Exactly. And I am sure that something analogous must occur in living processes generally, and on another level, in the creative processes of the mind itself. The great biologist Alexander Gurwitsch had some appreciation of this.

Todtkopf: What you say is amazing.

Lebensfroh: Not really. Imagine how stupid you would be right now, if Nicolaus of Cusa had not helped me get your mind moving.

Science and Life: The Importance of Keeping People in a Healthy, Unbalanced State

by Jonathan Tennenbaum

Part III

At the end of last week’s discussion, Prof. Todtkopf was amazed and a bit overwhelmed by the conception Lebensfroh came up with, that physical economy might provide the key to understanding living processes in general. But later, as he thought back on the conversation, his admiration turned to suspicion, then irritation, and finally rage. The more he thought about it, the more ridiculous it seemed to him to mix up economics and biology as Lebensfroh had done, comparing an economy to a living cell, for example. Todtkopf’s teachers had taught him to beware of sweeping analogies, which might excite our fantasy, but undermine the objectivity that are essential to professional scientific work. Todtkopf saw himself admonishing an audience of his colleagues: “In science the first step is to {define your terms}; and once you have done that, you have to stick to the definitions. If you start to play with metaphors and analogies, as Lebensfroh loves to do, then you can make anything into anything, as if you would say: the solar system is a living process, the galaxy is a living process, an atom is a living process, EVERYTHING is a living process?!! Then we would all feel happy, like Dr. Lebensfroh. Absurd! By throwing words around like that we accomplish nothing of any substance.”

Lebensfroh has to be cut down to size, thought Todtkopf. He should stop acting as if he were superior to us empiricists, just because he has a creative mind. I’ll give him a lesson on what science is all about. He started lecturing again:

“Science is based on empirical fact. That means observing and investigating the real objects in the world around us. To be able to arrange the facts, and to correlate facts in order to adduce general laws, you need to establish a division of the sciences. The sciences are divided according to the different kinds of objects you study. So, biology studies the living organisms which are divided into plant and animal. To determine what a living process is, you start concretely, by studying this specific plant, that specific animal. Nothing to do with economics or anything like that. You keep studying those plants and animals and then you correlate your observations and measurements and draw general conclusions. So, by painstaking investigations, molecular biologists discovered the common molecular basis of living organisms — the amino acids and proteins, the genetic code and so forth. Step by step, we unravelled the mechanisms and we discovered that in each case we examine carefully, we find everything occurs according to the known laws of physics and chemistry — laws verified in hundreds of thousands of laboratory experiments. At least, no one in academia dares refute us. The wispy dreams of the vitalists, have given way to piles of hard facts. This is the triumph science, the triumph of Aristotle, the first biologist and systems analyst!

“So don’t ever forget, Lebensfroh: We empiricists are the ones who do the real work. We know what functions and what doesn’t function in the real world. Don’t stand there and try to tell us how we should do things!” Professor Todtkopf was so preoccupied, that he emptied his coffee cup onto his trousers.

The next day Todtkopf sought out Dr. Lebensfroh.

Todtkopf: Our conversation last week was fun, Dr. Lebensfroh. But speaking as a professional scientist, I must say, it was a waste of time.

Lebensfroh (taken aback): Why that?

Todtkopf: You presented not a single solid scientific fact, but only wild, irrelevant analogies to economics and so forth. I was taken in for a moment, but now no more.

Lebensfroh: Oh, oh, I see you have decayed into your lower state!

Todtkopf: Lower state? Decayed?

Lebensfroh: Well you know, according to modern physics we find that atoms and molecules can exist in different modes or states, which form a discrete series or spectrum that is characteristic for the species of atom or molecule involved.

Todtkopf: Every chemistry student knows that.

Lebensfroh: In the so-called ground states or lower-energy states, atoms and molecules are typically inactive and inert. But if we irradiate them with photons of the right wavelength, for example, we can raise them into a higher-energy, excited state. They become highly reactive, they begin to emit radiation, they are more lively and interesting in every way. We can get lasing and all sorts of wonderful things to happen.

Todtkopf: And?

Lebensfroh: But if they are left to themselves, and taken out of the special environment we have created, the atoms and molecules tend to decay back to their lower-energy states, and become lazy and boring again. So it is with PEOPLE, too.

Todtkopf: There you go again with your analogies and metaphors! What does that have to do with me?

Lebensfroh: Because last week at the end of our discussion I had pulled you up to an excited state for a while and now you seem to have slipped back down. The difference is elementary and very easy to observe, when one knows what to look for. People in higher (creative) states of mind think of the Universe in terms of {change}, while in your lower state, you think of it in terms of arrangements of objects.

Todtkopf: What difference does that make? Thinking is thinking.

Lebensfroh: Not so. If you were to stay in your present state, you would be incapable of making any fundamental discovery.

Todtkopf: How do you know? I can look through a microscope as well as you!

Lebensfroh: Maybe even better than me, but you won’t {discover} anything. Because a fundamental discovery is not the discovery of some property of an object, but a {change} in the characteristics of our own mental processes, a change in the way we {think} about the Universe as a whole. It occurs entirely inside the mind. And that is the beginning of actually changing the Universe itself. But it can’t happen if your mind is in the deadened state, typified by a fixation on objects or object-like images.

Todtkopf: Challenge me. I will show you you’re wrong.

Lebensfroh: Fine. The other day you asserted molecular biology had for the first time identified the chemical basis for living processes?

Todtkopf: Yes, of course.

Lebensfroh: Then tell me, what is the difference between a living cell, and the same cell immediately after it has died? The molecules stay the same. Even many reactions keep going for a while, as they might in the non-living environment of a test tube.

Todtkopf: Um…Uh… Well, eventually the normal processes stop and the cell disintegrates. You can see this in a microscope.

Lebensfroh: I am not asking what {eventually} happens, as a {result} of the event of the cell dying. I mean the event itself. What is it {precisely}, that has happened at that moment?

Todtkopf: Obviously, there was some divergence from normal functioning, and the cell did not recover.

Lebensfroh: {Why} didn’t it recover? As the Russian biologist Gurwitsch and others showed, sometimes living cells can recover from the grossest sorts of disturbances. So, for example, Gurwitsch centrifuged fertilized egg cells until the visible structures in the cell had been destroyed, and yet the cells reorganized themselves and developed into adult organisms. What is it that occurs, at the moment when a living process, which was viable before, loses that capability?

Todtkopf: Actually, I must admit I don’t know. Maybe there is no simple general answer. Of course there are millions of papers about aging of tissue and various damage mechanisms which can lead to the death of cells. But actually, I don’t recall anyone having posed exactly the question you are asking, in such a straighforward way.

Lebensfroh: Isn’t that a bit strange? After all, you were just claiming the molecular biologists had uncovered the molecular basis for the main processes which occur in living organisms. But as for such a central issue in biology, as I now have raised, you haven’t even begun to address it. Doesn’t that suggest some problem with you thinking?

Todtkopf: I see what you mean. But maybe the answer is very complicated.

Lebensfroh: If you had studied how Gauss determined the orbit of Ceres, you would at least know how the question would have to be approached experimentally. What is the characteristic of the orbit of a comet, for example, which is headed for a collision with the Sun? What is the {change} in orbital {characteristics}, between a “healthy” orbit and an orbit which might differ at first only imperceptibly from the healthy one, but lead inexorably to the destruction of the comet?

Todtkopf: How can you compare the processes of a living organism with the orbit of a comet? Another of your wild analogies.

Lebensfroh: I am not comparing the two as objects. I am talking about how we have to {think} about two problems that share a common, crucial methodological feature.

Todtkopf: Well, it doesn’t help me to bring in the astronomical example. I saw that long article in Fidelio, but I didn’t work it through.

Lebensfroh: Why not?

Todtfroh: My friends all told me it is very difficult.

Lebensfroh: Why in the world, should it be regarded as an argument {against} doing something, to say it is difficult? If what Gauss accomplished were just trivial, so people could swallow it at one gulp, like a doggie cookie, then it wouldn’t be worth much, would it?

Todtkopf: I guess not.

Lebensfroh: And didn’t Gauss himself work on this for months, and other scientists spend years and decades or even lifetimes struggling to work through a crucial paradox and make a fundamental discovery of principle, coming back to it again and again from different angles until they had succeeded, for the benefit of mankind, in mastering it? Didn’t Beethoven oftentimes spend years developing a single composition?

Todtkopf: He did.

Lebensfroh: Then we should be happy when the essentials of a crucial discovery, and relevant materials, have been put together in such a way that we don’t have to waste time on non-essentials, but can get to the real issues directly. Because, truly, we live in a world where there is no time to waste. So we should concentrate on the difficult things, and brush trivial things aside.

Todtkopf: I agree. But can you at least tell me what Gauss’ work has to do with biology?

Lebensfroh: The oldest, classical problem in astronomy, is that when you observe the motion of the Sun or any planet in the sky, that motion actually results from many different motions, all acting during in any arbitrarily small interval of the observered motion. So, the motion of Mars in the sky, for example, involves Mars’ own orbital motion, the rotation of the Earth, the orbital motion of the Earth with respect to the Sun, the precession of the equinoxes, and even still other, more subtle and partly even not-yet-discovered cycles. The subtler point is, none of these motions is strictly independent from the other, but each one reacts to the existence of the others.

Todtkopf: Then, how is it possible to disentangle them?

Lebensfroh: There is no formal mathematical solution. But there does exist a method of {experimental measurement} based on so-called analysis situs, which Kepler applied in a masterful way to his founding of modern astronomy. The crucial point is, that the principles or “dimensionalities” of action we are looking for are axiomatically distinct, linearly incommensurable principles; each is characterized by a different characteristic curvature in the infinitesimally small. Their mutual action generates dense singularities. Secondly, the ensemble of such principles must be harmonically ordered according to a still higher principle.

Todtkopf: How do you know that?

Lebensfroh: That is Kepler’s higher hypothesis, that our Universe is ordered in that sort of way. He demonstrated that the harmonic organization of motions of our solar system is uniquely coherent with that hypothesis, and in his snowflake paper he did the same thing for the microscopic domain, too — at least provisionally.

Todtkopf: I will have to believe you. But get to my question: what does this have to do with biology?

Lebensfroh: Very much, obviously. But in our discussion the particular issue keeps coming up, that the processes in living tissue are determined by more than one fundamental ordering principle. We have one set of principles — the one you associate with “ordinary physics and chemistry”, and which you and your colleagues observe operating also within living organisms, at least to a very great extent. However, in living tissue another, higher set of principles — a higher geometry, in effect — is superimposed upon those “inorganic” principles. If fact, we can even say, that the higher principle {rules} the lower one, even though the effect of the higher geometry might only appear as a virtually infinitesimal displacement from the pathway, that the process would have followed, had only the lower, inorganic principles been active. Nevertheless, the overall cunulative effect of that “infinitesimal deviation”, is enormous. This sort of situation is quite familiar from astronomy. There, the most powerful, “tectonic” forces are the ones connected with what appear at first as barely perceptible, infinitesimal deviations or anomalies within otherwise well-determined orbits.

Todtkopf: What you say seems strange to me. How can it be that a “strong” force appears as the most infinitesimal?

Lebensfroh: Here is another case, where a key point of method can hardly be communicated effectively, without geometry. But this time maybe you will offer more patience than last time I tried.

Todtkopf: I am definitely in an excited state.

Lebensfroh: Good. Now take this piece of paper, and observe how I role it into a cylinder. No problem, eh?

Todtkopf: Very easy.

Lebensfroh: And now I role it into a conical shape.

Todtkopf: Also no problem.

Lebensfroh: And many other shapes are possible, obviously. But what about giving the paper a spherical shape, or even part of a sphere. See, here I have a globe and I am trying to bend the paper onto its shape.

Todtkopf: I see, it doesn’t work. You get creases all over and it still doesn’t really fit.

Lebensfroh: And what would happen if I tried to make part of the surface of the globe into a flat surface?

Todtkopf: You would tear it, for sure, if it were made of some material like paper which doesn’t stretch.

Lebensfroh: Is that problem a matter of how large the portions of surface I use?

Todtkopf: Evidently not.

Lebensfroh: So, then, the characteristic which causes these violent creases and tears — and I guess you will agree, these would be typical of “strong forces” — is manifested as a virtually {infinitesimal} difference at the level of a tiny section of the spherical surface vis-a-vis the flat surface. Of course, when I look at larger portions of the surfaces, the discrepancy in shape and characteristics becomes macroscopically evident.

Todtkopf: OK, I get it. So you want to say, for example, that we should think about the higher principle acting in living tissue as a kind of “curvature” imposed on otherwise relatively “flat” geometry of non-living physio-chemical processes.

Lebensfroh: Wonderful!

Todtkopf: So that, if we just examine a small, isolated aspect of the living process, the effect of that curvature might appear virtually infinitesimal. But, if this your approach is correct, somewhere in there we must find extremely intense forces of tearing or wrenching between the geometries. Because they are axiomatically incompatible. What form would those “creases and tears” take?

Lebensfroh: That question obviously takes us beyond mathematics, into the domain of experimental biophysics. This is exactly the area of Alexander Gurwitsch’s fundamental work, which led him to the discovery of the so-called “mitogenetic radiation”, or constant photon emission from living tissue. This radiation is so extremely weak, many orders of magnitude weaker than the metabolic energy of the tissue itself — so weak that most scientists today regard it as an irrelevant, mere curiosity devoid of biological or biochemical significance. This is because they don’t understand the elementary point you just grasped. Alexander Gurwitsch and his followers developed an elaborate series of unique experiments based on the characteristics of this very weak radiation, and all directed at disentangling and measuring the higher principles of ordering of living processes.

Todtkopf: What did they discover?

Lebensfroh: Well, this was literally a life’s work, and worth more than five minutes’ discussion. But without my going into the experimental method, perhaps you might in conclusion like to hear how one of Gurwitsch’s students summarized some of the main {conclusions} of that work. Actually, the conclusions are {questions}: they lead into an entirely new domain of biology, which has barely been explored up to this day. Here is the quote:

“The conclusion was that the harmonic movements observed in a normal cell are due to a certain factor related to the cell as a whole and this factor is not destroyed or inactivated by the destruction of the visible intracellular structures or processes. Hence, {space-time connections between separate intracellular structures or processes are not due to any properties of the structures themselves}. A further conclusion was, that together with stable structures in which the molecules are bound by means of various types of chemical bonds, there are {unstable} molecular constellations in which the molecules are not connected with each other by any of such bonds, but where their association is maintained by a continuous influx of energy… Such labile, energy-dependent molecular constellations were designated by A.G. Gurwitsch as {‘unbalanced molecular constellations’}. … However, the continuous influx of metabolic energy is a {necessary} condition, but {not the only one} for the existence of unbalanced molecular constellations. Their existence is elicited by a certain dynamic factor, whose action, although somehow connected to a continuous utilization of metabolic energy, is quite independent.”

Todtkopf: What are those “unbalanced molecular constellations”? I don’t know of such a thing in chemistry, even today.

Lebensfroh: Well first of all, you might have fun thinking about the last of Gurwitsch’s conclusions, mentioned above, in relation to physical economy. What is involved by the impact of scientific and technological progress on the investment cycle (metabolism) of free energy and energy-of-the-system of an economy? As for Gurwitsch’s “unbalanced molecular constellations”, I think we illustrated that principle in our very conversation today.

Todtkopf: How so?

Lebensfroh: Well obviously the living process is a constant battle to keep those molecules from slipping back into their accustomed, banal, stupid, boring inorganic state. What must be supplied, to accomplish that, is not “energy” in the ordinary sense, but rather something akin to what Nicolaus of Cues did for me the other day, and what I have tried to do for you in these last two talks. Don’t you think those great men are to be honored and emulated, who constantly raise people upward toward the passionate pursuit of truth. These are the real benefactors, fathers and leaders of the human race!

A note on the fate of Gurwitsch’s work:

The discovery of so-called “mitogenetic radiation” by the Russia biologist Alexander Gurvitsch, as a biproduct of Gurwitsch’s investigations into the higher principles of organization of living processes, was regarded by many leading scientists in the 1920s and early 1930s as one of the most far-reaching experimental discoveries in modern science. Among those was V.I. Vernadsky, a personal friend of Gurwitsch from the beginning of his researches at the Crimean town of Simferopol in 1918. Gurwitsch’s decisive 1923 experiments, established that 1) all living tissue is a source of sustained, though highly variable and (in scalar terms) {extremely weak} radiation in the ultraviolet range of the light spectrum; 2) the process of cell division (mitosis) can be triggered by the absorption of no more than a {single photon} of such light by a suitably disposed cell; and 3) the existence and function of such “mitogenetic radiation” is intimately connected with the manner in which all local processes in a living organism — e.g. on the cellular, molecular and even atomic scales — are subordinated to a principle of organization unique to the living organism as a whole. By using mitogenetic radiation as a crucial experimental method in embryology, physiology, the study of the nervous system and other areas, Gurwitsch and his collaborators made one remarkable discovery after another, continuing through Gurwitsch’s death in 1954.

Starting no later than the end of the 1920s, systematic operations were launched to “kill” the new area of research. These included a widely-publicized hatchet-job done on behalf of the Rockefeller Foundation by one A. Hollaender. By the end of the 1930s, Gurwitsch’s scientific reputation in Western countries had been significantly undermined, only to be virtually buried under the onslaught of ultra-reductionist currents of molecular biology after World War II. While the main lines of Gurwitsch’s work continued to be pursued in the Soviet Union — including in military-related domains –, the efforts of Hollaender et al. established the “consensus opinion” in the West, that Gurwitsch’s radiation did not exist; or in case it did exist, had no scientific importance. With the rapid overall decline in the quality of science in the Soviet Union from the 1960s and especially the 1970s on, the focus on the fundamental implications of Gurwitsch’s work was nearly lost there, too.

It was only in the mid-1970s that Gurwitsch’s work began to be revived in a serious way, with the work of Fritz Popp and is collaborators in Germany and other countries. From 1985 on, Lyndon LaRouche personally and his collaborators have played a crucial, indispensible role in keeping work in this and related areas alive internationally. In every case that has been examined so far, the results of Gurwitsch’s laboratory have been confirmed. In the meantime, technological developments make it possible to design new species of experiments which would not have been possible in Gurwitsch’s time.

In retrospect, it is obvious that a major motivation for burying Gurwitsch’s work, was that it threatened to derail the British plans, notably supported by the Harrimans, Rockefellers and others, to establish “race science” and eugenics as “authoritative scientific doctrines”. This program was resumed immediately after the war, and took the included form of a massive promotion of radical-mechanistic, reductionist forms of molecular biology and genetics, which had already begun to be developed by Max Delbrueck and others in the middle 1930s, with the support of the Rockefeller Foundation’s Warren Weaver. Of course, these operations went hand-in-hand with the promotion of behaviorist psychology, mechanistic theories of nerve function (Hodgkin-Huxley, John von Neumann, Norbert Wiener etc), the work of von Neumann and others on formal logic, “artificial intelligence”, “self-reproducing machines”, “information theory” and so on and so forth. The British side, often clothed in “holistic” trappings, included the Huxleys, Joseph Needham, J.S. Haldane, Waddington, Bernal, Russell of course etc. The Cambridge side of the British elite were predominantly biologists, following in the putative footsteps of Aristotle himself. Of course, the so-called “biologism” of Haekel et al. was an important current flowing into the Nazi movement, and into today’s New Age and green movements.

Incidentally, I personally had the occasion to visit Hollaender in his office together with Fritz Popp around 1985, not long before Hollaender’s death at the age of over 90. Hollaender admitted having being deployed by the Rockefeller Foundation to Russia with the sole purpose to “investigate” Gurwitsch and his laboratory, bringing back the story that Gurwitsch’s experimental technique was allegedly “sloppy” and his results “unreliable”. (Hollaeder subsequently carried out and published in 1937 his own botched series of experiments, allegedly failing to discover any evidence of Gurwitsch’s radiation.) Confronted with Popp’s detailed measurements of mitogenetic radiation using modern photomultiplier instruments, Hollaeder admitted without blinking an eyelash, that he “had always suspected Gurwitsch had been right.”

How Archimdedes Screwed the Oligarchy, Part 1

by Ted Andromidas

I began my investigation of the implications of the use a minimal surface by Brunelleschi, not merely as a theoretical or experimental investigation of physical principle, but as a “machine tool” breakthrough in constructing the cupola of Santa Maria de la Fiore, by investigating the historic scientific foundations upon which this breakthrough depended. I began, therefore, looking at the Classical Hellenic scientific tradition.

First let us re-acquaint ourselves with the physical principle used by Brunelleschi in the Dome’s construction. Why do we call a soap film bound by one or more wire hoops or boundaries a minimal surface? With amazing elegance and simplicity soap film solves an historic mathematical problem, namely, the soap film finds the least surface area amongst all imaginable surfaces spanned by the wire. For example, a “trivial” minimal surface which connects the interior of a circular hoop is a flat circular plain.

In a minimal surface the surface tension stabilizes the whole surface because the tension is in equilibrium at each point on the soap film. In other words, the tension at each point on the surface is equal to the tension at any other point on the surface. Just as the hanging chain or cable equally distributes the weight across its entire length, so the minimal surface also distriubtes the tension equally across its entire surface.

To see this for yourself, take a simple wide rubber band and begin stretching it. As you apply greater tension across the rubber band, you will notice that the middle of the rubber band is narrower, thinner, and almost translucent. The tension across the surface, at that point, is greatest. In fact, you know that the band will snap at that point if you continue to pull it apart. The stretched rubber band is not a minimal surface!

What a wonderful paradox! The surface which creates the minimal of all possible areas within any given set of boundaries also creates equal and minimal tension across the surface.

As we discovered last time, the current history of science indicates that the first non-trivial examples of minimal surfaces were the catenoid and helicoid found by J.B. Meusnier in 1776. Yet, as LaRouche discovered, Brunelleschi uses a minimal surface as a principle of physics in the construction of the Dome.

At this point I thought: Does this principle of least action, though not “proven’ mathematically go back to the classical Hellenic period? If the Archimedian screw has been described as a kind of helicoid, was, perhaps, the common bolt thread the first minimal surface studied? In reading a small article on the history of the bolt, I learned that the first comprehensive studies and development the screw or bolt thread are attributed to Archytas of Tarentum, the last and greatest of the Pythagoreans.

I went looking for Archytas.

A close friend and collaborator of Plato, it is if Plato had Archytas in mind when he says that “…those cities rejoice, whose kings philosophize and whose philosophers reign.” Archytas himself was so loved and respected in his native city that, though there was a one year “term limit” for anyone to act as chief executive of the city of Tarentum, the citizens suspended these rules and elected him to hold that position for seven consecutive years. We get a sense of his collaboration with Plato in the “Seventh Letter”.

Here, Plato discusses his various attempts, at the behest of his student and friend Dion, to teach the just anointed ruler of Syracuse, Dionysus the Second how to become a “philosopher king”. Plato says: “Dion persuaded Dionysios to send for me; he [Dion, ed.] also wrote himself entreating me to come by all manner of means and with the utmost possible speed, before certain other persons coming in contact with Dionysios should turn him aside into some way of life other than the best. What he said…was as follows: ‘What opportunities,’ he said, ‘shall we wait for, greater than those now offered to us by Providence?'” Archytas certainly helped Plato in this endeavor: “…it seems, Archytas came to the court of Dionysios. Before my departure I had brought him[Archytas, ed.] and his Tarentine circle into friendly relations with Dionysios.”

Plato makes clear his regard for Archytas when he says again in the “Seventh Letter”, that when Dionysus invited Plato to Syracuse a second time, he sent the invitation with one of the students of “…Archytas, and of whom he supposed that I had a higher opinion than of any of the Sicilian Greeks-and, with him, other men of repute in Sicily.”

Finally, when it becomes clear to all that not only is Dionysus deaf to Plato’s teaching, but, infact, the tyrant is determined to kill him, Plato turns to Archytas for help: “I sent to Archytas and my other friends in Taras, telling them the plight I was in. Finding some excuse for an embassy from their city, they sent a thirty-oared galley with Lamiscos, one of themselves, who came and entreated Dionysios about me, saying that I wanted to go, and that he should on no account stand in my way.”

Most of what we know about Archytas and his thoughts comes from either references from the writings Plato, Eudoxos, Plotinus, Eratostenes and others, and a handful of fragments his own writings. Nonetheless Archytas’ contributions seem to have been substantial and essential, to classical Hellenic science. In the following fragment Archytas writes of the science of mathematics: “Mathematicians seem to me to have excellent discernment…for inasmuch as they can discern excellently about the physics of the universe, they are also likely to have excellent perspective on the particulars that are. Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music. These seem to be sister sciences, for they concern themselves with the first two related forms of being [number and magnitude].”

Besides tutoring Eudoxos, some historians contend that Archytas also tutored Plato in mathematics at some point during the ten years that Plato spent in Sicily and Southern Italy.

Besides saving Plato’s life, itself no mean contribution to the future of humanity, Archytas’ is also known as the founder of scientific mechanics. Other numerous contributions were in the fields of music, astronomy, mathematics, and aerodynamics. He also provided the first solution the age-old problem of “doubling the cube”, i.e. constructing the side of a cube that is double the volume of a given cube.

As I said, Archytas speaks to us only through fragments, yet his thoughts on human creativity and resonate with our own when he says in one fragment: “To become knowledgeable about things one does not know, on must either learn from others or find out for oneself. Now learning derives from someone else and is foreign, whereas finding out is of and by oneself. Finding out without seeking is difficult and rare, but with seeking it is manageable and easy, though someone who does not know how to seek cannot find. ….”

In astronomy Archytas first put forward the notion of an infinite and boundless universe when in another fragment he says: “…since space is that in which body is or can be, and in the case of eternal things we must treat that which potentially is as being, it follows equally that there must be body and space extending without limit.” [This is not to be confused with the idea of simple extension of three linear extensions in space. Ed.]

As with all leading Pythagoreans, Archytas studied music. From these studies comes his discovery and development of the so-called “harmonic mean”.

Archytas is also credited with having developed a geometrical method for the famous “doubling of the cube” using a cylinder, cone and torus. Though not attributed to him there, some historians insist that Archytas approach to this problem can be found in Book VIII of Euclid’s “Elements” .

Since Archytas avowed that geometry was came from the study of physics, this particular solution to the “cube” problem could well have developed out of his work as an inventor and machine tool designer. As I said, Archytas is sometimes called the founder of mechanics.

As reported last week, General of the Revolution and student of Monge, Jean Baptiste Meusnier not only “discovered” the minimal surfaces of the helicoid and catenoid. But also designed and flew the proto-type of the first Dirigible.

In an historical parallel which is certainly not accidental, Archytas is credited with designing and flying the proto-type model of the first heavier than air aircraft.

According to Hero of Alexandria, Archytas designed and built an apparatus wherein a wooden bird was apparently suspended from the end of a pivoted bar, and the whole apparatus revolved by means of a jet of steam or compressed air.

Which takes us to the bolt or screw thread, in principle, the first use of a minimal surface. The which Archytas created and Archimedes then developed even further. Over the next week, why don’t you investigate this problem for yourself.

Construct a cylinder and a helix on that cylinder. You can do this by either constructing a paper of cardboard rectangle with a diagonal, and bending the rectangle into a cylinder; or get an empty paper towel role, which has the helical structure built in. Using the helix as a guide and the cylinder as your unthreaded “bolt”, with paper or any other “bendable” material, try to construct the “threads” of the “bolt” around your cylinder.

I urge you to take some time and try various ways of creating the appropriate shape of the surface that you will “bend ” around the cylinder. I actually spent several hours drawing and cutting various shapes out of paper and then trying to fit them around a cylinder. So give it a try See what you get. Find out for yourself.

Next installment we will look at exactly what kind of surface we need to construct.

How Archimedes Screwed the Oligarchy, Part 2

Once I determined to investigate the implications of LaRouche’s 1987 discovery of the use of “minimal surface” or ” least action’ physical principles in the design and construction of Fillipo Brunelleschi’s Dome of the Cathedral of Florence, I began to look at some of the history classical Hellenic and Hellenistic science.

Among the first “connecting references to minimal surfaces in Classical Hellenic and Hellenistic science was between the Archimedean Screw a water pumping device, though developed sometime in the 3rd century BC, still widely used today, and the helicoid surface as discovered by French Revolutionary General J.B. Meusnier.

Initial investigations of Archimdes’ invention, led to several references comparing the minimal surface helicoid to his invention. Yet none of these references noted the obvious paradox that the former discovery of the helicoid is attributed to Meusnier, the student of Monge, 2000 year later.

This in turn led me back to the 4th century BC founder of mechanics and rescuer of Plato, Archytas of Tarentum, as a way of coming back to the Archimedean principle two centuries later. It is important to note that, in principle, the “machine tool physics” as developed by Archimedes rested upon an historical foundation of at least two centuries or more. This in turn, and in steps, I’m convince will lead back to the implications of Brunelleschi’s Dome of Cathedral.

The problem of design faced by Archimedes would have been:

What kind of surface is the thread* of a bolt or screw?

How would I investigate and map such a surface? Put in another way: How would I “blueprint” the necessary specifics of a new machine tool product like the bolt thread?

Let me be clear. Despite what many historians assert, the engineering methods used by these early “machine tool” designers were not based on trial and error.

Let’s look at the “physics” we began investigating last week; the physics out of which the Archimedean Screw must have developed. As I indicated earlier, this device, invented sometime in the 3rd century BC, is still in use today. It is an ideal, relatively inexpensive means for pumping large volumes of water or other fluid like material, i.e. sand, fine gravel, ore, etc. Therefore improvements in design and development have continued to the present day.

In the latest study of “Optimal Design Parameters for the Archimedean Screw,” as printed in the Journal of Hydraulic Engineering, March 2000 edition, it has been determined that, given various critical parameters, the Archimedean Screw as designed by Archimedes and described by the Rome architect in Book VIII of the Architecture, is in fact, if not the optimal design…the best design! Given design parameters like angle of pitch of the tread surface to amount of thread rotation, or the width of the thread surface compared to the diameter of the overall structure, pumping screw as designed by Archimedes is 7% off from the optimal as determined by today’s engineering capabilities.

In other words, the Journal of Hydraulic Engineering concluded, there is no cost effective way to improve upon the original 2000 year old design. Yet that same Journal’s authors assert that the incredible success of this design is a result of mere experience with the technology over centuries. This is quite an arrogant assertion on the part of the Journal, as none of Archimedes thoughts on the invention of the screw are extant, owing in part to the Roman’s burning of the library of Alexandria. The only course left to the modern investigator, therefore, is to replicate Archimedes thinking, which, in no way, can be considered trial and error.

Two centuries earlier, Archytas was inventing the bolt and screw, whose function can be studied as the intersection of several different, intersecting and interacting surfaces. Archytas is also credited with providing a solution to the age old problem of doubling the cube, using the intersection of those surfaces, i.e. the cone, cylinder and torus.

Archimedes developed a machine tool of such efficient design that, to date it is the best design for doing the job, moving large volumes of fluids. This design also requires the intersection several different surfaces. Archimedes is the first to scientifically investigate volumes of spheres, cylinders and cones, and their inter-relationships. He studied the relationship of weight to volume, using water, to develop the idea of specific gravity. He was not only a mathematician, he was a master inventor and hydraulic engineer.

With this all said; what is the relationship between the “thread” of the bolt and the “cylinder of the bolt? What kind of surface is that thread?

We can “develop” a cylinder by “bending” rectangular plane such that two parallel sides are joined to form the side of the cylinder, while the other two parallel sides from the base and top circles. A cone can be “developed” from a circular plane. Simply cut an arc out of the circle in a “slice of pie” shape. Bend that circular slice of pie arc such that two radii of the circle meet forming the side “ray” of the cone; the point where the two sides of the pie meet, the center of the complete circle of the circular arc is the apex of the cone; the semi-circle forms in the circular base of the cone.

In both cases there is no “ripping” of the surface to make it fit. You just bend it. As you know, we can not “develop” the sphere from a plane; it is not a developable surface. If you did some experimentation last week you might have discovered that the surface of the thread is also not developable.

It is the case though that the circular place surface and the helicoid share common features: 1) They are both minimal surfaces. They define the least area connecting a set of boundaries. The circle, for example is the maximum area for the minimal circumference. The helicoid is a surface which in connecting the boundary defined by a helix also describes the minimal area.

As we pointed out last week, the easiest way to construct a minimal surface is to dip a wire in the shape of the boundary with which you wish to construct the surface, i.e. circle, two circles, cube, pyramid, etc. The soap film will quite beautifully “describe” the shape of the minimal surface connecting those boundaries. A helical wire with a central axis will “describe” the surface called a “helicoid”. 2. Both the circular plane and the helicoid are “ruled” surfaces. If you rotate a straight line such that one end is fixed at a point and the other end of the line rotates around tat point, the straight line become the radius of the circle which seeps out a circular plane surface.

Now look at the helix on your cylinder. The cylinder is bound by two circle whose radii are the radii of the helix as well. Now begin to wind one of those radii along the helix, keeping it perpendicular to the side of the cylinder. Think of a winding staircase inside a lighthouse or turret. Think of the edge of each step as the radius of the helix.

This process will describe the helicoid as “discovered” by Meusnier. Now this is fascinating. We’ve discovered the minimal least action surface of the helicoid as developed in the Archimedean screw in the 3rd century B.C. Now, while trying to convey the idea of constructing a helicoid, we discover that the spiral staircase, an ancient architectural and engineering feature, also describes the helicoid minimal surface. One of the best examples of this is Tycho Brahes observatory in Copenhagen Denmark.

It must be the case that for centuries, if not millennia, architects have been incorporating least action principles of minimal surfaces into their engineering techniques.

More next time.

The Simplest Discovery, Part I

by Jonathan Tennenbaum

The fundamental crisis of civilization is forcing the question: Where do the ideas come from, which we find in our own heads and those of our fellow human beings, and which determine Mankind’s ability or inability to survive the onrushing crisis?

The “short answer” is, that apart from the products of oligarchical manipulation, corruption, and decay, everything {positive} in our culture — not only science and technology, but the concepts of everyday life, and language itself — derives from nothing but the generation and assimilation of validated discoveries of principle, made by individual human minds, as measured on the metric of increase in the per-capita potential population density of the human species. The power of the oligarchy, of course, depends to a large extent on the success of its own massive efforts to cover up and distort the historical generation of culture (including science), while promoting popular belief in various varieties of empiricism and so-called “innate” or “self-evident” ideas.

It is worth stressing, that the battle against oligarchical obfuscation of the history of ideas, was a center of concern to the “American” republican circles around Schiller, Humboldt, and Gauss et al. at the beginning of last century. Moreover, exactly the concept of positive human culture as the result of integration of individual acts of resolution of fundamental paradoxes by the Platonic method of hypothesis, was key to the revolutionary work of (among others) Gauss, Weber, and Riemann on the “anti-entropic” geometry of physical space-time. Riemann discusses this explicitly in his posthumous fragments on epistemology, which provide a most useful background for comprehending his 1854 paper on “The hypotheses underlying geometry.” Consider, in particular, the following passage translated from Riemann’s posthumous fragment, entitled “Attempt at a theory of the fundamental concepts of mathematics and physics as the basis for the explication of Nature.” (Note, that in this location, Riemann employs the terms “Begriff,” concept, and “Begriffssystem,” system of concepts, in a sense congruent with Lyn’s use of “fundamental assumption” and “hypothesis,” respectively).

“On the basis of the concepts, through which we grasp the natural world, we not only constantly supplement our observations, but, in addition, we determine certain future observations in advance as necessary, or — in case our system of concepts is not sufficently complete — as probable; on this basis it is determined, what is `possible’ (i.e., also what is `necessary’ or that whose opposite is impossible); furthermore, the degree of possibility (the `probability’) of every single event so judged possible, can be mathematically determined, when the concepts are sufficiently precise.

“If an event occurs, which is necessary or probable according to the given system of concepts, then that system is thereby confirmed; and it is on the basis of this confirmation through experience, that we base our confidence in those concepts.

“But if something unexpected occurs, being impossible or improbable according to the given system of concepts, then the task arises, to enlarge the system, or, where necessary, to transform it, in such a way that the observed event ceases to be impossible or improbable according to the enlarged or improved system of concepts. The extension or improvement of the conceptual system constitutes the `explanation’ of the unexpected event. Through this process, our understanding of Nature gradually becomes more comprehensive and more true, while at the same time reaching ever deeper beneath the surface of the phenomena.

“The history of the exact sciences, as far as we can follow it backwards in time, demonstrates that this, in fact, is the pathway by which our knowledge of Nature has progressed. The systems of concepts, which form the basis of our present understanding of Nature, were generated by progressive transformations of older conceptual systems; and the reasons that pushed forward the generation of new modes of explanation, can in every case be traced back to contradictions or improbabilities arising in older modes of explanation.

“Thus, the generation of new concepts, insofar as it is accessible to observation, occurs by this process.

“Herbart, on the other hand, has provided proof, that those concepts, upon which our conceptualization of the world is based, but whose origins we can neither trace back in history, nor in our own development, because they are transmitted together with language without being noticed — all of those concepts, insofar as they are more than mere forms of connection between simple sense perceptions, can be derived from the above source; and need not be attributed to some special property of the human soul, assumed to predate all experience (as Kant claimed to do with his categories).”

Often it is most instructive, in exploring the implications of a fundamental principle such as Riemann’s, to focus attention on the most deceptively simple cases — cases of the sort fools would be likely to dismiss as being “too obvious to be worth thinking about.”

Take, for example, the everyday concept of a “day.” What could be more self-evident? Does Riemann actually mean to say that there is a real, creative {discovery} embedded in that idea? What would have been the paradox or paradoxes, whose resolution gave birth to the concept of “a day”? Evidently, the discovery involved predates history in the usual sense. What we might do, is try to “project” ourselves mentally back to a hypothetical, very, very distant point in time, at which the concept of “a day” did not exist, and then ask: What paradoxes {must intrinsically} confront a mind in the process of freeing itself from a naive, beast-like belief in the primacy of sense-perception? First, reflect on the following:

Could we discover anything without memory? Is a pot-headed Yahoo, who cannot remember what he saw or did 5 minutes earlier, able to make scientific discoveries? Would a Yahoo ever have been able even to discover the existence of a “day” as a recurring cycle of light and darkness? Or was the development of poetry, as a means of development of the powers of memory, crucial to the emergence of human civilization?

Pre-Socratic Greek tradition often spoke of the origin of the Universe in terms of the creation of Order (Cosmos) out of Chaos. Does this not exactly describe the subjective process by which a human mind frees itself from the blind impulses of “animal instinct” and “sense certainty”? The world of the existentialist Yahoo, or a newly-born infant, is a kind of Chaos, a “kaleidoscope of feelings” replacing each other in more or less rapid succession. Mankind could not survive, were it not possible to awaken a power of {creative discovery} in the infant or the supposedly infant-like, primitive man — a mental function energized by the most powerful form of human emotion, {Agape}. It is that agapic power, inseparable from the faculty of {memory} as understood by the Renaissance, which conquers the Chaos of bestiality and creates the Cosmos of human development as an ordering of successive acts of discovery.

Next, consider the elementary paradox of change, as it is addressed by the simplest of astronomical discoveries. The following exploration is hypothetical, but necessarily touches upon a discovery actually made (and in fact, made repeatedly in various forms) in human history.

You are a prehistoric human being, living perhaps 500,000 years ago. On a beautiful clear night, you seek a place to lie down under the open sky. Gaze up, from there, at the magnificent canopy of the heavens! The myriad stars shine down on you in majestical silence, like little lights afixed to a lofty dome. Here is peace, here is rest! You close your eyes and relax.

You awaken later that night. As your eyes once more open to the sky, you are struck with a sudden sense of strangeness. Something is different! Something has happened! The stars seem to have changed. Looking around, you recognize a group of bright stars, whose form you remember having remarked before you took your nap. That group of stars is no longer where it was before; the stars have changed position!

Changed? How is that possible? You stare intently at the stars. Not the slightest motion is perceptible; only a gentle twinkling while they remain, seemingly immovable, in their places.

A paradox! On the one hand, your faculty of sense perception, swears to you that the stars are fixed and motionless. On the other hand, you remember that the same faculty has earlier testified, no less insistently, of an arrangement of stars in the sky, which is different from the one it now reports!

Intrigued, you repeat the experiment, but with a variation: you ask a friend to keep watching the stars, without interruption, during the time your eyes are closed. The experiment is performed. Once again, you find an undeniable change in the positions of the stars, when you look at the sky again after a nap. Your friend, however, swears he never saw the stars move!

The paradox strikes deep into your mind. Whatever follows, will depend on how you respond to the paradox. However you respond — or even if you do {not} respond — that response will reflect some sort of {hypothesis}, an hypothesis generated nowhere but inside your own mind.

Shall you merely conclude that your eyes (or those of your friend) have lied to you in some arbitrary fashion? Or that the Universe itself is maliciously arbitrary? If so, then how would human existence be possible?

Or is there another way out? Perhaps we should not {completely} reject the evidence of our senses. Perhaps it were better to assume, that although our sense perceptions in themselves do not represent reality, still there must be some implicitly discoverable, lawful relationship between sense perception and reality. This is the pathway of science.

Choosing that pathway, the paradox moves us to hypothesize the existence of something, which our senses — in virtue of some lawful limitation of the same — cannot grasp: to hypothesize a concept of a {process of change}, which in itself is {invisible} to the senses, but yet efficiently accounts for the observed (or rather, remembered) {difference} in positions! That adduced concept, of an invisible — but efficient! — process of change, is an object of a different sort than a sense perception (including the paradoxical entity we commonly identify as the “perception of motion”). It is not sufficient to account for that new concept, by merely saying: “the stars move too slowly for our eyes to see.” The point is, that the paradox just presented, evokes the potential of a {new quality of relationship of our mind to the Universe}.

A change in the substance of our mind! Prior to the explosion of the paradox, you looked at the Universe (the starried heavens) as an object of sense perception. Now, you are looking at the Universe from the standpoint of a process of discovery, which stands in ironical contrast to naive belief in sense perception. To the reflecting mind, that {difference in mental attitude}, from before to now, provokes the hypothesis of {higher species of change} — a process of improvement of human cognitive powers, which is invisible to our senses, but real and earthshakingly powerful nonetheless.

Turning once more to our nightly observations, what shall be our next step? Does our power of discovery give us the capability to hypothesize, not only the existence, but also the {form} of the process of change of position of the stars? How would we discover the coherence between the paradox of the stars’ motion, and a similar paradox, posed by the behavior of the Sun? And how could we do that, using nothing more than the means which were available to prehistoric Man?

Lest the reader find the above discussion “too trivial” to be important, consider the following. Nearly everyone today is faced (or will be soon) with a congruent form of paradox: On the one hand, most people would claim that their most deeply held “values and beliefs,” being absolutely self-evident (to them!) are fixed and unchangeable. On the other hand, comparing those people’s “deeply held personal values” of today, with the corresponding values held “self-evidently” by those same people 30 years ago, we find almost nothing in common! If mankind is to survive, an increasing ration of leading and ordinary citizens must be brought to discover, as an “enemy image,” the process by which the oligarchy was able to induce that radical, downward “paradigm shift” in their own minds.

THE SIMPLEST DISCOVERY, PART II

by Jonathan Tennenbaum

Have you ever stopped to consider, how a human being, a “mere infinitesimal” on the scale of the world as a whole, could actually come to know the vast dimensions of the solar system, or to measure astronomical cycles hundreds or thousands of times longer than the brief span of his or her individual life? The existence of such powers of cognition, by which the “infinitesimal” can know the macrocosm within its own internal mental processes, is the central issue in the bitter, millenial conflict between the human species and the oligarchical “Gods of Olympus.” Witness the words of Aeschylus’ Prometheus (1):

“Believe not, that I from pride or stubbornness 
Keep silent. Heart-rending thoughts I nurture, 
Watching myself thus trodden under foot. 
And yet to the new Gods, they — Was it not I 
Who granted them their fitting honors? 
But, of this I’ll say nothing. Besides, it were to those who know 
That I would address you. But, of the dire need of Men 
Let me tell, how I made them, foolish at first, 
To be full of thought and empowered with Reason. 
I say this not to complain of them, 
But only to explain the goodly intention of my gifts. 
They, who had eyes from the first, but saw not, 
Who had ears, yet heard not; but like figments 
Of dreams, their entire life long 
Mixed all things blindly together, and knew nothing 
Of bricklaid houses and walls, 
But lived deep-down in sunless caves 
Like hords of ants, 
And knew nothing: no sign to fortell the winter storm, 
Nor the spring rich in flowers, nor the fruitful 
Summer, no sure measure. Without Reason did they act 
In everything, ’til I made them heed the rising and setting 
Of the stars, so difficult to distinguish. 
And number, a most ingenious invention, 
I created for them, and the invention of writing 
As a monument to all, and Mother of the Muses. 
And ’twas I that first put the wild beasts under yoke, 
That they do service to the plough and bear burdens, and so 
Lift many a heavy task from the backs of men. 
And to the wagons I hitched, eager willing to obey, 
Horses, the splendor of wealth. 
And to sail o’er the seas — none but I 
Invented the shipman’s winged sails. 
Yet I, who for mortals such things 
Created, can find nothing for myself 
To deliver me from my present plight.”

Not without cause did Aeschylus emphasize the earliest discoveries of astronomy, connected with the construction of a solar calender, as crucial events in the emergence of human reason as “the sure measure” of things.

Implicitly, the discoveries made by our pre-historic “colleague” in connection with the “invisible” motion of the stars, refute everything university students have been taught to believe about science and “liberal arts” since the mid-1960s. Astronomical cycles — beginning with the “day” — are neither objects of sense perception, nor are they “robust statistical correlations.” Rather, the astronomical cycles emerge as {conceptions} created in the human mind, through a process of generation and creative solution of paradoxes.

From this standpoint, let us push our exploration of prehistorical discoveries a few steps further, to identify paradoxes which {necessarily} must have arisen, even though we do not now know the specific historical circumstances.

Our prehistoric observer notes: (i) The positions of the stars appear to undergo a constant process of change. (ii) But at the same time, certain arrays of stars, identified and fixed in memory through poetic (mnemonic) devices already from earliest times, remain seemingly unchanged throughout the course of a night, reappearing every night with the same distinct form. Also, apart from the appearence and disappearance of stars on the horizon, the overall configuration of the constellations in relation to each other in the sky — “the constellation of constellations!” — remains unchanged.

This paradox of “change” combined with “no change” evokes the notion, that the “invisible” motion of the stars, has an implicitly intelligible {form}. That paradoxical idea becomes a specific thought-object, undergoing its own process of evolution in the direction of a notion of a {universal, rotational action} subsuming both the process of change in the night sky, and the daily motion of the Sun.

Indeed, observation of the rising and setting of the Sun, and studying the Sun’s overall using such means as observation of the shadows cast by poles (gnomon), demonstrates an overall {coherence} between the nightly motion of the “constellation of constellations” and the motion of the Sun during daytime. As singularities of the hypothesized universal action, we get (among other things) the differentiation of East, West, North and South as determinate directions on the Earth’s surface.

In this way, we revolutionize the empirical notion of a “day” as a mere “yin-yang” alternation of light and darkness. Instead, we conceive the day as an astronomical cycle, subsuming an {increasing density of distinct events} within a single ordered totality. Just as the gnomon’s shadow progressively transits the markings of a primitive sundial, including the meridean defined by the position of longest shadow; so the cycle of the “day” subsumes and orders the events of rising and setting of stars and constellations, and their transit across definable angular positions as defined by the sightings of a primitive stellar observatory. From the development of these methods, our predecessors established the regular division of the day, and an indispensible means for harmonically ordering the activities of society.

But, there is a far-reaching paradox embedded in this splendid hypothesis of the day’s rotational cycle as a universal ordering principle! Looming long on the horizon of our prehistoric astronomer’s mind, but now growing in urgency, is the realization, that the day itself is subject to {change}. For example, the array of constellations, which are visible in the sky just before sunrise, is strikingly different in winter than in summer. To investigate the origin of this difference, identify a star or constellation, whose setting in the West immediately preceeds the rising of the Sun in the East. Within a few days, we become aware of a slight delay in the appearance of the Sun, after the selected star or constellation sets in the West. The delay keeps growing: the Sun seems to be “slipping” backward in time relative to the stars! That apparent slippage constitutes a new, anomalous degree of change. Again, the question is posed: what is the exact {form} of this change?

Our prehistoric astonomer juxtaposes this solar anomaly with a whole cluster of paradoxes, connected with the empirical cycle of “the year.” The empirical notion of a year as a mere alternation of “hot” and “cold” seasons, or periodic recurrence of monsoons, floods or other natural phenomena, bespeaks the nearly bestial state of Man before Prometheus bestowed his gifts. The mere counting of days between recurrence of some terrestial event, leads to erratic results, falling far short of the “sure sign” promised by Prometheus.

Worse, was the attempt to arbitrarily impose upon society, a non-existent correlation between changes in season and the cycles of the Moon. So, the Babylonians (and others) insisted on a calender based on the socalled synodic lunar month, as defined by the recurrence of the full moon after approximately 29.5 days. After the passage of a mere 18 “years” of 12 synodic lunar months each, winter now occurs in the months where summer used to be, and vice versa! The attempt to “fix” this monstrous failure with the addition of special days and alternation of “longer” and “shorter” months, while rejecting the primacy of the solar cycles and insisting on the cult of the Moon, (or some “rotten compromise” between the two) is more than typical of the psychosis which dooms every oligarchical empires to collapse. Although the present Western calender is entirely solar-based, and our months have no correlation to the phases of the moon, the term “month” still remains as an apparent relic of Babylonian lunacy.

In contrast, by adducing a new, “solar long cycle” from the {anomaly} posed by the slight discrepancy between the solar motion and daily stellar motion, our prehistoric astronomer was eventually able to invent a “sure measure” of the seasonal cycle, which remains true over centuries and even millenia! The result is best demonstrated by the spherical sundials of ancient Greek times, which registered not only the daily trajectory of the Sun, but also the cycle of variation of the Sun’s (approximately circular) pathway in the sky, over a period of (approximately) 364 days. That cycle subsumes the cycle of change in the relative lengths of night and day, as well as the angles of inclination of the Sun’s rays to the Earth’s surface, providing in turn an intelligible basis for the variation of the seasons.

But, the manner in which the yearly solar cycle “modulates” the daily one, ordering the variations of the latter, implicitly poses a new array of paradoxes. For example: If the day is variable, might not the year be so also? And in fact, careful observation of the loci of rising or setting of the Sun and the stars, by means of suitable horizon markers and observation points, revealed a very slight — but distinct — anomaly in the solar cycle. From this, the ancient astronomers were able, thousands of years ago, to adduce an approximately 26,000 year-cycle of the so-called precession of the equinoxes! The result is a third, “long cycle” modulating the year. The latter, according to our best present knowledge, essentially determines the cycle of ice-ages, together with a fourth anomaly, namely the elliptical character of the earth-sun orbit.

Still another paradox exploded in the repeated, failed attempts to fit various among the multiply-connected solar cycles, as well as planetary and other cycles, into a single calender. This included the search for a single “great cycle” subsuming all the others, such that the end of such a “great cycle” would mark the simultaneous end of all the shorter cycles. The work of the Pythagorians, on the fallacy of “linear commensurability” put an end to such Babylonian “systems analysis,” and posed yet a new level of paradox:

If there exists no grand mathematical system which can combine and account for the various cycles, then how can we conceptualize the “One” which subsumes the successive emergence of new astronomical cycles as apparent new degrees of freedom of action in our Universe? How do we master the paradoxical principle of Heraclitus, that “nothing is constant except change?”

———————————————————

(1) This version is my first-draft translation of lines 436-471 in the “Reclam” series German translation of Aeschylus’ drama, which seemed somewhat clearer than an English translation I had seen earlier. Bruce Director, who first called my attention to this selection, told me there are better English translations. But, I think the point is well enough made for the present purpose.

The Spiral Of The Primes

by Ted Andromidas

When I presented the draft of our last discussion on prime numbers and the notion of indicative quota, to one of my closer collaborators, I was filled with a sense of satisfaction, and wonder, at having gotten a glimpse at what I thought was the idea of number, the generation and distribution of the prime numbers and their connection to the notion of indicative quota. She read it, looked up and said, not the “Ah-ha” for which I had so patiently waited, but “Yeah, so what?”

I was stunned! I sputtered: “What do you mean ‘SO WHAT?’? Are you confused.”

“No, not really.” she said. “I was confused for a moment when you tried to convince me that Eratosthenes and Earthshines was a clever play on words, until I realized that you screwed up the spell checker; AND, despite the fact that you then tried to convince me that there was some pedagogical significance to putting the footnotes out of order, rather than just sloppy editing, I do see how Eratosthenes’ method works; it’s just: What’s the significance of this to indicative quota? And, as a matter of fact, what’s the significance of this problem of prime numbers at all?”

I walked away baffled and disoriented. I thought it was all so clear.

If someone were to say that an “indicative” quota, i.e. a systematic approach to raising money, is just like any other idea of quota, that it’s just a number reached by adding the money raised in any given week, and that any change in the that quota is merely an process of adding or subtracting from that number, could we not characterize that as an “axiom of the system of quota”? Then couldn’t…

“You say to somebody, ‘Here is the axiomatic problem.’ Everybody in mathematics who has a terminal degree–which is what happens to you before they put you in a body-bag–knows the hereditary principle. Even Bertrand Russell knows the hereditary principle–or knew it, wherever he is today. Everybody knows that if you construct a logical system– and mathematics as usually defined is nothing but a logical latticework– everyone knows that if you start out with a system based only on axioms and postulates, and you develop only deductive theorems based on these axioms and postulates, that the entire latticework, which can never be closed, consists of nothing but echoes of the axiomatic assumptions with which you started. Therefore, if one of the axioms is false, the entirety of that field of knowledge collapses.

“An example: If you say that the only thing that exists in arithmetic is the integers, as counting numbers–that everything else is synthetic–therefore, so the argument goes, all mathematics must be derived from the counting numbers as the axiomatic foundation. So you start with an axiomatic counting system, 1 + 1, you construct that, and from that elaborated basis you must develop all mathematics. This is essentially what Russell and Whitehead demanded: radical nominalism. Therefore, as the case of prime numbers implicitly proves–the Euler-Riemann theorem, the work of Gauss on prime number sequences, the ingenious foresight of Fermat on this question, the work of Pascal on the question of differential number series–the entire history of mathematics, centering around this fantastic little problem of prime numbers…”  —Lyndon LaRouche, Schiller Institute Conference, 1984

If we look at the process of counting as iteration, as a function of a one dimensional manifold, we are confronted with “…this fantastic little problem of prime numbers…”: we can not determine the distribution or “density” of prime numbers between 1 and any given number N by any other means than that of Eratosthenes? We can not determine what the next prime number, or for that matter, any future prime number in a counting series, will be before it is actually generated by counting?

Begin counting 1, (2), (3), 4, (5), 6, (7)…;(all prime numbers are bracketed in parentheses) at first it seems that all the prime numbers are also all the odd numbers; that is by just adding 2 + one, then add 2 + 3, i.e. f(p)= (1+2x). We see, to generate the primes, but as we continue to count, (2), (3), 4, (5), 6, (7), 9, 10, (11), (13), 14, 15, 16, (17), (19), 20, 21, 22, (23) 2…,the “pattern” or function seems to change. For a while it seems to be f(p)= (6x +/- 1) till we reach (23), then it changes again. We seem unable to discover a successor function for, not just all the prime numbers, but any particular continuous series of prime numbers.

There have been innumerable functions, theorems, hypotheses, corollaries and conjectures written on this problem: The Prime Number theorem, the Reimann Hypotheses, the Twin Prime conjecture, the Goldbach Conjecture, and the Opperman Conjecture just to name a few.

Let us look at a conjecture referenced several times by LaRouche, that of Pierre de Fermat. Fermat conjectured that every number of the form (22^n +1) is prime. So we call these the Fermat numbers, and when a number of this form is prime, we call it a “Fermat prime”; the only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537.

In 1732 Euler discovered 641 divides F5 and F(n) has been extended to 31, (i.e. 22^31 + 1) and no other primes have been generated by this function. It is, therefore, likely, yet not proven, that there are only a finite number of Fermat primes.

You might remember that the Fermat primes were the subject of a recent “Reimann for Anti-Dummies”. Gauss proved that a regular polygon of n sides can be inscribed in a circle with Euclidean methods (e.g., by compass and straightedge) if and only if n is a power of two times a product of distinct Fermat primes. (Hopefully we will look at this problem from the vantage point of Riemann’s and Dirichlet’s correction of Euler, if I can figure it out by then.)

Anyway, for now, let’s just continue to investigate the phenomenon of counting and the primes.

The Spiral of Prime Numbers

We saw last week that, using the method first developed by Eratosthenes in the 3rd Century B.C., with the circle as 2 dimensional manifold, we could construct a cyclical or modulo approach to the determination of the distribution of the prime numbers in the one dimensional manifold of the number line. The limitations of that approach are obvious. Moreover, it actually tells us more about the process of generating the non-prime, composite numbers of the number field as, implicitly, an ongoing succession of prime number cycles, than revealing something about the generation of the prime numbers themselves.

As a prime number is generated it is implicitly the modulus of an ongoing cycle, which intersects all past and future cycles of previous prime numbers, transforming the entire number field past and future. Yet, we seem unable to account for the generation of that singular event in the number field, the generation of a prime number, till, in fact, it occurs.

Perhaps it is the way we count; let us “count” differently. Rather than imagining the number line as straight, one dimensional manifold, i.e. 1, 2, 3, 4, 5, 6, 7…etc., is it possible to count in two dimensions? As we’ve seen from the dialogues of Philosph and Cando, numbers in a 2 dimensional manifold are not necessarily what they are in one; but rather than looking at the characteristic differences between two dimensional and one dimensional measure, let us take a more simple construction, and see what happens. Let us generate the number field in 2 dimensions by using a simple a kind of “Archimedean spiral”.

In the center of a piece of note paper write the number 0. To the right of that write the number 1; above one write 2. Now count to the left 3, 4. Below 4 write 5(at the same level as 0); below 5 write 6. To the right of 6 write 7, 8, 9; above 9, at the same level as 1 write 10, and go up 11, 12, 13; now count left of 13, 14, 15, etc. As you count this way, you will generate a spiral of numbers.

Now, beginning with zero, start counting the numbers spiralling out from there.(see figure 1). Try it; it is really not difficult.

(figure 1)

4-(3)-2 | | (5) 0–1 |

6—(7)-8—9

What do you notice, almost immediately: a certain number of prime numbers are generated along various diagonals of the number field.(see footnote 1)

Now, if we begin counting with 5 as our first number in the center of our spiral, we notice that between 5 and its square, 25, all the numbers that lie on a diagonal connecting 5 and 25 are prime numbers. They are not all the prime numbers between 5 and 25, but they do define a successor function of prime numbers between 5 and 25.

Begin counting at 11 and we generate a diagonal of prime numbers along the axis between the prime number 101 through 11, to 121, the square of 11. If we start counting with our Archimedean spiral at 41, we discover the same generating characteristic: it is a line prime numbers which stretch along the 41 and 1681 diagonal. In fact, if we count in this manner from 41 to 10,000,000, half of the numbers on that diagonal will be prime.

When we count in the one dimensional manifold, we can, through the sieve of Eratosthenes, determine that the cyclical, modulo characteristics of the counting numbers, the integers is ordered through the two dimensions of circular action. Yet, there seems to be no “connectedness” at all to the ordering characteristics of the prime numbers, no “pattern” seems to emerge.

When we actually begin to count in a two dimensional manifold, a “connectedness” emerges almost immediately, numerical shadows on the wall of Plato’s cave. Why? Is there some ordering principle from a higher, perhaps 3 dimensional manifold, ordering the two dimensions of our spiral counting? It doesn’t provide us with an actual function for determining the distribution of the prime numbers, nor does it help us develop a successor function, but it does impel us on to a notion of a succession of ordering principles, as we will begin to see in our next discussion.

And finally, to my collaborator’s insistent, “So what? What’s the significance of the prime numbers, anyway”, Karl Friedrich Gauss would reply:

“The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and prolix that even for numbers that do not exceed the limits of tables constructed by estimated men, i.e. for numbers that do not yield to artificial methods, they try the patience of even the practiced calculator… The dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.”

— Karl Friedrich Gauss, Disquisitiones Arithmeticae (translation: A. A. Clarke)

Footnote 1.

Here is a list of the first prime numbers to aid you in your investigations.   2 3 5 7 11 13 17 19  23 29 31 37 41 43 47 53 59  61 67 71 73 79 83 89 97 101  103 107 109 113 127 131 137 139 149  151 157 163 167 173 179 181 191 193  197 199 211 223 227 229 233 239 241  251 257 263 269 271 277 281 283 293  307 311 313 317 331 337 347 349 353  359 367 373 379 383 389 397 401 409  419 421 431 433 439 443 449 457 461  463 467 479 487 491 499 503 509 521  523 541 547 557 563 569 571 577 587  593 599 601 607 613 617 619 631 641  643 647 653 659 661 673 677 683 691  701 709 719 727 733 739 743 751 757  761 769 773 787 797 809 811 821 823  827 829 839 853 857 859 863 877 881  883 887 907 911 919 929 937 941 947  953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 1117 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 1493 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693

Spring Cleaning For The Mind: On `Proof,’ Part I

by Jonathan Tennenbaum

 “You have to prove your case…” “Demonstrate to me on my own terms, that what you say is right…” “Where are your facts? Give me the hard facts!” “I agree with you, but my wife…” “That sounds exaggerated. How can you be so sure?…” “Why should I believe you? I heard something different from my friends and high-level contacts!”

Most arguments tend to be a waste of time, because they avoid the really sensitive issue, underlying everything else: What does it mean to KNOW something, as opposed to merely having an opinion, belief or strong impression? And how is actual knowledge communicated? By what sort of deliberative processes might human beings arrive at shared, valid judgements of reality? What is the authority, by which a scientist (for example) might uphold a truth, otherwise regarded as obviously wrong or absurd by the overwhelming majority — or even every single one — of his colleagues?

At first glance, the mathematical notion of “proof”, as historically associated with Euclid’s elements of geometry, would seem to provide an {ideal} model of rational argumentation on any subject. According to this method, one first seeks to identify, as a “common denominator” and foundation for argument, those most elementary propositions and facts, as are acknowledged to be self-evident and true by all thinking persons. Then one seeks to reduce all other truths to those elementary ones, by means of logical deduction.

Unfortunately, most people nowadays lack even a rudimentary acquaintance with the old-fashioned treatment of Euclidean geometry as a lattice-work of theorems deduced from a set of definitions, axioms and postulates. They are thereby deprived of a most useful means, with which both to conceptualize the devastating fallacy of deductive notions of knowledge, and to grasp the LaRouche-Riemann correction of that fallacy.

Accordingly, I propose to approach the problem, this time not through geometry per se, but with reference to the practice of political organizing, whereby the issues of “knowledge” and “proof” are posed again and again, on a daily basis.

The trouble starts, typically, when the organizer asks himself or herself: “How do I CONVINCE this person to do [what they should do]?

This almost instinctively leads to the question: “How do I acquire the necessary AUTHORITY to move the mind of this person?” Having difficulty locating his or her INTERNAL authority — and Lyn’s authority — in a rigorous process of discovery of universal principles, the organizer tends to fall back on a dangerous ruse: To appeal to the purported authority of certain, strongly-held beliefs and opinions in the minds of the people one is attempting to organize, as the basis for eliciting agreement with the proposed proposition

In other words: the organizer wittingly or unwittingly adopts, as the standard of “proof”, that which public opinion accepts as “convincing arguments” — RATHER than those processes, by which reality can actually be made known to the cognitive processes of the individual, sovereign human mind. This inevitably leads, in the form of argument too commonly practiced nowadays, to the following parody of formal deductive method:

1. Select a set of basic, commonly-accepted concepts and a set of basic propositions, which appear so self-evident, that they are generally regarded as true beyond any doubt (or at least taken to be so by the person you are trying to convince!). This selection of concepts and “self-evident truths” plays a role analogous to the set of definitions, axioms and postulates in Euclidean geometry.

2. Now formulate the proposition you wish to “prove,” in terms of the adopted system of basic concepts, fulfilling the demand of your interlocutor: “to express your point in terms I can understand.”

3. Supplement, as required, the array of “basic, self-evident truths,” with a complementary array of “facts” — perceptions of events, as expressed and interpreted in terms of the given basic concepts, axioms and postulates, and having a comparable, self-evident quality of “hardness” and authority. “Facts” of the form, typically: “I heard him say it myself,” “I saw it with my own eyes, on television,” and so forth.

4. Now construct a (more or less rigorous) chain of deductions, showing that the proposition you are putting forward is a logical consequence of the given array of “basic truths” and “hard facts.”

5. Now tell the person: “You see! My proposition is a theorem of the postulates and facts, whose authority you accept. Therefore you must accept what I am saying.”

A bit of honest reflection, will show that the mode of argument, used by most people most of the time, does indeed converge on a parody of the mathematical-deductive method, along the lines just sketched. And, when this method fails to achieve the desired result — as it most often does fail, and in a deeper sense ALWAYS fails –, then we tend put the blame on the “irrationality” of the person we are arguing with, or on a purported lack of a sufficient arsenal of “hard facts” to back up our argument. Yet, the essential folly was on the organizer’s own side.

In the spirit of “spring cleaning,” let us look more carefully at this purported solution to the dilemma of “proof.” A very simple observation reveals a devastating fallacy — a fallacy of such incredible virulence, that it can bring down entire civilizations!

The very nature of a DEDUCTIVE argument (by definition!) is that it systematically EXCLUDES from consideration, everything except the original array of definitions or concepts, axioms, postulates and purported “facts” — the latter being framed on the basis of those same, generally-accepted concepts and axioms. Our argument was confined entirely to the “virtual world” of our interlocutor’s concepts and assumptions. At no point did we ever address reality itself! At no point did we oblige, or even encourage our interlocutor to actually DISCOVER anything about the real world!

We were arguing as if at a blackboard, in a room without windows. The very form of our argument was such, that no universal physical principle could EVER be discovered or otherwise known by such means. To the extent we “succeeded” in convincing our interlocutor, we actually perpetrated a fraud. Because, the “agreement,” thus elicited, is purely accidental and has occurred in the absence of any real process of cognition.

All this brings us to an agonizing paradox: If we cannot base ourselves in the authority of generally-accepted, “self-evident” truths, then what can we start with? What do we do, when, in the now-typical case, our interlocutor’s basic axioms are incompatible with every essential principle of reality, as these have become known to us, above all through the work of Lyn?

Archimedes reportedly once said: “If you give me a lever and a place to stand, I can move the Earth.” Our task, indeed, is to move the Earth. Where, then, shall we stand?

SPRING CLEANING FOR THE MIND: ON PROOF, PART II

“Science is a manifestation of action in human society…. One cannot know a scientific truth by logic, but only by life. ACTION is characteristic of scientific thought.” –Vladimir Vernadsky

In the first installment of this discussion, I set forth an elementary paradox, arising constantly in political organizing, and which might be restated in brief as follows:

In the course of arguing any point with a person, most of us have a nearly instinctive tendency, to seek for some commonly-shared, fundamental beliefs, values or opinions — plus commonly-accepted “hard facts” — as a basis for an essentially deductive “proof” of the point we are arguing for. But, what if the person, we are arguing with, seems to have a completely opposite set of basic assumptions?

Sometimes, in this situation, organizers resort to a clever, but ultimately disastrous form of cheating. They think to themselves: “This guy here has some beliefs and opinions which are totally opposed to our axioms. He will freak out if I show him what LaRouche really stands for.

“So let me find some a couple of specific issues where he will agree with us.” In other words: avoid a confrontation on the axiomatic level, by carefully selecting a small {subset} of theorems derived from “LaRouche’s axioms,” which happen at the same time to be theorems in the other guy’s axiom-system, or at least to be {consistent} with the latter — even though the two sets of axioms are themselves mutually inconsistent!

Of course, the “common ground” secured this manner, is entirely spurious, and can fall apart at any moment. But exactly for this reason, it becomes a trap for the organizer foolish enough to have cheated in the first place! For, the unfortunate organizer now has a stake in maintaining the illusory “agreement,” and attempts to constantly “screen” the contact from any direct confrontation with what we really stand for. The result is profoundly demoralizing for everyone involved.

Only {apparently} opposite to the indicated tendency, is the “super hardline” tactic, typified by: “No compromises! If the guy doesn’t agree with then hang up on him immediately!” Either way, we are avoiding the real issue, which is how to confront and {change} the fundamental axioms of thinking in another person.

Before returning to our organizing situation and a proposed solution to the paradox, let’s stop to clarify in our minds the crucial distinction between a fundamental axiom, and a mere isolated opinion or theorem.

The characteristic of a fundamental axiom or principle of thinking (they signify essentially the same thing, in this context) is, that each such axiom or principle implicitly shapes the {entirety} of our thinking. A change in any axiom implicitly changes how we think about each and every other object of thought. To put it another way: such principles, imbedded in our mind as singularities in the form of fundamental ontological assumptions, shape the entire {geometry} of our mental processes. All other ideas, opinions, judgments etc. are determined by that geometry, but do not determine it.

What determines the outcome of a person’s action upon the Universe, is not the apparent, literal content of his or her individual judgments and opinions per se, not positions on this or that issue, but the geometry of their mental processes as a whole. And that geometry is what we are acting upon, with greater or lesser success, when we organize.

But is it really possible, in a fundamental sense, to change people’s minds in such a profound way? When we attack our own and others’ nagging doubts on this account, an epistemological monster invariably pops up in front of us: the Kantian paradox, in one or the other of its countless reincarnations. For example: If “the way a person thinks,” the form of conciousness, is given {a priori} — as “pure reason,” inborn and prior to all experience — then it would seem to be unassailable and impervious to any fundamental change. For, all a posteriori evidence, including our attempts to argue with the person, will simply be interpreted within the given geometry of the mind, without having any effect upon that geometry. (“I am as I am, so you can’t change me.”)

But if, on the other hand, there is no “pure reason,” but the fundamental axioms of thinking are a more or less arbitrary product of our upbringing, education, environment, etc. then where is the standard of truth? How could we ever know anything for sure? (“All judgments are relative. My opinion is as good as yours.”)

Now, as a matter of fact, the history of religious beliefs and cultures generally, as well as the historical development of physical science, demonstrates beyond any doubt, that sweeping changes in the pervasive “geometry” of human mental processes {do} in fact occur — both in individual persons, and in entire societies. They occur all the time in periods of rapid, generalized scientific progress, as typified by the Golden Renaissance.

On the other hand, we have direct experience, in recent decades, of a process of rapid cultural degeneration, which is not a consequence of this or that wrong opinion, but rather the result of negative changes in the sets of fundamental axioms underlying practically all cultures of the world.

Not only do such positive and negative changes occur, but (as LaRouche has demonstrated in most devastating fashion) they are, in every known case, ultimately the result of conscious intentions.

But changes in the prevailing axioms of society, have {physical} effects, effects that are manifested in gross terms on the historical scale of the rise and collapse of cultures and civilizations. These same effects are {measurable}, on even much shorter time-scales and more precisely, by the methods of physical economy. The possibility of judging, measuring, and forecasting in advance, the net impact of alternative choices of fundamental axioms of thinking, on the power of entire societies to maintain and improve their physical existence per capita, is inseparable from {cognition}. By correlating the measurable physical effects produced by successive such choices, with the quality of the human mental processes — implicitly reproducible in our own mind — which generate either positive or negative series of choices of axioms, we can judge the relative truthfulness of those mental processes, their degree of agreement with the laws of the Universe.

The Kantian paradoxes pop up, automatically, when the implications of cognition are ignored.

Now reflect on the quoted statement by Vernadsky. Take the case of a creative scientist, discovering a new universal physical principle. What is the subject of the discovery? Not a so-called “objective physical Universe,” not an object “out there,” supposed to exist as if independently of human activity! No! The creative scientist is deliberating on his or her own {thinking processes}, and those of his colleagues and society generally, with a view toward correcting or improving upon those fundamental principles that govern our thinking about how we act upon the Universe.

The judgment validating the discovery takes the form of an {inequality}: the demonstration, that the discovery of a superior principle, and the accompanying modification of the aggregate array of pre-existing principles, implies a {higher} potential rate of increase in Man’s per-capita powers to command the Universe, than was previously achievable. In the simplest case, the inequality is satisfied by detecting and correcting a systematic falllacy in our way of thinking about the world.

This is our model of a non-deductive “proof”: proof by {discovery}, proof by {improvement}. The ultimate criterion is not {logical} in nature, but {moral}: advancing the common good of Mankind.

To the extent we can account for the geometry of our mental processes, as the accumulated effect of an ordered series of such discoveries of improvement of geometry — each of which can be qualified as a discovery of universal principle relative to our own action in the world — then our knowledge and practice becomes fully intelligible in the form of a self-subsisting Riemannian manifold. Only then do we really know what we are talking about. Then we can {prove} what we know to any person, who is willing and capable of reenacting, in his or her mind, the process by which we came to know what we know.

Now turn back to our organizing situation. The preceeding train of thought points to a radically different approach, than the failed, pseudo-deductive procedure we examined in the first installment of this series.

First: don’t waste time on theorems, but get at the axiomatic issues immediately, by the least-time path. Use theorems only as vehicles to address the axioms. Second: “prove” by bringing people to discover how their own mental processes become more powerful, the world more intelligible and their ability to change it, stronger, when they adopt a superior axiomatic standpoint to the demonstrably flawed one, they held up to that point.

Somebody might retort: “Surely you don’t mean we have to go through all that epistemological stuff with our contacts! We have no time for that.”

But someone might think we had a lot of time to waste, considering the woefully low efficiency of much of our organizing, and the sheer man-hours expended! The most decisive thing to be conveyed, can be conveyed as if in an instant, by little more than a happy shift in attitude or mood on the part of the organizer. This is nothing unknown to an experienced organizer: we do succeed, part of the time. The problem is, to do what succeeds, {all} the time.

So, instead of getting tied up arguing the truth or falsity of proposition

, consider something like the following:

“Look, the reason you get taken in by the kind of nonsense you are telling me now, is, that you never studied what LaRouche has to say about the difference between human beings and animals.”

“What do you mean?”

“You voted for Bush (or Gore), didn’t you?”

(A moment of embarrassment.)

“Well, in view of what has happened to us all, as a result of that kind of mistake, wouldn’t you agree, that the difference between Man and beast should be the key issue in all politics?”

“Fantastic! I never thought of that.”

Contrary to Kant, the form of conciousness is not a God-given, fixed entity. It is subject to deliberate improvement, by our God-given powers to organize!

The Well-Tempered System: Kepler vs Ptolemy

by Fred Haight

Some of this material was presented in a recent cadre school, and some in a previous pedagogical: at this time I wish to emphasize a particular point. There are still many gaps to be filled in, and questions to be asked, about the history identified here, but I am convinced that I am on the right track, and that Kepler is identifying the right problem.

Lyndon LaRouche has always stressed the importance of Kepler for our music work, but in the past, two problems arose:

1. Professional musicians resented such “outside intrusions” into “their turf”.

2. For a while, only Book Five was translated. You cannot “look at the back of the book”, and expect to find the answer. You have to read the entire work, and pay special attention to the relation between Book Three, where Kepler lays out his own revolutionary musical ideas(fn1), and Book Five; a relation which Kepler himself cites in the Introduction to Book Five:

“I found it truer than I had even hoped, and I discovered among the celestial movements the full nature of harmony, in due measure, together with all its parts unfolded in Book Three – not in that mode wherein I had conceived it in my mind (this is not last in my joy) but in a very different mode which is also very excellent and very perfect.”

I am putting forth the contention, that Kepler, without having composed a single measure of music, may be the greatest musical revolutionary, and that Bach’s breakthroughs, would not have been politically possible, without Kepler.

No great discovery has ever been made without attacking lies, falsehood, and stupidity. Kepler’s discoveries, from Mysterium Cosmographicum on, were all inseparable from his attacks on the method of Ptolemy, Aristotle, Tycho Brahe , Copernicus(fn2) et al. A recent 21st Century article, reprints Kepler’s argument that Aristotle lied, and reinstated the idea of an earth-centered solar system, when he knew that the Pythagoreans had known its true Heliocentric nature much earlier. This, and the revival of the “flat earth” theory, after Erasthosthenes’ discoveries, set science back for centuries.

Humanity lost 17 CENTURIES between the Rata-Maui expeditions, and Columbus, Magellan, et al. SEVENTEEN CENTURIES, because of politically imposed, in fact, REINSTATED, false axiomatic assumptions! As late as 1616, the Counter-reformation, once again, condemned the heliocentric system. Even today, fundamentalist “Christians” will sometimes use the Bible to “prove” that the sun rotates around the earth.(fn3) Mankind must be freed of such arbitrary, but popular opinions, before progress may take place.

Think about the following quote from “Economics: At The End of a Delusion”:

“Kepler was the founder of the first successful effort to establish a comprehensive form of mathematical physics, the first to establish a method which freed science from the ivory tower mathematician’s blackboard, and to civilize mathematics by bringing it into the real world, the world of universal physical principles, rather than the purely imaginary world of abstract ivory-tower mathematical speculations.”

Kepler did the same for music, which had been held back for Centuries, by a similar Ptolemaic system. Out of the thousands of years of mankind’s existence, the period of great Classical masterworks, from Bach to Brahms, lasts just under two hundred years! In the Twentieth Century, under the evil influence of the Frankfurt School, humanity allowed its greatest gifts to be stolen, again.

THE PARADOX OF THE COMMA

In order to examine the problem that held back musical progress, we must examine a few things that Kepler understood, but are not present in his Harmonice Mundi. We shall do this through the posing of a paradox. What I shall present here, are sometimes known, as the discoveries of Pythagoras, and his school, but I suspect that they may have suffered the same sort of rewriting, as Aristotle did to Astronomy.

Boethius tells the story, that Pythagoras was walking by a blacksmith shop one day, and “noticed”, that the different sizes of hammers hitting anvils produced different tones. This sounds too much like Newton “noticing” getting hit on the head to me, and besides, I think it would be the size on the anvils that made the difference.

Anyhow, Pythagoras was said to have investigated this, and, supposedly, moved quickly to investigating string lengths on an instrument called the monochord. This is a box with two strings of the same length tuned to the same tone. You produce different tones by dividing the second string into different lengths, and comparing the tones produced, to the sound of the open, first string.

You can approximate the experiment yourself using a Cello, and substituting your finger for the bridge that was used to divide the monochord. We shall use modern terms like “fifth”, rather than “diapente”, etc. Tune the second string of the Cello down to C, so that it is in unison with the first string.

First ,divide the second string in half. If you place your finger so as to divide the sounding portion (from the scroll to the bridge) of the second string in half, and compare it to the open first string, the interval should be an octave. You have blocked off the upper half of the second string with your finger, so that only the lower half of the string is sounding. So, the string length is half of the string, but the sound is twice as high (C at 64 becomes C at 128), so the ratio of the interval is 2/1. The string lengths and frequencies are in inverted ratios.

Next, try a string length of 2/3 (blocking off one third with your finger, and letting two thirds sound). This approximates the fifth. The ratio of the interval is 3/2, so multiply 64 times 1.5 to get G at 96. Then, continue by dividing the second string into three parts, four parts, five parts and 6 parts. The sounding portions will be 3/4, 4/5, and 5/6 of the string, and ratios for the intervals will be, 4/3, 5/4 and 6/5. (Don’t skip the experiment – you will undermine the discovery).

These were said to correspond to: String length: 1/2 2/3 3/4 4/5 5/6 Intervallic 2/1 3/2 4/3 5/4 6/5 ratio: Interval: octave fifth fourth major third minor third

Now, isn’t this beautiful? Here you have an interesting ordering of number (as in the sequence of numerators and denominators), a series of arithmetic and harmonic means (fifth and fourth as those two means of the octave, and major third, and minor third, as the same means of the fifth), inversion of a sort, and musical intervals derived from a physical process.

So, what is wrong with this? Think from the standpoint of method, and take a few minutes from your busy schedule before proceeding.

Did you do what I asked, or are you like those clients, who, when challenged to think, say “I’m sure you’re going to tell me, so let’s get to the bottom line”? Go back!

Three things, all interrelated, stand out. Perhaps you will find more:

1. If there are three types of successively higher-order physical processes, non-living, living, and cognitive; then this determination is from the lowest level, non-living, which might reflect higher order processes, projected downwards, but as through a glass, darkly.

2. If you try to determine planetary orbits individually, they won’t fit together as a solar system. Kepler, in the Mysterium Cosmographicum, starts by seeking the highest, top-down, ordering principle for the entire solar system. We shall see how this problem arises in these musical intervals shortly, in the paradox of the comma.

3. This is not as obvious, but the axiomatic PREJUDICE, that intervals could only be represented by rational numbers (fractions), set music back for centuries; much as the prejudice, that planetary orbits could only be perfect circles, did Astronomy. Organizers can do the same thing. “This way is the best, because we have been doing it this way for centuries.”

Kepler recognizes how long this prejudice held court. From the introduction to Book Three of Harmonice Mundi:

“Having discovered definite proportions,” or “the fact that,” it remained to track down the causes as well or “the reason why” some proportions marked out consonant intervals, and others dissonant.

And in the course of two thousand years the opinion had been reached that the causes are to be looked for in the proportions themselves, as they are contained within the bounds of a discrete quantity, that is to say, of Numbers.(fn4)

Question: Is the monochord experiment itself, a tautology?

Boethius, in his fifth century “De institutione musica”, states that intervals can only be represented by rational numbers, as they are the best. How could an irrational number, which is not precise, represent something as specific as an interval, he asks? Boethius was considered THE AUTHORITY for a thousand years.

Even in the debates over tempering, it was often insisted that the “pure fifth” 3/2 was the best, the closest to perfection, and should be used whenever possible, or come as close to as possible.

Kepler, on the other hand, goes for the throat on this point. He acknowledges the use of incommensurables, preferring the Greek term, {alogoi}, which he translates as “inexpressible”, to the Latin term irrational (which CAN mean without reason, as well as without ratio). In Book One he elaborates their “degrees of knowability”. Everything beyond the third degree of knowledge is an “inexpressible”. What a beautiful concept: the incommensurable is ordered, in a knowable way! From Book One:

“People are always molesting inexpressibles, by trying to express them – as numbers!”

Let’s look at the problems that arise from this fixation on rational numbers:

1. The Lydian interval, even on a monochord, is represented by the square root of two, so it would be have to be banned (it was banned on its own merits as the Devil’s interval).

2. Since half-tones and tones cannot even be approximated as fractions, the system begins to break down at their determination. They invented different sizes of them: 9/8 was a major whole-tone, 10/9 a minor whole-tone – half-tones were at 16/15 (wide semitone), 18/17, 25/24 (diesis), even 256/243 (narrow semitone)! Since they were trying to add these intervals up, to the pure ratios of the above mentioned consonant intervals, they had to invent certain critters to fill the gaps, such as the same diesis, the limma (135/128), etc. Does not all this remind you of the way quantum mechanics sometimes makes up particles, to force experimental evidence to conform to a faulty theory? Doesn’t it remind you of the “made-up” epicycles in the planetary orbits? Imagine trying to teach singers to sing all these!

3. Supposedly, Pythagoras himself developed the paradox of the “comma”. If he did, one would have to admire a man who challenged his own system, but I’m not sure that was the way it worked.

I will demonstrate the “comma” from our modern terminology of tones, intervals and frequencies.

Take C at 256. Go down three octaves to C at 32. This is the lowest C on the piano. Play, on the piano, a series of 12 fifths (with one “enharmonic”): C G D A E B F#(or Gb) Db Ab Eb Bb F C. This takes you up 7 octaves to C at 4096, the highest C on the piano. However, if you take C at 32, and multiply by 3/2, or 1.5, the ratio of the “pure fifth, you will get 48 for G. Keep doing this twelve times, and instead of an octave of C, 4096, you will get 4151. The fifths are a little too large! If you have a diagram of the circle of fifths, imagine if it did not meet at the top, C, but there was a slight gap. This gap was dubbed the comma, and it had a precise measurement.

There is an abbreviated version of this. Three major thirds should comprise an octave: C E G# B#(C). An octave of C at 256 should be 512.

Multiply 256 by the ratio of the major third, 5/4, or 1.25, three times, and you will get, 256, 320, 400, 500. So the major thirds are too small! (If you wish to argue that the Greeks did not know frequencies, you can multiply the ratios and find the same problem. If you accept the octave, or diapason, as having a ratio of 2/1, then multiply 1 by 1.25 three times, and you will obtain,1, 1.25, 1.5625, 1.953125. Again, it falls short of an octave).

So, think back to point number two, the problem that arises when you try to determine the system as a whole, rather than one interval, or orbit, at a time. Kepler has great fun pointing out, that even if one rejects incommensurables for rational numbers, as the only representations of intervals, these rational numbers are themselves incommensurable – with one another!

Kepler, in Book Three, is polemical about the need to destroy this prejudice. Not only are rational numbers not the best; they are not a cause at all.

From the introduction to Book Three:

“… the causes of intervals have remained unknown to men…. I shall be the first, unless I am mistaken, to reveal them with such accuracy.”

Also from Book Three, next to a margin entitled “His error (Ptolemy’s) in treating a non-cause as a cause”:

“… since the terms of the consonant intervals are continuous quantities, the causes which set them apart from the discords must also be sought among the family of continuous quantities, and not abstract numbers, that is in discrete quantity; and since it is the Mind which shaped human intellects in such a way that they would delight in such an interval….the causes of such intervals being harmonious, should also have a mental and intellectual essence….

“if the cause was sought in abstract numbers. Yet it would still not be very clear why the numbers 1,2,3,4,5,6,etc conform with musical intervals but 7,11,13, and the like do not conform.”

In Chapter One of Book Three, Kepler states that he is using a geometrical method (the inscription of plane figures in a circular string), as a “substitute for the Pythagorean abstract numbers, which have been repudiated.”

Kepler was more opposed to the numbers being seen as a cause in themselves, than the division of strings; his division, however, is a very different, geometrical one. He inscribes the plane figures in a circular string, and orders the intervals according to the same degrees of knowability that he laid out in Book One. This is still not his highest determination.

Throughout Harmonice Mundi, he consciously UPLIFTS the cause of intervals to a cognitive one – from Chapter Sixteen of Book Three:

“The theme of that book (Five) is the sole object which I intend in this whole work. For, being an astronomer, just as I argue about the regular figures not so much geometrically….as astronomically and metaphysically, so also I write about the ratios of melodies not so much musically as geometrically, physically, and lastly, as before, astronomically and metaphysically.”

Footnotes:

1. In the title page to the entire work, Kepler’s description of Book Three includes:

“…and on the nature and distinguishing features of matters relating to music, contrary to the ancients;”

I.e., he is refuting the ancients. The translators, Duncan and Field, insist that the only “real discovery” in the entire work, is the so-called, Third Law.

But, Kepler challenged future musicians to act on his discoveries. He sought his Bach, as well as his Leibniz. The introduction to Book Five reveals his sense of what a revolution he was unleashing:

“I am free to taunt the mortals with the frank confession that I am stealing the golden vessels of the Egyptians, in order to build of them a temple for my God, far from the territory of Egypt. If you pardon me, I shall rejoice; if you are enraged, I shall bear up. The die is cast, and I am writing this book- whether to be read by my contemporaries or not. Let it await its reader for a hundred years, if God himself has been ready for His contemplator for six thousand years.”

2. Bruce Director, in a conference presentation, quoted Ptolemy on how, of the Theological, Physical, and Mathematical causes of something, mankind could only know the Mathematical “cause”. Years ago, Bruce had a pedagogical, quoting Copernicus on how it didn’t really matter, if your mathematical model corresponded to physical reality, only if it described it. If that quote is reliable, then that, plus his continued insistence on perfect circles, would tend to put Copernicus in the Ptolemaic camp, despite his acknowledgement of the Heliocentric nature of the Solar system.

3. Harmonice Mundi is completed in 1619, at the beginning of the German part of the Thirty Years War, and the same year as Kepler’s works were put on the Index of Prohibited Books.

4. Supposedly there was a difference between Ptolemy and what was represented as the Pythagorean view, on whether it even mattered how the intervals sounded, or whether the ratios alone determined consonance or dissonance. Kepler doesn’t think there is much difference. The two Venetians, Galilei, and Zarlino, took side in this matter. 5. The translator of Boethius into English, says that his work is basically just a translation of Ptolemy. I have to check this out. Ask yourself, what is the axiomatic prejudice built into the monochord experiment?

KEPLER VERSUS PTOLEMY, PART 2: TEMPERAMENT –

by Fred Haight

Is the question of tempering then, just a question of finding then right ratios, and correcting the errors, or is it something far more important?

Equal tempering arose, supposedly, as early as Aristoxenus, a pupil of Aristotle, as a mechanistic procedure of simply dividing the octave into a “chromatic scale”of twelve equal tones, based only on what sounded good, in disregard to any physical cause (I’m not sure that I am giving fair due to him that’s something I have to look into more closely). A kind of gang-countergang debate sprung up, between those who said that this was best, because the ear was the ultimate guide, and those who said that you cannot abandon the physical cause of the intervals, for what seems merely sensuously pleasing. This allowed Boethius to make a phony distinction between practicing musicians (whom he considered vulgar), and the superior, theoretical musicians, who only contemplated the beauties of the ratios! Centuries later the Venetian Vincenzo Galilei, in a phony debate with his deceased Venetian predecessor, Zarlino, proposed to divide the octave into twelve equal tones by the ratio 18/17. Kepler, who otherwise speaks positively of tempered intervals, rejects Galilei’s determination as “mechanistic”(6) ( he also points out that it doesn’t work-it generates a comma). The well-tempered system is not a matter of finding the right ratios, but of CHANGING YOUR THINKING, and starting from the TOP down, in terms of the actual processes governing the universe, as do LaRouche, and Kepler. Look back at the previous quotes from Kepler, on lifting the investigation of melodies to an astronomical, and metaphysical level, and on the cognitive nature of the causes of the intervals, as communicated from the Mind, to our minds.

An academic reader would have a hard time identifying Kepler as founder of the well- tempered system after all he keeps using these so-called “pure”ratios to represent the intervals, even in the Fifth Book.

But, in the Fifth Book, he does something different. The ratio between the aphelial, and perihelial angular velocities within a single orbit, he refers to as being like ancient plainchant, a single, primitive melody.

The ratios between planets though, he refers to in terms of polyphony, which blossomed in the Rennaissance, with the development of bel-canto. In Chapter Five, he sets up two scales, one from the set of convergent ratios, i.e., from the aphelion of the lower planet (farthest out from the sun), to the perihelion of the upper ( e.g. Saturn to Jupiter; Jupiter to Mars etc.) The other is the set of divergent ratios, where he inverts the process, by starting with the perihelion of the lower planet, and the aphelion of the upper. These two scales, which he calls hard, and soft, are not exactly our major and minor: the hard scale differs from the major by one tone, but that is enough to make the two scales inversions of one another!

In Chapter Seven of Book Five, Kepler examines the possibility of several planets being in tune, at these extreme ratios, at the same time, which he compares to four-part harmony. Here, he says that a “certain latitude of tuning,”is not only acceptable, but necessary. In his charts, in this chapter, he identifies the highest and lowest possible tunings for each of these measurements. (This latitude of tuning is not an arbitrary variance, as in equal tempering, but comes from different means of measuring these physical ratios, of perihelial, and aphelial angular velocities).

After the chart on the possibility of five planets being in tune, he states: “Here at the lowest tuning, Saturn and the Earth coincide at their aphelia; At the mean tuning, Saturn joins in at its perihelion, Jupiter at its aphelion; At the highest tuning, Jupiter joins in at its perihelion.”

Even a single pair of planets being located strictly at the “pure ratio”can exclude other planets from being “in tune” at all.

The same problem arises in polyphony. So-called “just intonation”was an attempt to construct a scale with as many “pure”fifths and thirds as possible. This doesn’t even work within a single scale (how many fifths are there in any “diatonic”scale? How many major, and minor thirds? Can they all correspond to the “pure ratios”)? In both cases, bel-canto polyphony, and the solar system, tempering arises not from some pragmatic evening out of the scale, but new discoveries of physical principle in the universe; and, in both cases, tuning is determined, not “at the blackboard,”but by the composition itself, whether it be by a human artist, or The Divine Architect himself! (see footnote 10)

Lyndon LaRouche has always insisted that music originates in the polyphonic, bel- canto vocalization of sung poetry (which itself “contains a score”in its prosodic elements such as the ordering of vowels, meter, etc.) Is bel-canto vocal registration living, or cognitive; or perhaps living participating in the higher level? The discovery of solutions to paradoxical problems: to ironic, polyphonic “dissonances”through inversion etc, is cognitive, and parallels the discovery of new physical principles as solutions to paradoxes in physical science, but, such cognitive ironies, are usually expressed, as ironies in the living harmonics of VOCAL REGISTRATION, and can only exist in the “physics”of actual polyphonic musical composition, not the “classroom mathematics”of formal systems of scales, keys etc. In other words, cognitive musical ideas do not exist as disembodied notes, as Heinrich Schenker, or a counterpoint text would imagine: but, perhaps we could follow Vernadsky and LaRouche, in saying that cognitive discoveries in music, create ironies in voice registration, as “natural products,” in the “living processes”of bel-canto, much as the biosphere creates “natural products”such as soil, water cycles etc.; and as cognitive processes, generate increased relative potential population density, as a natural product, in the biosphere.

Wait a minute! Does not all this sound like what has been presented in Volume One of the Music Manual (or projected for Volume Two?) But step back a bit. Did not Lyn, in the Music Manual, revolutionize musical theory, by finding the origins of music in the highest cognitive, and living levels of physical processes? Compare this to oligarchical theories, of music originating in, “bird songs,” “the dance,””hammers hitting anvils,”etc. They all wish to eliminate the idea that human cognitive activity originates in anything human! Now, you can begin to appreciate what Kepler actually did. (7)

THE LIVING HUMAN VOICE

As discussed in a previous pedagogical, but worth repeating, human voice registration produces an entirely different set of harmonics than a mere vibrating string, characterized by a series of Lydian intervals, and a chromatic scale of register shifts (when the down shifts are considered as well as the up). (8)

Now look at the Lydian intervals in the six voice species:

VOICE UP DOWN

Soprano-Tenor C F# B F

Mezzo-Baritone Bb E A Eb

Bass Ab D G C#

Here we have all six Lydian intervals organized in a series of descending half-steps (the next one would be F#-C), in a form where each of the two tones comprises the main register shifts for a specific voice type. C F# B F BbE A Eb Ab D G C#

This is , of course, not all that could be said about the harmonic ordering of the human voice, but here, in this series determined by voice register shifts, we have a “chromatic scale” of twelve half-tones, but from the true physical cause. Not only is every tone a register shift (as Eliane Magnan used to demand), they are not all equal. As Lyndon LaRouche first pointed out in “Beethoven as a Physical Scientist,”tones are not “point frequencies,”but more like regions of negative curvature. They can occupy an area, and move (as can, and must, Brunelleschi’s dome), according to the Analysis situs of actual composition (for this reason, well-tempering could not be derived from the keyboard, which is a FIXED tuning). (9) So, Gb can differ from F#, as Pablo Casals clearly understood, but it will differ more or less, as the composition itself, requires. (10)

Ironically though, there are still only twelve tones; all attempts to create quarter-tones etc, fail.

One of the most exciting ideas ever presented by Lyndon LaRouche on music, was in the famous footnote 65 to “The Becoming Death of Systems Analysis,”which applies perfectly to Mozart’s K.475: “The pivot of the entire composition so unfolding, is a conflict in tonality, derived lawfully from those simpler ironies of well-tempered counterpoint, but expressing a clash of ironies equivalent to an ontological paradox in physical science. Thus, it is a physical reality, as represented by the natural (i.e., bel-canto) composition of the natural-determined division of the human singing-voice . which imposes naturally generated ironies and paradoxes upon the formalist’s musical scale.”

Think of that idea: of the cognitive activity, of playfully imposing the Lydian centered harmonic series of the human voice, on a lower order, non-living, more formal harmonic species, and generating ironies and paradoxes throughout; much as the collapse of the real, physical economy, such as U. S. steel production is posing such paradoxes for the formal, utopian schemes of the Globalists, now. K 475 , by including the F# in the opening Bach statement, and the pedal point series, generates new modalities, unthinkable in, say, a pre-1782 Mozart Sonata.

So, the well-tempered system, is inseparable from human bel-canto voice registration, from the Classical principle in Art, and from the moral intent of actual classical musical composition, whereas equal tempering, implies nothing for composition. Not surprisingly, Schoenberg seized on it as the basis for his so-called twelve-tone system, which throws out the voice, the mind, etc.

Could the Greeks have known this? Could equal tempering have occurred on its own, or only as a Delphic operation against a real discovery? Polyphony is certainly natural (despite textbooks that say it wasn’t thought of until the 11th century A.D., and then as an annoying drone)! But, I’ll bet that human beings were born with natural bel-canto voices then, even as they are now. It’s true it has to be developed, but in the last century, some great singers were “naturals”for whom the “voice” was there. The nature of the best of Greek culture, and science, would suggest that they would investigate the right areas, and if you’re looking in the right place, it should not be that hard to find.

There is a lot more work to be done here, but let me ask another question: How many people, through a revolution in method, bring about such a fundamental change in science, and art? How many Leonardos, Keplers, and LaRouches are there in history? In our new Century, great works of art shall be made, by artists who absorb LaRouche’s ideas, as a whole.

Lastly, let me leave you with a beautiful thought by Kepler on tempering:

“There is an absurd arithmetic equality at banquets if everybody is seated indiscriminately, with no account taken of sex, condition, or age. On the other hand mere geometric similarity is insipid. For if the learned are put only next to the learned, what good will they do to the unenlightened? If women only next to women, what pleasure will there be? If the rowdy next to the rowdy, who will instill good behavior into them? But if you admit neither blind equality, nor peevish similarity, the proportion will be harmonic. For you will bring it out that the old rejoice to see the young,the men to see the women, the young are ruled by the wisdom of the old,the women by the authority of the men (sic), the sociable stimulate the unsociable …this is not a combination of intact kinds, but to a certain extent an infringement of them, to set up a harmonic proportion. Friendships are given life by harmonic tempering. For what concord is to proportion, that love, which is the foundation of friendship, is to the whole compass of human life.” Footnotes:

6. In the Nineteenth Century, Helmholtz divided the octave into twelve equal parts by the twelfth root of two. He follows Mersennes and Rameau in basing musical theory on the overtone series of a vibrating string, which is somewhat worse than a monochord (Mersennes and Descartes also had a phony debate over tempering going on, with Descartes taking the side against temperament).

In the late 1600s, Werckmeister, wrote that he had created one diatonic-chromatic-enharmonic scale. Diatonic vs Chromatic music was another phony debate. (Read “the Case against Rock,” and other writings from that time, and you will see how we fell into that trap). These three “genera,”chromatic, diatonic, and enharmonic, were considered incommensurable; the well-tempered system, created a Gauss-like congruence, and thus integrated them 7. Thought processes themselves are highly musical. This is an area that requires a lot of investigation, but, rather than the so-called Mozart effect, I find it very interesting that professionals who work with Alzheimer’s patients, and stroke victims, find that the musical memory persists, even when memory is otherwise impaired, or gone. 8. The F# centered voice register series is what determines C at 256. Without that, you have no defense against the arguments of “relative pitch,”which does exist. It is the intersection of the fixed, voice register values, with the “transposable”keys, which gives each key its unique “color,”and protects us against random transposition.

9. For this reason, Werckmeister had at least three tunings for his well-tempered Clavier. Though Kepler identified the difference between living and non-living processes in his “Snowflake”paper, it remained for Bach to discover all these questions of the living bel-canto voice. His son, Emmanuel, makes it clear that bel- canto was the basis, even of Bach’s keyboard technique.

10. Plato’s, and Kepler’s, Composer of the Universe, requires the same quality of change. In the Seventh Chapter, of Book Five, of Harmonice Mundi, Kepler must temper the intervals differently for the hard, and soft scales. Thus, the well-tempered system, is neither a fixed tuning, or a series of fixed tunings, but requires constant change, as generated by the composition itself!

{Dynamis} vs. {Energeia} — A Sketch

by Jonathan Tennenbaum

Since at least the time of Plato and Aristotle, and most likely even long before Pythagoras, the struggle between oligarchical and republican conceptions of physics has turned on the relationship between what the Greeks called {dynamis} and {energeia}. To a rough first approximation, the Greek {dynamis} might be rendered, in its broad usage, variously as “ability,” “potential,” “potency,” “power” (German {Vermoegen}, {Faehigkeit}, {Kraft}, etc.); whereas {energeia} corresponds (roughly) to “activity” (German {Taetigkeit}) and (in Aristotle, especially) to “actuality” in the sense of “actively existing.”

Plato’s dialogs demonstrate, however, that Plato and his circles possessed a precise and highly developed scientific conception of {dynamis}, having no direct equivalent in today’s degenerated modern language usage.

Perhaps the best illustration of that degeneration, and of its causes, is the freak-out by virtually every modern translator, at the implications of a celebrated passage in Plato’s {Theaetetus}, to which Lyn has often referred. This is the place where the young Theaetetus recounts to Socrates a preliminary discovery concerning the nature of the “powers” connected with the doubling, tripling etc. of a square, and which lie beyond the domain of simple linear magnitudes. Rejecting the implications of Plato’s actual term, {dynamis}, modern translators typically try to bring the passage into conformity with the “academic correctness” of textbook mathematics, using “root” or “surd” in place of “power” and apologizing in footnotes for the supposed “inappropriateness” of Plato’s choice of language!

Actually, as the {Theaetetus}, the Meno and other dialogs demonstrate, Plato’s conception of {dynamis} belongs uniquely to the domain of {physics}, not mathematics per se. In particular, the subject of Theaetetus’ account is not to solve an equation, but rather to discover the unseen principles of generation of the Universe — physical principles! –, focussing for this purpose on the paradoxical characteristics of the visual domain.

It is Plato’s conception of {dynamis}, as revived and developed by Nicholas of Cusa and Kepler, that leads to Leibniz’ founding of physical economy and what Leibniz called “the science of dynamics,” as opposed to Newton’s mechanics; the pathway leads from there into the work of Gauss and Riemann, and finally to Lyndon LaRouche’s discoveries in physical economy. It is not by accident that Lyn, in his book {In Defense of Common Sense}, for example, cites exactly the indicated passage of Plato’s {Theaetetus}, in the context of presenting his own conception of “rate of increase of relative potential population density” through the process of individual human discovery and the successive integration into social practice, of new physical “powers.” That latter conception constitutes, in my view, the highest development reached so far, in “unfolding” what was implicit in Plato’s {dynamis}.

To throw further light on these matters, I propose now to take a brief look at the oligarchical side of the coin, which goes very clearly back to Aristotle. What sticks out immediately, in examining Aristotle’s {Metaphysics}, is his insistence on the primacy of {energeia} as opposed to {dynamis}. That insistence went hand-in-hand with Aristotle’s attack on metaphor and the Platonic ideas. Aristotle writes, for example ({Metaphysics}, Book IX):

“Since all abilities (powers) are either inborn, as are our senses; or are acquired by practice, as the ability to play a flute; or are acquired by learning, as the powers of the sciences; in all cases one can gain such powers, as are acquired by practice or learning, {only} through the aid of something that was {already} realized (actualized)…

“For from the potentially existing, the actually existing is always produced by an actually existing thing, e.g. man from man, musician by musician; there is always a first mover, and the mover already exists actually. We have said in our account of substance that everything that is produced is something produced from something and by something, and that the same in species as it…

“Obviously, then, actuality ({energeia}) is prior both to potency ({dynamis}) and to every principle of change.”

Rather than get entangled in the ins and outs of Aristotle’s theory of existence and becoming, focus on the systematic, axiomatic flaw in Aristotle’s whole manner of argumentation: He rejects — or at least disregards, as if it were nonexistent — the power of human creative discovery, of human reason, and of a creative principle underlying the Universe as a whole. In other words, Aristotle denies the possibility of a {self-developing, or self-actualizing potential}, that which Nicholas of Cusa later called the {posse-est} ({posse} corresponding to Plato’s {dynamis}). Lurking behind Aristotle’s notion, that existence can only flow from what he calls “actually existing things,” is a mind-set, which can attribute “actual existence” only to such objects and motions, as have the quality of objects of sense perception.

These points require more elaboration. For the present purposes, however, as a short-cut and in order to throw the issue of “dynamis vs energeia” into strategic perspective, I propose turning to one of the more effective British operations of the 19th century, one which — as so much British wickedness — drew originally from Aristotle: The Cult of Energy

From the early decades to the middle of the 19th century, parallel with operations leading to the unleashing of the Confederacy and the US Civil War, a scientific cult was launched by Lord Kelvin and the Thomas Huxley-Herbert Spencer “X-Club” circles, Hermann Helmholtz, Rudolf Clausius et al, directed against the influence of Leibniz and his successors, including Gauss in particular. Although that cult involved several interrelated “theme parks” — such as the so-called Darwinian theory of evolution and Herbert Spencer’s fraudulent concept of an “iron law of progress” — we might fittingly refer to it as “the Cult of Energy.”

Crucial to the operation was the relative success, achieved by the conspirators, in foisting two fraudulent formulations on the scientific community: “First and Second Laws of Thermodynamics,” and their monstrous corollary, the supposedly inevitable “heat death of the Universe.”

The utopian political thrust of the operation was more or less obvious from the beginning, but became luridly explicit, among other things, in the “Energeticist Movement” associated with Wilhelm Ostwald around the turn of the 19th century. Ostwald advocated a World Government based on the use of “energy” as the universal, unifying concept not only for all of physical science, but also for economics, psychology, sociology and the arts.

Although the energeticists and the myriad, competing materialist (including “Diamat”), reductionist and positivist movements and countermovements of the late 19th century and early 20th century, are now mostly forgotten, the axiomatic germ of the Cult of Energy remains deeply embedded in European culture, like the modified genome left over in the tissues of a patient after an acute lentivirus infection has subsided. In particular, for over a century nearly everyone has been miseducated to believe, that “energy” is an objective scientific reality, and the First and Second Law constitute proven scientific truths.

Not accidentally, the Kelvin-Helmholtz doctrine of “energy,” became a key feature of Anglo-American geopolitics, from the British launching of Middle East “oil politics” at the beginning of the 20th century, to the orchestration of the so-called “energy crisis” of 1973-74, and, not least of all, the present march toward a new Middle East war. This is not to say that “energy” per se (or “oil supplies”) has anything really significant to do with the present war drive. Rather, the reasons, why people permit themselves to be manipulated into tolerating actions leading to perpetual war and a new “dark age,” are inseparably connected to those axiomatic flaws in thinking, that underlie popular belief in the cult doctrine of “energy.”

The common origins of the “energy” doctrine and utopian geopolitics go much further back than the launching of the modern energy cult itself, by Helmholtz, Kelvin et al. From the standpoint of economics, the energy doctrine represented nothing but a rewarming, under “scientific” guise, of old feudalist, and specifically physiocratic doctrines of supposedly fixed “natural resources,” ignoring the function of the human mind in discovering and realizing new physical principles. On the other hand, anyone who has thought through what Lyn and others have written on Gauss’ early work concerning the “Fundamental Theorem of Algebra,” should immediately recognize, in the so-called “First and Second Laws of Thermodynamics” exactly the same essential fallacy, that Gauss refuted in his 1799 attack on the “utopian” mathematics of Euler and Lagrange. Not accidentally, the Euler-Lagrange doctrine of “analytical mechanics” created the mathematical foundation for the Helmholtz-Kelvin energy doctrine. Conversely, the manner in which Gauss generates the algebraic “powers,” in the cited 1799 work, by principles lying entirely outside the mathematics of Euler and Lagrange, is characteristic of the way Man acts as an instrument of the anti-entropic development of the Universe.

On one level, the fallacy of the “First and Second Laws of Thermodynamics” is simply this: these laws have never been demonstrated to be properties of the real Universe, but only properties of certain closed mathematical-deductive systems, which ignorant or malicious physicists {claim} to represent the real Universe, but that manifestly do not. On this level, the fraud is identical to that of so-called economists, who claim to be able to deduce theorems about the real economy, from supposed self-evident properties of “money.” In fact, the elementary error, revealed in the very title of Newton’s famous “Principiae mathematica philosophiae naturalis” (“Mathematical Principles of Natural Philosophy”) finds itself reproduced, countless times, in textbooks dealing with non-existent “Financial Principles of Economics.”

Contrary to popular academic belief, there are no actual experiments establishing the validity of the “First and Second Laws of Thermodynamics” as {universal} physical principles. To the extent those “laws” have a certain empirical correlate at all, they are both circumscribed by a purely {negative} principle, identified already by Leibniz long before the Kelvin-Helmholtz gang came along: the impossibility of a so-called “perpetuum mobile” or “perpetual motion machine” — a hypothetical subsystem of the Universe, able to generate a net surplus of power in the course of a closed cycle, in which the system is supposed to return to its exact original state, without any other net change in the surrounding Universe.

Just as in the case of so-called “impossible” or “imaginary” numbers, the source of the supposed “impossibility” involved is not a limitation of the real physical universe. The limitation is located rather in the notion of a “machine,” as a system describable by the “utopian” Euler-Lagrange form of analytical mechanics. To put it another way: To the extent a physical system is either chosen or forced to mimic the characteristics of a “machine” in the indicated sense, it will appear to obey the First and Second Laws of Thermodynamics. But the Universe as a whole is not a machine; the Universe not only {never} returns to an earlier state, but its successive states are strictly {incomparable} with each other from a formal-mathematical standpoint. Thus, the extrapolation of the so-called “First and Second Laws” to the Universe as a whole constitutes the crudest, most elementary sort of scientific error.

If “Universe” refers to the most generalized form of Man’s action upon Nature — no other Universe could be known to us! — then the “state of the Universe” changes fundamentally with each discovery, by some human mind, of a new universal physical principle (power). A formal-mathematical system, which (to a first, “engineering” approximation) may have more or less adequately described Man’s physical-economic activity up to that point, now breaks down, as technologies, based upon the new principle, transform the physical economy to the effect of increasing the relative potential population-density of the human species beyond any apriori “limits.”

The very fact of the successful increase in human population potential, by some three orders of magnitude over documented history and prehistory, attests to the existence of a self-developing “power,” lying entirely outside the domain of visible or visible-like objects, but commanding the visible Universe to an increasing extent.

This brings us back to the fundamental flaw of Aristotle’s {energeia}.

Utopianism and the Enlightenment

Before the modern cult of energy could be created, Aristotle had first to be reincarnated in the so-called the “Enlightenment” of Paolo Sarpi et al., as a crucial component of the Venetian operation to destroy the influence of the Renaissance and the nation-state principle, and to plunge Europe into decades of religious war.

Sarpi’s “Enlightenment” based itself essentially on Aristotle, but with some differences that are relevant to the mindset of the Utopians to this day. The quarrel between the Enlightenment ideologues and Aristotle was not essentially a matter of substance. From their standpoint, Aristotle was excessively cautious and old-fashioned, wrapping his conclusions in endless distinctions and qualifications. Furthermore, Aristotle felt obliged to at least quote the existence of opposing views; while Locke, Descartes et al. went for a “clean break,” blatantly ignoring the entire preceding history of philosophy and science, and promoting the crudest, “post-modernist” sort of reductionism.

In this way, the creation of the modern cult of energy out of Aristotle’s {energeia}, represents just one more case of “putting lipstick on a pig.”