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- The Astronomical Origins of Number Theory -- Part I

- The Astronomical Origins of Number Theory -- Part II

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"!

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."

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The preceding article is a rough version of the article that appeared in

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