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.