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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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