# University of Virginia Library

Circularity. A circle is rich in symmetries. It admits
mirror symmetry with respect to everyone of its in-
finitely many diameters. As a mathematical conse-
quence of this the circle can also be rotated into itself
around the center by an arbitrary angle; in fact, if two
diameters form an angle α, then mirror reflection with
respect to one diameter followed by a reflection with
respect to the second diameter will rotate the circle
by the double angle 2α.

Apparently because of this wealth of symmetry, for
2000 years, from Plato to Tycho Brahe, and including
Copernicus, scientific astronomers somehow took it for
granted that a celestial orbit of the kind that came
under their observation is, or ought to be, a circle, or
a circle rolling off on a circle (epicycle), or a figure
mathematically equivalent to such a one. They
undoubtedly had it in their thinking that what is
aesthetically (and ontologically) appealing is also
kinematically distinguished and dynamically prefera-
ble. But a mechanical preference from outward math-
ematical symmetry, while frequently profitable, can
also be misleading, and in the present case it certainly
was the latter. It was miraculously divined by Kepler,
and then mathematically rationalized by Newton that,
under gravitation, the closed orbit of one celestial body
around another—in an “ideal” two-body setting—is not
just a circle, but an ellipse, any ellipse. The ellipse can
have any eccentricity, that is any measure of deviation
from a circle. A circle can also occur; but it occurs
then only as a case of an ellipse whose eccentricity
happens to be zero. But since the eccentricity can be

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any real number between 0 and 1, this is a most
unlikely value to occur. Even if, by an unlikely chance,
a pure circle does eventuate, that form undoubtedly
is very unstable; the smallest perturbation would
quickly make it into an ellipse of a small but non-zero
eccentricity. Thus, in this case, the figure with a wealth
of symmetries is exceptional within a large family of
figures each having only a few symmetries; and the
wealth of symmetries makes the exceptional figure very
unstable and most unlikely to occur.

Still, circular motion does play a role, in all parts
of physics, as a constituent of any wave-like event; of
an ordinary wave on the water or in the air; of an
electromagnetic wave in the propagation of light, as
a dual to the photon; and of a de Broglie wave, as
a dual to the corpuscular aspect of any elementary
particle of matter. A wave, wherever and however
occurring, is a composite bundle of “simple” waves,
so called “monochromatic” ones, and the mathematical
structure of a simple wave is always the same. The
pulse of a monochromatic ray of energy is rigorously
invariant in time and thus constitutes a most “depend-
able” clock (atomic clock). The Greeks were already
groping for such a clock. Aristotle reports that some
philosopher(s) before him not only measured time by
the daily rotation of the celestrial sphere but even
defined it quantitatively in this way. Surprisingly,
Aristotle frowns on this definition (Physica Book 4, Ch.
X, 218b 1-5).

Returning to gravitation we note that physicists
nowadays, out of their fertile imaginations, have
imputed a nuclear structure to gravitation too, com-
plete with corpuscles, hopefully named gravitons, and
with dual de Broglie waves, hopefully spoken of as
gravitational waves. Whenever these will be con-
clusively verified to “exist,” circularity will have finally
come to gravitation too; and how a Eudoxus and a
Ptolemy would welcome such a newcomer would be
worth knowing.