Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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Dictionary of the History of Ideas | ||

*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

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.

Dictionary of the History of Ideas | ||