In the
Ptolemaic system of
astronomy, the
epicycle
(literally:
on the circle in
Greek) was a geometric model used to explain
the variations in speed and direction of the apparent motion of the
Moon,
Sun, and
planets. It was first proposed by
Apollonius of Perga at the end of the
3rd century BC and formalized by
Ptolemy of Alexander in his 2nd-century AD
astronomical treatise the
Almagest. In particular it explained the
retrograde motion of the
five planets known at the time. Secondarily, it also explained
changes in the apparent distances of the planets from Earth.
In the Ptolemaic system, the
planets are
assumed to move in a small circle called an
epicycle, which in turn moves along a larger
circle called a
deferent. Both circles rotate
eastward and are roughly parallel to the plane of the Sun's orbit
(
ecliptic). The
orbits
of planets in this system are
epitrochoids.
Despite the fact that the Ptolemaic system is considered
geocentric, the planets' motion was not thought
to be actually centered on the Earth. Instead, the deferent was
centered around a point halfway between the Earth and another point
called the
equant. The
epicycle, meanwhile, rotated and revolved along the deferent with
uniform motion. The rate at which the planet moved on the epicycle
was fixed such that the angle between the center of the epicycle
and the planet was the same as the angle between the earth and the
sun.
Ptolemy did not predict the relative sizes of the planetary
deferents in the
Almagest. All of his calculations were
done with respect to a normalized deferent. This is not to say that
he believed the planets were all equidistant. He did guess at an
ordering of the planets. Later he calculated their distances in the
Planetary Hypotheses.
For
superior planets the planet
would typically move through in the night sky slower than the
stars. Each night the planet would "lag" a little behind the star.
This is
prograde motion.
Occasionally, near
opposition, the planet
would appear to move through e in the night sky faster than the
stars. This is retrograde motion. Ptolemy's model, in part, sought
to explain this behavior.
The
inferior planets were always
observed to be near the sun, appearing only shortly before sunrise
or shortly after sunset. To accommodate this, Ptolemy's model fixed
the motion of Mercury and Venus so that the line from the equant
point to the center of the epicycle was always parallel to the
earth-sun line.
History
When ancient astronomers viewed the sky, they saw the Sun, Moon,
and stars moving overhead in a regular fashion. They also saw the
"wanderers" or "planetai" (our
planets). The
regularity in the motions of the wandering bodies suggested that
their positions might be predictable.
The most obvious approach to the problem of predicting the motions
of the heavenly bodies was simply to map their positions against
the star field and then to fit mathematical functions to the
changing positions.
The ancients worked from a geocentric perspective because the Earth
was the platform on which they stood. Some Greek astronomers (e.g.,
Aristarchus of Samos) had
speculated that the planets (Earth included) orbited the Sun but
the mathematics needed to transform geocentric observations to a
heliocentric perspective didn’t exist in Ptolemy’s time.
Furthermore,
Aristotelian
Physics was incapable of supporting such notions.
The apparent motion of the heavenly bodies with respect to time was
cyclical in nature. Apollonius of Perga discovered that the
cyclical variation could be represented mathematically by circles,
or epicycles, running on a larger circle, or deferent. Deferents
and epicycles in the ancient models didn’t represent orbits in the
modern sense. The ancients didn’t know about orbits or any kind of
connections between the heavenly bodies. They simply saw lights
moving about the sky. (In fact, it wasn’t until Galileo saw the
moons of
Jupiter and the phases of
Venus that astronomers began to accept the notion that
the planets were individual worlds orbiting the Sun.)
Claudius Ptolemy refined the deferent/epicycle concept and
introduced the equant as a mechanism for accounting for velocity
variations in the motions of the planets. The
empirical methodology he developed proved to be
extraordinarily accurate for its day and was still in use at the
time of Copernicus and Kepler.
Owen Gingerich describes a planetary conjunction that occurred in
1504 that was apparently observed by Copernicus. In notes bound
with his copy of the
Alfonsine Tables, Copernicus
commented that “Mars surpasses the numbers by more than two
degrees. Saturn is surpassed by the numbers by one and a half
degrees.” Using modern computer programs, Gingerich discovered
that, at the time of the conjunction, Saturn indeed lagged behind
the tables by a degree and a half and Mars led the predictions by
nearly two degrees. Moreover, he found that Ptolemy’s predictions
for Jupiter at the same time were quite accurate. Copernicus and
his contemporaries were therefore using Ptolemy’s methods and
finding them trustworthy more than a thousand years after Ptolemy’s
original work was published.
When Copernicus transformed Earth-based observations to
heliocentric coordinates , he was confronted with an entirely new
problem. The Sun-centered positions displayed a cyclical motion
with respect to time but without retrograde loops in the case of
the outer planets. In principle, the heliocentric motion was
simpler but with new subtleties due to the yet-to-be-discovered
elliptical shape of the orbits. Another complication was caused by
a problem that Copernicus never solved: correctly accounting for
the motion of the Earth in the coordinate transformation. In
keeping with past practice, Copernicus used the deferent/epicycle
model in his theory but his epicycles were small and were called
“epicyclets”.
In the Ptolemaic system the models for each of the planets were
different and so it was with Copernicus’ initial models. As he
worked through the mathematics, however, Copernicus discovered that
his models could be combined in a unified system. Furthermore, if
they were scaled so that Earth’s orbit was the same in all of them,
the ordering of the planets we recognize today literally fell out
of the math. Mercury orbited closest to the Sun and the rest of the
planets fell into place in order outward, arranged in distance by
their periods of revolution.
Whether or not Copernicus’ models were simpler than Ptolemy’s is
moot. Copernicus eliminated Ptolemy’s somewhat-maligned equant but
at a cost of additional epicycles. Various 16th-century books based
on Ptolemy and Copernicus use about equal numbers of epicycles. The
idea that Copernicus used only 34 circles in his system comes from
his own statement in a preliminary unpublished sketch called the
Commentariolus. By the time he published
De revolutionibus orbium
coelestium, he had added more circles. Counting the total
number is difficult, but estimates are that he created a system
just as complicated, or even more so. The popular total of about 80
circles for the Ptolemaic system seems to have appeared in 1898. It
may have been inspired by the
non-Ptolemaic system
of
Girolamo Fracastoro, who used
either 77 or 79 orbs in his system inspired by
Eudoxus of Cnidus.
Copernicus’ theory was at least as accurate as Ptolemy’s but never
achieved the stature and recognition of Ptolemy’s theory. In
scarcely more than a hundred years, Copernicus would be overcome by
events set in motion by
Johannes
Kepler and
Galileo Galilei.
Copernicus’ work provided explanations for phenomena like
retrograde motion, but really didn’t prove that the planets
actually orbited the Sun.
Ptolemy’s and Copernicus’ theories proved the durability and
adaptability of the deferent/epicycle device for representing
planetary motion. The deferent/epicycle models worked as well as
they did because of the extraordinary orbital stability of the
solar system. Either theory could be used today and might still be
in use had Isaac Newton not invented
Physics
and the
Calculus.
The first
planetary model without any epicycles was that of Ibn Bajjah (Avempace) in 12th century Andalusian Spain
, but epicycles were not eliminated in Europe until
the 17th century, when Johannes Kepler's model of elliptical orbits
gradually replaced Copernicus' model based on perfect
circles.
Newtonian or
Classical Mechanics
eliminated the need for deferent/epicycle methods altogether and
produced theories many times more powerful. By treating the Sun and
planets as point masses and using
Newton’s law of
universal gravitation, equations of motion were derived that
could be solved by various means to compute predictions of
planetary orbital velocities and positions. Simple
two-body problems, for example, can be
solved analytically. More-complex
n-body
problems require
numerical
methods for solution.
The power of Newtonian mechanics to solve problems in
orbital mechanics is illustrated by the
discovery of
Neptune. Analysis of observed
perturbations in the orbit of
Uranus produced
estimates of the suspected planet’s position within a degree of
where it was found. This could not have been accomplished with
deferent/epicycle methods.
Epicycles on epicycles
According to a school of thought in the history of astronomy, minor
imperfections in the original Ptolemaic system were discovered
through observations accumulated over time. More levels of
epicycles (circles within circles) were added to the models, to
match more accurately the observed planetary motions. The
multiplication of epicycles is believed to have led to a nearly
unworkable system by the 16th century.
Copernicus created his
heliocentric system in order to simplify
the Ptolemaic astronomy of his day, and he succeeded in drastically
reducing the number of circles, a term which included both
epicycles and (eccentric) deferents.
- With better observations additional epicycles and
eccentrics were used to represent the newly observed phenomena till
in the later Middle Ages the universe became a 'Sphere/With Centric
and Eccentric scribbled o'er,/Cycle and Epicycle, Orb in
Orb'--
As a measure of complexity, the number of circles is given as 80
for Ptolemy, versus a mere 34 for Copernicus. The highest number
appeared in the
Encyclopaedia Britannica on
Astronomy during the 1960s, in a discussion of King
Alfonso X of Castile's interest in
astronomy during the 13th century. (Alfonso is credited with
commissioning the
Alfonsine
Tables.)
- By this time each planet had been provided with from 40 to
60 epicycles to represent after a fashion its complex movement
among the stars. Amazed at the difficulty of the project,
Alfonso is credited with the remark that had he been present at the
Creation he might have given excellent advice.
A major difficulty with the epicycles-on-epicycles theory is that
historians examining books on Ptolemaic astronomy from the Middle
Ages and the Renaissance have found no trace of multiple epicycles
being used for each planet. The Alfonsine Tables, for instance,
were apparently computed using Ptolemy's original unadorned
methods.
Another problem is that the models themselves discouraged
tinkering. In a deferent/epicycle model, the parts of the whole are
interrelated. A change in a parameter to improve the fit in one
place would throw off the fit somewhere else. Ptolemy’s model is
probably optimal in this regard. On the whole it gave good results
but missed a little here and there. Experienced astronomers would
have recognized these shortcomings and allowed for them.
Slang for Bad Science
In part, due to misunderstandings about how deferent/epicycle
models worked, "adding epicycles" has come to be used as a
derogatory comment in modern scientific discussion. The term might
be used, for example, to describe continuing to try to adjust a
theory to make its predictions match the facts. According to this
notion, epicycles are regarded by some as the paradigmatic example
of Bad Science.
See also
Notes
- For an example of the complexity of the problem, see Owen
Gingerich, The Book Nobody Read, Walker, 2004, p. 50
- ibid., Chapter 4
- One volume of de Revolutionibus was devoted to a
description of the trigonometry used to make the transformation
between geocentric and heliocentric coordinates.
- Owen Gingerich, The Book Nobody Read P. 267
- ibid.. p. 54
- Robert Palter, Approach to the History of Astronomy,
in Studies in the History and Philosophy of Science 1
(1970): 94.
- Owen
Gingerich, Alfonso X as a Patron of Astronomy, in
The Eye of Heaven: Ptolemy, Copernicus, Kepler (New York:
American Institute of Physics, 1993), p. 125.
- Gingerich, Crisis versus Aesthetic in the Copernican
Revolution, in Eye of Heaven, pp. 193-204.
- The popular belief that Copernicus' heliocentric system
constitutes a significant simplification of the Ptolemaic system is
obviously wrong...the Copernican models themselves require about
twice as many circles as the Ptolemaic models and are far less
elegant and adaptable. O. Neugebauer, The Exact Sciences in
Antiquity, 2nd ed. (New York: Dover, 1969), p. 204. This is an
extreme estimate in favor of Ptolemy.
- Palter, Approach to the History of Astronomy, pp.
113-14.
- Interestingly, a deferent/epicycle model is used to compute
Lunar positions needed to define modern Hindu calendars. See Nachum
Dershovitz and Edward M. Reingold: Calendrical
Calculations, Cambridge University Press, 1997, Chapter 14.
(ISBN 0-521-56474-3)
- Bernard R. Goldstein (March 1972). Theory and Observation
in Medieval Astronomy, Isis 63 (1),
p. 39-47 [40-41].
- Dorothy Stimson, The Gradual Acceptance of the Copernican
Theory of the Universe (New York, 1917), p. 14. The quotation
is from John
Milton's Paradise Lost, Book 8, 11.82-85.
- Robert Palter, An Approach to the History of Early
Astronomy
- Encyclopaedia Britannica, 1968, vol. 2, p. 645. This
is identified as the highest number in Owen Gingerich, Alfonso
X. Gingerich also expressed doubt about the quotation
attributed to Alfonso. In The Book Nobody Read (p. 56),
however, Gingerich relates that he challenged Encyclopaedia
Britannica about the number of epicycles. Their response was
that the original author of the entry had died and its source
couldn’t be verified.
- Gingerich, The Book Nobody Read (p. 57)
- See e.g., Kolb, Rocky, Blind Watchers of the Sky,
Addison-Wesley, 1996. P. 299 (ISBN 0-201-48992-9)
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