Deferent and epicycle
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The basic elements of Ptolemaic astronomy, showing a planet on an epicycle (A) with a deferent (C) and an equant (B). |
In the
Ptolemaic system of
astronomy, the
epicycle (literally:
on the circle in
Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the
Moon,
Sun, and
planets. It was designed by
Apollonius of Perga at the end of the
3rd century BC. In particular it explained
retrograde motion. Secondarily, it also explained changes in the distance of the planet 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 counterclockwise and are roughly parallel to the Earth's plane of orbit (
ecliptic). The
orbits of planets in this system are
epitrochoids.
The deferent was a circle centered around a point halfway between the equant and the earth. The epicycle rotated on the deferent with uniform motion, not with respect to the center, but with respect to the off-center point called the
equant. The rate at which the planet moved on the epicycle was fixed such that the angle between the center of the of 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 rotate 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 rotate 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.
According to one 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 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 eccentric 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'--"[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.]Most commonly the number of circles is given as 80 for Ptolemy, versus a mere 34 for Copernicus.
[Robert Palter, "An Approach to the History of Early Astronomy," Studies in the History and Philosophy of Science 1 (1970): 94.] 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. (Among his other activities, Alfonso is supposed to have commissioned 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."[Encyclopaedia Britannica, 1968, vol. 2, p. 645. This is identified as the highest number in 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.]The addition of epicycles, a gradual process of going from a simple model to a complex model is sometimes used as an allegory for some modern technical
boondoggles.
The difficulty with this account is that historians examining books on Ptolemaic astronomy from the Middle Ages and the Renaissance have not found any trace of multiple epicycles being used for each planet. The Alfonsine Tables, for instance, were actually closer to Ptolemy's original calculations than the older
Tables of Toledo, while 16th-century books based on Ptolemy and Copernicus use about equal numbers of epicycles.
[Palter, "Approach to the History of Astronomy"; Gingerich, "Alfonso X"; Gingerich, "'Crisis' versus Aesthetic in the Copernican Revolution," in Eye of Heaven, pp. 193-204. Gingerich also expresses doubt about the quotation attributed to Alfonso.] 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 belief that Copernicus's 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.] 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.
[Palter, "Approach to the History of Astronomy," pp. 113-14.] Epicycles were finally eliminated in Europe in the 17th century, when
Johannes Kepler's model of elliptical orbits gradually replaced Copernicus' model based on perfect circles.
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A flash animation showing epicycles with adjustable parameters and presets for various planets.*
An Applet showing the principle of the epicycle
