"Deferent" redirects here. For the acknowledgement of the legitimacy of the power of one's superior or superiors, see Deference.




In both Hipparchian and Ptolemaic systems, 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 clockwise and are roughly parallel to the plane of the Sun's orbit (ecliptic). Despite the fact that the system is considered geocentric, the planets' motion was not centered on the Earth but on what is called the eccentric. The orbits of planets in this system are epitrochoids.

The epicycle rotated and revolved along the deferent with uniform motion. However Ptolemy found that the rate at which the deferent revolved was not constant unless he measured it from another point at an equal distance from the eccentric, which he called the equant. It was the angular rate at which the deferent moved around the equant that was constant. It was the use of the equant that distinguished the Ptolemaic system.







The ancients worked from a geocentric perspective for the simple reason that the Earth was where they stood and observed the sky, and it is the sky which appears to move while the ground seems still and steady underfoot. Some Greek astronomers (e.g., Aristarchus of Samos) speculated that the planets (Earth included) orbited the Sun, but the optics (and the specific mathematics – Newton's Law of Gravitation for example) necessary to provide data that would convincingly support the heliocentric model did not exist in Ptolemy's time and would not come around for over fifteen hundred years after his death. Furthermore, Aristotelian Physics was not designed with these sorts of calculations in mind, and Aristotle's philosophy regarding the heavens was entirely at odds with the concept of heliocentrism. It was not until Galileo Galilei observed the moons of Jupiter on January 7, 1610, and the phases of Venus in September, 1610 that the heliocentric model began to receive broad support among astronomers, who also came to accept the notion that the planets are individual worlds orbiting the Sun (that is, that Earth is a planet and is one among several.) Johannes Kepler was able to formulate his famous three laws of planetary motion, which described the orbits of the planets in our solar system with incredible accuracy; Kepler's three laws are still taught today in University physics and astronomy classes, and the wording of these laws has not changed since Kepler first formulated them four hundred years ago.




When Copernicus transformed Earth-based observations to heliocentric coordinates,[3] 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.[4] In keeping with past practice, Copernicus used the deferent/epicycle model in his theory but his epicycles were small and were called "epicyclets".


Although Copernicus' models reduced the magnitude of the epicycles considerably, whether they 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.[6][7][8] 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.[9] Koestler, in his history of man's vision of the universe, equates the number of epicycles used by Keplar at 48.[10] 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.[11] Copernicus in his works exaggerated the number of epicycles used in the Ptolemic system; although original counts ranged to 80 circles, by Copernicus's time the Ptolemic system had been updated by Peurbach towards the similar number of 40; hence Copernicus effectively replaced the problem of retrograde with further epicycles.[12]


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.[13]

The first planetary model without any epicycles was that of Ibn Bajjah (Avempace) in 12th century Andalusian Spain,[14] 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.





As a measure of complexity, the number of circles is given as 80 for Ptolemy, versus a mere 34 for Copernicus.[17] 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.)




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.[20] Part of the problem may be due to the misconception of the epicycle as an explanation of a body's motion rather than merely a description. Toomer explains as follows,

"Whereas we use 'hypothesis' to denote a tentative theory which is still to be verified, Ptolemy usually means by ύπόθεσις something more like 'model', 'system of explanation', often indeed referring to 'the hypotheses which we have demonstrated'."[21]

Mathematical formalism

According to the historian of science Norwood Russell Hanson:
There is no bilaterally-symmetrical, nor excentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent.[22]

This is because epicycles can be represented as a complex Fourier series; so, with a large number epicycles, very complicated paths can be represented in the complex plane.[23]

Let the complex number

z_0=a_0 e^{i k_0 t},

where a_0 and k_0 are constants, i=\sqrt{-1} is an imaginary number, and t is time, correspond to a deferent centered on the origin of the complex plane and revolving with a radius a_0 and angular velocity


where T is the period.

If z_1 is the path of an epicycle, then the deferent plus epicycle is represented as the sum

z_2=z_0+z_1=a_0 e^{i k_0 t}+a_1 e^{i k_1 t}.

Generalizing to N epicycles yields

z_N=\sum_{j=0}^N a_j e^{i k_j t},

which is a particular type of complex Fourier series known as a Besicovitch almost periodic function. Finding the coefficients a_j to represent a time-dependent path in the complex plane, z=f(t), is the goal of reproducing an orbit with deferent and epicycles, and this is a way of "saving the phenomena" (σώζειν τα φαινόμενα).[24]

This parallel was noted by Giovanni Schiaparelli.[25][26] Pertinent to the Copernican Revolution debate of "saving the phenomena" versus offering explanations, one can understand why Thomas Aquinas, in the 13th century, wrote:

See also


External links

  • Ptolemaic System – at Rice University's Galileo Project
  • Eccentrics, Deferents, Epicycles, and Equants - at MathPages

Animated illustrations

  • A flash animation showing epicycles with adjustable parameters and presets for various planets.
  • An Applet showing the principle of the epicycle, with a side-by-side comparison of the geocentric and heliocentric models.