Caption: A cartoon of an epicycle model with the equant and eccentric Earth. Such epicycle models were used in the Ptolemaic system of Ptolemy (c.100--c.170 CE) which is described in Ptolemy's book the Almagest. For reasons we discuss below, they are the crime of Ptolemy.

Features:

1. In an epicycle model, a planet orbits on an epicycle which orbits on a deferent.

   The eccentric Earth and the equant are special features of the epicycle models of the Ptolemaic system.

2. Ptolemy in his book Planetary Hypotheses (c.150) added celestial spheres (adopted with some modifications from Aristotelian cosmology) to tightly bound the epicycles. The celestial spheres were NOT needed for the Ptolemaic system as calculational device, but allowed Ptolemy to develop a distance scale for the Solar System. The celestial spheres of the Planetary Hypotheses (c.150) were a wrong theory, but the theory is a credit to Ptolemy's genius and its limitations. For more information on the theory, see file ptolemaic_physical_model.html.

3. The geometric center of the deferent is the center of the Solar System which was the whole universe to Ptolemy.

   The Earth is displaced from the geometric center making the deferent an eccentric orbit. The model is thus NOT strictly geocentric.

4. The equant is displaced from the geometric center on the opposite side from the Earth by an equal distance.

   The angular velocity of the epicycle on the deferent is constant as viewed from the equant.

   Hereafter when we say equant we mean the theory of the equant which includes the constant angular velocity of the epicycle on the deferent as viewed from the equant, the eccentric orbit of the epicycle as seen from the Earth, and the lack of strict geocentrism.

5. Ptolemy found that he needed the equant to get a good fit of his epicycle models to the observations.

   However, the equant violated Ptolemy's own stated principle of uniform circular motion for the elementary astronomical motions since the tangential speed of the epicycle on the deferent is NOT uniform with the device of the equant.

   The physical law of uniform circular motion had been emphasized by Plato (428/427--348/347 BCE) and been followed by all or almost all ancient Greek astronomers.

   The equant also violated strict geocentrism which was also assumed to be a physical law by all or almost all ancient Greek astronomers.

6. There are two significant points about the equant:
   1. First, as a violation of physical laws, it made Ptolemaic system less realistic and more of just a calculation tool if you believed that the physical laws were true.

      Because of the violation, later astronomers up to and including Nicolaus Copernicus (1473--1543) judged the equant to be the crime of Ptolemy.

      Copernicus was quite proud of having NO equant in his Copernican system. He believe that his system was more physical, and so more likely to be true than the Ptolemaic system.

   2. Second, the equant can be recognized as Ptolemy feeling his way toward elliptical orbits (Kepler's 1st law) obeying Kepler's 2nd law (equal areas swept out in equal times by an orbiting planet's orbital radius as measured from a focus point of the elliptical orbit).

      In fact, Ptolemy must have been aware that the Moon (which does really orbit the Earth geometrically) does have an eccentric orbit since its angular diameter on the sky does vary by ∼ 11 % and it does have varying angular velocity as seen from the Earth. These facts were probably a clue leading him to the equant, but he never explained this??? and no one else ever did either it seems until circa 1600 or later.

7. From the point of view of modern science, what Ptolemy should have done was openly state that uniform circular motion and strict geocentrism may NOT be a true physical laws and that one had to investigate more general motions.

   Perhaps, Ptolemy did think this, but decided that it was a road too long for him to go down. He may have thought he had to bring the Almagest (his magnum opus book on the Ptolemaic system) to closure without open-ended questioning of basic principles: e.g., uniform circular motion and strict geocentrism.

   But Ptolemy may have been on the verge of a conceptual breakthrough when exhausted from his labors in the Heavens he fell to rest on Earth.

   Maybe he did write his Geographia (c.150 CE) later than the Almagest (c.150 CE).

   It's unfair of us to complain that Ptolemy should have gone further in his work since we should all do so in ours, but we will complain anyhow.

8. See the animation of Ptolemy's (c.100--c.170 CE) epicycle model (with equant) for Jupiter in the figure below (local link / general link: ptolemy_epicycle_equant_animation.html).

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9. See also:

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Credit/Permission: © David Jeffery, 2003 / Own work.
Image link: Itself.
Local file: local link: ptolemy_epicycle_equant.html.
File: Ptolemy file: ptolemy_epicycle_equant.html.

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