Cycle Counts
You may have heard that Ptolemaic systems grew to have 80 spheres or cycles, while the Copernican system had only 34. This is a myth.
Let’s do a cycle count for the geometric models of the Almagest. The Sun has one cycle, the Moon three: deferent, epicycle, and inner small circle (or epicyclet). Each outer planet has four: deferent, epicycle, equant, and latitude wheel. (Recall from post 4 that Ptolemy used an equant circle to implement his speed law.) Venus has two more latitude wheels than the outer planets, so six cycles. Mercury has all of Venus’s cycles plus an epicyclet, or seven. We have the sphere of the fixed stars, and another sphere to account for the precession of the equinoxes. Total: 31.
If we adopt the simpler (and more accurate) latitude theory of the Planetary Hypotheses, we eliminate all those latitude wheels, saving 9 cycles. You might also argue that the equant circles really shouldn’t count as cycles, getting rid of 5 more. New total: 17 cycles. We will look at Ptolemy’s “celestial body count” in the next section.
Nor did Ptolemaic astronomy become more complicated over time. Evans:
One often hears it said that during the Middle Ages, the old astronomy of the Greeks became more and more complicated, as epicycles were added to epicycles, until the system collapsed of its own intellectual weight. This is quite untrue. Almost everywhere in Islam and Christendom computation of planet positions continued to follow standard Ptolemaic models. The planetary theories underlying the Alfonsine Tables, for example, are standard Ptolemaic models.1 The one big exception to this rule is trepidation theory2. The addition of this oscillatory motion to the sphere of stars did make the universe more complicated—but not all that much more complicated.
—Evans (p.280)
As for the notion that Copernicus greatly reduced the cycle count, Gingerich disposes of that in this passage:
Many simple historical accounts of the Copernican revolution emphasize not the accuracy but the simplicity of the new system, generally in contrast to the horrendous complex scheme of epicycles-upon-epicycles supposedly perpetrated by pre-Copernican astronomers. This tale reached its most bizarre heights in the 1969 Encyclopaedia Britannica [7], where the article on astronomy states that by the time of Alfonso in the thirteenth century, 40 to 60 epicycles were required for each planet! More typically, we find what Robert Palter has called the “80-34 syndrome”—the claim that the simpler Copernican system required only 34 circles in contrast to the 80 supposedly needed by Ptolemy.[8] The Copernican count derives from the closing statement of his Commentariolus: “Altogether, therefore, 34 circles suffice to explain the entire structure of the universe and the entire ballet of the planets.” [9] By the time Copernicus had refined his theory for his more mature De revolutionibus, he had rearranged the longitude mechanism, thereby using six fewer circles, but he had added an elaborate precession-trepidation device as well as a more complicated latitude scheme for the inner planets. Even Copernicus would have had difficulty in establishing an unambiguous final count.[10] A comparison between the Copernican and the classical Ptolemaic system is more precise if we limit the count of circles to the longitude mechanisms for the Sun (or Earth), Moon, and planets: Copernicus requires 18, Ptolemy 153. [11] Thus, the Copernican system is slightly more complicated than the original Ptolemaic system.
—Gingerich (p.196–197), “‘Crisis’ Versus Aesthetic in the Copernican Revolution”
And here are the footnotes to this passage:
- “Astronomy. I. History of astronomy. B. Mediaeval astronomy”, Encyclopaedia Britannica, vol. 2 (Chicago, 1969); p. 645:
“King Alfonso X of Castile kept a number of scholars occupied for ten years constructing tables (the Alphonsine tables, c. 1270) for predicting positions of the planetary bodies. 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. After surviving for more than a millennium, the Ptolemaic system had failed; its geometrical clockwork had become unbelievably cumbersome and without satisfactory improvements in its effectiveness.”
- Robert Palter, “An Approach to the History of Early Astronomy”, History and Philosophy of Science, vol. 1 (1970), pp. 93–133. Palter traces the 80–34 myth back as far as Arthur Berry’s A Short History of Astronomy (London, 1898).
- Edward Rosen, “Nicholas Copernicus, a Biography”, in his Three Copernican Treatises, 3rd ed. (New York, 1971), p. 90.
- According to Ernst Zinner, Entstehung und Ausbreitung der Copernicanischen Lehre (Erlangen, 1943), pp. 186–87, Copernicus should have included precession, the regression of the lunar nodes, and the change of solar distance in his count in the Commentariolus, thus getting a total of 38 circles. Arthur Koestler, in The Sleepwalkers (London, 1959), pp. 572-73, attempted to count the circles in De revolutionibus, but he overlooked the fact that Copernicus had by then replaced the so-called Tusi couple in the longitude mechanisms by an eccentric, thereby listing at least six unnecessary circles; on the other hand, he could have claimed that the motion of the apsidal lines for Mercury and the superior planets each required a circle.
- Copernicus replaced the Ptolemaic mechanism for varying the size of Mercury’s orbit with a Tusi couple, and he also accounted for the apsidal motion of the Earth’s orbit with two circles. If the apsidal motions for Mercury and the superior planets are counted, then Copernicus required 22 circles for the motions in longitude.
The 80 sphere figure probably originally referred to the system of Girolamo Fracastoro, expounded in his work Homocentrica (1538). This is a descendent of Aristotle’s system, in turn based on the work of Eudoxus. These are so-called homocentric systems: rotating spheres, all concentric with the center of the Earth, but with different axes. Thus eccentrics and epicyles are not allowed. Aristotle made do with 56 spheres (including the sphere of fixed stars), but Fracastoro required 79. Dreyer (pp.296–301) provides a detailed description.
Finally, if you want to count the cycles yourself, here are some references. Toomer is the standard English translation of the Almagest. Pedersen’s Survey is more readable. Dreyer Ch.9 is a good brief summary of the models, and Linton Ch.3 provides a more recent treatment.
None of these furnish an explicit cycle count. You’ll have to add them up yourself. Here are some pointers, corresponding to my posts as indicated. (Ptolemy page numbers refer to Toomer’s translation.)
- Post 9. Ptolemy has three lunar models; he rejects the first two for observational discrepancies. See Ptolemy Book V Ch.2 (pp.220–222), Ch.5 (pp.226-233), and Ch.9 (pp.237–239). See Pedersen Ch.6, especially p.186 (Fig.6.9) and p.193 (Fig.6.14).
- Post 10, Post 11. Ptolemy introduces the basic Deferent-Epicycle-Equant model in Book IX Chs.5&6 (pp.426, 442-444; Fig.9.1). He applies this to all the planets except Mercury in other chapters of Books IX–XII, computing parameters, retrogradations, and the like. But these don’t matter for the cycle count. See Pedersen Ch.9, especially p.285 (Fig.9.7), and Ch.10, especially p.303 (Fig.10.3).
- Post 12. Mercury is a special case; Ptolemy gives the basics of the model in Book IX Ch.6 (p.445 Fig.9.2), right after the model for the other planets. See Pedersen Ch.10, p.316 (Fig.10.7). See also Gingerich, “Mercury Theory from Antiquity to Kepler”.
- Post 13. Book XIII covers the latitude mechanisms. See Pedersen Ch.12, especially p.361 (Fig.12.4) and p.366 (Fig.12.7). See also Swerdlow.
[1] See Gingerich, “Alfonso X as a Patron of Astronomy”, for more about this.
[2] Trepidation is a supposed oscillation in the precession of the equinoxes. It doesn’t exist, but was a feature of various astronomical theories.
[3] This agrees with our reduced count of 17 cycles once you add in the fixed stars and the precession of the equinoxes.
Diagonal Argument 