Return to Vignettes of Ancient Mathematics

Eudoxos of Knidos was born approximately 395-390 BCE and lived 53 years. A polymath, he made important contributions to geography, metaphysics, and ethics. However, his most important work was in geometry, the theory of proportion, and astronomy. Our principal sources for his astronomy are Aristotle, Aratus (3rd cent. BCE), Hipparchus (2nd cent. BCE), and Simplicius (6th cent. CE), although the last is our principal source for his astronomical models. The first to present a general, geometrical model of celestial motion, Eudoxos started with five basic principles.

  1. The earth is the center of the universe.
  2. All celestial motion is circular.
  3. All celestial motion is regular.
  4. The center of the path of any celestial motion is the same as the center of its motion.
  5. The center of all celestial motion is the center of the universe.

Since the apparent motion of all celestial bodies is neither circular nor regular, though it is about the earth, Eudoxos needed to construct models that preserved the five principles and the appearances. This means constructing the apparent motions as combinations of circular motions, the basic idea behind most subsequent Greek mathematical astronomy and the basis of mathematical astronomy from Ptolemy up to Kepler.

  1. The apparent irregular and non-circular motions of celestial bodies is a result of the combinations of real, circular motions that satisfy principles (1) - (5). In other words, the models must preserve or save the phenomena.

The qualification in (4) is significant only in the context of Ptolemy's later introduction of models which distinguish the center of the motion from the center of the path of the motion. It would have been implicit in the astronomy of Eudoxos.

There is no way of knowing why Eudoxos adopted these five principles. Most historians have seen a more fundamental metaphysical principle in making celestial motion regular and circular, that the universe is perfect and that this is the simplest and most perfect motion. Certainly, such a view may be found in the justification for such principles in the writings of his junior contemporary and colleague, Aristotle. It should also be kept in mind, however, that regular, circular motion is mathematically more tractable than any alternative available in the fourth century BCE. So any geometrical theory of somewhat circular motion would have to be resolved into regular circular motion if it were to be treated mathematically with any reasonable depth.

The basis for principle (5) is harder to discern. However, it might well be grounded in a view that (1) is meaningless without (5). Nonetheless, it was the first to be rejected by Greek astronomers, probably some time in the late 3rd or early 2nd century BCE. True, the earth is kept as the center of the motion of the first stars, and the general area where the earth is located contains the centers of the principal motions of the the stars with the earth also inside those motions; yet, other motions are later introduced which do not even have the earth within them (epicycles). Again, we have no way of knowing whether Aristotle's justifications for (5), that the element of the celestial bodies has a natural motion in a circle about the center of the universe, has any bearing on the thought of Eudoxos.

The model that Eudoxos produces to satisfy (1) - (5) has each motion being physically the rotation of a celestial sphere. The fixed stars, planets, sun, and moon, as we see them, may be thought of as knots or pimples that occur on some of the celestial spheres.

  1. Each circular celestial motion is physically the motion of a single body.
  2. Each individual body producing a circular motion is a sphere that rotates about a pair of poles.

Why does Eudoxos think that these bodies must be spheres? For Aristotle, it is pertinent that there is no void, so that the the planets must move where there is body that does not impede them, with the result that the body in the path of a planet needs to move with the planet, but also that spheres are perfect shapes. Again, we do not know whether either of these are relevant to the thought of Eudoxos.

For the fixed stars, it is enough to have one sphere rotating daily about the earth. Their apparent motion, at least until Hipparchus (ca. 150 BCE), is circular and regular.

For each of the planets, sun, and moon, whose apparent motion is neither circular nor regular, Eudoxos introduces a series of nested or homocentric spheres. Each nested sphere is attached at its poles to the next sphere up. So the governing sphere will determine the rotation of the poles of the lower sphere. However, every sphere will rotate about its poles independently of any sphere above or below. The only condition is that its motion be regular.

[Note: in recent years, there has arisen a controversy about what the regular motion will be. Traditionally, the regular motion has been regarded as relative to the next up governing sphere. Ido Yavetz, however, has proposed that for spheres below the sphere of the ecliptic, the motion must be regular in relation to the sphere of the ecliptic. He also does not necessarily place the planet on the equator circle of the last sphere. This leads to models that preserve the phenomena better, but not the textual evidence for the models. No one has yet proposed that the regular motion be in relation to the staid earth, but one would expect this to be the reference frame for a model that does not treat the next higher, governing sphere as the reference frame for the regular motion. This model would also conflict even more with the textual evidence.]

The combination of the poles being rotated by regularly rotating, governing spheres with the regular motion of the lowest sphere produces an apparent irregular motion. The phenomena to be explained, though crucial to understanding Eudoxos' models, is outside the present discussion. For the moment, we shall only look at a simple version of the standard model (see previous note) for the planets. For each of the planets Eudoxos provided four spheres:

  1. The sphere corresponding to the sphere of the fixed stars (here gray). This sphere rotates daily, east/west about the celestial equator with North and South poles. It is identical in its movement and position of its poles to the sphere of the fixed stars. It rotation will actually have to be slightly less than a day, one sidereal day, as the combination of motion of the sun's spheres will have to add up to one year and 365+ solar days (the precise length of the Eudoxan year is unknown). This will be 1 rotation west/east (the combination of the motion of the sun's second and third sphere) and 366+ rotations of the first sphere to yield the required 365+ rotations.
  2. The sphere of the ecliptic (here blue). This sphere has its poles on the sphere of the fixed stars, each possibly 1/15 circular-arc from the corresponding pole of the first sphere. Its motion is west/east, but differs for every planet (one zodiacal month for the moon, one year for the sun, Mercury, and Venus, two years for Mars, twelve years for Jupiter, and thirty for Saturn). For the planets (but not the sun and moon), the motion of the sphere is the zodiacal period of the planet, i.e. its motion relative to the fixed stars, e.g. the approximate time it takes to return to conjunction with the same star.

The next spheres only pertain to the five planets (Mercury, Venus, Mars, Jupiter, and Saturn).

  1. The first sphere for the synodic period (here green). This sphere has as its period the time it takes for the planet to return to the same phase, e.g. last appearance in the evening or first appearance in the morning. Its poles are on the ecliptic (that is the equator of the second sphere) and it rotates south/north (the meaning here is a little unclear, since any south/north motion relative to some point on earth is also a north/south motion 1/2 day later from the same location on the earth, but our source, Simplicius, may just be bringing out the point that it must be of this sort). These are 110 days for Mercury, 19 months (570 days) for Venus, 8 months and 20 days or 260 days for Mars (almost certainly an error in the report of Simplicius, as the number should be 24-26 months), approximately 13 months (390 days) for Jupiter and Saturn.
  2. The second sphere for the synodic period (here red). This spherσe has the same period as the first. It rotates opposite to the first sphere, i.e., approximately north/south, although Simplicius actually describes this as east/west. The same issues arise as for the third sphere. The poles of the fourth sphere are distant from the poles of the third, but our sources do not say how distant. Since there was no standard way of measuring angles in the time of Eudoxus, except as parts of a circular-arc or a right angle, this is hardly surprising.



The combination of spheres 3 and 4 constitutes a figure, according to Simplicius, called the hippopede, which is either a horse fetter (a loop of ropes for restraining a pair of feet of a horse) or, metaphorically, an exercise in horse training to keep the horse from leaning to one direction. It is almost certainly a figure-eight or two loops connected by a line or lines, or something of this sort.

We can see this in the animation here. Think of the BLACK points and circle as the poles and equator of the SECOND, ecliptic sphere. We won't rotate it yet. The GREEN, THIRD sphere's poles are attached to the ecliptic circle, so that its equator is at right angles to it. We will rotate it south/north for one privileged orientation of the viewer on the earth (also Green) facing our supra celestial eye view. As we rotate this sphere, the green equator will thicken, so that we can see how much it has rotated.

If the star were on the third sphere, "the star would have come to the poles of the zodiacal circle and would come to be near the poles of the universe," as Simplicius says. The fourth sphere prevents this.

Then, attached to this sphere are the RED poles of the FOURTH sphere. In the example, each pole is 40 degrees from the corresponding pole of the third sphere (as is, therefore, the angle between he equators), but this is a convenience. The third sphere will carry these poles about in a circle. However, the fourth sphere will rotate at the same rate as the third sphere in the opposite direction, in effect, east to west. As Simplicius says, this "will excuse [the star] from traveling way beyond the zodiacal circle and provides the star with the means to trace out the hippopede..." Again, the equator in the diagram will thicken as it proceeds from the chosen intersection of the two equators (that of the third and fourth spheres). The angular distance is the same.

Observe that the (angular) length of each loop is the same as the (angular) distance between the poles. The (angular) width increases as the distance between the poles does. However, the size of these is not readily determinable within the state of Greek geometry of the 4th century BCE.

The width of the curve is greatest when the defining pair of circle has each rotated 1/2 right angle. This can be seen on some reconstructions of the curve. Again, there is the same difficulty about measuring angles.

Right click here to download a Quicktime™ of EudoxusHomocentricDiag2.mov (File Size: 3.2 megs)


Here is another view that shows more clearly the rotations of the great circles connecting the poles. Again, the outermost, 2nd sphere is black, the next, 3rd, sphere is green, with the 4th sphere red, and the planet the white point.

Right click here to download a Quicktime™ of EudoxusHomocentricDiag3.mov (File Size: 3.2 megs).

We now turn to what happens when this model is inserted into a sphere for the ecliptic motion.

We can rotate the ecliptic west/east to see the overall effect of planet moving on the hippopede and the hippopede moving along the ecliptic. Here the angle between the poles of spheres 3 and 4 is 30 degrees and the periods of spheres 3 and 4 are 1/3 the period of the ecliptic. The motion of the planet is irregular. Again, what irregularities need to be modeled require a separate discussion, an issue for which there is no historically certain answer.

Right click here to download a Quicktime™ of EudoxusHomocentricDiag4.mov (File Size: 1.1 megs)


The primary difficulty in discussions of this reconstruction of the models depends on two claims.

  1. The primary purpose of spheres 3 and 4 is to account for retrograde motion of the planets.
  2. It impossible to model retrograde motion for Venus and for a reasonable value of the synodic period of Mars.

It is now known that (2) is false, but the resulting models have movement in latitude that are extreme. More important, we do not know if modeling retrograde motion was a goal of the models. There is much evidence that maximum elongation (angular distance) of Mercury and Venus had to be preserved in the models, and weaker evidence that variations in invisibility periods (periods from last evening to first morning for all planets, and from last morning to first evening for Mercury and Venus) could have been phenomena to be preserved in the model. The evidence for modeling retrograde motion is much weaker. It has also been suggested that modeling latitudes was a goal. If so, the model does a poor job (unlike the version of Yavetz), but the only evidence here is a complaint reported by Simplicius that the models did a poor job with latitudes. This could be interpreted in many ways, some consistent with the standard model.

A NOTE ON RETROGRADE MOTION:

It is impossible to find stations and determine the arc length of a retrograde motion for the model within the confines of 4th century mathematics. There are some approximations possible, however. Neugebauer suggested the following. Let us take γ as the inclination of the two poles. Then the hippopede is of length 2 γ. Therefore, it takes 1/2 a synodic period to travel the arc length 2 γ, e.g., in the opposite direction of the planets travel along the ecliptic. So, if the ecliptic motion is less than 2γ (the length of the curve) in 1/2 synodic period, there will be a retrograde motion. Let σ be the synodic period, ζ the period of the zodiacal period, that is the period for the ecliptic motion, and C one circular-arc. Then, the ecliptic motion in this time will be C σ/2ζ. If one assumes that the curve has an inregular motion which is fastest at its center,

the retrograde motion ≥ 2γ - C σ/2ζ.

Hence, the model will provide some retrograde motion if

σ/2ζ ≤ 2γ/C

Or

set γ =1/2 (σ/2ζ + desired retrograde motion), where a little less is okay.

In fact, the assumption that the curve has the fastest longitudinal motion at its center is only true where γ < 65.53 degrees. As a result we can construct models that have a retrograde motion, followed by a forward motion, and followed by a second retrograde motion, all during the backwards motion on the curve. However, these will have a maximum latitude over 17 degrees.

Here's a table of the planetary periods according to Simplicius with possible values for γ. Note that the alternative values for the synodic period of 260 days for Mars are guesses based on the view that the report of Simplicius must be in error.

Planet Zodiacal Period (ζ) Synodic Period (σ) C σ/ 2ζ desired retrogradation possible γ retrograde in model maximum latitude (degees)
Mercury 1 year = 365 days 3 2/3 months = 110 days 11/73 1/24 (15 degrees) 1/3 right angle 15.3 3.84
Mercury 1 year = 365 days 19 months = 570 days   elongation as the basis for γ 2/9 right angle

1.6

1.7
Venus 1 year = 365 days 19 months = 570 days 57/73 1/20 (18 degrees) 1 2/3 right angle

54.8-40.3 (forward)+54.8, retrograde angle = 14.1

68.6
Venus 1 year = 365 days 19 months = 570 days   elongation as the basis for γ 1/2

none

8.6
Mars 2 years = 730 days 8 2/3 months = 260 days 13/73 1/9 - 2/9 1 right angle 25.4 6.7
Mars 2 years = 730 days 2 years = 730 days 1/2   1 1/12 right angle

20.3-.004 (forward)+20.3, retrograde angle = 20.2

34.4
Mars 2 years = 730 days 760 days 38/73   1 1/7 right angle 22.7-.9 (forward)+22.7, retrograde angle 21.9 37.7
Mars 2 years = 730 days 780 days 39/73   1 1/6 right angle 23.2-1.66 (forward)+23.2, retrograde angle 21.5 39
Jupiter 12 years = 4380 days 13 months = 390 days 29/876 1/36 (less than 10 degrees) 1/9 right angle 6.6 0.44
Saturn 30 years = 10950 days 13 months = 390 days 13/730 1/60 (about 7 degrees) 1/15 right angle 6.3 0.16

Here is a quick look at the sizes and data for different hippopedes at 5 degree intervals (5 to 180).

Again, the outermost, 2nd sphere is black, the next, 3rd, sphere is green, with the 4th sphere red, and the planet the white point. The curve is blue. The rotation is depicted as being 2/3 right angle.