- Eudoxos of Knidos (Eudoxus of Cnidus): astronomy and homocentric spheres and Venus© by Henry Mendell (Cal. State U., L.A.)

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We can construct a standard version of the Eudoxan model for Venus. Keep in mind, however, that this is pure speculation. These are the considerations that should go into any reconstruction, even if not all of them become part of the actual model we use.

- Invisibility occurs when the planet is 1/2 a zodiacal sign from the sun. The evidence for this principle is that this is the theory we find in Autolycus (about 3/4 century after Eudoxus).
- Venus is isodromic with the sun (has the same ecliptic period and stays with the sun) and so has a zodiacal period of 1 year, so that sphere 1 rotates about 366+ times east/west for 1 rotation of sphere 2 west/east.
- Tbe synodic period is 570 days. Here it is indeterminable whether our source, Simplicius, intends sidereal or solar days. The difference will be insignificant in any case, i.e., less than two days.
- Venus has a retrograde motion of between 1/2 and 2/3 sign.
- The invisibility period of Venus for last evening appearance to first morning is less than 10 days.
- The invisibility period of Venus for last morning appearance to first evening 55 days to 80 days.
- The maximum elongation of Venus is about 1 1/2 signs.
- Venus changes its latitude by over 8 1/2 degrees from the ecliptic. How one might observe this is tricky, as one needs to measure this either off a fixed star whose position relative to the ecliptic is known or by knowing where the sun would set were it at the position that Mercury is now or by some other method appropriate to the world of Eudoxus.

No version of Eudoxan models can account for the discrepancy between the two profoundly different invisibility periods, since the curve is essentially symmetric. At least, one would have to adjust the center of the hippopede, which would require more spheres. A 1 1/2 sign curve seems reasonable as it will capture the most important feature, the maximum elongation from the sun.

Assume that the poles of spheres 2 and 3 are 1 1/2 signs (45 degrees) apart, i.e., that the angles between the equators are 1 1/2 signs. Hence, the loops are each 1 1/2 signs in length. The maximum latitude will be 9.74 degrees, but in the wrong part of the cycle. It should be clear from the diagram that there is no retrograde motion on this model.

Right click here to download a Quicktime™ of EudoxusVenus45Hippopede.mov (File Size: 1.1 megs)

Right click here to download a Quicktime™ of EudoxusVenus150Hippopede.mov (File Size: 1.1 megs)

Some modern readers have been sceptical about this assumption. Alan Bowen thinks that the primary purpose of the model was to account for latitude variation. I have suggested that invisibility periods were an important consideration. However, it is possible that the only purpose of the models for Venus and Mercury was to account for elongation.