A very brief introduction to the models and an equally brief User’s Guide.

For more detail and background, try Six Easy Lectures on Ancient Mathematical Astronomy.

I. The following links point to animations which run in web browsers, and thus need the latest Flash plug-in from http://www.macromedia.com. Note that right-clicking a link will generally offer the option of opening the animation in a new window, thus allowing you to easily open several animations simultaneously if you choose.

• The Sun (or try this version if you are using Netscape).

• See how the Season Lengths change as you vary the eccentricity and apsidal line direction of the Sun (or try this version if you are using Netscape). You might want to zoom in to get a clear view of the eccentricity.

• The concentric equant (or try this version if you are using Netscape), wherein the motion is on a concentric deferent but is uniform with respect to an offset point (the equant). The animation shows that the concentric equant is equivalent to either an eccentre with varying eccentricity or an epicycle with varying radius.

• The simple Moon (or try this version if you are using Netscape). These first two are not to scale. The eccentricities are made larger to show more clearly.

• The final Almagest Moon (or try this version if you are using Netscape) and with korny soundtrack (or try this version if you are using Netscape).

• The Ibn ash-Shatir model for the Moon (or try this version if you are using Netscape), also adopted by Copernicus. This exaggerated model is somewhat clearer (or try this version if you are using Netscape).

• A variety of models relevant to my paper The Second Lunar Anomaly in Ancient Indian Astronomy, , (in press, 2007). The article is also available at www.springerlink.com.

  Archive for History of Exact Sciences

• Mercury (or try this version if you are using Netscape), with a greatly exaggerated eccentricity for the sake of clarity

• The Ibn ash-Shatir model for Mercury (or try this version if you are using Netscape), also adopted and modified by Copernicus.

• Venus (or try this version if you are using Netscape).

• Regiomontanus moving eccentric models for Mercury and Venus (Netscape versions: Mercury Venus). The slider takes you from the usual epicycle model, to the moving eccentric model (which Ptolemy claims in XII.1 does not apply to inferior planets), and finally to the positions (first geocentric, then heliocentric) that Swerdlow proposes as a factor in Copernicus considerations of heliocentric models (see Proc. Amer. Phil. Soc. 177 (1973) p 476).

  Almagest

• Mars (or try this version if you are using Netscape).

• Jupiter (or try this version if you are using Netscape).

• Saturn (or try this version if you are using Netscape).

• Arabic models for replacing the equant for the outer planets and Venus (or try this version if you are using Netscape) compared to the equant model. These are the models of Nasir al-Din al-Tusi, Muayyad al-Din al-Urdi, and Ibn ash-Shatir. The models of al-Urdi were also used at a later date by Qutb al-Din al-Shirazi, and it is not known if al-Shirazi was aware of the al-Urdi models. For the outer planets Copernicus adopted the version of al-Urdi or al-Shirazi, while for Venus and Mercury he adapted other ash-Shatir models. In no case is it known how Copernicus became aware of any of these models. In all cases the epicycle of the planet is optionally included for clarity, and of course is not needed in any event for the heliocentric Copernican models. The eccentricity is also greatly exaggerated for clarity.

  Almagest

• An interactive Tusi couple (or try this version if you are using Netscape). The Tusi couple is a way to produce linear simple harmonic motion using only combinations of uniform circular motions (i.e. just the inverse of the usual method of producing uniform circular motion by combining two orthogonal simple harmonic motions), and as far as is known, using the couple to produce linear motion was the use by Arabic astronomers. In this modern, and hence ahistorical, version you may vary the relative radii of the two circles, which will change the path to an ellipse, or you may vary the relative frequencies of the rotations to get other patterns (try the values 25 and 75, and do not even think about asking if there is any connection to the Da Vinci Code). See http://mathworld.wolfram.com/Hypocycloid.html for more information.

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• The transformation between a geocentric model and a heliocentric model for an outer planet (Jupiter/Mars) and an inner planet (Venus) (or try this version for Jupiter/Mars and this version for Venus if you are using Netscape). The primary eccentricities are neglected.