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.
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
Regiomontanus moving eccentric models for Mercury and Venus
(Netscape versions: MercuryVenus). The slider takes you from the usual
epicycle model, to the moving eccentric model (which Ptolemy claims in Almagest
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).
Arabic models for replacing
the equant for the outer planets and Venus (or try this
version if you are using Netscape) compared to the Almagest
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.
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 only 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.
Motion in latitude of an outer planet (or try this
version if you are using Netscape) and an inner
planet ((or try this version if you are
using Netscape). The inclinations are greatly exaggerated, and some of the
minor details in the Almagest models are omitted, but the
animations are qualitatively correct.
Ptolemy’s Cosmology (or try this version if you are using Netscape). Make
scale smaller to bring in Jupiter and Saturn, and make it larger to see
Mercury and the Moon. Zoom far in and see Ptolemys geography of the Earth.
Tycho Brahe’s Cosmology (or try this version if you are using Netscape). Note how
Mars’ orbit intersects the orbit of the Sun. However, after observing the
comet of 1577, Tycho became convinced that there are no solid celestial
spheres, and so hence no reason not to prefer his geocentric version of
Copernicus’ heliocentric model (for which, ignoring the difference between
the mean and real Sun, see Kepler’s 3rd Law below).
A comparison of Kepler motion
and equant motion (or use this version if you
are using Netscape). You will have to use a rather large eccentricity to
see much difference at this scale.
The above files have the advantage
that you may save the executable files locally on your computer and thus avoid
any dependence on a network connection when you want to use them. They have the
disadvantage that if you do that, you might not have the latest
version of the animations.
III. Some technical details
which might be useful for anyone who wants to understand how the models work,
or to create similar models.
Please consider all the animations as works in progress.
Anyone is welcome to use them freely for any non-commercial purpose. They are
particularly intended to be useful for teaching, independent study by students,
and perhaps contemplation of just how clever the ancient astronomers were.
Please with any suggestions for improvements, and especially if you notice
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