Keith's Mars Applet using
Aux, Equant, Deferent and Epicycle

When the applet is first seen, the yellow circle showing the position of the mean Sun, the epicycle and the red circle showing the True Position of Mars will be rotating.
Clicking anywhere in the applet window (except on a button, of course) will stop the action.
Clicking on one of the buttons will change the action to be faster or slower, backwards (clockwise) or forwards (anticlockwise). You can stop it at any date after 1500 A.D..

As the date changes from October to December 1501, for example, the retrograde motion of Mars can be seen. At the bottom-right, the 1st and 2nd stationary points are indicated. You must have a slow positive motion for this to work reasonably well. There is another example starting in November 1503.

Have fun.

The Mars applet shows the following:
A scale of the Zodiac has a point marked 'Earth' in the centre.
A line from 'Earth' to the Zodiac has a yellow circle at the end. This circle represents the Sun. It roughly shows the position of the MEAN Sun along the Zodiac on the date shown at the top-centre of the window. (The mean Sun indicates the position of the Sun assuming it moves through the same angle every day.)
A line from 'Earth' to the Zodiac has a small green circle at the end representing Mars. It roughly shows the mean position of Mars around the Zodiac on the date shown at the top-centre of the window.

A line marked 'Aux' passes over the Zodiac at an angle about which Mars exhibits symmetry. This line is commonly drawn right across the circle of the zodiac and is known as the line of apsides. The line is roughly shown in the position it had in the year of 1500 A.D..
A little way along this line from the Earth point is a point marked 'centre of deferent'.
At the same distance further along the line is a point marked 'equant'.
There is a circle marked 'Deferent' which is drawn centred on the point marked 'centre of deferent'.

A red line from the 'Earth' point is drawn to the point marked 'equant'.
Another red line is drawn from the point marked 'equant' to the deferent circle which is parallel to the line with the small green circle.
The point where this line touches the deferent circle is at the centre of the epicycle circle.
The epicycle circle is graduated in signa values (angles of 30degrees) from the starting point to 12.
A red radius line from the centre of the epicycle circle to the circle is drawn parallel to the line from the Earth point to the yellow circle showing the position of the mean Sun. The angle around the epicycle is known as the mean argument of Mars.
A line from the 'Earth' point has a red circle at the end. It passes through the point at the end of the epicycle radius line. The line with the red circle marks Ptolemy's 'true position' of Mars on the scale of the Zodiac.

The above applet demonstrates the method used in the middle ages to predict the position of Mars for any date. Mars was chosen for the applet because the set of values necessary for Mars best illustrates the underlying technique.

In about 140 A.D. Claudius Ptolemy integrated the aux, equant, deferent and epicycle ideas devised by his predecessors into a single technique. This allowed him to predict the position around the Zodiac of four of the planets more accurately than had been accomplished by any of his predecessors.

He used exactly the same principles to account for the retrograde motions of Mars, Saturn, Jupiter and Venus but for each of these he used different values for:

the aux,
the ratios of the equant distance and epicycle length to the deferent radius,
the starting point at some epoch,
the average rate of movement (mean motion).

His technique was used by astronomers until (and well beyond) the date when Copernicus published his revolution--ary theories in 1542, about 1400 years after Ptolemy wrote his Almagest, which described this technique.

Purpose

I achieved my objective, and hope that after conscientiously studying the applet, others will benefit in the same way.

In late 2010 I was studying Schöner's Aequatorium Astronomicum (1521) and wanted to understand precisely the way his tables were used to set his equatoria. (The examples in his Opera (1551) are not clear on this point.) In particular I wanted to know precisely how the mean position of the Sun was incorporated within the minimal arithmetic required to set the equatoria. (Schöner only provides tables of the mean motion of each planet, and includes no tables for each planet which give the mean argument.) Although Evans (1998) pointed me in the right direction, it was my applet which clarified the method for me.

I was intrigued by Schöner's technique of using parallel threads to set the position of the offset mean longitude [Evans 1998 p405] which eliminated the need for a scale around the equant point, A similar procedure might have eliminated the need for the calculation of the mean argument. Presumably Schöner decided to use a very simple arithmetical calculation to derive the mean argument because of its improved accuracy [Evans 1998 p406].

I also tried to relate the position of the stationary points to the lines from the Earth point to the line on the epicycle circle, as shown by Pedersen (1974) on, for instance, p185 and p313.

Accuracy

Note that the values of the retrograde arc at the bottom-right assume that the daily rotation is set to plus 1 day and the full retrograde arc is witnessed from the first to the last stationary points. Clicking on the 0day button doesn't upset the count, if you want to write down the appropriate data. I recognise that the angles for the retrograde motion when occurring on either side of the First Point of Aries, are incorrect.

However, I achieved my objective. The applet gave me a real feel for how Ptolemy's system worked in practice. I think I started it on the 15th November 2010 and completed it at issue 0.5 on 24th November 2010. Issue 0.5 displayed the angles of the first and last retrograde points, and the retrograde arc. I worked on it again at the beginning of March 2011, producing issue 1.0

References:

Evans, James (1998). History and Practice of Ancient Astronomy'.
Note: the tables on page 368, the prime tables in this wonderful book, are for an epoch at the beginning of the year 1900.

North, John (1976). Richard of Wallingford in 3 volumes.

North, John (2005). God's Clockmaker.
Warning: John North's books are written in such an interesting style that you can read onwards for many pages following the page you had found using the index before it dawns on you that you don't have time for such enjoyment.

Gingerich, Owen (1971). Apianus's Astronomicum Caesareum and its Leipzig Facsimile. Published in the Journal for the History of Astronomy ii (1971), pages 168-177.
Although primarily intended to advise people who had bought the facsimile that the equatoria in them were incorrectly assembled, it includes basic instructions explaining how to set the (repaired) equatoria for Mars and the Moon.

Pedersen, Olaf (1974). A Survey of the Almagest. Edidit Bibliotheca Universitatis Hauniensis Vol.30. Odense University Press, Denmark. ISBN 87 7492 087 1.

Ptolemy, Claudius (approx.140 A.D.). Almagest.
Translation: Toomer, G.J. (1984). Ptolemy's Almagest. Duckworth, London. ISBN 0 7156 1588 2.

Schöner, Johanne (1521/1534). Aequatorium Astronomicum.
The copy in the British Library, shelfmark Maps C.1.d.10.(2.) was perfectly adequate for my studies although the instrument for the Sun is incorrectly assembled and the one for the Moon is incorrectly pivotted.

Apian, Peter (1540). Astronomicum Caesareum. Examples for his Mars longitude instrument are given on D2v and D3v, the instrument being on D3. Although Apian introduced extra discs which eliminated the arithmetic, his basic technique is the same as had been used since the time of Ptolemy (trepidation excepted).