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## Keith's Moon Applet using Apogee, Deferent and Epicycle

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The above applet is intended to show how a technique introduced by Claudius Ptolemy in the second century was used to predict the location of the Moon along the Zodiac at any time and date. Because the method is so complicated, a newcomer to Ptolemy's methods may find it difficult to understand.

(Fortunately, Ptolemy's system for Mars is much simpler to comprehend. His method for the Mercury is different again and also rather complicated.)

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 the Moon
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..

I principally use the buttons marked 1day and -1day per second to move the position of the Moon to an interesting position, clicking in the window to stop it. I use the button indicating 1 year per second when I want to advance the date by several years.

Have fun.

The Moon 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 almost at the end.
This circle shows the MEAN position of the Moon
at the date shown at the top of the window.
(The mean position of the Moon indicates its position
assuming it moves through the same angle every day.)

A line whose end is marked 'DA'
passes through 'Earth' to the Zodiac, and
is extended a small amount "backwards"
away from the centre.

'DA' stands for the Apogee of the Deferent.
The angle between the mean Sun and the mean Moon is
the same angle as that between
the Apogee of the Deferent and the mean Sun.
In other words, the line marked DA is at
an angle so that it forms
a mirror image of the line to the mean Moon
(marked with a small green circle)
about the line to the mean Sun,
The mean Sun and mean Moon move anticlockwise.
Consequently, the Apogee of the Deferent moves clockwise.

Around the 'Earth' point is a small circle. At one point where this circle crosses the 'DA' line it is marked 'DC' standing for Deferent Centre and at the other point it is marked 'X' and is at an equal distance from the Earth as the Deferent Centre.

The point marked 'DC' is at the centre of a large circle marked 'Deferent'.

The point where the mean position of the Moon
crosses the deferent circle
is at the centre of the epicycle circle.
The epicycle circle is surrounded by
the epicycle signa scale.

A line from the point marked 'X' on the line marked 'DA'
passes through the epicycle centre and
is extended to touch the far side of the
epicycle circle scale.
This is the zero point of the epicycle scale
and the scale is marked clockwise from this point
with 12 Signa (each of 30°).

The point where the line from the epicycle centre crosses the origin of the epicycle circle is referred to as the mean apogee of the epicycle. The point close by where the line from the Earth point through the epicycle centre crosses the epicycle circle is known as the true apogee. (See B&T p389)

A radius line to the epicycle scale is drawn to the angle of the mean argument. Where this line crosses the inner circle, of the epicycle circle is a large red dot. A line from the 'Earth' point through this large dot to the Zodiac scale has a large red circle at the end, indicating the true position of the Moon.

### Purpose

I wrote the applet mainly because I wanted to understand how Ptolemy's method adjusted the position of the Moon at the octants. When studying it, I also learned quite a lot about the relative rates of movement of the true Moon.

Unfortunately, the resolution constraints of Java applets limit the accuracy of the presentation. Accuracy is also limited by my not allowing for leap days. However, accuracy wasn't important to me when I wrote the program (Christmas, 2010). The accuracy and the clarity could be increased by creating a larger diameter applet, but that isn't feasible for an applet intended to be displayed on the small screen of a laptop.

I suspect that the applet will mainly be of interest to people interested in roughly verifying the Moon's positions provided in books published during the 1500's.

### References:

Apian, Peter (1540).
*Astronomicum Caesareum*. F1v/F2/F2v

Regiomontanus, Joannes (Various calendars and ephemerides, 1474 onwards. Those with predictions of eclipses are very interesting.)

Schoner, (1521). *Aequatorium Astronomicum*. A4v/A5.

Schoner (1551). *Opera Mathematica*.
Section 13, *Aequatorium Astronomicum.* 20v/21.

- - -

Bengamin, F.S. and Toomer G.J. (1971).
*Campanus of Novara and Medieval Planetary Theory*. p45 and p389.

Evans, James (1998).
*History and Practice of Ancient Astronomy.*
Sadly, the book doesn't cover the Moon directly.

Gingerich (1971).
*Apianus's Astronomicum Caesareum
and its Leipzig Facsimile*.
Journal for the History of Astronomy (1971) II, 174/175/176.

Linton, (2004). *Eudoxus to Einstein*, 72-74.

Neugebauer, O. (1952). *The Exact Sciences in Antiquity*. p196 (and 197/198).

North, John (1976). *Richard of Wallingford*
III Appendix 29, 171/172/173.

Pedersen, Olaf (1974).
*A Survey of the Almagest*. p193 and p316.

You can email me at (Sorry, you can't click on this.)