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Ancient Theories of the Sun:
1. Eccentric Model Applet
3. Eccentric and
Equant Model Applet

Select from
the Details menu. 
Select the time interval.  
The angular
position of the apogee of the Sun is slowly moving
with time (about 1.71° per cencury, this is not the
precession of the equinoxes). 
There are
two mathematically
equivalent models of ancient Greek
astronomy explaining the unequal motion of the Cun:


Apogee A P Perigee 
Apogee A P Perigee 
The
Sun moving on the epicycle
with center D rotating on the
deferent circle (center C) at the
same angular speed. e = distance from D to Sun rMin=re, rMax=r+e eccentricity=(rMaxrMin)/(rMax+rMin)=e/r 
The
Sun moving on the circle (radius r) centered at C seen
from the eccentric
point E. e = CE rMin=EP, rMax=EA eccentricity=(rMaxrMin)/(rMax+rMin)=e/r 
Lenghts
of the seasons:

Ptolemy (90 AD168 AD) 
100 BC 
Meeus 0 
Meeus 2000 


vernal
equinox  summer solstice 


93.96 d  92.76 d 

summer
solstice  autumnal equinox 


92.45 d 
93.65 d 

autumnal
equinox  winter solstice 


88.69 d 
89.84 d 
Winter  winter
solstice  vernal equinox 
90 1/8 d  90.07 d  90.13 d 
88.99 d 
Sum  365
1/4 d 
365.24 d  365.23 d  365.24 d 
The
rounded
number
of days in zodiac signs agree with those of Geminus:
Ari  Tau  Gem  Can  Leo 
Vir  Lib  Sco  Sag  Cap  Aqu  Pis  
days  31  32  32  31  31  30  30  29  29  29  30  31  365 
days  95  92  88  90  365 
The equation of the center
(EoC) is the difference between the actual position of the Sun and the
position it would have if its angular motion were
uniform. maxEoC = arcsin(e)
e=1/24.0 maxEoC = 2° 23.3'
e=1/24.1 maxEoC = 2° 22.7' e=1/24.04 maxEoC = 2° 23.0' Ptolemy: In the appendix 2 "Calculation of the EccentricQuotient for the Sun" of Thurston's book, e is computed to be 143/3438 = 24.04, using the lengths of the seasons and 365 d ^{14}/_{60} h ^{48}/_{3600} min = 365.2467 d for the length of the tropical year given by Ptolemy. 

Hipparchus:
Orbit of the Sun (Wikipedia) Gemini Elementa Astronomiae, editit C.
Manitius (PDF, Greek/German) Des Claudius Ptolemäus Handbuch
der Astronomie (Übers. Karl Manitius) 
Books 
James Evans: The History and Practice of
Ancient Astronomy, Oxford University Press, 1998, Chapter Five: Solar Theory. Hugh Thurston: Early Astronomy, Springer, Berlin, New York 1994. Jean Meeus: Astronomical Tables of the Sun, Moon and Planets. 2nd ed., WillmannBell, Richmond 1995. Gemini Elementa Astronomiae, editit C. Manitius (Greek/German), Teubner, Stuttgart 1974. 
Updated:
2012, Jul 15