Improvement on the Observations of the Sun's Inclination

We know that Ptolemy used averaging to improve the values for the mean motion of the Sun. Making observations of the altitude of the Sun at local noon each day would allow one to directly compare the observations since they are made under similar conditions. The times and position on the horizon of sunrise and sunset vary throughout the year and observations near the horizon also are subject to more atmospheric diffraction. Could Ptolemy have used some sort of averaging to improve on the position of the Sun at the Solstice? When the altitude of the Sun is plotted against the day of year one gets a sinusoidal curve. One needs to know the declination vs right ascension in order to determine the plane of the Sun's apparent motion and determining the right ascension is complicated by the Sun's nonuniform motion along the Ecliptic. Plolemy did determined the Sun's anomaly which measures its deviation from uniform circular motion along the Ecliptic. So he knew both the Sun's altitude and its right ascension.

Plotting these positions on a sphere would show that observations are coplanar or nearly so. Using spherical trigonometry one could estimate the Pole of the Ecliptic from two points on it. How would one improve on the estimate? One could plot the distance of the observations from the equator of the estimated pole and see how linear the fit is using Menelaus' theorem. One knows the right ascension where the Earth's Equator and the Ecliptic cross each other since these points are at the Equinoxes. So only the Inclination of the Ecliptic is unknown.

One would then try to find an Inclination for which the deviation of the Equator of the corresponding Pole would best fit Menelaus' theorem for combinations of three points. Averaging over a number of three point fits would give a better result for the average deviation from the theorem.

One might try a curve fit of this sort to improve on an estimate for the Pole of the Ecliptic. Finding the Inclination for which the sum of the magnitudes of the deviations from an assumed equator is a minimum might also work. Hero of Alexandria showed that the path of light reflecting from a mirror was a minimum. With the right ascensions of the Equinoxes and those for the times of the observations known projecting nearby altitudes of the Sun onto the right ascension of the Solstice and averaging might provide the simplest method for determining the the altitude of the Sun at time of the Solstice. It's easy to do and only involves a little spherical trigonometry. In effect one would be measuring the local zenith or horizon relative to the Ecliptic on a Celestial Sphere with the Ecliptic as the Equator.