Sunday, February 28, 2016
Comparison of Ptolemy's Observations With Those of Horizons
One can compare Ptolemy's observations of the solsticial and equinoctial points with data from JPL's HORIZONS online ephemeris. To avoid comparing apples and oranges we need to use methods similar to those of Ptolemy. One has to set HORIZONS to compute azimuth and elevation using the standard atmosphere to get the apparent positions of the Sun and then determine the time in Alexandria when the Sun crosses the local meridian on days near equinoxes and solstices. One can then use linear fits to determine the local time of an equinox or solstice. The computed times are sensitive to the precision used to measure angles and an error of three minutes of arc can shift the time for an event by a couple of hours.
Although the time of the autumnal equinox for 132 AD differs slightly from stated time of Sep 25, 2pm on p. 168 of Toomer, Ptolemy's Almagest, the results compare fairly well with the intervals for the passage through a quadrant considering the accuracy with which Ptolemy worked.
The difference between the altitudes of the midday Sun at the solstices can be used to estimate the obliquity of the ecliptic which appears to be about 23°40'.
Ptolemy gives a value for the difference in the Sun's altitude at the solstices of 11/83rds of a complete circle which corresponds to the value shown above. It's difficult to determine exact times since the Sun altitude is relatively stationary at the solstices and one needs to estimate the altitude of the equator to get the times for the equinoxes. The errors for the intervals are twice those of the estimates of the time of an individual event. Ptolemy used Hipparchus' times of passage for the quadrants and method to deduce the anomaly. They may not have been correct for the year 132 AD.