The longitude formula assumed in the last blog gives incorrect ecliptic longitudes and we need to be more careful about the Julian dates of the equinoxes, solstices and perigee. The times for the quadrant passages were correct since they involved 2nd differences.

Since the time of the autumnal equinox is known, the Julian dates of the vernal equinox and solstices can be computed. The epoch is chosen to be the beginning of the year 132 and is given for comparison.

The anomaly is measured from the time of perigee, t

_{p}, and this longitude is given by Ptolemy. Since the time of the autumnal equinox is also given we can determine the date and time of perigee.

We now have the correct longitude formula and can compute the true anomaly for a set of times of the year.

This data can then be used to fit an ellipse to Ptolemy's anomaly. As one can see there is a systematic error of about half a degree for the residuals of the linear formula involving the true anomaly and its rate of change but this is below the 1 minute of arc precision that Ptolemy appears to have been working at.

The values of σ and τ allow us to compute the Keplerian elements as before.

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