Effect of Refraction on the Inclination of the Ecliptic

If our goal is to determine the Inclination of the Ecliptic the effect of refraction is not as bad a one might initially believe. One needs to measure the altitude of the Sun on both solstices and each measurement is subject to some refraction. That of the Winter Solstice would have a little more since it is lower in the sky but we take half the difference of the two altitudes to get the Inclination. Since the refractions will be of the same sign they will partially cancel out and only the difference in the refractions contributes to the error in the determination of the Inclination. The effect is more significant at higher latitudes. Ptolemy seems to be aware that a shift in altitude is greater near the horizon than near the zenith (see Toomer p. 421) and shows concern about it in his Optics.

Using 31°, the latitude of Alexandria, in Bennett's formula for the refraction gives a correction of 0° 3' 13.6" if the environmental conditions are unchanged between the Summer and Winter Solstices and along with the correction for vertical deflection it would reduce Ptolemy's Inclination of the Ecliptic to 23° 37' 1.4". Making corrections for changes in altitude from sea level should have a negligible effect on the corrections to the Inclination since they would cancel out on subtraction of the two solstice measurements.