Geodetic vs Geocentric Latitudes

The latitude used for maps and GPS is also known as the geodetic latitude. It is very close in value to the geocentric latitude and its deviation from that value is the vertical deflection. Ptolemy was not familiar with the ellipsoid shape of the Earth's surface and the latitudes he uses reflect this. In his Geography Ptolemy lists the latitude of Alexandria as being 31°. If we add the vertical deflection to this we get the modern geodetic latitude for Alexandria is about 31° 11'. In the Greek text of Ptolemy's Geography one finds common fractions used to represent angles less than a degree. The smallest fraction used is 1/12 of a degree which is 5 minutes of arc. Round off errors would again be about 2' 30". Since Ptolemy's Inclination was off by the vertical deflection his latitudes are probably geocentric latitudes.

I found a formula for calculating the Obliquity of the Ecliptic, the modern term used for the Inclination of the Ecliptic, in the Explanatory Supplement to the Astronomical Almansc (p. 171). The Almagest was probably written between 150 AD and 168 AD, the time if Ptolemy's death. Using the beginning of 159 AD as a nominal date the formula gives an inclination of 23° 40' 32.5" which is close to Ptolemy's value corrected for the vertical deflection.

If Ptolemy's angular scale was only accurate to 1/12th of a degree, how could he get a value for of the Inclination of the Ecliptic accurate to 1/3 of a minute of arc? What may have happened is that he measured the difference between the values of the Sun's declination at the Summer and Winter Solstices over a number of years and averaged the results which would give an improved estimate. If his scale indicated twelveths of a degree it would be possible for him to estimate an angle to an accuracy of 1/24th of a degree. This would mean that his measurements were then accurate to at most 2 minutes of arc. While his estimates of the inclination may have approached an accuracy of one minute of arc, his practice seems to have been to do calculations to the second sexagesimal place in order to minimize round off errors.