The last fit for the Sun's declination wasn't a fair test for uniform circular motion. One can get a better fit to declination that results from using a Keplerian eliptical orbit for the Earth. To fit a sine function to the declination of Kepler's orbit one can choose it be symmetric about the Summer Solstice and let the fit parameters be a constant term and the coefficient of a cosine term. If one assumes the average error is zero then this gives one linear equation with two unknowns which can be solved for the constant term, ε_0. One can then look at the maximum of the absolute value of the errors for each value for the magnitude of the declination, ε_1, and plot this as a function of ε_1. This function has a clear minimum which is easily determined. The solution for the two unknowns is given above. This is a much better fit than before with the maximum error being less than half a degree. The green solid line is the declination using Kepler's method of solution and the dotted yellow line might be called that of Eudoxus. The presence of a constant term might add a little error to the determination of the equator and as a result add an error to the determination of one's latitude. Actual measurements would add additional errors which could also skew the results. One would have to accurately measure the declination to something on the order of a minute of arc to get a good measure of the errors in uniform circular motion. Ptolemy may have been the first to do this. Tycho Brahe's observations at the end of the 16th Century were intended to be accurate to 1 minute of arc but may have had errors of 3 minutes in some cases but they allowed Kepler to deduce his laws of motion. The telescope greatly improved the accuracy of measurements.
Edit (30 Mar 2011): A slightly better fit was found by doing a 2D search which gave the values 173.289, 0.379° & 23.360° and a maximum error of 0.468°.