Hipparchus determined the chord for small angles (see Chord (geometry)). His values for the latitudes of the Hellespont (41 degrees) and Alexandria (31 degrees) are known from Strabo's work on Geography. To determine the declination, it is necessary to know which eclipse Hipparchus used. Because he knew the value which Hipparchus eventually gave for the distance to the Moon (71 Earth radii) and the rough region of the eclipse, Toomer was able to determine that Hipparchus used the eclipse of March 14, 190 BC. This eclipse fits all the mathematical parameters very well, and also makes sense from a historical point of view. The eclipse was total in Nicaea, Hipparchus' birthplace, so he may have heard stories of it. There is also an account of it in Strabo's Ab Urbe Condita VIII.2. The declination of the Moon at this time was . Hence, using chord trigonometry, gives Now using Hipparchus' chord tables, and hence This agrees well with the value of 71 Earth radii that Pappus reports. This analysis assumed that the eclipse took place in the middle of the day, with the Sun and Moon on the meridian. But this was not the case for the eclipse of 190 BC, which took place in the early morning of February 28th. Conclusion Assuming that these reconstructions accurately reflect what Hipparchus wrote in On Sizes and Distances, then this work was a remarkable accomplishment. This approach of setting limits on an unknown physical quantity was not new to Hipparchus (see Aristarchus of Samos. Archimedes also did the same with pi), but in those cases, the bounds reflected the inability to determine a mathematical constant to an arbitrary precision, not uncertainty in physical observations. Hipparchus appears to have eventually resolved the contradiction between his two results. His aim in calculating the distance to the moon was to obtain an accurate value for the lunar parallax, so that he might predict eclipses with more precision. To this, he had to settle on a particular value for the distance/parallax, not a range of values. There is some evidence that he did this. Combining the calculations of Book 2 and the account of Theon of Smyrna yields a lunar distance of 60.5 Earth radii. Doing the same with the account of Cleomedes yields a distance of 61 Earth radii. These are remarkably close to both Ptolemy's value and the modern one. According to Toomer, This