On Sizes and Distances - Conclusion

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 procedure, if I have constructed it correctly, is very remarkable... What is astonishing is the sophistication of approaching the problem by two quite different methods, and also the complete honesty with which Hipparchus reveals his discrepant results... which are nevertheless of the same order of magnitude and (for the first time in the history of astronomy) in the right region.