Editing Hipparchus (section)

===Distance, parallax, size of the Moon and the Sun===
{{main|On Sizes and Distances (Hipparchus)}}
[[File:HipparchusEclipse.png|thumb|400px|Diagram used in reconstructing one of Hipparchus's methods of determining the distance to the Moon. This represents the Earth–Moon system during a partial solar eclipse at A ([[Alexandria]]) and a total solar eclipse at H ([[Hellespont]]).]]
Hipparchus also undertook to find the distances and sizes of the Sun and the Moon, in the now-lost work ''On Sizes and Distances'' ({{langx|grc|Περὶ μεγεθῶν καὶ ἀποστημάτων}} {{transliteration|grc|Peri megethon kai apostematon}}). His work is mentioned in Ptolemy's ''Almagest'' V.11, and in a commentary thereon by [[Pappus of Alexandria|Pappus]]; [[Theon of Smyrna]] (2nd century) also mentions the work, under the title ''On Sizes and Distances of the Sun and Moon''.

Hipparchus measured the apparent diameters of the Sun and Moon with his ''[[alidade|diopter]]''. Like others before and after him, he found that the Moon's size varies as it moves on its (eccentric) orbit, but he found no perceptible variation in the apparent diameter of the Sun. He found that at the ''[[mean]]'' distance of the Moon, the Sun and Moon had the same apparent diameter; at that distance, the Moon's diameter fits 650 times into the circle, i.e., the mean apparent diameters are {{frac|360|650}} = 0°33′14″.

Like others before and after him, he also noticed that the Moon has a noticeable [[Lunar parallax|parallax]], i.e., that it appears displaced from its calculated position (compared to the Sun or [[star]]s), and the difference is greater when closer to the horizon. He knew that this is because in the then-current models the Moon circles the center of the Earth, but the observer is at the surface—the Moon, Earth and observer form a triangle with a sharp angle that changes all the time. From the size of this parallax, the distance of the Moon as measured in Earth [[radius|radii]] can be determined. For the Sun however, there was no observable parallax (we now know that it is about 8.8", several times smaller than the resolution of the unaided eye).

In the first book, Hipparchus assumes that the parallax of the Sun is 0, as if it is at infinite distance. He then analyzed a solar eclipse, which Toomer presumes to be the eclipse of 14 March 190&nbsp;BC.<ref>{{cite web |url=http://eclipse.gsfc.nasa.gov/SEcat5/SE-0199--0100.html |title=Five Millennium Catalog of Solar Eclipses |access-date=11 August 2009 |url-status=live |archive-url=https://web.archive.org/web/20150425070114/http://eclipse.gsfc.nasa.gov/SEcat5/SE-0199--0100.html |archive-date=25 April 2015  }}, #04310, Fred Espenak, NASA/GSFC</ref> It was total in the region of the [[Hellespont]] (and in his birthplace, Nicaea); at the time Toomer proposes the Romans were preparing for war with [[Antiochus III]] in the area, and the eclipse is mentioned by [[Livy]] in his ''[[Ab Urbe Condita Libri]]'' VIII.2. It was also observed in Alexandria, where the Sun was reported to be obscured 4/5ths by the Moon. Alexandria and Nicaea are on the same meridian. Alexandria is at about 31° North, and the region of the Hellespont about 40° North. (It has been contended that authors like Strabo and Ptolemy had fairly decent values for these geographical positions, so Hipparchus must have known them too. However, Strabo's Hipparchus dependent latitudes for this region are at least 1° too high, and Ptolemy appears to copy them, placing Byzantium 2° high in latitude.) Hipparchus could draw a triangle formed by the two places and the Moon, and from simple geometry was able to establish a distance of the Moon, expressed in Earth radii. Because the eclipse occurred in the morning, the Moon was not in the [[meridian (astronomy)|meridian]], and it has been proposed that as a consequence the distance found by Hipparchus was a lower limit. In any case, according to Pappus, Hipparchus found that the least distance is 71 (from this eclipse), and the greatest 83 Earth radii.

In the second book, Hipparchus starts from the opposite extreme assumption: he assigns a (minimum) distance to the Sun of 490 Earth radii. This would correspond to a parallax of 7′, which is apparently the greatest parallax that Hipparchus thought would not be noticed (for comparison: the typical resolution of the human eye is about 2′; [[Tycho Brahe]] made naked eye observation with an accuracy down to 1′). In this case, the shadow of the Earth is a [[conical surface|cone]] rather than a [[cylinder (geometry)|cylinder]] as under the first assumption. Hipparchus observed (at lunar eclipses) that at the mean distance of the Moon, the diameter of the shadow cone is {{frac|2|1|2}} lunar diameters. That apparent diameter is, as he had observed, {{frac|360|650}} degrees. With these values and simple geometry, Hipparchus could determine the mean distance; because it was computed for a minimum distance of the Sun, it is the maximum mean distance possible for the Moon. With his value for the eccentricity of the orbit, he could compute the least and greatest distances of the Moon too. According to Pappus, he found a least distance of 62, a mean of {{frac|67|1|3}}, and consequently a greatest distance of {{frac|72|2|3}} Earth radii. With this method, as the parallax of the Sun decreases (i.e., its distance increases), the minimum limit for the mean distance is 59 Earth radii—exactly the mean distance that Ptolemy later derived.

Hipparchus thus had the problematic result that his minimum distance (from book 1) was greater than his maximum mean distance (from book 2). He was intellectually honest about this discrepancy, and probably realized that especially the first method is very sensitive to the accuracy of the observations and parameters. (In fact, modern calculations show that the size of the 14.03.190&nbsp;BC solar eclipse at Alexandria must have been closer to {{frac|9|10}}ths and not the reported {{frac|4|5}}ths, a fraction more closely matched by the degree of totality at Alexandria of eclipses occurring on 15.08.310 and 20.11.129&nbsp;BC which were also nearly total in the Hellespont and are thought by many to be more likely possibilities for the eclipse Hipparchus used for his computations.)

Ptolemy later measured the lunar parallax directly (''Almagest'' V.13), and used the second method of Hipparchus with lunar eclipses to compute the distance of the Sun (''Almagest'' V.15). He criticizes Hipparchus for making contradictory assumptions, and obtaining conflicting results (''Almagest'' V.11): but apparently he failed to understand Hipparchus's strategy to establish limits consistent with the observations, rather than a single value for the distance. His results were the best so far: the actual mean distance of the Moon is 60.3 Earth radii, within his limits from Hipparchus's second book.

[[Theon of Smyrna]] wrote that according to Hipparchus, the Sun is 1,880 times the size of the Earth, and the Earth twenty-seven times the size of the Moon; apparently this refers to [[volume]]s, not [[diameter]]s. From the geometry of book 2 it follows that the Sun is at 2,550 Earth radii, and the mean distance of the Moon is {{frac|60|1|2}} radii. Similarly, [[Cleomedes]] quotes Hipparchus for the sizes of the Sun and Earth as 1050:1; this leads to a mean lunar distance of 61 radii. Apparently Hipparchus later refined his computations, and derived accurate single values that he could use for predictions of solar eclipses.

See Toomer (1974) for a more detailed discussion.{{r|toomer1974-sunmoon}}