Editing Hipparchus (section)

==Lunar and solar theory==
[[File:HipparchusConstruction.svg|thumb|400px|Geometric construction used by Hipparchus in his determination of the distances to the Sun and Moon]]

===Motion of the Moon===
{{further|Lunar theory|Orbit of the Moon}}

Hipparchus also studied the motion of the [[Moon]] and confirmed the accurate values for two periods of its motion that Chaldean astronomers are widely presumed to have possessed before him. The traditional value (from Babylonian System B) for the mean [[synodic month]] is 29&nbsp;days; 31,50,8,20 (sexagesimal) = 29.5305941... days. Expressed as 29&nbsp;days + 12&nbsp;hours + {{sfrac|793|1080}}&nbsp;hours this value has been used later in the [[Hebrew calendar]]. The Chaldeans also knew that 251 [[synodic month]]s ≈ 269 [[anomalistic month]]s. Hipparchus used the multiple of this period by a factor of 17, because that interval is also an eclipse period, and is also close to an integer number of years (4,267 moons : 4,573 anomalistic periods : 4,630.53 nodal periods : 4,611.98 lunar orbits : 344.996 years : 344.982 solar orbits : 126,007.003 days : 126,351.985 rotations).{{efn|These figures use modern [[dynamical time]], not the solar time of Hipparchus's era. E.g., the true 4267-month interval was nearer 126,007 days plus a little over half an hour.}} What was so exceptional and useful about the cycle was that all 345-year-interval eclipse pairs occur slightly more than 126,007 days apart within a tight range of only approximately ±{{frac|1|2}} hour, guaranteeing (after division by 4,267) an estimate of the synodic month correct to one part in order of magnitude 10 million.

Hipparchus could confirm his computations by comparing eclipses from his own time (presumably 27 January 141&nbsp;BC and 26 November 139&nbsp;BC according to Toomer{{r|toomer1980}}) with eclipses from Babylonian records 345 years earlier (''Almagest'' IV.2{{sfn |Jones |2001}}). 

Later [[al-Biruni]] (''Qanun'' VII.2.II) and [[Nicolaus Copernicus|Copernicus]] (''de revolutionibus'' IV.4) noted that the period of 4,267 moons is approximately five minutes longer than the value for the eclipse period that Ptolemy attributes to Hipparchus. However, the timing methods of the Babylonians had an error of no fewer than eight minutes.{{r|sf1993}}{{r|ssm1997}} Modern scholars agree that Hipparchus rounded the eclipse period to the nearest hour, and used it to confirm the validity of the traditional values, rather than to try to derive an improved value from his own observations. From modern ephemerides{{r|ccf2002}} and taking account of the change in the length of the day (see [[ΔT (timekeeping)|ΔT]]) we
{{Who|date=September 2024}} estimate that the error in the assumed length of the synodic month was less than 0.2 second in the fourth century&nbsp;BC and less than 0.1 second in Hipparchus's time.

===Orbit of the Moon===
It had been known for a long time that the motion of the Moon is not uniform: its speed varies. This is called its ''anomaly'' and it repeats with its own period; the [[anomalistic month]]. The Chaldeans took account of this arithmetically, and used a table giving the daily motion of the Moon according to the date within a long period. However, the Greeks preferred to think in geometrical models of the sky. At the end of the third century BC, [[Apollonius of Perga]] had proposed two models for lunar and planetary motion:
# In the first, the Moon would move uniformly along a circle, but the Earth would be eccentric, i.e., at some distance of the center of the circle. So the apparent angular speed of the Moon (and its distance) would vary.
# The Moon would move uniformly (with some mean motion in anomaly) on a secondary circular orbit, called an ''epicycle'' that would move uniformly (with some mean motion in longitude) over the main circular orbit around the Earth, called ''deferent''; see [[deferent and epicycle]].

Apollonius demonstrated that these two models were in fact mathematically equivalent. However, all this was theory and had not been put to practice. Hipparchus is the first astronomer known to attempt to determine the relative proportions and actual sizes of these orbits. Hipparchus devised a geometrical method to find the parameters from three positions of the Moon at particular phases of its anomaly. In fact, he did this separately for the eccentric and the epicycle model. Ptolemy describes the details in the ''Almagest'' IV.11. Hipparchus used two sets of three lunar eclipse observations that he carefully selected to satisfy the requirements. The eccentric model he fitted to these eclipses from his Babylonian eclipse list: 22/23 December 383&nbsp;BC, 18/19 June 382&nbsp;BC, and 12/13 December 382&nbsp;BC. The epicycle model he fitted to lunar eclipse observations made in Alexandria at 22 September 201&nbsp;BC, 19 March 200&nbsp;BC, and 11 September 200&nbsp;BC.
* For the eccentric model, Hipparchus found for the ratio between the radius of the [[eccenter]] and the distance between the center of the eccenter and the center of the ecliptic (i.e., the observer on Earth): 3144 : {{frac|327|2|3}};
* and for the epicycle model, the ratio between the radius of the deferent and the epicycle: {{frac|3122|1|2}} : {{frac|247|1|2}} .

These figures are due to the cumbersome unit he used in his chord table and may partly be due to some sloppy rounding and calculation errors by Hipparchus, for which Ptolemy criticised him while also making rounding errors. A simpler alternate reconstruction{{sfn|Thurston|2002}} agrees with all four numbers. Hipparchus found inconsistent results; he later used the ratio of the epicycle model ({{frac|3122|1|2}} : {{frac|247|1|2}}), which is too small (60 : 4;45 sexagesimal). Ptolemy established a ratio of 60 : {{frac|5|1|4}}.{{r|toomer1968}} (The maximum angular deviation producible by this geometry is the arcsin of {{frac|5|1|4}} divided by 60, or approximately 5° 1', a figure that is sometimes therefore quoted as the equivalent of the Moon's [[equation of the center]] in the Hipparchan model.)

===Apparent motion of the Sun===
{{More citations needed section|date=July 2024}}
Before Hipparchus, [[Meton]], [[Euctemon]], and their pupils at [[Athens]] had made a solstice observation (i.e., timed the moment of the summer [[solstice]]) on 27 June 432&nbsp;BC ([[proleptic Julian calendar]]). [[Aristarchus of Samos]] is said to have done so in 280&nbsp;BC, and Hipparchus also had an observation by [[Archimedes]]. He observed the summer solstices in 146 and 135&nbsp;BC both accurately to a few hours, but observations of the moment of [[equinox]] were simpler, and he made twenty during his lifetime. Ptolemy gives an extensive discussion of Hipparchus's work on the length of the year in the ''Almagest'' III.1, and quotes many observations that Hipparchus made or used, spanning 162–128&nbsp;BC, including an equinox timing by Hipparchus (at 24 March 146&nbsp;BC at dawn) that differs by 5 hours from the observation made on [[Alexandria]]'s large public [[equatorial ring]] that same day (at 1 hour before noon). Ptolemy claims his solar observations were on a transit instrument set in the meridian.

At the end of his career, Hipparchus wrote a book entitled ''Peri eniausíou megéthous'' ("On the Length of the Year") regarding his results. The established value for the [[tropical year]], introduced by [[Callippus]] in or before 330&nbsp;BC was {{frac|365|1|4}} days.{{r|leverington2003}} Speculating a Babylonian origin for the Callippic year is difficult to defend, since Babylon did not observe solstices thus the only extant System B year length was based on Greek solstices (see below). Hipparchus's equinox observations gave varying results, but he points out (quoted in ''Almagest'' III.1(H195)) that the observation errors by him and his predecessors may have been as large as {{frac|1|4}} day. He used old solstice observations and determined a difference of approximately one day in approximately 300 years. So he set the length of the tropical year to {{frac|365|1|4}} − {{frac|1|300}} days (= 365.24666... days = 365&nbsp;days 5&nbsp;hours 55&nbsp;min, which differs from the modern estimate of the value (including earth spin acceleration), in his time of approximately 365.2425 days, an error of approximately 6&nbsp;min per year, an hour per decade, and ten hours per century.

Between the solstice observation of Meton and his own, there were 297 years spanning 108,478 days; this implies a tropical year of 365.24579... days = 365&nbsp;days;14,44,51 (sexagesimal; = 365&nbsp;days + {{sfrac|14|60}} + {{sfrac|44|60{{sup|2}}}} + {{sfrac|51|60{{sup|3}}}}), a year length found on one of the few Babylonian clay tablets which explicitly specifies the System B month. Whether Babylonians knew of Hipparchus's work or the other way around is debatable.

Hipparchus also gave the value for the [[sidereal year]] to be 365 + {{sfrac|1|4}} + {{sfrac|1|144}} days (= 365.25694... days = 365&nbsp;days 6&nbsp;hours 10&nbsp;min). Another value for the sidereal year that is attributed to Hipparchus (by the physician [[Galen]] in the second century AD) is 365 + {{sfrac|1|4}} + {{sfrac|1|288}} days (= 365.25347... days = 365&nbsp;days 6&nbsp;hours 5&nbsp;min), but this may be a corruption of another value attributed to a Babylonian source: 365 + {{sfrac|1|4}} + {{sfrac|1|144}} days (= 365.25694... days = 365&nbsp;days 6&nbsp;hours 10&nbsp;min). It is not clear whether Hipparchus got the value from Babylonian astronomers or calculated by himself.{{sfn|Neugebauer|1975|loc=Vol. 1, pp. 293, 294}}

===Orbit of the Sun===
Before Hipparchus, astronomers knew that the lengths of the [[season]]s are not equal. Hipparchus made observations of equinox and solstice, and according to Ptolemy (''Almagest'' III.4) determined that spring (from spring equinox to summer solstice) lasted 94{{frac|1|2}} days, and summer (from summer solstice to autumn equinox) {{frac|92|1|2}} days. This is inconsistent with a premise of the Sun moving around the Earth in a circle at uniform speed. Hipparchus's solution was to place the Earth not at the center of the Sun's motion, but at some distance from the center. This model described the apparent motion of the Sun fairly well. It is known today that the [[planet]]s, including the Earth, move in approximate [[ellipse]]s around the Sun, but this was not discovered until [[Johannes Kepler]] published his first two laws of planetary motion in 1609. The value for the [[eccentricity (orbit)|eccentricity]] attributed to Hipparchus by Ptolemy is that the offset is {{frac|1|24}} of the radius of the orbit (which is a little too large), and the direction of the [[apogee]] would be at longitude 65.5° from the [[March equinox|vernal equinox]]. Hipparchus may also have used other sets of observations, which would lead to different values. One of his two eclipse trios' solar longitudes are consistent with his having initially adopted inaccurate lengths for spring and summer of {{frac|95|3|4}} and {{frac|91|1|4}} days.{{sfn |Thurston |2002 |loc=p. 67, note 16}}{{failed verification|date=September 2023}} His other triplet of solar positions is consistent with {{frac|94|1|4}} and {{frac|92|1|2}} days,{{sfn |Jones |2001}}{{sfn |Thurston |2002 |loc=note 14}}{{failed verification|date=September 2023}} an improvement on the results ({{frac|94|1|2}} and {{frac|92|1|2}} days) attributed to Hipparchus by Ptolemy. Ptolemy made no change three centuries later, and expressed lengths for the autumn and winter seasons which were already implicit (as shown, e.g., by A. [[Aaboe]]).{{citation needed|date=April 2023}}

===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}}

===Eclipses===
[[Pliny the Elder|Pliny]] (''Naturalis Historia'' II.X) tells us that Hipparchus demonstrated that lunar eclipses can occur five months apart, and solar eclipses seven months (instead of the usual six months); and the Sun can be hidden twice in thirty days, but as seen by different nations. Ptolemy discussed this a century later at length in ''Almagest'' VI.6. The geometry, and the limits of the positions of Sun and Moon when a solar or lunar eclipse is possible, are explained in ''Almagest'' VI.5. Hipparchus apparently made similar calculations. The result that two solar eclipses can occur one month apart is important, because this can not be based on observations: one is visible on the northern and the other on the southern hemisphere—as Pliny indicates—and the latter was inaccessible to the Greek.

Prediction of a solar eclipse, i.e., exactly when and where it will be visible, requires a solid lunar theory and proper treatment of the lunar parallax. Hipparchus must have been the first to be able to do this. A rigorous treatment requires [[spherical trigonometry]], thus those who remain certain that Hipparchus lacked it must speculate that he may have made do with planar approximations. He may have discussed these things in ''Perí tēs katá plátos mēniaías tēs selēnēs kinēseōs'' ("On the monthly motion of the Moon in latitude"), a work mentioned in the ''[[Suda]]''.

Pliny also remarks that "he also discovered for what exact reason, although the shadow causing the eclipse must from sunrise onward be below the earth, it happened once in the past that the Moon was eclipsed in the west while both luminaries were visible above the earth" (translation H. Rackham (1938), [[Loeb Classical Library]] 330 p.&nbsp;207). Toomer argued that this must refer to the large total lunar eclipse of 26 November 139&nbsp;BC, when over a clean sea horizon as seen from Rhodes, the Moon was eclipsed in the northwest just after the Sun rose in the southeast.{{r|toomer1980}} This would be the second eclipse of the 345-year interval that Hipparchus used to verify the traditional Babylonian periods: this puts a late date to the development of Hipparchus's lunar theory. We do not know what "exact reason" Hipparchus found for seeing the Moon eclipsed while apparently it was not in exact [[astronomical opposition|opposition]] to the Sun. Parallax lowers the altitude of the luminaries; refraction raises them, and from a high point of view the horizon is lowered.