Hipparchic cycle
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The Greek astronomer
Hipparchus Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equi ...
introduced two cycles that have been named after him in later literature.


Calendar cycle

Hipparchus proposed a correction to the 76-year-long
Callippic cycle The Callippic cycle (or Calippic) is a particular approximate common multiple of the tropical year and the synodic month, proposed by Callippus in 330 BC. It is a period of 76 years, as an improvement of the 19-year Metonic cycle. Description A ce ...
, which itself was proposed as a correction to the 19-year-long
Metonic cycle The Metonic cycle or enneadecaeteris (from grc, ἐννεακαιδεκαετηρίς, from ἐννεακαίδεκα, "nineteen") is a period of almost exactly 19 years after which the lunar phases recur at the same time of the year. The recu ...
. He may have published it in the book "On the Length of the Year" (Περὶ ἐνιαυσίου μεγέθους), which has since been lost. From solstice observations, Hipparchus found that the
tropical year A tropical year or solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky of a celestial body of the Solar System such as the Earth, completing a full cycle of seasons; for example, the time fro ...
is about of a day shorter than the days that Calippus used (see ''Almagest'' III.1), so he proposed to make a 1-day correction after 4 Calippic cycles, such that 304 years = 3,760 lunations = 111,035 days. This is a very close approximation for an integer number of lunations in an integer number of days (with an error of only 0.014 days). However, it is in fact 1.37 days longer than 304 tropical years: the mean tropical year is actually about day (11 minutes 15 seconds) shorter than the Julian calendar year of days. These differences cannot be corrected with any cycle that is a multiple of the 19-year cycle of 235 lunations: it is an accumulation of the mismatch between years and months in the basic Metonic cycle, and the lunar months need to be shifted systematically by a day with respect to the solar year (''i.e.'' the Metonic cycle itself needs to be corrected) after every 228 years. Indeed, from the values of the tropical year (365.2421896698 days) and the synodic month (29.530588853) cited in the respective articles of Wikipedia, it follows that the length of 228=12×19 tropical years is about 83,275.22 days, shorter than the length of 12×235 synodic months, namely about 83,276.26 days, by one day plus about one hour. In fact, an even better correction would be to correcting by two days every 437 years, rather than one day every 228 years. The length of 437=23×19 tropical years, about 159,610.837 days, is shorter than that of 23×235 synodic months, about 159612.833 days, by almost exactly two days, up to only six minutes.


Eclipse cycle

An
eclipse cycle Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series. Eclipse conditions Eclipses ...
constructed by Hipparchus is described in
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importanc ...
's ''Almagest'' IV.2. Hipparchus constructed a cycle by multiplying by 17 a cycle due to the
Chaldea Chaldea () was a small country that existed between the late 10th or early 9th and mid-6th centuries BCE, after which the country and its people were absorbed and assimilated into the indigenous population of Babylonia. Semitic-speaking, it was ...
n astronomer
Kidinnu Kidinnu (also ''Kidunnu''; possibly fl. 4th century BC; possibly died 14 August 330 BC) was a Chaldean astronomer and mathematician. Strabo of Amaseia called him Kidenas, Pliny the Elder Cidenas, and Vettius Valens Kidynas. Some cuneifor ...
, so as to closely match an integer number of
synodic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Europ ...
s (4,267),
anomalistic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Euro ...
s (4,573), years (345), and days (126,007 + about 1 hour); it is also close to a half-integer number of
draconic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Europ ...
s (4,630.53...). By comparing his own eclipse observations with Babylonian records from 345 years earlier, he could verify the accuracy of the various periods that the Chaldean astronomers used. The Hipparchic eclipse cycle is made up of 25
inex The inex is an eclipse cycle of 10,571.95 days (about 29 years minus 20 days). The cycle was first described in modern times by Crommelin in 1901, but was named by George van den Bergh who studied it in detail half a century later. It has been sugg ...
minus 21 saros periods. There are only three or four eclipses in a series of eclipses separated by Hipparchic cycles. For example, the
solar eclipse of August 21, 2017 The solar eclipse of August 21, 2017, dubbed the "Great American Eclipse" by the media, was a total solar eclipse visible within a band that spanned the contiguous United States from the Pacific to the Atlantic coasts. It was also visible as a ...
was preceded by one in 1672 and will be followed by one in 2362, but there are none before or after these.See It corresponds to: *4,267
synodic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Europ ...
s *4,630.531
draconic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Europ ...
s *363.531 eclipse years (727
eclipse season An eclipse season is the period, roughly every six months, when eclipses occur. Eclipse seasons are the result of the axial parallelism of the Moon's tilted orbital plane ( tilted five degrees to the Earth's orbital plane), just as Earth's wea ...
s) *4,573.002
anomalistic month In lunar calendars, a lunar month is the time between two successive syzygies of the same type: new moons or full moons. The precise definition varies, especially for the beginning of the month. Variations In Shona, Middle Eastern, and Euro ...
s


References

{{DEFAULTSORT:Hipparchic Cycle Ancient Greek astronomy Time in astronomy Calendars fr:Calendrier grec