The Greek astronomer Hipparchus introduced two cycles that have been named after him in later literature.

Hipparchus proposed a correction to the Calippic cycle (of 76 years), which itself was proposed as a correction to the Metonic cycle (of 19 years). He may have published it in the book "On the Length of the Year" (Περὶ ἐνιαυσίου μεγέθους), which is lost. From solstice observations, Hipparchus found that the tropical year is about 1⁄300 of a day shorter than the 365+1⁄4 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 = 3760 lunations = 111035 days. This is a very decent approximation for an integer number of lunations in an integer number of days (error only 0.014 days). But it is in fact 1.37 days longer than 304 tropical years: the mean tropical year is actually about 1⁄128 day (11 minutes 15 seconds) shorter than the Julian calendar year of 365+1⁄4 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.^{[ citation needed ]} 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 83275.22 days, shorter than the length of 12×235 synodic months, namely about 83276.26 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 159610.837 days, is shorter than that of 23×235 synodic months, about 159612.833 days, by almost exactly two days, up to only six minutes.

An eclipse cycle constructed by Hipparchus is described in Ptolemy's *Almagest* IV.2. Hipparchus constructed a cycle by multiplying by 17 a cycle due to the Chaldean astronomer Kidinnu, so as to closely match an integer number of synodic months (4267), anomalistic months (4573), years (345), and days (126007 + about 1 hour); it is also close to a half-integer number of draconic months (4630.53...). 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. ^{[ citation needed ]}

The Hipparchic eclipse cycle is made up of 25 inex 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 was preceded by one in 1672 and will be followed by one in 2362, but there are none before or after these.^{ [1] }

It corresponds to:

- 4267 synodic months
- 4630.531 draconic months
- 363.531 eclipse years (727 eclipse seasons)
- 4573.002 anomalistic months

**Hipparchus of Nicaea** was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 to 127 BC.

The **Metonic cycle** or **enneadecaeteris** is a period of approximately 19 years after which the phases of the moon recur at the same time of the year. The recurrence is not perfect, and by precise observation the Metonic cycle defined as 235 synodic lunar months is just 1 hour, 27 minutes and 33 seconds longer than 19 tropical years. Meton of Athens, in the 5th century BC, judged the cycle to be a whole number of days, 6,940. Using these whole numbers facilitates the construction of a luni-solar calendar.

**Kidinnu** was a Chaldean astronomer and mathematician. Strabo of Amaseia called him Kidenas, Pliny the Elder Cidenas, and Vettius Valens Kidynas.

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**.

The **saros** is a period of exactly 223 synodic months, approximately 6585.3211 days, or 18 years, 10, 11, or 12 days, and 8 hours, that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle. A **sar** is one half of a saros.

The **inex** is an eclipse cycle of 10,571.95 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 suggested that the cycle was known to Hipparchos. One inex after an eclipse of a particular saros series there will be an eclipse in the next saros series, unless the latter saros series has come to an end.

An **exeligmos** is a period of 54 years, 33 days that can be used to predict successive eclipses with similar properties and location. For a solar eclipse, after every exeligmos a solar eclipse of similar characteristics will occur in a location close to the eclipse before it. For a lunar eclipse the same part of the earth will view an eclipse that is very similar to the one that occurred one exeligmos before it. The exeligmos is an eclipse cycle that is a triple saros, three saroses long, with the advantage that it has nearly an integer number of days so the next eclipse will be visible at locations and times near the eclipse that occurred one exeligmos earlier. In contrast, each saros, an eclipse occurs about eight hours later in the day or about 120° to the west of the eclipse that occurred one saros earlier.

As a moveable feast, the **date of Easter** is determined in each year through a calculation known as * computus*. Easter is celebrated on the first Sunday after the Paschal full moon, which is the first full moon on or after 21 March. Determining this date in advance requires a correlation between the lunar months and the solar year, while also accounting for the month, date, and weekday of the Julian or Gregorian calendar. The complexity of the algorithm arises because of the desire to associate the date of Easter with the date of the Jewish feast of Passover which, Christians believe, is when Jesus was crucified.

The **epact**, used to be described by medieval computists as the age of a phase of the Moon in days on 22 March; in the newer Gregorian calendar, however, the epact is reckoned as the age of the ecclesiastical moon on 1 January. Its principal use is in determining the date of Easter by computistical methods. It varies from year to year, because of the difference between the solar year of 365–366 days and the lunar year of 354–355 days.

For astronomy and calendar studies, the **Callippic cycle** is a particular approximate common multiple of the year and the synodic month, that was proposed by Callippus during 330 BC. It is a period of 76 years, as an improvement of the 19-year Metonic cycle.

**Callippus** was a Greek astronomer and mathematician.

**Babylonian mathematics** denotes the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. With respect to time they fall in two distinct groups: one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia.

**Babylonian astronomy** was the study or recording of celestial objects during the early history of Mesopotamia.

A **total lunar eclipse** will take place on 3 March 2026.

A **total lunar eclipse** will take place on June 26, 2029. A dramatic total eclipse lasting 1 hour and 42 minutes will plunge the full Moon into deep darkness, as it passes right through the centre of the Earth's umbral shadow. While the visual effect of a total eclipse is variable, the Moon may be stained a deep orange or red colour at maximum eclipse. This will be a great spectacle for everyone who sees it from most of the Americas and western Europe and Africa. The partial eclipse will last for 3 hours and 40 minutes in total.

A **total lunar eclipse** will take place on June 6, 2058. The moon will pass through the center of the Earth's shadow.

A **total lunar eclipse** will take place on February 11, 2055.

A **total lunar eclipse** will take place on May 6, 2069. The eclipse will be a dark one with the southern tip of the moon passing through the center of the Earth's shadow. This is the first central eclipse of Saros series 132.

A partial solar eclipse will occur on Wednesday, February 27, 2036. A solar eclipse occurs when the Moon passes between Earth and the Sun, thereby totally or partly obscuring the image of the Sun for a viewer on Earth. A partial solar eclipse occurs in the polar regions of the Earth when the center of the Moon's shadow misses the Earth.

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.

- ↑ See "Five Millennium Catalog of Solar Eclipses". NASA.

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