# Meton cycle

Meton cycle ( Greek Μέτωνος κύκλος) or Meton period (also Enneakaidekaeteris , Enneadekaeteris ; Greek εννεαδεκαετηρίς: " nineteen years old ")

• a period that is both 19  solar years and 235  lunar months each 29.53 days long,
• a presumed calendar system that the ancient Greek astronomers Euktemon and Meton used in the fifth century BC. In which 19 years contained 6940 days,
• and thus the possibility of a 19-year calendar cycle that includes 12 years with 12 lunar months and 7 years with 13 lunar months.

The fact that 19 solar years and 235 lunar months are the same length (difference 2 hours and 5 minutes) was already known in ancient times and was the basis of their lunar calendar among the Babylonians . Along with Euctemon, Meton was one of the first Greek astronomers and probably one of the first Greeks to find out about it. It is not known why later Greek historians exclusively associated his name with it. A neutral term is the lunisolar cycle .

## history

### Meton as a contemporary of Euctemon

Neither Euktemon nor Meton have written records. Both are only mentioned individually from a century after their work. The name Euktemons occurs more often. Meton is used in connection with the finding of the summer solstice in 432 BC. And the establishment of an astronomical and weather forecast calendar ( Parapegma ) called. It is not certain that both worked together. Knowledge of Greek astronomy up to the first century BC BC has summarized Geminos of Rhodes . He does not mention the name Metons and traces the “nineteen-year cycle” in a calendar system back to “astronomers from the school of Euctemon, Philip and Callippus ”. The latter two only lived a century after Euctemon and Meton.

### The historical sources

In the fourth century BC Eudoxus of Knidos mentions in Greek literature a "nineteen-year period of the Euctemon ". A short time later, Meton is mentioned for the first time by Theophrastus of Eresos in this context. A century later, Aratos von Soloi names the astronomer Kallippos, who in 330 BC. Modified the "nineteen-year Meton calendar system". Philochorus reports that Meton built a heliotropion on the Pnyx , which was used to mark the summer solstice . In the Eudoxus papyrus published in Giza around 190 BC And in the writings of Geminos of Rhodes, which dates from about 70 BC. Originated in BC, Democritus , Eudoxus, Euktemon and Callippus are mentioned, but not Meton.

Only the historian Diodor mentioned a parapegma in connection with the summer solstice as a "nineteen-year cycle (έννεακαιδεκαετηρίδα), which Meton, son of Pausanias, introduced". Vitruvius names among the astronomers who had developed a calendar (parapegma) linked to weather forecasts , in addition to Euctemon and Meton, all the names already mentioned. Claudius Ptolemy reports in his astronomical notes of the Almagest only once about a "nineteen-year period" with the background of a summer solstice observed by Meton and Euktemon.

From the ancient traditions it is not clear who designed the model of the "Meton calendar system" and what it looked like. A common work of Meton and Euktemon is often assumed. The modern critical assessment by Otto Neugebauer reduces this calendar system to the equation of 19 years with 6940 days made by its namesake. Meton would have only indirectly determined the length of the solar year and created a pure solar calendar so that the annual weather report contained in his parapegma would have eternal validity. It remains uncertain whether Meton equated the period of 19 solar years or 6940 days with 235 lunar periods and derived an application from it.

The use of a period in the Easter calculation ( computus ) that is both 19 solar years and 235 lunar periods is nonetheless inextricably linked to the name Metons, regardless of the fact that the astronomical background of the Meton cycle was known before Meton and that the Meton period not with 6940, but since Callippus is equated with 6939.75 days.

### Meton and the summer solstice in 432 BC Chr.

Diodor writes that in the same year that Apseudes held the office of Archon eponymos in Athens, Meton put the beginning of his calculated nineteen-year calendar system on the 13th day of the month Skirophorion . This month was the last of the fourth year of the 86th  Olympiad , which ran from 433 to 432 BC. Chr. Was enough. Claudius Ptolemy remarked that Meton and Euktemon in this context the summer solstice in 432 BC. Observed. In the same text he gives the 21st Phamenoth in the Egyptian calendar as the day . This is June 27 in the Julian calendar (June 22 in the Gregorian calendar ) and from today's perspective is considered a reliable date for the summer solstice in 432 BC. The new light fell on June 16 in the Julian (June 11 in the Gregorian) calendar. Assuming that the months used in Athens coincided with the phases of the moon, Meton would have found the summer solstice one day too late: 1. Skirophorion on June 16; 13. Skirophorion on June 28th (Julian dates).

Otto Neugebauer draws the conclusion from the mention of the 13th Skirophorion as the start date that Meton did not try to create a new annual calendar, but only looked for a clear starting point in the solar year for the creation of a parapegma. He should have started an annual calendar on the day of a new lunar month (new light). One such fell in 432 BC. Does not coincide with the summer solstice. Meton's nineteen-year calendar, which, according to Diodorus, the Greeks still used in his day, he considers this parapegma.

### Distribution of 235 lunar months over 6940 days

Calendar months always consist of a whole number of days. Months that are supposed to follow the phases of the moon (mean period: about 29.53 days) are approximately 29 days ( hollow months ) or 30 days ( full months ) long. The Meton period with 6940 days can clearly only be composed of 110 hollow months and 125 full months.
Control calculation: 110 · 29 + 125 · 30 = 3190 + 3750 = 6940.
Because the number of hollow and full months is not the same, a regular change in the sequence is out of the question. It is not known whether the astronomers of ancient Greece, following the Babylonian model, formed normal years of six hollow and six full months and occasionally leap years with an additional month. Geminos assigns the following indirect method to Euktemon, which leads to a favorable sequence of hollow and full months: 30 days are first formally assigned to all 235 months. At 7050 days, the total is 110 days too large. Therefore, every 64th day of the 7050 days is skipped, resulting in 6940 days, with a hollow month usually following a full month. Two full months follow each other fifteen times. That this approach would actually have been applicable is supported by two papers by Fotheringham (1924) and van der Waerden (1960).

Geminos also reports, however, that the calculations of the euctemon were not in agreement with the assumed length of 365.25 days for the solar year at his (Geminos') time, and finally mentions that the “erroneous excess was later reported by astronomers from the school of Callippus was corrected through an improved nineteen year cycle. "

## Differences in a nineteen year calendar system of 6940 days

Calculation with today's values ​​for the solar year and the lunar period ( lunation ):

${\ displaystyle {\ text {solar year}} = 365 {,} 24219 \ {\ text {days}}}$
${\ displaystyle {\ text {lunar period}} = 29 {,} 53059 \ {\ text {days}}}$

### 19 calendar years

${\ displaystyle {\ begin {matrix} 19 \, {\ text {calendar years}} & = & 6940 {,} 00000 \, {\ text {days}} \\ 19 \, {\ text {solar years}} & = & 6939 {,} 60161 \, {\ text {days}} \\ {\ text {difference}} & = & 0000 {,} 39839 \, {\ text {days}} \ end {matrix}}}$

19 calendar years are 0.39839 days too long. After less than 48 years, the calendar advances the solar year by one day.

### 235 lunar months

${\ displaystyle {\ begin {matrix} {} \ 235 \, {\ text {lunar months}} & = & 6940 {,} 00000 \, {\ text {days}} \\ 235 \, {\ text {lunar periods}} & = & 6939 {,} 68865 \, {\ text {days}} \\ {\ text {difference}} & = & 0000 {,} 31135 \, {\ text {days}} \ end {matrix}}}$

235 lunar months are 0.31135 days too long. The calendar moves forward by one day after about 755 lunar months (about 61 calendar years) compared to the lunar periods.

## A nineteen year calendar system of 6939.75 days

A century after Meton, Callippus indirectly corrected the 19-year period to 6939.75 days. The Callipean cycle given in whole days assigns 27,759 days to 76 years. The first known use of the length of 365.25 days contained therein for the individual year occurred in one in Egypt at the time of Ptolemy III. in the third century BC Chr. Briefly used solar calendar . It is not known whether the knowledge of the year length of 365.25 days was adopted from Callippos. The inclusion of an additional day every four years was later adopted by the Julian calendar after Julius Caesar had found out about it personally in Egypt. At the time of Jesus Christ , a bound lunar calendar was used in Palestine . Reminding Christians to Easter , "the day of the resurrection of Jesus Christ," based on this calendar, which will continue to be applied within the Julian solar calendar (and improved Gregorian calendar). The link to the lunar months is expressed by the fact that Easter Sunday follows the first full moon in spring, i.e. it is in the first lunar month of the religious Jewish calendar . In the determination of the (now in the Gregorian) other year in the Julian calendar Easter date, the 19-year-period plays an important role.

Calculation with today's values ​​for the solar year and the lunar period (lunation):

${\ displaystyle {\ text {solar year}} = 365 {,} 24219 \ {\ text {days}}}$
${\ displaystyle {\ text {lunar period}} = 29 {,} 53059 \ {\ text {days}}}$

### 19 calendar years

${\ displaystyle {\ begin {matrix} 19 \, {\ text {calendar years}} & = & 6939 {,} 75000 \, {\ text {days}} \\ 19 \, {\ text {solar years}} & = & 6939 {,} 60161 \, {\ text {days}} \\ {\ text {difference}} & = & 0000 {,} 14839 \, {\ text {days}} \ end {matrix}}}$

19 calendar years are 0.14839 days too long. After a little over 128 years, the calendar advances the solar year by one day.

With the Gregorian calendar reform, this difference was almost eliminated by a leap day in the calendar on average every 133.3333 years ( solar equation ), i.e. in real terms every 400 years a leap year takes place (1600, 2000, 2400, ...), during the entire centuries that cannot be divided by 400 without a remainder (e.g. 1700, 1800, 1900, 2100, ...), the leap day is canceled.

### 235 lunar months

${\ displaystyle {\ begin {matrix} {} \ 235 \, {\ text {lunar months}} & = & 6939 {,} 75000 \, {\ text {days}} \\ 235 \, {\ text {lunar periods}} & = & 6939 {,} 68865 \, {\ text {days}} \\ {\ text {difference}} & = & 0000 {,} 06135 \, {\ text {days}} \ end {matrix}}}$

235 lunar months are 0.06135 days too long. The calendar advances by one day after about 3,830 lunar months (about 310 calendar years) compared to the lunar periods.

This difference was almost eliminated in the Gregorian calendar reform, in that it was also determined that a lunar month should be one day shorter on average every 312.5 years ( lunar equation ).

## Difference between 19 solar years and 235 lunar periods

Calculation with today's values ​​for the solar year and the lunar period (lunation):

${\ displaystyle {\ text {solar year}} = 365 {,} 24219 \ {\ text {days}}}$
${\ displaystyle {\ text {lunar period}} = 29 {,} 53059 \ {\ text {days}}}$
${\ displaystyle {\ begin {matrix} 19 \, {\ text {solar years}} & = & 6939 {,} 60161 \, {\ text {days}} \\ 235 \, {\ text {lunar periods}} & = & 6939 {,} 68865 \, {\ text {days}} \ end {matrix}}}$

difference

{\ displaystyle {\ begin {aligned} 0 {,} 08704 \, {\ text {days}} & = & 2 {,} 08896 \, {\ text {hours}} & = & 2 \, {\ text {hours} } \ dots 5 \, {\ text {minutes}} \ dots 20 \, {\ text {seconds}} \ end {aligned}}}

## Application in the Julian calendar

A historically very important application of the Metonic cycle in the Alexandrian calendar and the Julian calendar was the Metonic 19-year lunar cycle. Around AD 260, the Alexandrian computist Anatolius was the very first to construct a version of this efficient computist instrument for determining the date of Easter Sunday. However, it was Annianus 'version (around AD 400) of the Metonic 19-year lunar cycle that would ultimately get the upper hand as the basic structure of Beda Venerabilis ' Easter table (AD 725) in all of Christianity for a long time, at least until 1582, when the Julian calendar was replaced by the Gregorian calendar .

## Footnotes

1. ^ Alfred Fleckeisen: Yearbooks for Classical Philology . Teubner, Leipzig 1860, p. 345.
2. ^ Wilhelm Friedrich Rinck: The religion of the Hellenes, from the myths, the teachings of the philosophers, and the cult . Meyer and Zeller, Zurich 1855, p. 35.
3. In the historical sciences, the strict scientific distinction between a cycle and the duration ( period ) between cyclical events is generally not usual.
4. Otto Neugebauer: The Metonic and the Callippic Cycle . P. 622 f.
5. ^ Heinz Zemanek : Calendar and Chronology. Oldenbourg 1990, p. 43: The lunar circle was known to Babylonian astronomers as early as 747 BC. Known.
6. a b
7. Since, according to Meton, the stars meet again after 19 years , Diodorus reported about four centuries after Euctemon and Meton that the 19-year period was also called the " Year of Meton ".
8. Theophrastus of Eresos: About weather signs , 4.
9. ^
10. page no longer available , search in web archivesInfo: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.
11. ^ Friedrich Karl Ginzel: Handbook of mathematical and technical chronology, Vol. 2 . Pp. 392-394.
12. Diodor: Bibliothéke historiké ' , 12, 36, 1.
13. Evans, J .; Berggren, JL: Geminus, Introduction to the Phenomena. Princeton University Press 2006, VIII 52, p. 184.
14. Evans, J .; Berggren, JL: Geminus, Introduction to the Phenomena. Princeton University Press 2006, VIII 53 to 55, p. 184.
15. Van der Waerden, BL: Greek astronomical calendars. II. Callippos and his calendar. Archive for History of Exact Sciences 29 (2), 1984, pp. 121-124.
16. Evans, J .; Berggren, JL: Geminus, Introduction to the Phenomena. Princeton University Press 2006, VIII.
17. Zuidhoek (2019) 16-17
18. ^ Declercq (2000) 65-66
19. Zuidhoek (2019) 70

## literature

• FK Ginzel : Handbook of mathematical and technical chronology. Timekeeping of the Nations . Volume 2: Chronology of the Jews, the primitive peoples, the Romans and the Greeks as well as addenda to the 1st volume. (Reprint of the original Leipzig edition in 1906). sn, Innsbruck 2007 ISBN 3-226-00428-X ( Austrian literature online 54).
• Helmut Groschwitz: Moon times. On the genesis and practice of modern lunar calendars. Waxmann, Münster et al. 2008, ISBN 978-3-8309-1862-2 ( Regensburg writings on folklore - comparative cultural studies 18), (At the same time: Regensburg, Univ., Diss., 2005).
• Otto Neugebauer , William Kendrick Pritchett : The calendars of Athens. Harvard University Press, Cambridge MA 1947.
• Otto Neugebauer: The Metonic and the Callippic Cycle. In: O. Neugebauer: A history of ancient mathematical astronomy. Springer, Berlin et al. 1975, ISBN 3-540-06995-X ( Studies in the History of Mathematics and Physical Sciences 1), (reprint. Ibid 2006).
• W. Kendrick Pritchett: Athenian Calendars and Ekklesias. Gieben, Amsterdam 2001, ISBN 9-0506-3258-0 .
• Carl Christian Redlich: The astronomer Meton and his cycle. Meißner, Hamburg 1864, online .
• Jan Zuidhoek (2019) Reconstructing Metonic 19-year Lunar Cycles (on the basis of NASA's Six Millenium Catalog of Phases of the Moon): Zwolle ( ISBN 9789090324678 )
• Georges Declercq (2000) Anno Domini (The Origins of the Christian Era): Turnhout ( ISBN 9782503510507 )