Abstract
A lunisolar calendar counts months by the phases of the Moon and inserts an occasional leap month to stay with the seasons. This paper confirms how the Metonic cycle, $235$ months in $19$ years, comes out of the continued fraction of the ratio of year to month, and then counts how the miss changes when the days are made whole. No new theorem or law is claimed.
First, $235/19$ is a convergent of the ratio of year to month: the tropical year of $365.24219$ days divided by the synodic month of $29.530589$ days is $12.368266342$, and the convergents of its continued fraction run $12/1$, $25/2$, $37/3$, $99/8$, $136/11$, $235/19$, $4131/334$. A search of all denominators up to $19$ also found $235/19$ best (Section 2).
Second, $235$ months and $19$ years differ by only $0.086805$ days: the eight-year cycle $99/8$ differs by $1.5908$ days, and the Metonic drift accumulates to $1$ day in $218.881$ years. There are $7$ leap months in $19$ years (Section 2).
Third, fitting the days too makes the miss larger: the difference between $235$ months and the nearest whole number of days, $6940$, is $\mathbf{0.311585}$ days, $\mathbf{3.589482}$ times the miss between year and month. The difference between $19$ years and the nearest whole number of days is $0.398390$ days (Section 3).
Fourth, the Callippic cycle refits the days: setting $76$ years and $940$ months to $27759$ days, four times $6940$ less $1$, leaves a difference of $\mathbf{0.246340}$ days from the months and $0.593560$ days from the years (Section 3).
Fifth, the separator is whether two clocks are fitted or three: with year and month alone, the convergents of the continued fraction choose the best cycle. Adding the day as a third clock imposes the separate condition that the cycle be a whole number of days, which the convergents do not look after (Section 4).
Keywords: Metonic cycle, synodic month, tropical year, continued fraction, convergent, lunisolar calendar.
Scope (what is not claimed)
No new theorem or law is claimed: that convergents of a continued fraction give best approximations, and the Metonic, Callippic and eight-year cycles, are all standard. The Antikythera Mechanism is not decoded: proposals for its gear tooth counts or how its dials are read are not entered. That the mechanism kept calendar cycles is only mentioned by name. History is not judged, which calendar was used when, and who found the cycles, are not treated. The lengths are mean representative values: a tropical year of $365.24219$ days and a synodic month of $29.530589$ days are used, without tidal change or month-to-month fluctuation. Relation to earlier papers: Paper 319, in its scope note, did not treat other calendars such as lunar and lunisolar ones; this paper sits in that seat of the lunisolar calendar (no earlier occupant). Paper 539 showed that the tooth counts of Huygens’s planetarium are convergents of the continued fraction of the ratio of the periods of Saturn and Earth. Here the same tool is applied to the ratio of year to month, and it is counted that with three clocks the convergents alone do not decide. Paper 812 showed that how easily a number is approximated is decided by whether its partial quotients are bounded. Here a partial quotient of $17$ appears in the ratio of year to month, and the convergent just before it, $235/19$, fits especially well. Paper 406 showed that seeing the same face is the evidence that the Moon rotates. Here only the synodic month, the period of the Moon’s phases, is used. What is added: setting down the continued fraction and convergents of the ratio of year to month, confirming by exhaustive search that $235/19$ is best for denominators up to $19$, finding that the miss when fitted to whole days is $3.589482$ times the year-month miss, counting the Callippic refitting of the days, and placing the separator on “two clocks fitted, or three”.
1. Notation and the Question
Setting: with tropical year $Y$ and synodic month $M$, the ratio $r=Y/M$ is expanded as a continued fraction. A convergent $h/k$ stands for “$h$ months in $k$ years”, with drift $hM-kY$ days.
$$r=\frac{Y}{M}=12+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cdots}}},\qquad \Delta=hM-kY$$The check uses $Y=365.24219$ days and $M=29.530589$ days, compares all fractions with denominators up to $19$, and finds the differences from whole numbers of days.
Question: where do $235$ months in $19$ years come from, and how does the miss change when the days are made whole too?
2. Year and Month
Computation 2.1: the partial quotients of $r=12.368266342$ were $[12; 2,1,2,1,1,17,3,14]$. The drift of year against month for each convergent:
| convergent (months/years) | years (days) | months (days) | drift (days) |
|---|---|---|---|
| $12/1$ | $365.2422$ | $354.3671$ | $-10.8751$ |
| $25/2$ | $730.4844$ | $738.2647$ | $+7.7803$ |
| $37/3$ | $1095.7266$ | $1092.6318$ | $-3.0948$ |
| $99/8$ | $2921.9375$ | $2923.5283$ | $+1.5908$ |
| $136/11$ | $4017.6641$ | $4016.1601$ | $-1.5040$ |
| $235/19$ | $6939.6016$ | $6939.6884$ | $+0.0868$ |
| $4131/334$ | $121990.8915$ | $121990.8632$ | $-0.0283$ |
| $12628/1021$ | $372912.2760$ | $372912.2779$ | $+0.0019$ |
Among denominators up to $19$, the fraction with smallest $|h-kr|$ was $235/19$ ($0.002939$ months). The Metonic drift is $+0.086805$ days, accumulating to $1$ day in $218.881$ years. There are $235-12\times19=7$ leap months.
Observation 2.2 $235/19$ is a convergent of the ratio of year to month. It is the fraction cut off just before the partial quotient $17$ appears, so it fits well for the size of its denominator. The product of denominator and miss, $k|h-kr|$, is $0.055850$ for $235/19$, an order of magnitude below the other convergents ($0.314397$ to $0.560227$) and close to $1/17=0.058824$: with year and month as the only two clocks, the continued fraction chooses the best cycle.
3. Adding the Day
Computation 3.1: differences from whole numbers of days.
| compared | difference (days) |
|---|---|
| $235$ months and $19$ years | $+0.086805$ |
| $6940$ days and $235$ months | $+0.311585$ |
| $19$ years and the nearest whole number of days | $+0.398390$ |
The miss when fitted to whole days was $\mathbf{3.589482}$ times the miss between year and month.
Computation 3.2: the Callippic cycle ($76$ years, $940$ months, $27759$ days). $940$ months are $27758.753660$ days and $76$ years $27758.406440$ days; the differences from $27759$ days are $\mathbf{0.246340}$ days for the months and $0.593560$ days for the years.
Observation 3.3 fitting the days too makes the miss larger than with year and month alone. Year and month agree within $0.086805$ days, but the length of that cycle lies $0.311585$ days from a whole number of days: the Callippic cycle, taking $1$ day from the $27760$ days of four Metonic cycles, brings the day miss against the months back to $0.246340$ days.
4. The Separator
Observation 4.1 the separator is whether two clocks are fitted or three. With year and month, the convergents of the continued fraction of their ratio choose the best cycle for each denominator, and at $19$ years that is $235/19$: adding the day as a third clock brings the separate condition that the cycle be a whole number of days, which the convergents do not look at. The miss therefore became $3.589482$ times as large.
Observation 4.2 what is not said. The gear teeth of the Antikythera Mechanism, the history of calendars, and changes in the lengths of the year and month are not treated. What is shown is that the Metonic cycle is a convergent of year over month, and that adding the day brings a separate miss.
5. Roll Call and Closing
- $r=12.368266342$ (Computation 2.1): year over month.
- $235/19$ is a convergent, best for denominators up to $19$ (Computation 2.1): chosen by the continued fraction.
- year-month drift $0.086805$ days (Computation 2.1): $1$ day in $218.881$ years.
- difference from whole days $\mathbf{0.311585}$ (Computation 3.1): $\mathbf{3.589482}$ times.
- Callippic cycle $\mathbf{0.246340}$ days (Computation 3.2): refitting the days.
- two clocks or three (Observation 4.1): the separator; the scope note of Paper 319; Papers 539, 812, 406.
Closing: The Metonic cycle is a convergent of year over month, and fitting whole days too makes the miss over three and a half times larger. Among the convergents of the ratio $12.368266342$ is $235/19$, best for denominators up to $19$, with a drift of $0.086805$ days. Yet its difference from $6940$ days is $0.311585$ days, $3.589482$ times the year-month miss. The Callippic cycle takes away $1$ day and brings the miss against the months back to $0.246340$ days. With year and month alone the convergents choose the cycle, but adding the day imposes the separate condition of whole days: the separator is whether two clocks are fitted or three. Placed among the earlier papers: the lunisolar calendar left untreated by Paper 319 was expanded with the same continued fractions as the tooth counts of Paper 539, and the size of the partial quotients of Paper 812 was seen inside the ratio of year to month. To be honest: no mechanism has been decoded. The difficulty of fitting three clocks at once was placed in one number, and no further.
References
- Freeth, T., et al. (2006). Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Nature.
- Neugebauer, O. (1975). A History of Ancient Mathematical Astronomy. Springer.
- Yamagishi, Y. (2026). A Calendar Is Three Approximations Stacked [Paper 319]. YOI: https://www.yamagishiyuuki.com/yoi/319; DOI: 10.5281/zenodo.22208933
- Yamagishi, Y. (2026). That the Same Face Is Seen Is the Evidence That the Moon Rotates [Paper 406]. YOI: https://www.yamagishiyuuki.com/yoi/406; DOI: 10.5281/zenodo.22699455
- Yamagishi, Y. (2026). What Fixed the Tooth Counts Was Not Astronomy but a Continued Fraction [Paper 539]. YOI: https://www.yamagishiyuuki.com/yoi/539; DOI: 10.5281/zenodo.23014129
- Yamagishi, Y. (2026). What Decides How Easily a Number Is Approximated Is Whether Its Partial Quotients Are Bounded [Paper 812]. YOI: https://www.yamagishiyuuki.com/yoi/812; DOI: 10.5281/zenodo.23015472
Reference
Yamagishi, Y. (2026). The Metonic Cycle Is a Convergent of Year over Month, and Fitting Whole Days Too Makes the Miss Over Three and a Half Times Larger [Paper 916]. YOI: https://www.yamagishiyuuki.com/yoi/916
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 916. This paper is a preprint and has not been peer reviewed.