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## 1. Introduction

For calendars and for predictions it is important to be able to approximate an arbitrary number with a ratio of two whole numbers. For example, a solar calendar approximates the length of the year of seasons (for example the tropical year), and a lunar calendar approximates the length of the month of the phases of the Moon (the synodical month). For predicting solar eclipses and lunar eclipses it is important to be able to approximate the ratio of the length of the synodical month and the draconitic month.

If you use an approximation that isn't very good, then a calendar or prediction based on that approximation will quickly get out of step with the thing that the approximation was for. You want an approximation that is as good as possible. How you find such good approximations is explained on the page about the Extended Euclidean Algorithm.

Suppose that we approximate the value $$X = P/Q$$ with the ratio $$a/b$$ of whole numbers $$a$$ and $$b$$. Then we call

\begin{equation} s = bP − aQ \end{equation}

\begin{equation} v = \frac{P}{Q} − \frac{a}{b} = \frac{s}{bQ} \end{equation}

the difference, and

\begin{equation} l = aQ \end{equation}

the length.

The length $$l$$ says how much of what you measure with $$X$$ is, according to the approximation, equal to a whole number of $$Q$$ and also equal to nearly a whole number of $$P$$. The spread $$s$$ says how much difference there is between the $$P$$s and $$Q$$s in $$l$$. The difference $$v$$ says how much difference there is between the target $$P/Q$$ and the approximation $$a/b$$.

Suppose that we observe a regular phenomenon and notice that it occurs 7 times in nearly exactly 22 days, so our estimate for the period of the phenomenon is $$a/b = 22/7$$ days.

Further suppose that the real period of that phenomenon is $$X = P/Q = 3.14159265/1$$ days per phenomenon. Then the spread is

$s = 7×3.14159265 − 1×22 = −0.00885145$

days, so that is the size of the error in the assumption that 22 days are equal to 7 periods of the phenomenon.

The difference is

$v = 3.14159265 − \frac{22}{7} = −0.001264493$

days per phenomenon, so that is the error in the length of the period of the phenomenon if we assume that period to be 22/7 days.

The length is

$l = 1×22 = 22$

days, that is the time to which the spread $$s$$ applies.

## 2. The Year and the Day

The year that runs in step with the average of all seasons is the tropical year, which at the beginning of the 21st century had an average length of 365.2421898 dagen (365 days, 5 hours, 48 minutes, and 45.20 seconds). We seek good approximations of that value for making calendars with whole numbers of days.

1 365 1 −5h49m −5h49m Egyptian
2 1461 4 +44m59s +11m15s Julian
3 10592 29 −33m51s −1m10s
4 12053 33 +11m08s +20s
5 46751 128 −26s −0.2s
6 1227579 3361 +3s +0.0008s

The "offset" shows by how much that many days are offset from that many tropical years. The "difference" is the difference between the approximation and the true year. For comparison, the offset for the average Gregorian calendar year (146097 days equals 400 years) is 2 hours and 59 minutes, and the difference is 27 seconds. A calendar with 12053 days in 33 years follows the seasons better than the Gregorian calendar does.

The year that runs in step with the beginning of spring in the northern hemisphere now has a length of on average 365.24237404 days (365 days, 5 hours, 49 minutes, and 1.12 seconds). Good approximations for that year are:

1 365 1 −5h49m −5h49m Egyptian
2 1461 4 +43m56s +10m59s Julian
3 10592 29 −41m32s −1m26s
4 12053 33 +2m23s +4s
5 215493 590 −59s −0,1s
6 443039 1213 +25s +0,02s

The Gregorian year tries to follow the March equinox, but a calendar based on 12053 days for 33 years follows the March equinoxes more closely.

## 3. The Month and the Day

The month of the lunar phases, the synodical month, is now 29.530588853 days (29 days, 12 hours, 44 minutes, and 2.88 seconds) long. Good approximations of that month with calendars with whole days only are:

1 29 1 −12h44m −12h44m
2 30 1 +11h16m +11h16m
3 59 2 −1h28m −44m03s
4 443 15 +59m17s +3m57s
5 502 17 −28m49s −1m42s
6 1447 49 +1m39s +2,02s
7 25101 850 −45s −0,053s

The month of the stars, the sidereal month, is now 27.321661548 days (27 days, 7 hours, 43 minutes, and 11.56 seconds) long. Good approximations for this are:

1 27 1 −7h43m −7h43m
2 82 3 +50m25s +16m48s
3 765 28 −9m24s −20s
4 3907 143 +3m27s +1,45s
5 8579 314 −2m29s −0,48s
6 12486 457 +58s +0,13s

## 4. The Year and the Month

If we want to combine with synodical month and the tropical year, then the approximations from the following table are good. For convenience I've also included the average number of days: the average of the numbers of days that correspond to the given numbers of months and years. That number does not get ever closer to a whole number, which indicates that it is in general not possible to find a period that very accurately corresponds to a whole number of days, months, and years.

1 12 1 −10,9d −10,9d 359.80
2 25 2 +7,8d +3,9d 734.38
3 37 3 −3,1d −24h46m 1094.18
4 99 8 +38,2h +4h46m 2922.73
5 136 11 −36,1h −3h17m 4016.91
6 235 19 +2h04m +6m35s 6939.65
7 4131 334 −41m32s −7,7s 121990.88

## 5. SolarEclipses and Lunar Eclipses

For a solar eclipse or a lunar eclipse, the Moon must be in a node of its orbit and it must at the same time be New Moon (for a solar eclipse) or Full Moon (for a lunar eclipse). Between two eclipses of the same kind, there is a whole number of synodical months (now $$Q = 29.530588853$$ days) and also a multiple of one half of a draconic month (now $$P = 13.60611041$$ days). This leads to the following approximations, with length $$l$$ measured in years:

$${a}$$ $${b}$$ $${s}$$ $${v}$$ $${l}$$ name
2 1 +2d7h38m +0.17 0.075
11 5 −2d0h20m −0.030 0.41
13 6 +7h17m +0.0037 0.48 semester
89 41 −4h33m −0.00034 3.3 hepton
102 47 +2h44m +0.00018 3.8 octon
191 88 −1h48m −6.3 × 10−5 7.1
293 135 +56m22s +2.1 × 10−5 10.9 tritos
484 223 −52m01s −1.2 × 10−5 18.0 saros
777 358 +4m20s +6.2 × 10−7 28.9 inex
9031 4161 −4m10s −5.1 × 10−8 336.4
9808 4519 +10s +2.0 × 10−9 365,4

For example, 484 times half a draconic month (i.e., 242 whole draconic months) are nearly equal to 223 synodic months. The difference is only 52 minutes. That period is called the Saros and is roughly 18 years long. 242/223 as approximation for the number of draconic months in a synodic month deviates only −1.2 × 10−5 from the true ratio.

## 6. Libration of the Moon

The Moon shows always nearly the same side to the Earth, but appears to wobble a bit, as if it is slowly nodding yes and at the same time shaking no. Those wobbles are called librations. The shaking no is called libration in longitude and is associated with the anomalistic month (between two passages through the perigee), and the nodding yes is called libration in latitude and is associated with the draconic month (between two passages through the same node of the orbit of the Moon).

The anomalistic month is (at 1 January 2000) on average 27.55454988 days long, and the draconic month is on average 27.21222082 days long. If you want to get nearly exactly the same wobble condition of the Moon (so the same crater is exactly in the center of the lunar disk again) then you must wait nearly a whole number of anomalistic months and at the same time nearly a whole number of draconic months.

The best approximations for this are shown below, with $$l$$ measured in years.

$${a}$$ $${b}$$ $${s}$$ $${v}$$ $${l}$$
1 1 +8h12m +0.013 0.075
80 79 −4h02m −7.8 × 10−5 5.9
161 159 +8m27s +1.4 × 10−6 11.8
4588 4531 −5m12s −2.9 × 10−8 337.6
4749 4690 +3m15s +1.8 × 10−8 349.4
9337 9221 −1m56s −5.4 × 10−9 687.0

For example, 161 draconic months are nearly the same length as 159 anomalistic months; the difference between those periods is only 8 minutes and 27 seconds. That period of 161 draconic months or 159 anomalistic months corresponds to about 11.8 years. The ratio between the lengths of the anomalistic and draconic months is nearly equal to 161/159; the deviation from the true ratio is only about 1.4 × 10−6. languages: [en] [nl]

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Last updated: 2021-07-19