Accuracy of the Gregorian calendar

Years	Numbers	Years	Numbers	Years	Numbers
before.e.	March	n.e.	March	n.e.	in March
1001	30.70	100	22.00	900	15.76
601	27.53	200	21.22	1000	14.98
501	26.73	300	20.43	1100	14.21
401	25.93	400	19.66	1200	13.45
301	25.14	500	1887	1300	12.68
201	24.35	600	18.10	1400	11.90
101	23.57	700	17.32	150E	11.14
1	22.78	800	16.53	1600	10.36
Note. The table is constructed for leap years
.When determining the date of the equinox in other
dahs, follow the	interpolation for the year of the search; add	to the	amendment	0.25;0,50	or 0,75	day co
responsibly for	1st, 2nd	or 3rd year after	of a leap year,
with	for years	BC.e.they are those for which the
residue after dividing R-1	by 4( R-	is the year number) is equal to
, respectively, 3,	2 and 1. In this case, 0.1 days = 2	h 24 min,
0.01 days = 14.4	min.

The history of our calendar is yet to be discussed. Here we will dwell on the question of its accuracy, since this relates precisely to the "arithmetic of calendars."To start such an analysis is appropriate from the calendar that was used in Europe for 1600 years and on the dates of which all events of world history, which took place before the Gregorian reform, are projected.

Arithmetic of the Julian calendar. The attractive side of the Julian calendar is its simplicity and strict rhythmicity of the change of simple and leap years. Each period of time in four years has( 365 + 365 + 365 + 366 =) 1461 days, each century 36,525 days. Therefore, it was convenient for measuring long time intervals.

But, as already noted, the average duration of the Julian calendar year is more than the tropical year by 0.0078 days. Therefore, for every 128 years, any particular phenomenon of the tropical year( for example, the vernal equinox) in such a calendar is shifted by one day to earlier dates. Let us explain this with a drawing( Fig.).

Fig. Comparison of the Julian calendar with the tropical years

If in the beginning of the count of years the transition of the Sun through the point of the vernal equinox( point B on the time scale) occurred on March 21 in the Julian calendar, then 400 years later it will happen three days earlier;It is therefore customary to say that the Julian calendar is moving forward relative to certain seasons, whereas in relation to the dates of this calendar, one or another of the annual astronomical phenomena are shifting backwards.

The speed of moving the date of the vernal equinox according to the Julian calendar dates was calculated by F. Ginzel. The results of these calculations are partially shown in Table.

Table. Dates of the vernal equinox in the Julian calendar( according to the world time)

Let's define here with her the dates of the spring equinox for several years, which played a decisive role in the fate of the Julian calendar - for 45 BC.e., 325 g.e.and 1582 AD.e.

In the first case, the year number is R = 45. Since R-1 = 44 is divisible by 4 without remainder, this year was a leap year and the calendar correction is zero. The change in the date of the vernal equinox for a hundred years was 23.57 - 22.78 = 0.79 days, for 44 years( preceding the 1st millennium BC) - 0.79 / 100 * 44 = 0.35 days. Consequently, in 45 BC.E., when the Julian calendar was introduced, the vernal equinox was 22.78 + 0.35 = 23.13 March. We also find that for the years of the 44th, 43rd, 42nd and 41st this date is accordingly: 23.37;23.61;23.85 and 23.09 of March.

For 325 g.e.the change in the date of the equinox for 100 years 20.43-19.66 = 0.77 days, for 25 years, 0.19 days. This year is the 1st after the leap, so the calendar correction is 0.25 days. Consequently, the spring equinox in 325, when the Council of Nice was convened, came 20.43 - 0.19 + 0.25 = 20.49 March, that is, on March 20 at 12 noon on Greenwich time or at 2 pm onAlexandrian time. For the years 321, 322, 323 and 324, we find this date accordingly: 20.52;20.76;21.00 and 20.24 March. Let's notice, that just in 323 for the last time the vernal equinox in the Julian calendar was on March 21( !).

Similarly, for 1582 we find: 11.14 - 10.36 = 0.78 = 0.78;0,78 / 100 * 82 = 0,64, the calendar amendment 0,50( the second year after the leap), and the date of the vernal equinox 11,14 - 0,64 + 0,50 = 11,00 March. For the years nearest to it 1580, 1581, 1583 and 1584, respectively, the dates of the vernal equinox are 10.52;10.76;11.24 and 10.48 March.

The rules of these calculations are very simple. If the moment of the vernal equinox is known in a particular year, then in the subsequent simple calendar year it moves to 0d, 2422 forward, and in the leap forward it moves back to 0d, 7578.By the end of each four-year period, the moment of the spring equinox is moved back to 0d, 0312, which is 400 years and gives an error in 3d, 12.

The Gregorian calendar. In the Gregorian calendar, a simple year also has 365 days, a leap year 366. As in the Julian calendar, every fourth year is a leap year, the one whose ordinal number is divided into 4 without residue in our calendar. However, those centuries of the calendar, the number of hundreds of which are not divided without a remainder by 4, are considered to be simple( for example, 1500, 1700, 1800, 1900, etc.).The leap years are 1600, 2000, 2400, etc. Thus, the full cycle of the Gregorian calendar consists of 400 years;By the way, the first such cycle ended very recently, on October 15, 1982, and it contains 303 years for 365 days and 97 years for 366 days. The total number of days in the 400-year period is 303 X 365 + 97 X 366 = 146 097. The average duration of the calendar year is 146097/400 = 365.24205 - it is longer than the duration of the tropical year by 0.00030 days, that is, only 26seconds. The error of this calendar in one day runs for 3300 years. Therefore, in terms of the accuracy and clarity of the leap system( which facilitates its memorization), this calendar should be recognized as very successful.

However, if you look more closely at the distribution of leap years in a 400-year cycle, it turns out that the situation is not so good, and the calendar itself looks less attractive. Take for example the 400-year cycle that began in 1600. The duration of the first 96 years in it averages 365.25 days. But the year 1700 was a simple, leap year was only 1704 year. Thus, the average duration of each of these eight years( from 1697 to 1704) is only 365 days. The same can be said about the years 1797-15.04 and 1897-1904.Therefore, the calendar error( which should be corrected by inserting an extra day in a leap year) is distributed unevenly from year to year. And this leads, in particular, to the fact that the beginning of spring( the moment of passing the center of the disk of the Sun through the point of the vernal equinox) in each 400th anniversary shifts by 1.6954 days and ranges from 19( 1) to 21 March.

In fact, having started the account since 1601, we find that the first year of the 400-year cycle is simple. Therefore in it in comparison with the initial moment( 1600th year) the equinox will move on 0,2422 days ahead, for three years it will make 0,7266 days. The fourth year is a leap year( 366 days), and the equinox is moved back to 365d, 2422 to 366d = -0d, 7578, that is, to 0.7578 days ago. In general, for four years the equinox in comparison with the initial moment is moved back by 0.0312 days. For 96 years this will give 0.7488 days. And if in 1600 the vernal equinox occurred on March 20, 36, then in 1696 it took place 20.36 - 0.75 = 19.61 March. Each of the next seven years is simple, so that the moment of the vernal equinox shifts seven times on 0d, 2422 annually, and by 1703 it reaches the limit of 21.31( March).The difference between the dates of the moments of 1703 and 1696.and is 1,6954 days.

A similar phenomenon occurs "on the brink" of the 18th-19th and 19th-20th centuries: in 1796 and 1803,the dates of the spring equinox were respectively 19.83 and 21.53 March, in 1896 and 1903 respectively.- on 20.05 and 21.75 March. All this is shown in Fig.

Fig. Displacement of the moments of the vernal equinox from year to year in the XVII-XX centuries;in each subsequent 400 years the picture repeats, shifting, however, as a whole down to 0d, 12

It can be added that in the second half of the XVII century.every fourth, and at the end of every second year the vernal equinox happened on March 19, there it was and every fourth year at the end of the XVIII century. And, on the contrary, on March 21 it happened only in the first decade of the 17th century.and every first and fourth year in the XVIII century. In the first half of the XX century. The equinox was more often on March 21, in the second - on March 20.Of course, such a large error( 1.5 days!) Noted above in setting the beginning of spring and other seasons in the calendar would be impossible if it were based on, say, a period of 128 or even 33 years, since they have leapyears can be distributed so that the deviation from the average position does not exceed half a day.

It is also obvious that in fact the equinox does not return to the initial moment of the Gregorian calendar. After all, the average for 400 years of this calendar is 0.0003 days longer than the tropical year. Over 400 years this will be 0.12 days or 2 hours 52 minutes 48 seconds. The spring equinox in 2000 will come earlier than in 1600.

For ages or millennia? Next, we will pay attention to the discussion, which once flared around the calendar reform of 1582 All these disputes have long been the property of history. In our time, hardly anyone doubts that the mentioned calendar reform was necessary. It is enough to look at the figure to see this one more time.

Fig. Displacement of the average date of the vernal equinox in: 1- Julian, 2-Gregorian calendars, taking into account the change in the duration of the day.

. Despite all the advantages of the Julian calendar, it still had a serious flaw: the calendar dates are not too fast matched with the specific times of the year. For every( 128 X 30 =) 3800 years, it would have lagged behind them for a whole month, and after about 41,000 years, the spring equinox, having bypassed all the seasons, would have returned to the original date. Thus, the Julian calendar as a solar calendar is quite acceptable for using it for several hundred years, but not millenniums. ..