Our Calendar Part 4

RULE.

Multiply the difference between the Julian and the solar year by 100, and we have the error in 100 years. Multiply this product by 4 and we have the error in 400 years. Now, 400 is the tenth of 4,000; therefore, multiply the last product by 10, and we have the error in 4,000 years. Now, as the discrepancy between the Julian and Gregorian year is three days in 400 years, making 3-400 of a day every year, so by dividing 365-1/4, the number of days in a year, by 3-400, we have the time it would take to make a revolution of the seasons.

SOLUTION.

(365 d, 6 h.) - (365 d, 5 h, 48 m, 49.62 s.) = (11 m, 10.38 s.) Now, (11 m, 10.38 s.) 100 = 18 h, 37.3 m, the gain in 100 years. This is, reckoned in round numbers, 18 hours, or three-fourths of a day. Now, (3/4 4) = (1 3) = 3: the Julian rule gaining three days, the Gregorian suppressing three days in 400 years. (3 10) = 30, the number of days gained by the Julian rule in 4,000 years. 365-1/4 3 400 = 48,700, so that in this long period of time, this falling back 3/4 of a day every century would amount to 365-1/4 days; therefore, 48,699 Julian years are equal to 48,700 Gregorian years.

CHAPTER II.

ERRORS OF THE GREGORIAN CALENDAR.

By reference to the preceding chapter it will be seen that there is an error of 37.3 minutes in every 100 years not corrected by the Gregorian calendar; this amounts to only .373 of a minute a year, or one day in 3,861 years, and one day and fifty-two minutes in 4,000 years.

RULE.

To find how long it would take to gain one day: Divide the number of minutes in a day by the decimal .373, that being the fraction of a minute gained every year. To find how much time would be gained in 4,000 years, multiply the decimal .373 by 4,000, and you will have the answer in minutes, which must be reduced to hours.

SOLUTION.

(24 60) .373 = 3,861, nearly; hence the error would amount to only one day in 3,861 years.

(.373 4,000) 60 = (24 h, 52 m,) = (1 d, 0 h, 52 m), the error in 4,000 years.

This trifling error in the Gregorian calendar may be corrected by suppressing the intercalations in the year 4,000, and its multiples, 8,000, 12,000, 16,000, etc., so that it will not amount to a day in 100,000 years.

RULE.

Divide 100,000 by 4,000 and you will have the number of intercalations suppressed in 100,000 years. Multiply 1 d, 52 m, (that being the error in 4,000 years) by this quotient, and you will have the discrepancy between the Gregorian and solar year for 100,000 years. By this improved method we suppress 25 days, so that the error will only amount to 25 times 52 minutes.

SOLUTION.

100,000 4,000 (1 d, 52 m,) = (25d, 21 h, 40 m.) Now, (25d, 21 h, 40 m,) - 25 d = (21 h, 40 m,) the error in 100,000.

CHAPTER III.

DOMINICAL LETTER.

Dominical (from the Latin _Dominus_, Lord,) indicating the Lord's day or Sunday. Dominical letter, one of the first seven letters of the alphabet used to denote the Sabbath or Lord's day.

For the sake of greater generality, the days of the week are denoted by the first seven letters of the alphabet, A, B, C, D, E, F, G, which are placed in the calendar beside the days of the year, so that A stands opposite the first day of January, B opposite the second, C opposite the third, and so on to G, which stands opposite the seventh; after which A returns to the eighth, and so on through the 365 days of the year.

Now, if one of the days of the week, Sunday for example, is represented by F, Monday will be represented by G, Tuesday by A, Wednesday by B, Thursday by C, Friday by D, and Sat.u.r.day by E; and every Sunday throughout the year will have the same character, F, every Monday G, every Tuesday A, and so with regard to the rest.

The letter which denotes Sunday is called the Dominical or Sunday letter for that year; and when the dominical letter of the year is known, the letters which respectively correspond to the other days of the week become known also. Did the year consist of 364 days, or 52 weeks invariably, the first day of the year and the first day of the month, and in fact any day of any year, or any month, would always commence on the same day of the week. But every common year consists of 365 days, or 52 weeks and 1 day, so that the following year will begin one day later in the week than the year preceding. Thus the year 1837 commenced on Sunday, the following year, 1838, on Monday, 1839 on Tuesday, and so on.

As the year consists of 52 weeks and 1 day, it is evident that the day which begins and ends the year must occur 53 times; thus the year 1837 begins on Sunday and ends on Sunday; so the following year, 1838, must begin on Monday. As A represented all the Sundays in 1837 and as A always stands for the first day of January, so in 1838 it will represent all the Mondays, and the dominical letter goes back from A to G; so that G represents all the Sundays in 1838, A all the Mondays, B all the Tuesdays, and so on, the dominical letter going back one place in every year of 365 days.

While the following year commences one day later in the week than the year preceding, the dominical letter goes back one place from the preceding year; thus while the year 1865 commenced on Sunday, 1866 on Monday, 1867 on Tuesday, the dominical letters are A, G and F, respectively. Therefore, if every year consisted of 365 days, the dominical cycle would be completed in seven years, so that after seven years the first day of the year would again occur on the same day of the week.

But this order is interrupted every four years by giving February 29 days, thereby making the year to consist of 366 days, which is 52 weeks and two days, so that the following year would commence two days later in the week than the year preceding, thus the year 1888 being leap-year, had two dominical letters, A and G; A for January and February, and G for the rest of the year. The year commenced on Sunday and ended on Monday, making 53 Sundays and 53 Mondays, and the following year, 1889, to commence on Tuesday. It now becomes evident that if the years all consisted of 364 days, or 52 weeks, they would all commence on the same day of the week; if they all consisted of 365 days, or 52 weeks and one day, they would all commence one day later in the week than the year preceding; if they all consisted of 366 days, or 52 weeks and two days, they would commence two days later in the week; if 367 days or 52 weeks and three days, then three days later, and so on, one day later for every additional day. It is also evident that every additional day causes the dominical letter to go back one place. Now in leap-year the 29th day of February is the additional or intercalary day. So one letter for January and February, and another for the rest of the year. If the number of years in the intercalary period were two, and seven being the number of days in the week, their product would be 2 7 = 14; fourteen, then, would be the number of years in the cycle. Again, if the number of years in the intercalary period were three, and the number of days in the week being seven, their product would be 3 7 = 21; twenty-one would then be the number of years in the cycle. But the number of years in the intercalary period is four, and the number of days in the week is seven, therefore their product is 4 7 = 28; twenty-eight is then the number of years in the cycle.

This period is called the dominical or solar cycle, and restores the first day of the year to the same day of the week. At the end of the cycle the dominical letters return again in the same order, on the same days of the month. Thus, for the year 1801, the dominical letter is D; 1802, C; 1803, B; 1804, A and G; and so on, going back five places every four years for twenty-eight years, when the cycle, being ended, D is again dominical letter for 1829, C for 1830, and so on every 28 years forever, according to the Julian rule of intercalation.

But this order is interrupted in the Gregorian calendar at the end of the century by the secular suppression of the leap-year. It is not interrupted, however, at the end of every century, for the leap-year is not suppressed in every fourth centurial year; consequently the cycle will then be continued for two hundred years. It should be here stated that this order continued without interruption from the commencement of the era until the reformation of the calendar in 1582, during which time the Julian calendar, or Old Style was used.

It has already been shown that if the number of years in the intercalary period be multiplied by seven, the number of days in the week, their product will be the number of years in the cycle. Now, in the Gregorian calendar, the intercalary period is 400 years; this number being multiplied by seven, their product would be 2,800 years, as the interval in which the coincidence is restored between the days of the year and the days of the week.

This long period, however, may be reduced to 400 years; for since the dominical letter goes back five places every four years, in 400 years it will go back 500 places in the Julian and 497 in the Gregorian calendar, three intercalations being suppressed in the Gregorian every 400 years.

Now 497 is exactly divisible by seven, the number of days in the week, therefore, after 400 years the cycle will be completed, and the dominical letters will return again in the same order, on the same days of the month.

In answer to the question, "Why two dominical letters for leap-year?" we reply, because of the additional or intercalary day after the 28th of February. It has already been shown that every additional day causes the dominical letter to go back one place. As there are 366 days in leap-year, the letter must go back two places, one being used for January and February, and the other for the rest of the year. Did we continue one letter through the year and then go back two places, it would cause confusion in computation, unless the intercalation be made at the end of the year. Whenever the intercalation is made there must necessarily be a change in the dominical letter. Had it been so arranged that the additional day was placed after the 30th of June or September, then the first letter would be used until the intercalation is made in June or September, and the second to the end of the year. Or suppose that the end of the year had been fixed as the time and place for the intercalation, (which would have been much more convenient for computation,) then there would have been no use whatever for the second dominical letter, but at the end of the year we would go back two places; thus, in the year 1888, instead of A being dominical letter for two months merely, it would be continued through the year, and then pa.s.sing back to F, no use whatever being made of G, and so on at the end of every leap-year. Hence it is evident that this arrangement would have been much more convenient, but we have this order of the months, and the number of days in the months as Augustus Caesar left them eight years before Christ. The dominical letter probably was not known until the Council of Nice, in the year of our Lord 325, where, in all probability, it had its origin.

CHAPTER IV.

RULE FOR FINDING THE DOMINICAL LETTER.

Divide the number of the given year by 4, neglecting the remainders, and add the quotient to the given number. Divide this amount by 7, and if the remainder be less than three, take it from 3; but if it be 3 or more than 3, take it from 10 and the remainder will be the number of the letter calling A, 1; B, 2; C, 3, etc.

By this rule the dominical letter is found from the commencement of the era to October 5th, 1582. O. S. From October 15th, 1582, till the year 1700, take the remainder as found by the rule from 6, if it be less than 6, but if the remainder be 6, take it from 13, and so on according to instructions given in the table on 49th page. It should be understood here, that in leap-years the letter found by the preceding rule will be the dominical letter for that part of the year that follows the 29th of February, while the letter which follows it will be the one for January and February.

EXAMPLES.

To find the dominical letter for 1365, we have 1365 4 = 341 +; 1365 + 341 = 1706; 1706 7 = 243, remainder 5. Then 10 - 5 = 5; therefore E being the fifth letter is the dominical letter for 1365.

To find the dominical letter for 1620, we have 1620 4 = 405; 1620 + 405 = 2025; 2025 7 = 289, remainder 2. Then 6 - 2 = 4; therefore, D and E are the dominical letters for 1620; E for January and February, and D for the rest of the year. The process of finding the dominical letter is very simple and easily understood, if we observe the following order:

1st. Divide by 4.

2d. Add to the given number.

3d. Divide by 7.