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Nones the ninth day before the Ides.

Ides was near the middle of the month, either the 13th or the 15th day.

The first day of each month was invariably called the Calends. The Nones were the fifth, and the Ides the thirteenth, except in March, May, July, and October, in which the Nones occurred on the seventh day and the Ides on the fifteenth.

From these three points the days of the month were numbered--not forward, but backward--as so many days before the Nones, the Ides, or the Calends, the point of departure being counted in the reckoning, so that the last day of every month was the second of the Calends of the following month.

It will be seen by the Roman and English calendar found on the following pages, that there are six days of Nones in March, May, July and October, and four of all the other months; also that all the months have eight days of Ides. The number of days of Calends depend upon the number of days in the month, and the day of the month on which the Ides fall.

If the month has thirty-one days and the Ides fall on the thirteenth, there are nineteen days of Calends; but if the Ides fall on the fifteenth, there are only seventeen days of Calends. As the Ides fall on the thirteenth of all the months of thirty days, they have eighteen days of Calends. February, the month of twenty-eight days, has only sixteen, except in leap-year, when the sixth of the Calends is reckoned twice.

It may also be seen from the calendar that the Romans, after the first day of the month, began to reckon so many days before the Nones, as 4th, 3d, 2d, then Nones; after the Nones, so many days before the Ides, as 8th, 7th, 6th, etc., and after the Ides, so many before the Calends of the next month, the highest numbers being reckoned first.

In reducing the Roman calendar to our own, it should be remembered that in reckoning backward from a fixed point, that the point of departure is counted; also, that the last day of the month is not the point from which the Calends are reckoned, but the first day of the following month. We have then this rule for finding the English expression for any Latin date:

RULE.

In the table the three months are taken to illustrate how easily the change may be made from Roman to English, or from English to Roman date. A complete calendar for 1892, both in Roman and English, which will be very convenient for reference, may be found on the four following pages.

The first seven letters of the alphabet, used to represent the days of the week, are placed in the calendar beside the days of the week. The letter that represents Sunday is called the dominical or Sunday letter. The letter that represents the first Sunday in any given year represents all the Sundays in that year, unless it be leap-year, when two Sunday letters are used. The first represents all the Sundays in January and February, while the letter that precedes it represents all the Sundays for the rest of the year.

PART SECOND.

MATHEMATICAL.

ERRORS OF THE JULIAN CALENDAR.

It will be necessary in the first place to understand the difference between the Julian and Gregorian rule of intercalation. If the number of any year be exactly divisible by four it is leap year; if the remainder be 1, it is the first year after leap-year; if 2, the second; if 3, the third; thus:

And so on, every fourth year being leap-year of 366 days.

This is the Julian rule of intercalation, which is corrected by the Gregorian by making every centurial year, or the year that completes the century, a common year, if not exactly divisible by 400; so that only every fourth centurial year is leap-year; thus, 1,700, 1,800, and 1,900 are common years, but 2,000, the fourth centurial year, is leap year, and so on.

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.

ERRORS OF THE GREGORIAN CALENDAR.

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.

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.

SOLUTION.

DOMINICAL LETTER.

Dominical 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 Saturday 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.

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, 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 passing 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.

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.

EXAMPLES.

We divide by 4 because the intercalary period is four years; and as every fourth year contains the divisor 4 once more than any of the three preceding years, so there is one more added to the fourth year than there is to any of the three preceding years; and as every year consists of 52 weeks and one day, this additional year gives an additional day to the remainder after dividing by 7. For example, the year

Hence the numbers thus formed will be 1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and so on.

We divide by 7, because there are seven days in the week, and the remainders show how many days more than an even number of weeks there are in the given year. Take, for example, the first twelve years of the era after being increased by one-fourth, and we have

From this table it may be seen that it is these remainders representing the number of days more than an even number of weeks in the given year, that we have to deal with in finding the dominical letter.

Did the year consist of 364 days, or 52 weeks, invariably, there would be no change in the dominical letter from year to year, but the letter that represents Sunday in any given year would represent Sunday in every year. Did the year consist of only 363 days, thus wanting one day of an even number of weeks, then these remainders, instead of being taken from a given remainder, would be added to that number, thus removing the dominical letter forward one place, and the beginning of the year, instead of being one day later, would be one day earlier in the week than in the preceding year.

From the commencement of the Christian era to October 5th, 1582, take the remainders, after dividing by 7, from 3 or 10; from October 15th,

RULE FOR FINDING THE DAY OF THE WEEK OF ANY GIVEN DATE, FOR BOTH OLD AND NEW STYLES.

At Dover Dwells George Brown, Esquire, Good Carlos Finch, and David Fryer.

Now if A be dominical or Sunday letter for a given year, then January and October being represented by the same letter, begin on Sunday; February, March and November, for the same reason, begin on Wednesday; April and July on Saturday; May on Monday, June on Thursday, August on Tuesday, September and December on Friday. It is evident that every month in the year must commence on some one day of the week represented by one of the first seven letters of the alphabet. Now let

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