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12 months, each containing 30 days, adding 5 to the numnber in ordinary years and 6 in leap years, lo complete the true length of the year. In this way however the advantages of the Greek division were lost, for the months did not correspond with the revolutions of the moon, neither were the 12 months sufficient to make up the year. The French month was also divided into three decades or periods of 10 days each.
The Romans in the time of Romulus the founder of their state, are said to have employed a year consisting of only ten months, viz. Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September, October, November, December; of which the fuur first have appropriated names, but the rest have names expressing their numeral order. Of these months the 1st, 3d, 5th, and 8th consisted of 31 days, and all the others of 30, so that the whole year contained bot 307 days, a period very discordant with the annual course of either the sun or the moon, as well as with the regular returns of the different seasons. This irregularity was observa ed and in some ineasure remedied by Numa the successor of Romulus, who introduced two new months, Januarius of 31 days, and Februarius of 28 days, before Martius, formerly the beginning of the year, which from his time was reckoned to commence on the 1st of Januarius. This improvement however not perfectly corresponding to the course of the sun and moon, many temporary expedients were adopted to bring the year to coincide with the regular appearances of the seasons and the heavenly bodies. At dast Julius Cæsar, to bring forward the several months to their proper places, formed a year of 15 months, or 415 days, which on account of the changes it produced in ordinary affairs was called the year of confusion. This year being ended, the new Julian year began on the 1st of Janua. ry of the year 45 before Christ. Lrom this time the civil year and the months were regulated by the course of the
sun. Numa's year being 10 days shorter than the solar year, two days were added by Cæsar to each of the months of January, August, and December, and one to April, June, September, and November. He directed also an additional day to be introduced every 4th year after the 23d of February; or that the 6th day before the Calends, or 1st day of March, should be twice reckoned ; from which circumstance of repeating the sixth day before the Calends, this 4th year was termed bissextile : amongst us it is usually called leap year, as going beyond the bounds of ordinary years.
Sundry small inaccuracies in this Julian year made it too Jong, so that the vernal equinox, which in Cæsar's time happened on the 25th of March, had in the year 1582 gained somuchon the Calendar as to happen on the 11th of March. In this year, the reigning Pope Gregory XIII. at the instigation of many learned men, published a fresh reformation of the Calendar, in which 10 days were cut off after the 4th of October, and the 5th was reckoned the 15th of that month. To prevent the seasons from again falling back, Gregory ordered a day to be intercalated or introduced in February every 4in year; and that the 1600th year of the Christian era, and every 4th century thereafter, should be a bissextile or leap year; one day is therefore to be intercalaied in the year 2000, 2400, 2800, &c. but in other centuries, as 1700, 1800, 1900, 2100, &c. it is suppressed, these being reckoned common years.
This manner of reckoning time was immediately intro. duced into all countries acknowledging the papal authority, and was called the Gregorian or new style : but in other countries professing the doctrines of the reformation which had by that time made considerable progress in Europe, no immediate alteration was adopted, and even in this country the new style was not employed until the year 1752, when by an act of parliament the 3d day of September was declared
to be the 14th of that month, and the beginning of the year was removed from the 25th of March to the 1st of January. In Russia the old style is still in use.
But after all, the Gregorian calendar is not absolutely correct : the required correction however is so inconsiderable as to amount only to a day and a half to be suppressed in the course of about 5000 years.
Besides the foregoing natural divisions of time others have been devised, called cycles, from a Greek word signifying a circle, because by the revolution of these poriions of time certain appearances of the heavens regularly recur. Of these periods the most inportant are the cycles of the sun and the moon. The solar cycle consists of 28 years, in which time the days of the month return to the same days of the weck, as at the beginning of the cycle; the sun's placeto the same point of the ecliptic, and the leap years to the same course with respect to the days of the week, on which the days of the month fall. The lunar cycle is a revolution of 19 years, at the end of which time the new and full moons return very nearly to the days on which they occurred at its commencement. This period of 19 years was by the antient Greeks marked in letters of gold, for which reason, as also on occount of its utility in chronological computation, it was called the golden number.
The year of our Saviour's birth, according to the common calculation, was the 9th of the solar cycle: when therefore it is required to kuow what year of that cycle is any giien year of Christ, we add 9 to that year and divide the sum by 28, the years in the cycle, the quotient showing how many cycles have elapsed since his birth, and the remainder the year of the remaining cycle.
The first year of Christ being also the first of the lunar cycle, to find the year of this cycle corresponding to any given year of the Christian era, we add 1 to that year
and divide the sum by 19, when the quotient shows the number of cycles elapsed, and the remainder the years
of the current cycle.
If therefore to the year 1809, we add 9, and divide the the sum by 28, we have in the quotieni 64 cycles of the sun, and the remainder 26 is the year of the current 65th solar cycle: again, if to the same year 1809, we add i, and divide the sun by 19, we obtain a quotient of 95 lunar cycles; and the remainder 5 is the year of the current 96th cycle, or the golden number for the year 1809.
The epact is a number denoting the excess of the common solar year above the lunar, by which the age of the moon every year may be found: consequently a table of epacts is only a table of differences between these two sorts of years; and the epact of any year is the number indicating the moon's age in the beginning of that year according to the Calendar. If the new moon fall on the 1st of January, the epact of the ensuing year is nothing; bnt in the beginning of the following year it will be 11, because the lunar Pear is 11 days shorter than the solar; the moon must therefore have changed on the 20th of December preceding, and she will be 11 days old on the 1st January following. In the second year, the epact will be 22 days, in the third year 33 days, and so on : but as 30 days are reckoned for a month, this number being subtracted from 33, will leave 3 for the epact of the third year: hence the epacts will be the following, 0, 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, 18, 29, 10, 21, 2, 23, 24, 5, 16, 27, 8, 19, 0. This series of epacts would be correct if two lunar months were exactly 59 days, and if the civil ycar contained precisely 365 days, with 366 on every lcap year; but this is not precisely the case.
The epact is found in this way: multiply the golden number of the year by 11, add 19 to the product, and divide the sum by 30, when the remainder is the cpact: thus the
epact of the year 1809 is 14. If instead of adding 19 to the above product, we subtract 11 from it, and divide the remainder by 30, the quotient will still be 14 for the epact of the year 1809.
By means of the epact and the golden number, we are enabled to calculate the time of the new and the full moon, and her age on any given day. As the period of the moon's course round the earth, or one lunation, may in round numbers be reckoned 29 days, it is evident that every month of the year, excepting February, must exceed the length of a lunation: if therefore we suppose the moon to be new at the commencement of the ist day of January, the excess of that month over the lunation (or the epact for January,) will be led day, and February containing 28 days in commion years, this excess added to 28 will give 294 days, equal to another Junation. The following new moon coinciding therefore with the beginning of March, that month which consists of 31 days, must again exceed a lunation by 14 day, which is the epact for March. In this manner the following table is formed, showing the expact for every month of the year.
To find the time of the new moon in June, 1809; for instance, we add to the epact of that year, already found ta be 14 days, the epact or number for May, the inonth inmediately preceding June, vix. 3, days; and the sum