CHRONOLOGY. The excess of the solar year above the lunar is 11 days ; or the epact of any year expresses the num ber of days from the last new moon of the old year, which was the beginning of the present lunar year, to the first of January. The first year of the cycle of the moon, the epact is 0, because the lunar year begins with the solar. On the second, the lunar year has begun 11 days before the solar year, therefore the epact is 11. On the third, it has begun twice 11 before the solar year, therefore the epact is 22. On the fourth, it begins three times 11 days sooner than the solar year, the epact would therefore be 33 ; but 30 days, being a synodical month, must that year he intercalated ; or that year must be reckoned to consist of thirteen synodical months, and there re mains three, which is the true epact of the year ; and so on to the end of the cycle, adding 11 to the epact of the last year, and always rejecting 30, gives the epact of the present year. Thus, to ad just the lunar year to the solar through the whole of 19 years, 12 of them must consist of 12 synodical months each, and 7 of 13, by adding a month of 30 days to every year when the epact would exceed 30, and a month of 29 days to the last year of the cycle, which makes in all 209 days, i, e. 19x11; so that the inter. ciliary or embolimxan years in this cycle are 4, 7, 10, 12, 15, 18, 19.
If the new moons returned exactly at the same time after the expiration of nineteen years, as the council of Nice supposed they would do (when they fix ed the rule for the observation of Easter, and marked the new moons in the calen dar for each year of the lunar cycle) then the golden number, multiplied by 11, would always give the epact. But in a Julian century, the new moons antici pate, or happen earlier, than that coun cil imagined they would by of a day.
In a Gregorian common century, which is one day shorter than a Julian century, they happen H. of a (lay later, (1 day— = Now 4i X 3 for the vci three common centuries, but oeing subtracted, on account of the Gregorian bissextile century, there will remain Therefore, in four Gregorian centuries, the new moons will happen later by #4, of a day, and the epacts must be de creased accordingly.
At present the Gregorian epact is 11 days short of the Julian epact ; but the quotient of the number of the centuries divided by 4, which at this time is 4, mul tiplied by 23, with the addition of the re mainder 1 multiplied by 44, makes in all but IN, or 7 days 4-14 ; therefore 41, z. e. 3 days + if must be added to com plete the 11 days. Whence we have the following General rule for finding the Gregorian, Epact for ever. Divide the centuries of any year of the Christian wra by 4, (re jecting the subsequent numbers ; multi ply the remainder by 17, and to this pro duct add the quotient multiple(' by 43: divide the product + 86 by 25 : multiply the golden number by 11, from which subtract the last quotient ; and rejecting the thirties, the remainder will be the epact.
Example for 1808.
18=9=2 2 X 17=34 43 x 4 34 =206 206+ 86 25 = 11 11 X 4 (Gold. No.) = 44 44-11 = 1— 3 = Epact.
30 A shorter rule for the epact until the year 1900. Subtract 1 from the gold en number, and multiplying the der by 11, reject the thirties, and you have the epact.
Example for the year 1808.
Golden Number 4. 4 — 1 x 11 —30 = 3 = Epact.