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of the week will be the first day of the succeeding year. Now it has been customary to place the first seven letters of the alphabet opposite to the days of the month, and as there are but seven days in the week, the same letter that stands opposite to the Sunday of any week, will stand opposite to all the Sundays of that year, and is therefore called the dominical letter. Now the first of January having the letter A placed opposite to it, in every year, should this day be Monday, the letter G will be the dominical letter for that year, and Monday will be the last day of the year; so that the next year beginning on Tuesday, and having the letter A placed opposite to it, F will be opposite to all the Sundays of that year. For the same reason, E will be the dominical letter for the third year. Thus in seven years all the seven letters would be dominical letters in their turn, beginning at the last and going on in a retrograde order to the first, if they were all common years of 365 days. But as this order is interrupted every fourth year, which contains 366 days, or 52 weeks and two days, the next year must begin two days further in the week than the last, thereby leaping over one, for which reason the fourth is called leap-year. This series cannot return until it be as often interrupted as there are days in the week, that is, seven times. Now as this interruption happens every fourth year, and thereby one day in the week is passed over, it will require 28 years to pass over all the days of the week, after which time the same days of the month will happen on the same days of the week, and the dominical letter be the same that it was 28 years before. As the 28th and 29th of February are suppo. sed to have the same letter affixed to each of them, as was formerly the case with the 23d and 24th, when

the intercalary day was inserted after the 23d of the month, it is evident that the letter which pointed out Sunday before, will not stand opposite to the Sundays after that day. So that in leap-years there will be two dominical letters, the one serving for January and February, and the other, being the letter preceding it in the alphabet, for the remainder of the year. At the birth of Christ, 9 years of this cycle of the sun were elapsed: and therefore if 9 be added to any given year of the christian era, and the sum be divided by 28, the quotient will show how many complete cycles have elapsed, and the remainder will show what year of the cycle, the given year is.

To find the dominical letter; to the given year add its 4th part, rejecting fractions, divide the sum by 7, and the remainder taken from 7, leaves the number of the letter from the beginning of the alphabet:* in leapyears, it and the foregoing are the dominical letters.

The Metonic Cycle, so called from Meton an Athenian, who invented it about 432 years before Christ, is a period of 19 years, containing all the variations of the days in which the new and full moons, with the eclipses of the sun and moon can happen; after which they occur again on the same days of the month, on which they occured 19 years before; only with a variation of 1h 28′ 15" sooner. So that we may expect a new or full moon on the same day of the month, on which it happened 19 years before; only that it will be nearly an hour and a half sooner in the day. This anticipation will in a little more than 16 lunar cycles, or 310.7 years, amount to a whole day, and it has now anticipated five whole days since the council of Nice.

* In the 19th century, 1800 being a common instead of a leapyear, I must be added to this number. Ed.

If you place a unit opposite to the days of the months, in which the new moon happens, through the whole of the first year of the cycle, the number 2 opposite to the like days in the second year of the cycle; and so on, through the whole 19 years, you will have by these numbers, properly disposed in a table, the days of the month opposite to them, in which the new moons happen in each year of the cycle. These 19 numbers for their use and excellence, were denominated the primes or golden numbers. But on account of the beforementioned defect of 1h 28′ 15′′, this table can answer the purpose intended only for 310.7 years, and then it will require that these numbers be set one day higher in the table, by which a pretty regular correspondence may be preserved between the lunar and solar years. These golden numbers thus disposed, must indicate the number of the current year of the cycle at any time. Now as one year of it was elapsed at the birth of Christ, and the prime was 2 in the year in which he was born; if we add 1 to any year of the christian era, and the sum be divided by 19, the remainder will show the year of the current cycle.

At the time of the Nicene council, and for a long time after, it was thought that 19 years, or 228 solar months, were exactly equal to 235 lunar or synodical months, and therefore that the golden numbers, set in their calendars opposite to the days on which the new moons happened through one lunar cycle, would invariably serve for the new moons of every successive lunar cycle. Therefore they ordained that Easter should be kept on the first Sunday after the full moon, which in common years happened after the 21st of March, or after the 20th of March in bissextile years. Hence it can never happen after the 25th of April.

The epact, which depends upon this cycle of 19

years, indicates the age of the moon at the end of the year. As the time of a lunation is 29 days in the nearest whole numbers, there are 354 days=12x29.5 in the lunar year; but this being 11 days less than the solar year, it follows, that if these years should begin together at a new moon, the moon would at the end of it be 11 days old. For the same reason, it would be 22 days old at the end of the second, and 33 days old at the end of the third; but because she will have changed 3 days before the expiration of the year, she will only be 3 days old, and at the end of the next, 14, &c. to the end of the cycle of 19 years; always rejecting 30 for an intercalary month added to the lunar year, when the epact would exceed 30. At the end of the cycle, the epacts would run over again, if the cycle were perfect, and they would always be 11 times the primes.

To find the epact; multiply the golden number for the year by 11, divide the product by 30, and from the remainder take 11, if it be greater than 11, or add 19 to it if it be less, and the difference or sum will give the epact, till the year 1900.

By the intercalations of seven months in 19 solar years, to the lunar year, each month consisting of 30 days, except the last of 29 days, the lunar and solar years would keep pace with each other, if the Metonic cycle had been perfect.

The Dionysian Cycle or period, is the product of the cycles of the sun and moon =28×19=532 years; after which time, not only the new and full moons return on the same days of the month, but also the days of the month return on the same days of the week. Being a compound of the former two cycles, it must have the properties of both. So that after the expira

tion of this period, the dominical letters, days of the week, and of the month, with the new and full moons, return again in the same order. For this reason it is denominated the great Paschal Cycle, and the Dionysian Period.

The Cycle of Indiction is a period of 15 years, used by the Romans for indicating the times of taxation in that empire; but as it has no connexion with the motion of the heavenly bodies, we would have nothing to do with it in these lectures, had it not been that it made an element in another cycle which is of much importance in chronology, and which is called the Julian Period, from Julius Scaliger, who has eternized his name by the invention. Three years of this cycle were elapsed at the birth of Christ.

The Julian Period is a cycle of 7980 years, arising from the continued multiplication of the cycles of the sun, moon, and indiction. As it consists of the other three cycles, it must commence when they all began together, being carried backwards. Therefore its commencement is placed in that imaginary point of time, which was 710 years before the creation of the world; so that 4714 years of it were elapsed at the birth of our Saviour. Being of such extent, it comprehends all other periods, epochas and cycles, with all their memorable transactions, and is not yet completed; so that it is well adapted to the purposes of chronology.

SECONDARY PLANETS.

FOUR of the primary planets belonging to the solar system, viz. the Earth, Jupiter, Saturn and Herschel, are attended in their annual course round the sun by secondary planets, usually denominated moons. The Earth has one, Jupiter four, Saturn five, and Her

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