Certain precautions, however, are necessary for avoiding mistakes, when we recur to remote astronomical observations, where the accurate interval of days is required, however large. For, 1. Although 100 Julian years contain 36,525 days, we must keep in mind, that from A.D. 300, to A.D. 1600, there is a deficiency of ten days from the whole amount; eleven days from A.D. 1600, to A.D. 1900; and twelve days from A.D. 1900, to A.D. 2200.

2. In astronomical tables, the years before the Christian Era are counted backwards: the supposed year of the Nativity being marked 0 *, the foregoing, 1; the year before that, 2; &c. But Chronologers generally reckon the year of the Nativity, B.C. 1; the foregoing, B. C. 2, &c. as follows:

Julian years 41, 42, 43, 44, 45,
4, 3, 2, 1, 0,
5, 4, 3, 2, 1, 0,

Astronom.
Chronol.

46, 47, 48, 49.

1, 2, 3, 4.
1, 2, 3, 4.

The Astronomers are wrong, and the Chronologers right. For the commencement of the Era, or the fixed point from which we ought to reckon, both forwards and backwards, is midnight; which divides the old from the new year: it is the limit between both, and as not belonging to either, should be marked zero, or O: but the Julian 45, terminating therein, is a real quantity, as much as the year 46, and should be marked -1, or the first of the regressive series, as the latter, +1, is the first of the progressive.

To prevent such ambiguities chiefly, the Julian Period, of 7980 years, whose 4714th year corresponded to A.D. 1, was invented by Joseph Scaliger, who called it so †, as being accommodated to the Julian year, beginning the first of January. By the help of this, the years of the regressive series may be critically ascertained. As an Astronomical Cycle, indeed, it is of considerable use.

CYCLES.

Of the various Cycles, or recurring periods of years, noticed by Chronologers, the oldest and most celebrated is

*This is the case, not only in the old astronomical tables of Calvisius, &c. but in the modern, Pingrè, &c.

† Novam nostram periodum-Julianam vocavimus, quia ad annum Julianum accommodata, et quidem à Kalendis Januarii. p. 361.

I. THE CYCLE OF THE MOON.

The ancient Greeks were required, by the Laws and by the Oracles, to sacrifice according to Months, Days, Years; which they all interpreted, to regulate their years by the Sun; their months and days by the Moon. And their principal Games, the Olympic, were required to be celebrated every fifth year, (like the Roman lustra,) at the full moon next after the summer solstice. But as the lunar year was eleven entire days shorter than the solar, their great object was to bring both to a conformity, as near as may be, by occasionally intercalating months in the former, to fill up the deficiency. Meton, therefore, a celebrated Astronomer, found out, B. C. 430, that by intercalating 7 lunations in 19 lunar years, they were brought so nearly to a correspondence in length with 19 solar years, that the times of celebrating their games and festivals could be adjusted, both to the new and full moons, and to the equinoxes and solstices, with little variation, by the help of his tables. For a public service so important, he was declared victor in the first class, at the Olympic games, a statue was decreed to him, and his Tables were inscribed upon a marble pillar, in letters of gold. Hence the current years of his Cycle were called the Golden number.

But it was found that his Cycle was imperfect: for in 19 years there are sometimes five leap years, at other times only four. Hence the new and full moons computed thereby, sometimes varied a day from the truth, in leap years, after February. The only way to remedy this error, was by extending the Cycle to 4 × 1976 years, in which there are 19 leap years exactly. This, we may suppose, led Calippus, in the ensuing century, to publish the Calippic period of 76 years, as an improvement of the Metonic, beginning B. C. 330.

But the lunar and solar Tables, constructed on the Lunar Cycle, will require correction, after 310 years. For 235 lunations, amounting to 6939 d. 16 h. 32 m. 28 sec. are deficient from 19 Julian years, or 6939 d. 18 h. by a remainder of 1h. 27 m. 31 sec. or within an hour and half: and this interval in about 310, or 312 years, will amount to a whole day: so that the new moon, at the end of this period, will anticipate its time at the beginning of it, an entire day, earlier.

Not adverting to this, in our Prayer-books, before the altera

tion of the style, are given "Tables for finding Easter for ever," by the help of the Golden Number. In consequence of which, the Ecclesiastical Easter has several times been a week different from the true Easter, before A.D. 1752. But since the New Style, that inconvenience has been remedied, by giving Tables to find Easter only for limited times: 1. from 1765 to 1899, inclusive; and 2. from 1900 to 2199, inclusive: by shifting the Golden Numbers a day later in the latter Table than in the former.

To find the Golden Number, see the Rules given in the Prayer-Book, Beverege, Ferguson, &c.

II. THE CYCLE OF THE SUN.

This is, strictly speaking, the Cycle of the Sunday letter. Since a common year consists of 52 weeks, and one day over; if the first year begins on Sunday, the second will begin on Monday, the third on Tuesday, &c. the seventh on Saturday, and the eighth on Sunday again. And so, if all the years were common, it would be a cycle of seven years. But this order is interrupted by the Bissextile every fourth year, in which there are two days over: and this will prevent the recurrence of Sunday again, at the beginning of the year, until after a cycle of 4x7=28 years. The same will hold of any other day of the week.

To find the Dominical, or Sunday letter, see the rules in the foregoing books.

III. THE CYCLE OF INDICTION.

This was a cycle of 15 years, used only by the Romans, for appointing the times of certain public taxes; as appears from the title in the Code, De tributo indicto. It was established by Constantine, A.D. 312, in the room of the heathen Olympiads; and was used in the acts of the General Councils, Emperors, and Popes.

IV. THE JULIAN PERIOD.

This celebrated period was formed by the continued multiplication of the Cycles of the Sun, 28 years; of the Moon, 19 years; and of Indiction, 15 years; whose product gives 7980 years. In its first year, the Cycle of the Sun was 1, of the Moon 1, and of Indiction 1; but the three Cycles can never so correspond again, until the end of it. Every intermediate year