of the most remarkable achievements of that most remarkable man. The Roman year had previously been a lunar year, which of course requires constant readjustment by intercalation, to keep it in practical harmony with the solar year. In B.C. 46, it was found that the months were occurring far from the seasons with which they were supposed to be connected. It was necessary to make this "year of confusion year of confusion" to consist of 445 days to get things right again. Caesar wisely abandoned the lunar year altogether, but so far deferred to usage (it is said) as to fix the commencement of his first reformed year on the day of the following new moon instead of on the day of the winter solstice. At all events a new moon actually occurred on January 1, B.C. 45. The mean Julian year consists of 3651 mean solar days; and as a year suitable for everyday purposes cannot contain fractions of a day, the rule adopted was that three years in succession should consist of 365 days, and that every fourth year should consist of 366 days. Thus the average length of each of the four years is 365 days. The year of 366 days is called "bissextile" because the additional, or intercalated day, was inserted after February 24, and, in the Roman method of reckoning, this day is the sixth day before the Kalends of March. So that in every fourth year there were two "sixth days" before the Kalends of March, and hence the name bissextile." Leap year," the other and more familiar name for the year of intercalation, is so called because the day of the week corresponding to any particular day of the month, after the intercalary day, advances two places with reference to its position in the preceding year, instead of one place as in ordinary cases. Thus January 1, 1916, is a Saturday; but since 1916 is a leap year, January 1, 1917, is a Monday, instead of being a Sunday, as it would have been had 1916 been a common year.

The Julian calendar has thus the merit of great simplicity, but unfortunately, as time went on, it was found to be subject to considerable inaccuracy, and it was considered that reformation was desirable. In the middle of the 16th century it appeared that the spring equinox, which ought to have occurred on March 21 (the day on which it was assumed to have occurred at the time of the Council of Nica) actually occurred on March 11. Luigi Lilio, a native of Calabria, found the error of the mean Julian year to amount to about three days in 400 years. His scheme, submitted to Pope Gregory XIII, was that ten days should be dropped, so as to bring the equinox up to March 21 again, and that a more accurate length of the mean

year should be adopted. The Pope referred the matter to a commission, the principal member of which was a German Jesuit named Schlüssel, better known by his Latinised name of Clavius. It was decided, in order to bring up the spring equinox to what was considered to be the proper date, that the day after October 4, 1582, should be called October 15, and in order to correct for the assumed error in the length of the mean Julian year, of three days in 400 years, that the centennial years should be counted as leap years only when the number of centuries is divisible by four. Thus the years 1700, 1800, and 1900, which in the Julian calendar are leap years, are common years in the reformed calendar, whilst the year 2000 is a leap year in both calendars. The Gregorian calendar was immediately adopted in Roman Catholic countries, but the old style remained in force in England until 1752. The accumulation of error in the Julian reckoning having by that time amounted to eleven days, it was decided that the day after September 2 in that year should be called September 14. It will be noted that this change does not involve any change in the week-days, but only in their numeration as days of the month. Wednesday, September 2, was followed by Thursday, September 14. in Russia and Greece, where the old style is still continued, the day of the week is the same as with us, only the day of the month is different. Thus Monday, March 15, new style, corresponds to Monday, March 2, old style, the difference of the styles now amounting to 13 days.

And

It will be found that the mean length of the Gregorian year is 365-2425 days. The actual length of the tropical year being 365-2422 days, the error of the mean Gregorian year amounts to 3 ten-thousanths of a day, or 26 seconds, per annum, or to one day in about 3,300 years. This is sufficiently accurate for practical purposes. It may, however, he pointed out that as the error of the mean Julian year amounts, with great exactness, to one day in 128 years, greater accuracy would have been attained by following the rule that one intercalary day should be dropped in every such period. But the practical inconvenience of this arrangement would be much greater than that of the Gregorian rule, for which the increased accuracy would scarcely be a sufficient compensation.

It must be understood that the difference of styles causes a great deal of trouble, and is always a possible source of confusion to those who have to take account of it. And many a time astronomers and chronologists are constrained to wish that Pope Gregory and his advisers had adopted the alternative

scheme of assigning the spring equinox to March 11, instead of dropping ten days of the year. But the idea that the spring equinox had been assigned to March 21 by a Church Council was too firmly rooted in men's minds to be disregarded, and the opportunity of effecting a simple and natural reformation of the calendar was lost for ever. That great astronomer, the late Professor Newcomb, boldly asserted that, in his opinion, the so-called reformation of the calendar was a mistake; that it would have been far better to have adhered to the Julian style rather than that people should be worried by the inconvenience caused by the break of continuity. His view was that the change of the seasons relatively to the civil date, consequent on adherence to the old style, would progress so slowly as not to cause any practical inconvenience to the general public.

It is worth noting that our calendar does not rigidly fix the actual spring equinox to March 21; there is an oscillation backwards and forwards extending over two days. At the present time the equinox frequently occurs on March 20.

The next point to engage our attention is the determination of the day of the week corresponding to a given day of the civil month in a given year. To find Easter Day we must know what days of the year are Sundays. This is accomplished by means of the Dominical Letters, the use of which, as adopted in the Prayer Book calendar, we must now consider.

The Dominical, or Sunday, Letters are the first seven letters of the alphabet attached to the several days of the year: A to January 1, B to January 2, C to January 3, and so on, over and over again, throughout the year. No letter is attached to February 29, the intercalary day in the English Ecclesiastical and Civil Calendar. To find the Sundays throughout the year (for a common year) it is then only necessary to note what letter is attached to the first Sunday in the year, and every day throughout the year to which that letter is attached is a Sunday, and the letter is called the Dominical, or Sunday, Letter for the year. Thus January 3, 1915, was a Sunday, therefore C is the Sunday Letter for 1915, and every day in the year to which the letter C is attached in the calendar is a Sunday. In leap years. the same letter (D) applies to February 29 and to March 1, so that after February 29 the Sunday Letter for the year retrogrades one place. There are thus two Sunday Letters in a leap year: one from the beginning of the year up to February 29, and the other for the remainder of the year. For example, in 1916 the Sunday Letters are B A. As a common year consists of 52 weeks plus one day, and a leap year of 52 weeks plus two

days, it is evident that from one common year to the next, the Sunday Letter retrogrades one place, whilst after a leap year the Sunday Letter retrogrades two places. It appears, then, that knowing the Sunday Letter for any year-knowing for instance (as all chronologists ought to know) that January 1, A.D. 1, wast a Saturday, with corresponding Sunday Letter B-it is easy to write down a formula from which the Sunday Letter for any other year may be found. A number, occurring in this formula, has to be modified from time to time so as to adapt it to cases of the occurrence, or non-occurrence, of leap years in centennial years of the Gregorian calendar. This formula, translated into ordinary language, with the necessary modifications during successive periods, and the corresponding scale, is given in the Prayer Book calendar. It is not necessary, therefore, to dwell further on this point, except to note that in leap years the Sunday Letter so found will be the second letter for the year, the first being the preceding one in the Prayer Book scale referred to above.

We now come to the most complicated of the problems connected with the determination of Easter Day. To carry into effect the decree of the Council of Nicea it was necessary to determine the fourteenth day of the moon. But the Council did not say how this fourteenth day was to be found, the duty of determining it being assigned to the Bishop of Alexandria. This arrangement naturally caused a good deal of dissatisfaction to the ecclesiastical authorities at Rome. It was considered derogatory to the Papal See, and efforts were made to render the Western Church independent of Alexandria. This eventuated, in A.D. 437, in the decision arrived at by Hilarius (afterwards Pope), that the moon which governed the date of Easter should not be the real moon of the heavens, but should be an artificial moon supposed to move regularly, and that the full moon should be accounted as occurring on the fourteenth day. The phases of this artificial moon were to be computed by means of the Golden Numbers of the Metonic Cycle, on the assumption that 235 lunations are equivalent to 19 solar years. This artificial moon, and the corresponding Golden Numbers, are still used in the reformed ecclesiastical calendar in the way that must now be explained.

The Golden Numbers are the numbers attached to each year of a cycle of nineteen years, after which the calendar new moons fall on the same days of the Julian year. Thus, if a new moon falls on January 1 in any year, it will again fall on January 1 after a lapse of nineteen Julian years, and to each

of these years the same Golden Number would be attached. This cycle is said to have been discovered by Meton, a celebrated Athenian astronomer, about the year B.C. 433, and was called from him the Metonic Cycle; and the successive years of the cycle, with the dates of the new moons corresponding to each year, were inscribed in characters of gold upon the walls of the temple of Minerva. Hence the origin of the name Golden Numbers." In the distribution of the Golden Numbers over the successive years of the Metonic Cycle, it was assumed (as indeed was an actual fact at the date of the Council of Nicaea) that a new moon fell on January 1 in the third year of the cycle. The year 0 (or B.C. 1) of our era is reckoned the first year of the cycle; therefore, to find the Golden Number for any year, “add one to the year of our Lord, and then divide by 19: the remainder, if any, is the Golden Number: but if nothing remaineth, then 19 is the Golden Number," to quote the words of the Prayer Book rule.

The determination of Easter by this system made it recur, under the Julian calendar, after each period of 28 x 19, or 532 years. This period was called the Paschal Cycle. It was used as a practical means of finding the date of Easter, for a long time before the introduction of the Gregorian calendar.

Before the change of style was introduced into the ecclesiastical calendar it was the practice to attach their proper Golden Number to each of the 235 days of the year which were the computed first days of lunations. Twelve of the Numbers appeared twelve times, and seven appeared thirteen times. This left 130 days in a common year, and 131 in a leap year, without any Golden Number. There are, therefore, this number of days in the year upon which the first day of an artificial lunation does not occur. But in the reformed calendar, as now given in the Prayer Book, a different plan is adopted. It was considered more convenient to indicate the fourteenth day of the calendar moon (being the day of "full" moon) rather than the first day, and it was considered unnecessary to indicate other fourteenth days except those, nineteen in number, which fall in the respective years between March 21 and April 18, both inclusive. It was found that the fourteenth day of the Easter moon must fall between these limits-hence called the "Paschal Limits"-and that Easter Day must consequently fall on one of the thirty-five days, March 22 to April 25, both inclusive. There are thus only thirty-five possible forms of the ecclesiastical almanac. With regard to the accuracy of the Metonic Cycle as a practical means of