Page:EB1911 - Volume 04.djvu/1021

when it is divided by the three numbers 28, 19, and 15 respectively the three remainders shall be 10, 2, and 4.

Let x, y, and z be the three quotients of the divisions; the number sought will then be expressed by 28 x + 10, by 19 y + 2, or by 15 z + 4. Hence the two equations

28 x + 10 = 19 y + 2 = 15 z + 4.

To solve the equations 28 x + 10 = 19 y + 2, or y = x + 9x + 8/19, let m = 9x + 8/19, we have then x = 2 m + m − 8/9.

Let m − 8/9 = m′; then m = 9 m′ + 8; hence

x = 18 m′ + 16 + m′ = 19 m′ + 16

(1).

Again, since 28 x + 10 = 15 z + 4, we have

15 z = 28 x + 6, or z = 2 x − 2x − 6/15.

Let 2x − 6/15 = n; then 2x = 15n + 6, and x = 7n + 3 + n/2.

Let n/2 = n′; then n = 2 n′; consequently

x = 14 n′ + 3 + n′ = 15 n′ + 3

(2).

Equating the above two values of x, we have

15 n′ + 3 = 19 m′ + 16; whence n′ = m′ + 4 m′ + 13/15.

Let 4 m′ + 13/15 = p; we have then

4 m′ = 15 p − 13, and m′ = 4 p − p + 13/4.

Let p + 13/4 = p′; then p = 4 p′ − 13;

whence m′ = 16 p′ − 52 − p′ =15 p′ − 52.

Now in this equation p′ may be any number whatever, provided 15 p′ exceed 52. The smallest value of p′ (which is the one here wanted) is therefore 4; for 15 × 4 = 60. Assuming therefore p′ = 4, we have m′ = 60 − 52 = 8; and consequently, since x = 19 m′ + 16, x = 19 × 8 + 16 = 168. The number required is consequently 28 × 168 + 10 = 4714.

Having found the number 4714 for the first of the era, the correspondence of the years of the era and of the period is as follows:—

Era,	1,	2,	3, ...	x,
Period,	4714,	4715,	4716, ...	4713 + x;

from which it is evident, that if we take P to represent the year of the Julian period, and x the corresponding year of the Christian era, we shall have

P = 4713 + x, and x = P − 4713.

With regard to the numeration of the years previous to the commencement of the era, the practice is not uniform. Chronologists, in general, reckon the year preceding the first of the era −1, the next preceding −2, and so on. In this case

Era,	−1,	−2,	−3, ...	−x,
Period,	4713,	4712,	4711, ...	4714 − x;

whence

P = 4714 − x, and x = 4714 − P.

But astronomers, in order to preserve the uniformity of computation, make the series of years proceed without interruption, and reckon the year preceding the first of the era 0. Thus

Era,	0,	−1,	−2, ...	−x,
Period,	4713,	4712,	4711, ...	4713 − x;

therefore, in this case

P =4713 − x, and x = 4713 − P.

Reformation of the Calendar.—The ancient church calendar was founded on two suppositions, both erroneous, namely, that the year contains 3651/4 days, and that 235 lunations are exactly equal to nineteen solar years. It could not therefore long continue to preserve its correspondence with the seasons, or to indicate the days of the new moons with the same accuracy. About the year 730 the venerable Bede had already perceived the anticipation of the equinoxes, and remarked that these phenomena then took place about three days earlier than at the time of the council of Nicaea. Five centuries after the time of Bede, the divergence of the true equinox from the 21st of March, which now amounted to seven or eight days, was pointed out by Johannes de Sacro Bosco (John Holywood, fl. 1230) in his De Anni Ratione; and by Roger Bacon, in a treatise De Reformatione Calendarii, which, though never published, was transmitted to the pope. These works were probably little regarded at the time; but as the errors of the calendar went on increasing, and the true length of the year, in consequence of the progress of astronomy, became better known, the project of a reformation was again revived in the 15th century; and in 1474 Pope Sixtus IV. invited Regiomontanus, the most celebrated astronomer of the age, to Rome, to superintend the reconstruction of the calendar. The premature death of Regiomontanus caused the design to be suspended for the time; but in the following century numerous memoirs appeared on the subject, among the authors of which were Stoffler, Albert Pighius, Johann Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length Pope Gregory XIII. perceiving that the measure was likely to confer a great éclat on his pontificate, undertook the long-desired reformation; and having found the governments of the principal Catholic states ready to adopt his views, he issued a brief in the month of March 1582, in which he abolished the use of the ancient calendar, and substituted that which has since been received in almost all Christian countries under the name of the Gregorian Calendar or New Style The author of the system adopted by Gregory was Aloysius Lilius, or Luigi Lilio Ghiraldi, a learned astronomer and physician of Naples, who died, however, before its introduction; but the individual who most contributed to give the ecclesiastical calendar its present form, and who was charged with all the calculations necessary for its verification, was Clavius, by whom it was completely developed and explained in a great folio treatise of 800 pages, published in 1603, the title of which is given at the end of this article.

It has already been mentioned that the error of the Julian year was corrected in the Gregorian calendar by the suppression of three intercalations in 400 years. In order to restore the beginning of the year to the same place in the seasons that it had occupied at the time of the council of Nicaea, Gregory directed the day following the feast of St Francis, that is to say the 5th of October, to be reckoned the 15th of that month. By this regulation the vernal equinox which then happened on the 11th of March was restored to the 21st. From 1582 to 1700 the difference between the old and new style continued to be ten days; but 1700 being a leap year in the Julian calendar, and a common year in the Gregorian, the difference of the styles during the 18th century was eleven days. The year 1800 was also common in the new calendar, and, consequently, the difference in the 19th century was twelve days. From 1900 to 2100 inclusive it is thirteen days.

The restoration of the equinox to its former place in the year and the correction of the intercalary period, were attended with no difficulty; but Lilius had also to adapt the lunar year to the new rule of intercalation. The lunar cycle contained 6939 days 18 hours, whereas the exact time of 235 lunations, as we have already seen, is 235 × 29.530588 = 6939 days 16 hours 31 minutes. The difference, which is 1 hour 29 minutes, amounts to a day in 308 years, so that at the end of this time the new moons occur one day earlier than they are indicated by the golden numbers. During the 1257 years that elapsed between the council of Nicaea and the Reformation, the error had accumulated to four days, so that the new moons which were marked in the calendar as happening, for example, on the 5th of the month, actually fell on the 1st. It would have been easy to correct this error by placing the golden numbers four lines higher in the new calendar; and the suppression of the ten days had already rendered it necessary to place them ten lines lower, and to carry those which belonged, for example, to the 5th and 6th of the month, to the 15th and 16th. But, supposing this correction to have been made, it would have again become necessary, at the end of 308 years, to advance them one line higher, in consequence of the accumulation of the error of the cycle to a whole day. On the other hand, as the golden numbers were only adapted to the Julian calendar, every omission of the centenary intercalation would require them to be placed one line lower, opposite the 6th, for example, instead of the 5th of the month; so that, generally speaking, the places of the golden numbers would have to be changed every century. On this account Lilius thought fit to reject the golden numbers from the calendar, and supply their place by another set of numbers called Epacts, the use of which we shall now proceed to explain.

Epacts.—Epact is a word of Greek origin, employed in the calendar to signify the moon’s age at the beginning of the year.