Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/743

years. In the course of this long period no two years can be expressed by the same numbers in all the three cycles. Hence, when the number of any proposed year in each of the cycles is known, its number in the Julian period can be determined by the resolution of a very simple problem of the indeterminate analysis. It is unnecessary, however, in the present case to exhibit the general solution of the problem, because when the number in the period corresponding to any one year in the common era has been ascertained, it is easy to establish the correspondence for all other years, without having again recourse to the direct solution of the problem. We shall therefore find the number of the Julian period corresponding to the first of our era.

We have already seen that the year 1 of the era had 10 for its number in the solar cycle, 2 in the lunar cycle, and 4 in the cycle of indiction; the question is therefore to find a number such, that when it is divided by the three numbers 28, 19, and 15 respectively, the three quotients 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 resolve the equation 28 x + 10 = 19 y + 2, or y = x + 9 x + 8/19, let m = 9 x + 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 − x − 6/15.

Let x − 6/15 = n; then 2 x = 15 n + 6, and x = 7 n + 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 Nice. 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 John of Sacrobosco, in a work published under the title 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 Sextus 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 Stöffler, Albert Pighius, John Schöner, Lucas Gauricus, and other mathematicians of celebrity. At length Pope Gregory XIII. perceiving that the measure was likely to confer a great eclat 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 commencement of the year to the same place in the seasons that it had occupied at the time of the Council of Nice, 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 present century is twelve days. From 1900 to 2100 inclusive it will be 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