Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/740

668 

Of the and - Periods.—The  year, consisting of twelve lunar months, contains only 354 days; its commencement consequently anticipates that of the solar year by eleven days, and passes through the whole circle of the s in about thirty-four  years. It is therefore so obviously ill-adapted to the computation of, that, excepting the modern and s, almost all s who have regulated their months by the  have employed some method of  by means of which the beginning of the year is retained at nearly the same fixed place in the s.

In the early ages of the year was regulated entirely by the. divided the year into twelve months, consisting alternately of twenty-nine and thirty days, the former of which were called deficient months, and the latter full months. The year, therefore, contained 354 days, falling short of the exact  of twelve s by about 8⋅8 hours. The first expedient adopted to reconcile the and solar years seems to have been the addition of a month of thirty days to every second year. Two years would thus contain 25 months, or 738 days, while two solar years, of 365$1⁄4$ days each, contain 730$1⁄2$ days. The difference of 7$1⁄2$ days was still too great to escape observation; it was accordingly proposed by of, who flourished shortly after the  of, to omit the   every eighth year. In fact, the 7$1⁄2$ days by which two years exceeded two solar years, amounted to thirty days, or a full month, in eight years. By inserting, therefore, three additional months instead of four in every period of eight years, the coincidence between the and  year would have been exactly restored if the latter had contained only 354 days, inasmuch as the period contains 354× 8+ 3× 30= 2922 days, corresponding with eight solar years of 365$1⁄4$ days each. But the true of 99 s is 2923⋅528 days, which exceeds the above period by 1⋅528 days, or thirty-six hours and a few s. At the end of two periods, or sixteen years, the excess is three days, and at the end of 160 years, thirty days. It was therefore proposed to employ a period of 160 years, in which one of the months should be omitted; but as this period was too long to be of any practical use, it was never generally adopted. The common practice was to make occasional corrections as they became necessary, in order to preserve the relation between the period and the ; but these corrections being left to the care of incompetent persons, the calendar soon fell into great disorder, and no certain rule was followed till a new division of the year was proposed by  and, which was immediately adopted in all the s and  of. The mean motion of the in, from the mean, during a Julian year of 365⋅25 days (according to Hansen's Tables de la Lune, , 1857, pages 15, 16) is, at the present date, 13× 360°+ 477644″⋅409; that of the being 360°+ 27″⋅685. Thus the corresponding relative mean geocentric motion of the from the  is 12× 360°+ 477616″⋅724; and the duration of the mean synodic  of the, or lunar month, is therefore $360°⁄12× 360°+ 477616″⋅724$× 365⋅25= 29⋅530588days, or 29days, 12hours, 44. 2⋅8.

The ic Cycle, which may be regarded as the chef-d'œuvre of, is a period of nineteen solar years, after which the new s again happen on the same days of the year. In nineteen solar years there are 235 s, a number which, on being divided by nineteen, gives twelve s for each year, with seven of a remainder, to be distributed among the years of the period. The period of, therefore, consisted of twelve years containing twelve months each, and seven years containing thirteen months each; and these last formed the third, fifth, eighth, eleventh, thirteenth, sixteenth, and nineteenth years of the cycle. As it had now been discovered that the exact length of the is a little more than twenty-nine and a half days, it became necessary to abandon the alternate succession of full and deficient months; and, in order to preserve a more accurate correspondence between the civil month and the, divided the cycle into 125 full months of thirty days, and 110 deficient months of twenty-nine days each. The number of days in the period was therefore 6940. In order to distribute the deficient months through the period in the most equable manner, the whole period may be regarded as consisting of 235 full months of thirty days, or of 7050 days, from which 110 days are to be deducted. This gives one day to be suppressed in sixty-four; so that if we suppose the months to contain each thirty days, and then omit every sixty-fourth day in reckoning from the beginning of the period, those months in which the omission takes place will, of course, be the deficient months. The number of days in the period being known, it is easy to ascertain its accuracy both in respect of the and motions. The exact length of nineteen solar years is 19× 365⋅2422= 6939⋅6018 days, or 6939days 14hours 26⋅592s; hence the period, which is exactly 6940 days, exceeds nineteen s of the by nine and a half hours nearly. On the other hand, the exact of a synodic  of the  is 29⋅530588days; 235s, therefore, contain 235× 29⋅530588= 6939⋅68818 days, or 6939days 16hours 31s, so that the period exceeds 235 s by only seven and a half hours.

After the ic cycle had been in use about a century, a correction was proposed by. At the end of four cycles, or seventy-six years, the accumulation of the seven and a half hours of difference between the cycle and 235 s amounts to thirty hours, or one whole day and six hours., therefore, proposed to quadruple the period of, and deduct one day at the end of that by changing one of the full months into a deficient month. The period of, therefore, consisted of three ic cycles of 6940 days each, and a period of 6939 days; and its error in respect of the , consequently, amounted only to six hours, or to one day in 304 years. This period exceeds seventy-six true solar years by fourteen hours and a quarter nearly, but coincides exactly with seventy-six Julian years; and in the of  the length of the solar year was almost universally supposed to be exactly 365$1⁄4$ days. The period is frequently referred to as a date by.

.—The calendar, which is adopted in all the, and most of the   of , is -, being regulated partly by the solar, and partly by the lunar year,—a circumstance which gives rise to the distinction between the  and s. So early as the 2d century of , great disputes had arisen among the  respecting the proper  of celebrating , which governs all the other s. The  celebrated their  on the 14th day of the first month, that is to say, the lunar month of which the fourteenth day either falls on, or next follows, the day of the. 