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part, is extremely modest, and the seats assigned by him to the new moons tally with strict astronomi- cal findings in a degree which he seems never to have anticipated. The impossibility of taking the astro- nomical moons as our sole guide in finding the date of Easter will be best understood from an example: Let us suppose that Easter is to be kept (as is at least im- phed by the British Act of Parliament regulating its date) on the Sunday after the astronomical full moon, and that this full moon, as sometimes happens, occurs just before midnight on Saturday evening in the western tlistricts of London or New York. The full moon will therefore happen a little after midnight in the eastern districts, so that Easter, if regulated strictly by the paschal full moon, must be kept on one Sunday in the western and on the following Sunday in the eastern districts of the same city. Lest it be thought that this is carrying astronomical exactness to extremes, we may say that, if Easter were dependent on the as- tronomical moons, tlie feast could not always be kept on the same Sunday in England and America. Seeing, therefore, that astronomical accuracy must at some point give way to convenience and that an arbitrary decision on this point is necessary, the Church has drawn up a lunar calendar which maintains as close a relation with the astronomical moons as is practicable, and has decreed that Easter is to be kept on the Sunday after the fourteenth day of the paschal moon as indi- cated by this calendar.

Metonic Lunar Cycle or Cycle of Golden Num- bers. — In the year now known as 432 b. c, Meton, an Athenian astronomer, discovered that 235 lunations (i. e. lunar months) correspond with 19 solar years, or, as we might express it, that after a period of 19 solar years the new moons occur again on the same days of the solar year. He therefore divided the calendar into periods of 19 years, which he numbered 1, 2, 3, etc. to 19, and assumed that the new moons would always fall on the same days in the years indicated by the same number. This discovery found such favour among the Athenians that the number assigned to the current year in the Metonic Cycle was henceforth written in golden characters on a pillar in the temple, and, whether owing to this circumstance or to the impor- tance of the discovery itself, was known as the Golden Number of the year. As the 19 years of the Metonic Cycle were purely lunar (i. e. each contained an exact number of lunar months) and contained in the aggre- gate 23.3 lunations, it was clearly impossible that all the years should be of equal length. To twelve of the 19 years 12 lunations were assigned, and to the other seven 13 lunations, the thirteenth lunation being known as the embolismic or intercalary month.

Length of the Lunations. — The latest calculations have shown that the average duration of the lunar month is 29 days, 12 hours, 44 mins., 3 sees. To avoid the difficulty of reckoning fractions of a day in the cal- endar, all computators, ancient and modern, have as- signed 30 and 29 day s alternately to the lunations of the year, and regarded the ordinary lunar year of 12 luna- tions as lasting 354 days, whereas it really lasts some 8 hours and 48 mins. longer. This under-estimation of the year is compensated for in two ways: (1) by the insertion of one extra day in the lunar (as in the solar) calendar every fourth year, and (2) by assigning 30 days to six of the seven embolismic lunations, although the average lunation lasts only about 29i days. A comparison of the solar and lunar calendars for 76 years (one cycle of 19 years is unsuitable in this case, since it contains sometimes 4, sometimes 5, leap years) will make this clearer: — 76 solar years=76X 365+19, i. e. 27,759 days. Therefore 940 calendar lunations (since 19 years equal 235 lunations) contain 27,759 days (29 d., 12 hrs., 44 mins., 3 secs.X940= 27,758 d,, 18 hrs., 7 mins.). But9401unationsaverag- ing29V days equal only 27,730 days. Consequently, if we assign 30 and 29 days uninterruptedly to alternate V— 31

lunations, the lunar calendar will, after 76 years, antici- pate the solar by 29 days. The intercalation of the extra day every fourth year in the lunar calendar re- duces the divergence to 10 days in 76 years, i. e. 2j days in 19 years. The divergence is removed by assigning to the seven embolismic months (which would other- wise have contained 7X29V, or 206i, days) 209 days, 30 days being assigned to each of the first six and 29 to tlie seventh.

M.\NNER OF Insertion of the Embolismic Months. — As the Gregorian and Jletonic calendars differ in the manner of inserting the embolismic months, only the former is spoken of here. It has just been said that seven of the 19 years of the lunar cycle contain a thir- teenth, or embolismic, month, consisting in six cases of 30 days and in the seventh of 29 days. Granted that the first solar and lunar years begin on the same day (i. e. that the new moon occurs on 1 January), it is evident that, as the ordinary lunar year of 12 lunations is 1 1 days shorter than the solar, the lunar calendar will, after three years, anticipate the solar by 33 days. To the third lunar year, then, is added the first embolismic month of 30 days, reducing the divergence between the calendars to three days. After three further years, i. e. at the end of the sixth year, the divergence will have mounted to 36 (3X11 + 3) days, but, by the insertion of the second embohsmic lunation, will be reduced to six days. Whenever, then, the divergence between the calendars amounts to more than 30 days, an embolismic month is added to the lunar year; at the end of the nineteenth lunar year, the divergence will be 29 days, and, as the last embolismic month con- sists of 29 days, it is clear that after the insertion of this month the nineteenth solar and lunar years will end on the same day and that the first new moon of the twentieth (as of the first) year will occur on 1 January. The divergence, therefore, at the end of the 19 succes- sive years of the lunar cycle is: 11, 22, 3, 14, 25, 6, 17, 28, 9, 20, 1, 12, 23, 4, 15, 26, 7, IS, and days.

Cycle of Ep.^cts. — We have defined an epact as the age of the moon on 1 January, i. e. at the beginning of the year. If, then, the new moon occurs on 1 January in the first yearof the LunarCyde, the Epact of the year is or, as it is more usually expressed, *; and, since the lunar year always begins with the new moon, it is clear that the divergence between the solar and lunar calendars, of which we have just been speaking, gives the Epacts of the succeeding years. Thus, after the first year, the divergence between the calendars amounts to 11 days; therefore, the new moon occurs 11 days before 1 January of the second solar year, which is expressed by saying that the Epact of the second solar year is XI. Granted, then, that the new moon occurs on 1 January in the first year of the Lunar Cycle, the epacts of the 19 years are as follows:

Golden Numbers 1 2 3 4

Epacts * XI xxii III

5 6 7 8 9 10

XIV XXV VI XVII XXVIII IX

Golden Numbers 11 12 13 14

Epacts XX I XII xxin

15 16 17 18 19

IV XV XXVI VII XVIII

Inaccuracy' of the Metonic Cy'cle. — Meton's theory, as adopted by the Church until the year 1582, might be briefly expressed as follows: —

'19 lunar years aver- Days aging 354i days,

The average Lunar Cycle consists of

.6730?

6 extra, or embolis- mic, months of 30 days, i. e. 180

1 embolismic month

of 29 days 29

Total 6939}