160 A. M. W. DOWNING, M.A., D.SC, F.R.S., ON 



representing the elates of phases of the moon, it is assumed 

 that 235 calendar lunations (of thirty or twenty-nine days' 

 duration, combined in a certain proportion) are equal to 6,939 j 

 days, which, again, are equal to nineteen mean Julian years ; 

 whence a mean calendar lunation equals 29 days 12 hours 

 44 minutes 25*5 seconds ; being 22 - 7 seconds in excess of the 

 mean astronomical lunation. But in adapting the cycle to the 

 Gregorian style we have to take account of the assumed error 

 of the mean Julian year, viz., three days in 400 years ; and so 

 (allowing for the centennial years not made bissextile in the 

 new style) we find that the time of calendar full moon will 

 advance (i.e., fall later) three days in 400 years. Also it must 

 be noted that 6,939J days are 1\ hour longer than 235 mean 

 astronomical lunations, and therefore (on account of this error 

 in the adopted length of the mean calendar lunations), the 

 calendar full moons occur 1J hour too late at the end of each 

 cycle of nineteen years, or 1 day too late in 308 years. In 

 the calendar it is assumed that the error from this cause 

 amounts to 8 days in 2,500 years. And the correction 

 necessary to keep the calendar full moons in fair agreement 

 with the actual full moons is applied by subtracting 1 day 

 from the date of calendar full moon whenever the error amounts 

 to this quantity. 



If we now examine the Prayer Book tables (which were 

 drawn up by Bradley, and extend to the year 8500 of our era), 

 we shall see that the Golden Numbers are affixed to different 

 days at different periods of time, e.g., the first Prayer Book 

 table holds good until the year 2199, and after that a readjust- 

 ment is required. This readjustment is really the application to 

 the cycle of Golden Numbers of the two corrections referred to 

 above. The first, i.e., that depending on the difference between 

 the Gregorian and the Julian style, consists in adding one day 

 to the date of full moon, or shifting the Golden Numbers to a 

 position one day later in each of the years 1700, 1800, 1900, 

 2100, etc., which are leap years in the Julian calendar, but are 

 common years in the Gregorian style. The second correction 

 referred to, i.e., that depending on the error in the assumed 

 length of the calendar lunation, consists in subtracting one day 

 from the dates of full moon, or shifting the Golden Numbers to 

 a position one day earlier in each of the years 1800, 2100, etc. 

 So that the same system of Golden Numbers holds good from 

 1700 to 1899, another system holds good from 1900 to 2199, 

 whilst yet another holds good from 2200 to 2299. An 

 examination of the distribution of the nineteen Golden Numbers, 



