YOL. XLVI.] PHILOSOPHICAL TRANSACTIONS. 35 



again be set one day back at the end of 400 years, and so on, as in the preceding 

 2800 years. By which means the golden numbers would always point out the 

 mean times of the new moons, within a day of the truth. 



It is plain however that the lunar year will have lost one diiy more than ordi- 

 nary, with respect to the solar year, whenever the new moons shall have antici- 

 pated a whole day ; as they will have done at those times, when it is necessarj' 

 that the golden numbers should, by the rule just now given, be set back one 

 day : and consequently the epact, for that and the succeeding years, must ex- 

 ceed by an unit the several corresponding epacts of the preceding 1 Q years. 



For the epact is the difference, in whole days, between the common Julian 

 solar and the lunar year ; the former being reckoned to consist of 365, and the 

 latter of only 354 days. If therefore the solar and the lunar year at any time 

 should commence on the same day, the solar would, at the end of the year, 

 have exceeded the lunar by 1 1 days ; which number 1 1 would be the epact of 

 the next year : 22 would be the epact of the year following, and 33 the epact of 

 the year after that, the epacts increasing yearly by 1 1 . But as often as this 

 yearly addition makes the epact exceed 30, those 30 are rejected as making an 

 intercalary month, and only the excess of the epact above 30 is accounted the 

 true epact for that year. Thus when the epact would amount to 31, 32, 33, 34, 

 &c. the 30 is rejectetl, and the epact becomes 1, 2, 3, 4, &c. 



Since therefore the lunar year will have lost a day more than ordinary, in re- 

 spect of the solar year, whenever it is necessary to set the golden numbers one 

 day back, as before observed ; it follows, that the epact must at the same time 

 be increased by a unit more than usual ; the difference between the solar and the 

 lunar year having been just so much greater than usual. That is, 12 must be 

 added, instead of 1 1 , to the epact of the preceding, in order to form what will 

 be the epact of the then present year. Which addition of a unit extraordinary 

 to one epact, will occasion all the subsequent epacts (which will follow each other 

 in the usual manner, each exceeding the foregoing by U) to be greater by 1, 

 than their respectively corresponding epacts of the preceding IQ years. 



If therefore, instead of the golden numbers, the epacts of the several years 

 were prefixed, in the. manner the Gregorians have done, to the days of the ca- 

 lendar, in order to denote the days on which the new moons fall in those years 

 of which those numbers are the epacts ; there would never be occasion to shift 

 the places of those epacts in the calendar ; since the augmentation by 1 extraor- 

 dinary of the epacts themselves would answer the purpose, and keep all tolerably 

 right. Thus in a very easy method may the course of the new moons be pointed 

 out, either by the golden numbers, or by the epacts, according to the Julian 

 account or manner of adjusting the year, which goes on regular and uniform 

 without any variation. 



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