42 



ROYAI- SOCIETY OF CANADA 



If secular acceleration were taken into account, the results would 

 be slightly better for the first of the above, in ancient times. But the 

 epochs at which either of these cycles would become exact, owing to 

 secular acceleration, are extremely remote. 



The eclipse cycle of 18 years 10 days, in which eclipses recur, was 

 known to the ancient Chaldeans; and it is related to the draconitic 

 month dependent on the revolution of the moon's nodes, with which 

 we are not now dealing. A cycle of the synodic month can be deduced 

 from it however, as follows: — 



The eclipse cycle may be stated very closely as 183^ years = 223 

 lunations; or clearing of fractions, 649 years = 8,028 lunations. This 

 proves to be just one month too long; and the true synodic cycle is 

 therefore 649 years = 8,027 lunations. The value and error of this 

 cycle are as follows :^ — 



Solar years 649 x 365.2421961 = 237042.1853 day.«. 

 Lunations 8027 x 29.5305S870 = 237042.0355 " 



Lunar year, Epoch 1900 = 354.3670644 



Cycle-value of lunar \ear, 

 or (649 years ^ 8027) x 12 = 354.36728S3 



Error per century. 

 0.02.108 day. 



Error of one day in : 

 4,333 years. 



This cycle is twenty times more accurate than the Metonic cycle; 

 but yet it has only one-third of the accuracy of the next following,, 

 deduced from the numbers in the Book of Daniel. 



This cycle was discovered by M. de Cheseaux in the eighteenth 

 century. He found that the prophetic numbers 1,260 and 2,300, taken 

 a& tropical years, proved to be soli-lunar cycles of remarkable accu- 

 Yacy; and as their small outstanding errors were almost the same, he 

 concluded that the difference of these periods, or 1,040 yca-rs, should 

 be a perfect cycle. This discovery, with further researches based upoo 

 it, are given in " Mémoires posthumes de M. de Cheseaux,^' published 



