Sunday, March 29, 2009

On Cycle-Accuracy

The Lunar Cycle
We have a tradition that the average Molad interval is 29 days, 12 hours, and 793 chalakim, where there are 1080 chalakim in an hour.  Written differently, 29 days, 12 hours, 40 minutes, 73 chalakim (this is the way it is written in Pirkei d'Rebbi Eliezer).  
In Pirkei d'Rebbi Eliezer chapter 7 it is written:
The total of the days of the lunar year is 354 days, 8 hours and 876 chalakim.  All the hours of the lunar month are 708 hours and two thirds of an hour.  All the hours of a lunar year are 8504 hours.
That is to say that it gives the precise length of the lunar year, and then goes on to get imprecise.  First it drops the 876 chalakim, and then it reworks the calculations without the "pesky" chalakim.  If all the hours of a lunar month are 708 hours and 2/3 of an hour, then that means that a lunar month is 29 days, 12 hours, and 40 minutes.  That means that the imprecision is 73/1080 of an hour, or approximately 4 minutes.
The Solar Cycle
The year that we use for Birkat HaChamah is 365 days and 6 hours.  The difference between this and the solar year of Rav Adda is 365 days, 5 hours, and approximately 997 chalakim, or 365 days, 5 hours, 2/3 of an hour, and 277 chalakim.  The difference between the two is about 83 chalakim per year, or about 4.6 minutes per year.
In both places, we have inaccuracies on the "small" scale of less than 5 minutes per unit (lunar month or solar year). 

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