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This arrangement totals 2,930 days in the first two quadrennia and 2,906 days in the second, for a total of 5,836 days. This figure results in an estimate of the year length of 5,836/16 = 364.75 days, which is about half a day short of the actual length of the solar year.
Interestingly, Macrobius claims that this scheme is a Roman adaptation of the Greek practice of octaeteris, which intercalated a 13th month in three of every eight years, but you have no way of verifying this allegation.
Thus, the exact scheme of the Republican calendar eludes us. It was undoubtedly a lunar calendar and employed a scheme of intercalation of a month on a frequent basis. The error was sufficiently great that officials were designated to administer the process of intercalation. These pontifices greatly abused the power given to them, apparently as the consequence of gross incompetence in some cases and the result of political cunning in others. These abuses eventually created chaos in the Roman calendar.
9.1.3
Julian Calendar
In this section, I examine the Julian calendar which, with the essentially minor correction introduced by Gregory, is the calendar accepted as a global standard today.
9.1.3.1
The Pontifices
The three principal priests, or sacerdotes, of Rome were the rex sacrorum, the flamen Dialis, and the pontifex maximus, or chief pontiff of the college of pontifices. The pontifices were a class of priests, four to 16 in number at different times in history, who met in the regia, or official residence of the pontifex maximus and rex sacrorum. The pontifices were permitted to hold civil office as well, and the pontifex maximus was automatically a member of the Senate. Among the duties of the pontifices was the administration of the calendar.
The pontifices, by arbitrarily altering the lengths of years to favor their political friends, eventually made a complete mess of the calendar. It was in his capacity as Pontifex Maximus that Julius Caesar introduced the calendar reform that now bears his name. Centuries later, Pope Gregory would reform the calendar again, also in his capacity as Pontifex Maximus, a title passed from the Roman Empire to the Roman Catholic Church. This title is the origin of the title Supreme Pontiff.
9.1.3.2
The Julian Reform
Calendar reform, when it is prompted by a calendar that is not synchronized with the solar year, almost invariably is a two-step process:
1.) A one-time correction to re-establish certain desired relationships between calendar dates and astronomical events.

 
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