Proleptic Julian calendar
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The proleptic Julian calendar is produced by extending the Julian calendar backwards to dates preceding AD 4 when the quadrennial leap year stabilized. The leap years that were actually observed between the implementation of the Julian calendar in 45 BC and AD 4 were erratic: see the Julian calendar article for details.
A calendar obtained by extension earlier in time than its invention or implementation is called the "proleptic" version of the calendar. Likewise, the proleptic Gregorian calendar is occasionally used to specify dates before the introduction of the Gregorian calendar in 1582. Because the Julian calendar was used before that time, one must explicitly state that a given quoted date is based on the proleptic Gregorian calendar if that is the case.
Historians since Bede have traditionally represented the years preceding AD 1 as "1 BC", "2 BC", etc. Bede and later Latin writers chose not to place a "year zero" (nulla in Latin) between the years 1 BC and AD 1. Thus the year 1 BC would be a leap year, being four years before AD 4. However to help to determine an interval in years across the BC/AD boundary, it is more convenient to use a slightly different convention that includes a year zero and to represent earlier years as negative numbers. This is the convention used in "astronomical year numbering". In this system the year 0 is equivalent to 1 BC in Bede's system, and is a leap year, though in actuality there were no leap years in the actual calendar at this time due to the leap year error.
The Roman calendar was somewhat erratic especially before 45 BC. The best-known example is the year 46 BC: as part of the reform that initiated the Julian calendar of that name, 46 BC was allotted 445 days by Julius Caesar. Before then, the Romans added whole intercalary months in an unsystematic way. Between 45 BC and AD 4, the leap day was also unsystematic. Thus there is no simple way to find an equivalent in the proleptic Julian calendar of a date quoted using the Roman pre-Julian calendar (AUC or by reference to consuls).
See also