« A.U.C. 710 = 44 B.C. »
While the Julian calendar began operation on Kal. Ian. A.U.C. 709 = 45, leap years were initially inserted every third year instead of every fourth, until the error was corrected by Augustus. The most detailed description of this reform is given by the fifth century author Macrobius Saturnalia 1.14.13. He states that after Caesar's death the pontiffs caused the leap day to be inserted "at the beginning of every fourth year instead of at its end" (i.e., since the Romans counted inclusively, every third year instead of every fourth) for 36 years, after which time there had been 12 leap days in a period that should have had 9.
The analysis of A.U.C. 730 = 24 shows that A.U.C. 746 = 8 was the leap year that ended the 12th triennial cycle. Hence the first ended in A.U.C. 713 = 41, which is also documented as a leap year, and must have begun in A.U.C. 710 = 44.
Dio Cassius 48.33.4 implies that prid. Kal. Ian. A.U.C. 713 = 41 was a market day, while Dio Cassius 40.47 states that Kal. Ian. A.U.C. 702 = 52 was also a market day. These two dates were therefore a multiple of 8 days apart. It can be shown that the intervening Republican years were all regular, i.e. 355 days each, except for A.U.C. 702 = 52 itself, which was 378 days long, and that Caesar added 90 days to A.U.C. 708 = 46. Hence the length of this period, as otherwise accounted, is 378 + 5×355 + 445 + 5×365 = 4,423 = 7 (mod 8) days. Therefore there was precisely one extra leap day between A.U.C. 709 = 45 and A.U.C. 714 = 40.
The 36th year before A.U.C. 746 = 8, counting inclusively, is A.U.C. 711 = 43. This suffices to rule out A.U.C. 712 = 42 and A.U.C. 713 = 41. Also Dio states that a leap day was inserted into A.U.C. 713 = 41 in order to avoid a market day on Kal. Ian. A.U.C. 714 = 40, which was ill-omened. If the extra leap day was in A.U.C. 711 = 43 or either of the next two years then Kal. Ian. A.U.C. 711 = 43 would also have been a market day.
Therefore the extra bissextile day was in A.U.C. 709 = 45 or A.U.C. 710 = 44. Both candidates are before Caesar's assassination on Id. Mart. A.U.C. 710 = 14 March 44. Therefore this bissextile day was a leap day that Caesar intended. If it was in A.U.C. 709 = 45 then the leap day in A.U.C. 713 = 41 was not have been "against the rule" as Dio states. Hence it was in A.U.C. 710 = 44, which is also the beginning of the first triennial cycle. It follows that A.U.C. 710 = 44 was 366 days long and A.U.C. 709 = 45 was 365 days long.
This argument depends critically on the analysis of the length of the years between A.U.C. 702 = 52 and A.U.C. 713 = 41. Perhaps the weakest point is A.U.C. 708 = 46, which relies on secondary evidence, although primary evidence shows that the total intercalation of this year must have been much in excess of 67 days. Scaliger thought this year was 444 days long, which would imply two Julian leap days. This solution is only possible on Scaliger's triennial phase, in which A.U.C. 709 = 45 and A.U.C. 712 = 42 were both leap years. However, the evidence considered here is against this phase. The MS tradition of Macrobius makes A.U.C. 708 = 46 443 (or 440) days long, which would require three Julian leap days (which is impossible) or at least one error in the analysis of the late Republican calendar (which is very unlikely).
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