Ab Urbe Condita... b.C.

The datings conversion of the historical events, changing from a time reference to a different one, is often source of mistakes and bad interpretations. In particular, as far as the history of Rome, the events are dated, according with the origin source, starting from the Rome foundation, i.e. "ab Urbe condita" [a.U.c.], or with reference to the birth of Christ, then "before Christ" [b.C.] and "after Christ" or "Anno Domini" [A.D.].
It is just the conversion between the two references that usually generates mistakes, also in authoritative publications; we try therefore to deal with the matter supplying at last practical rules in order to simplify the calculation.

First of all some assumptions have to be agreed:
1) Rome has been founded on April 21st 753 b.C. (that is the 753rd year before the birth of Christ).
2) The years referred to the birth of Christ start from hours 00.00 of January 1st and end at midnight (hours 24.00) of December 31st.
3) In order to simplify, we postpone the birth of Christ from the midnight of December 25th to the midnight of December 31st of the same years; such approximation of six days only will simplify the calculations without distorting the reasoning.
The celebration of the birth of Christ on December 25th  is however conventional as it was  established by the Church in the IV century in order to replace the pagan celebrations to the Sun after the winter solstice; moreover the beginning of the Christian era  is itself a convention as it has been established as December 25th of  753 a.U.c. by Dionysius Exiguus only about in 523 A.D.
4) As a result of the above, the year 1 A.C. starts in the moment of the birth of Christ (hours 00,00 of January 1st), and the year just ended in that moment in which Christ was born is the year 1 b.C.

In the following diagram it is shown a timeline in which the years referred to the birth of Christ are indicated in the upper part while those referred to the Rome foundation are in the lower one; the two scales have a 110 days offset (or 111 in leap years) elapsing between January 1st and April 21st. All the years are considered as time periods of 365 (or 366 in leap years) days from January 1st to December 31st, the same included, for the upper part of the diagram and from April 21st to April 20th of the next solar year, the same included, for the lower part.

We consider the most overlapped part of the year between the different dating systems, that is the one from April 21st and December 31st (this is the normally used convention, mainly when the references about the day and the month of the event are missing): as shown in the diagram, for all the years before the birth of Christ the sum of the numbers of the two corresponding datings (b.C. year  and a.U.c. year) is constant and equal to 754, while for all the years after Christ the two datings differ by 753 years.

To sum up, for the events whose dating is referred to the foundation of Rome (normally they are events before the birth of Christ) the b.C. dating is obtained subtracting from 754 the a.U.c. one and vice versa.
E.g.: 245 a.U.c. (expulsion of the king Tarquinio il Superbo from Rome) corresponds to 754-245=509 b.C.

Conversions to a.U.c. datings are not normally performed for events after Christ's birth, other than in particular cases when an anniversary celebration is referred to monuments or events relevant to Rome history or, nearly as a curiosity, in websites dedicated to Rome.
In such cases you have only to add 753 to the current year (but, strictly speaking, only from the April 21st onwards) to obtain the corresponding one. However, if an anniversary is to be celebrated, so referred to a particular month and day, you'll have to pay attention to the exact time location of the event, that it is a defined moment, within the year, that it is an extended period of time.
On this subject it is enough to say that the Rome Council celebrated 2750 years from the foundation of Rome (event that happened, of course, just at the completion of the full year 2750 a.U.c.) on April 21st 1997, that is one year in advance, as shown in the diagram below.

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