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According to tradition, the founding of Rome by the legendary figures Romulus and Remus occurred in the year "753 BC". Of course, ancient Roman scholars would not have referenced that date using the (supposed) birth date of Jesus Christ as a starting point.

I have never seen that date in any other form than as a "BC" year. How do ancient Roman sources specify the founding date, and how did modern historians determine it to be equivalent to 753 BC?

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I am not sure about legends (are there any specifying Rome's foundation year? I suspect it might be in the form of "X years since the sack of Troy"). But since your question also asked about "ancient Roman scholars" and "ancient Roman sources"...

During most of the Roman Republic, years were named based on who had been elected to the consulship for that year. Since Rome was a monarchy, obviously the consular year system did not reach far enough back to the city's foundation.

When Quintus Fabius Pictor began working on historical annals of Rome, he created a new dating method as an alternative to Consular years. Instead, it tracked the number of years since the foundation of Rome: ab urbe condita, lit. "from the foundation of the city". Not all ancient writers agreed on the same foundation year, however. The popular 753 B.C. cited in the question was calculated by Varro Reatinus. Using his system, the year 753 B.C. could then be given as "1 ab urbe condita" or "1 a.u.c.". Varro's epoch became the accepted standard for giving dates in a.u.c. after the mid 1st century or so.

Another system was the Greek Olympiad dating. The first Olympics was thought to have been held in 776 B.C.; this, and subsequent Olympics, provided a conveniently universal (to Greeks) epoch for dating. From about the 3rd century B.C. onward, some Greek writers began giving dates on that basis. For instance, the foundation of Rome has been given as "the first year of the seventh Olympiad" by Dionysius - although that would be 752 B.C., so 753 B.C. would be something like "the fourth year of the sixth Olympiad".

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There is the legend of Romulus and Remus; They were children of gods, raised by a wolf mother and founded Rome in 753 BC –  user45891 Jul 23 at 21:14
    
@user45891 Yes, I know; but I'm feel confident saying that the legend of Romulus and Remus did not include the figure "753 B.C." in itself. Which is the point I was making. –  Semaphore Jul 23 at 21:51
    
It was however the beginning of the legend of the seven kings for which exact dates of reign exists(/were invented). After the last king the Roman Republic was established and I assume that from that point on years could be reliably told. So when that legend was invented during the Roman Republic the birth of the republic was a known fact and then 243 years were assumed as the time Kings ruled Rome (of which Romulus was the first) –  user45891 Jul 23 at 22:04
    
@user45891 that is however entirely offtopic to the question of "how is that year 'given'" - you're instead inferring the year from other sources. Moreover your assumptions are wrong. As I mentioned, different writers had calculated different years. –  Semaphore Jul 24 at 3:20

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