What Socratidion is saying is that the Romans reckoned dates much differently than we do. Instead of calling today the 21st of June, they would call it "11 days before the calends of July".
Instead of the year being 2013 AD, it would be 2766 AUC, the number of years since the traditional founding of Rome.
Random trivia: Leap day is actually on February 24, not the 29th. The Romans actually inserted a day between the 23rd and the 24th on leap years instead of putting it at the end of the month. I seem to recall finding evidence that the 24th was legally leap day in England some few hundred years ago, but I don't know where I would look to find that again...
I am pleased to hear that I am correct.
Thirteen years ago, fourteen inclusive: annum quartum decimum
Using the accusative case, I thought this might mean the period of years to the fourteenth year, compared to using the ablative case for on a year or day. This might explain 'diem quintum' in the date, meaning the period of days before the next key day to the fifth day. Am I correct?