I don't recall exactly how the conversation turned towards leap days, but it did. I think it may have been something to do with birthdays and being born on February 29th, the 29th being the leap day once every four years.
“But it's not,” I said. “It's actually February 24th.”
“How is it the 24th and not the 29th?” asked Spring.
“It has something to do with the Roman calendar,” I said. “But I'll have to find it on the Calendar FAQ.”
And here it is, from the Calendar FAQ, § 2.7.1:
2.7.1. How did the Romans number days?
The Romans didn't number the days sequentially from 1. Instead they had three fixed points in each month:
“Kalendae” (or “Calendae”), which was the first day of the month.
“Idus”, which was the 13th day of January, February, April, June, August, September, November, and December, or the 15th day of March, May, July, or October.
“Nonae”, which was the 9th day before Idus (counting Idus itself as the 1st day).
The days between Kalendae and Nonae were called “the 5th day before Nonae”, “the 4th day before Nonae”, “the 3rd day before Nonae”, and “the day before Nonae”. (There was no “2nd day before Nonae”. This was because of the inclusive way of counting used by the Romans: To them, Nonae itself was the first day, and thus “the 2nd day before” and “the day before” would mean the same thing.)
Similarly, the days between Nonae and Idus were called “the Xth day before Idus”, and the days after Idus were called “the Xth day before Kalendae (of the next month)”.
Julius Caesar decreed that in leap years the “6th day before Kalendae of March” should be doubled. So in contrast to our present system, in which we introduce an extra date (29 February), the Romans had the same date twice in leap years. The doubling of the 6th day before Kalendae of March is the origin of the word “bissextile”. If we create a list of equivalences between the Roman days and our current days of February in a leap year, we get the following:
7th day before Kalendae of March 23 February 6th day before Kalendae of March 24 February 6th day before Kalendae of March 25 February 5th day before Kalendae of March 26 February 4th day before Kalendae of March 27 February 3rd day before Kalendae of March 28 February the day before Kalendae of March 29 February Kalendae of March 1 March
You can see that the extra 6th day (going backwards) falls on what is today 24 February. For this reason 24 February is still today considered the “extra day” in leap years (see section 2.3). However, at certain times in history the second 6th day (25 Feb) has been considered the leap day.
Why did Caesar choose to double the 6th day before Kalendae of March? It appears that the leap month Intercalaris/Mercedonius of the pre-reform calendar was not placed after February, but inside it, namely between the 7th and 6th day before Kalendae of March. It was therefore natural to have the leap day in the same position.
So there you go … February 24th is the leap day, not the 29th.