Why is there a leap day?

Why is there a leap day? In short: Because of custom. The origins of our calendar trace back to Ancient Rome, back then the calculations were more imprecise, adjustments(that is, adding from days to a whole month) were necessary more often, and were customarily inserted before the beginning of each year, and that was the period of time correspondent to February. Our current leap day date is reminiscent of that. Why is there a leap day? Wikipedia says!

In the earliest version of the Roman Calendar, the year started in March and ended in December(only 304 days), leaving most of the winter as a month less period. That changed on subsequent reforms which added Ianuarius and Februarius to the calendar. Back then, the calendar based its calculations solely on the phases of the moon, not the sun, which made it much more irregular/imprecise, and most occasional necessary corrections(i.e. adding days) were performed before the beginning of each year, that is, February.
It was under Julius Cesar that improvements turned it into a fixed 365-days solar calendar with a leap day each four years being instituted, and February continued to be the month chosen to carry on adjustments. The calendar was also renamed Julian Calendar. Further improvements, such as taking out a few leap years by adding the rule that years ending in 00 aren’t to be leap years unless divisible by 400, were instituted during the papacy of Pope Gregor XIII, and that’s why our calendar nowadays is called Gregorian.

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