How Many Days Are There in a Year?

Astronomy of antiquity was founded in solving a problem that we don't really consider today to be astronomical at all. Ancient peoples noticed the regularity of the heavens and found that they were the most accurate way they had of setting their calendars. But how many days are there in a year? Well, we enlightened moderners would say that's easy, there are 365. Well, unless it's a leap year, in which case it's 366. And finally, how do we keep time? The principle issue is that there is not an even number of days in a year.