Re: How is a solar year (365 days, 5 hours, 48 mins, 45+ seconds determined?

A solar day is the time it takes between when the sun is at its highest point one 
day and when it is at its highest point the next day. It is fixed by the speed of 
the Earth's rotation, and it is a few minutes longer than the time it takes the 
Earth to make a full rotation (the sidereal day), because it has moved further 
around its orbit, so it has to turn just a little bit further to put the sun back 
into the same apparent position.

A year is the time it takes the Earth to make a complete circuit of its orbit 
around the sun.

Both of these lengths of time are independently fixed by the way that the Earth 
moves. There is no reason why there should be a simple ratio between them, and in 
fact there is not. 365 days, 5 hours, 48 minutes, 45+ seconds is just what it 
happens to come out at!

When we start talking about Pope Gregory, we have changed from talking about 
astronomical phenomena like the motions of the Earth to human choices and 
cultural phenomena like the calendar. For the calendar, we need a year with a 
whole number of days. We do not want to have a new year start at midnight one 
year, 5.49 a.m. the next, 11.38 a.m. the next, and so on! So instead we start 
each new year at midnight, but on the fourth year, when the new year should start 
at 11.15 p.m., we wait until the following midnight -- that is, we put in an 
extra day.

You will notice, though, that putting in an extra day every fourth year has not 
quite fixed the problem. The new year is starting 45 minutes too late in that 4th 
year! After 32 lots of 4 years = 128 years, we are starting the new year a whole 
day late! We can only fix it by leaving out one of those extra days!

Roughly speaking, that is what happened with the calendar. 

Having a leap year each 4th year is usually associated with the name of Julius 
Caesar, and known as the Julian calendar. The Roman year started in March (which 
is why we get the Latin words for 7,8,9, & 10 in the months Sep, Oct, Nov, and 
Dec), so the extra day got tacked onto February at the end of the year. 

But because of that extra 45 minutes, the calendar was out by about 12 days in 
the time of Pope Gregory. This mattered, because all of the seasons seemed to 
have gotten earlier. To fix it up, his advisers came up with a new calendar, 
which was immediately adopted in catholic Europe, with protestant Western Europe 
following shortly afterward. But some parts of Europe, notably Russia, did not 
adopt the new calendar for a long time, or ran things with two rival calendars!

The date was immediately shifted about 12 days later, which caused riots in the 
streets, because people felt that part of their lives had been stolen from them! 
And from then onward, the leap year day was to be left out on 3 out of 4 exact 
century years -- only century years divisible by 400, like this year, would have 
a leap year day. This Gregorian calendar is the calendar we use today. It is 
still not exactly right, but it is close enough that it will be several thousand 
years before the error amounts to a full day.