The Gregorian Calendar eventually replaced the Roman calendar at some point, I know this was something to do with the days not being quite accurate so specific days (I think equinox and solstice) were not consistent after some point. So what was the motivation behind this? If the change came centuries later wasn't it already known?
Who determined the actual time scale used, how was this established and determined, especially leap years? Was there any analysis over the method used?
Essentially the extra quarter of a day that the Julian leap year added was slightly longer than the 0.242 of a day left over in the actual solar year. It affected Pope Gregory XIII because the Christian holidays were being celebrated on the wrong days. This was noticed by the Pope's astronomers and prompted the need for change.
What's the science behind this?
The calendar currently in worldwide use for secular purposes based on a cycle of 400 years comprising 146,097 days, giving a year of average length 365.2425 days. The Gregorian calendar is a modification of the Julian calendar in which leap years are omitted in years divisible by 100 but not divisible by 400. By this rule, the year 1900 was not a leap year (1900 is divisible by 100 and not divisible by 400), but the year 2000 will be a leap year (2000 is divisible by 400). The total number of days in 400 years is therefore given by
The following National Geographic Article does a great job of describing the advent of the Gregorian Calendar. Also included, below, is a snippet from the Vatican's official website and more from Science World.
...
Leap Year Needed to Correct Calendar Drift
We observe the modern leap year because Earth orbits the sun every 365.242 days—not an easy number for a calendar to accommodate.
...
The so-called Julian calendar reorganized the 12 Roman months into a 365-day year with a leap year every four years. It was a tremendous improvement but with a lingering flaw: The extra quarter of a day that the leap year added was slightly longer than the 0.242 of a day left over in the actual solar year. This seemingly small difference made the solar year about 11 minutes too long, resulting in an entire day of discrepancy every 128 years. Because of this glitch, the Julian calendar had drifted ten days by the late 16th century. "Finally it became so ridiculous that Pope Gregory XIII was convinced by his astronomers that basically all the Christian holidays were being celebrated on the wrong days," Duncan said. The pope introduced his Gregorian calendar in 1582, which determined that only one out of every four "century years" would observe a leap year. Thus while the years 2000 and 2400 are leap years, 2100, 2200, and 2300 are not. The Gregorian calendar was gradually, and sometimes grudgingly, adopted by much of the world and remains in common use.
...
Source: National Geographic News - Leap year (why)
Brian Handwerk for National Geographic News Updated February 29, 2012
As noted the Gregorian calendar was not wholly accepted directly after the Pope's decree. In fact, the rest of Europe did not follow suit for more than a century.
NOTE: The Vatican Observatory can be traced back to Pope Gregory XIII. At the time it was referred to as Tower of the Winds. It was from this structure that Jesuit astronomers and mathematicians studied the scientific data which would later aid in the reform of the calendar which occurred in 1582.
The Protestant rejection: (as noted by mgb in comment)
One of the original reasons the Gregorian calendar was designed was to restore the spring equinox to the March 21st date (Alexandrian Easter) that had been traditionally accepted since the Council of Nicaea. However, because this was mandated by the Pope many Protestant countries rejected the change.
The Gregorian Conversion In an age of intense religious passion, the simple fact that the Pope instituted the reform was enough to make Protestant countries reject the change. The greater part of protestant Germany did not switch to the Gregorian calendar until 1700, the Protestant Cantons of Switzerland and Protestant Netherlands until 1701. ...
Additional resources:
