Corrections to the Julian calendar

After Julius Caesar’s death, there was still confusion about the placing of leap years. This was corrected by his adopted son, Emperor Augustus Caesar, who also renumbered the days of the latter months of the year. In his honour, (as for Julius), the sixth month of the original Roman calendar was renamed Augustus.

The Julian calendar had a major flaw: the Julian calendar was 11 minutes 14 seconds longer than the solar year. This problem was corrected by Pope Gregory XIII in 1582, who dropped ten days from the calendar, and decreed that only centenary years exactly divisible by 400 were to be leap years (e.g. 2000, but not 1900). Though not immediately adopted, it is now almost universally accepted.
Exercise 1                              Check

Given that the Julian calendar was too long by 11 Minutes 14 seconds from 150 BC to 1582 AD, how many days too long was it by 1582? Compare your answer with that obtained by Pope Gregory XIII. How might you account for the difference?		 Exercise 2                          Check

Given that the Gregorian calendar is too long by 24 seconds (one of its faults), how long will it be before the difference between the calendar year and the solar year is one day? How might you correct the calendar to take account of this?

  
Approximately 13.5 days. The reason for Pope Gregory getting it wrong is speculative e.g. he may have forgotten to take account of his own rule about leap years.
3600 years: drop one day from the calendar
every 3600 years!