Why did September 1752 have fewer days?
Short answer: In 1582, the Julian calendar was replaced by the Gregorian calendar by skipping ten days: October 4, 1582 was followed by October 15, 1582. Most of Catholic Europe adopted the change soon, but the British Empire adopted it much later, in 1752. By 1752, the error was of 11 days, not 10 days, so September 2, 1752 was followed by September 14, 1752. And hence, the September 1752 was not 30 days long, and there is nothing like September 3 to September 13, 1752 in the history of dates in the British Empire.
Long answer: Before 1582, the calendar in use was the Julian calendar. It assumed a year to be 365.25 days long. The fractional day was facilitated by having 365 days every year, with one extra day in February once every four years—thus effectively having 365.25 days every year on an average.
However, the actual length of the solar year (the time taken by the earth to complete one revolution around the sun) is not 365.25 days, but approximately 365.2422 days. The small difference doesn't cause much of a difference in a short span of 100 years (365.25 – 365.2422 = 0.0078 days per year, i.e. we're ahead by just 0.78 days per century). But, by 1582, it was realized that the over the centuries, the erstwhile calendar had calculated 10 extra days due to this malpractice!
Needless to say, this way, a time would come when the seasons would not be in sync with the calendars, and so a change was needed as early as possible. So, Pope Gregory XIII established the Gregorian calendar in 1582. To launch the calendar, ten days were dropped from the calendar. October 4, 1582 was followed by October 15, 1582.
Catholic Europe gradually accepted the new calendar. But British Empire (UK and its colonies) adopted it as late as 1752. By 1752, an error of about 10 + 0.0078 × (1752 – 1582) = 11.326 days had crept up. Therefore, eleven days had to be dropped this time, and September 2, 1752 was immediately followed by September 14, 1752. So September 1752 was shorter than other Septembers.
September 1752 was short only in the countries that adopted the Gregorian calendar in 1752. The ones which adopted it in 1582 had a short October 1582 and a normal September 1752. Moving to the Gregorian calendar was basically the decision to drop off some dates because they were causing an error, and introducing a new system to prevent more extra dates from creeping up in future.
The new system:
To prevent extra days from getting added in the future, the Gregorian calendar recommended that the leap years would still come every 4 years, but we'd need to drop 3 extra days every 400 years. This would be done by having 365 days in century years such as 1700, 1800 and 1900, which were not divisible by 400.
Effective number of days per year = 365 per year + 1 per 4 years – 3 per 400 years = 365 + 1/4 – 3/400 = 365.2425 days per year. Further refinement was done by making a year divisible by 4000 a non-leap year. So, effective year becomes 365.2425 – 1/4000 = 365.24225, an error of 0.00005 days per solar year, or, 1 day per 20000 years.
