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Calendar History

England and her colonies were way out of date when they finally made two calendar changes in the same year, in 1752.  Most countries/regions did not make both of the changes in the same year; indeed, they were more often made centuries apart.  Seeing England as a kind of norm has lead to much unfortunate confusion and misinformation.
The two separate changes were: 
1. moving the first day of the new year from 25 March to 01 January 
2. adopting the Gregorian calendar to replace the earlier Julian calendar
When is New Year's Day? -- "Double Dating" 
In the 230 years preceding 1752, starting with Venice in 1522, region after European region changed the start of the year from 25 March (the date the angel came to the Virgin Mary) back to 01 January.  Germany made the change in 1544, Spain and Portugal in 1556, the Scandinavian countries in 1559, France in 1564 and Scotland in 1600, to name a few. 
A year lasting from Jan.-Dec. or a year lasting 25 Mar.-24 Mar., caused no problem within most countries/regions.  Communication between two regions with different systems ran into ambiguity with regard to dates within the overlapping period in January-March. 
For example 15 February 1715, near the end of the year in England, was 15 February 1716 in Scotland, near the beginning of the (next) year.  The unambiguous ways of writing this date are:
15 Feb. 1715/16 or 15 Feb. 1715/6 (called Double Dating), 
15 Feb. 1715 OS (for Old Style) and 
15 Feb. 1716 NS (for New Style). 
Note that "Double Dating" does not exist for the dates 25 March through 31 December.  These dates will have the same year no matter which date is used for New Year's Day. 
In England itself both dating systems were in use for different purposes.  "The people" had always celebrated the first of January as New Year's Day, even though the legal year began in late March.
The terms, "Old Style" and "New Style", are sometimes used with regard to the other calendar reform (below) as well, but this should be avoided.  One must consider the two reforms separately for most countries, as they did not occur at the same time.
Pope Gregory Reforms the Calendar
The Julian calendar, named for Julius Caesar, was in use from 45 BC to at least AD 1581.  In Roman Catholic areas, the Gregorian calendar, named for Pope Gregory XIII, was instituted in 1582 when the days 05 through 14 October simply disappeared.  The reason for Pope Gregory's reform was that the calendar (and, not incidentally, the cycle of Christian holidays) was out of step with the sun and the seasons.  This was rectified by removing ten days in one short year and allowing fewer leap years thereafter so that the same displacement wouldn't happen in the future.
[It is safest to refer to the two calendars by their names, not mixing in the terms "Old Style" and "New Style" which refer to a different reform (above).]
Accepting the Gregorian calendar meant:
shortening by 10 (later 11) days the year in which it was adopted, 
adopting new rules for leap year, 
adopting new rules for figuring the date of Easter Sunday and 
(at least for England which hadn't done it earlier) changing the start of the year to the first of January. 
Except for the loyal Roman Catholic areas (including Spanish America) which made the changes in 1582-1584, adoption of the Gregorian calendar was a long process.  Protestant Germany, the Netherlands and Denmark/Norway adopted the new calendar in 1700-1701.  Scotland, England with its colonies and Sweden adopted in 1752 and 1753.  Russia, Jugoslavia, Rumania and Greece made the change after World War One.
New Confusion
Moving the first day of the new year from March to January had caused ambiguity as to which year a date in the period 01 Jan through 24 March occurred.  In addition to these Double Dating problems, one now had dates which appeared to be ten or more days apart
If we look at the aforementioned year of 1715:
England used the Julian calendar and the year started on 25 March. 
France used the Gregorian calendar with the year starting on 01 Jan. 
In England, 1700 had been a leap year, in France it had not. 
10 June 1715 in England appears to be eleven days "earlier" than the same day, 21 June 1715, in France. 
England kept the same year going until far into March, so 15 Feb. 1715 in England was 26 Feb. 1716 in France. 
In Scotland, still using the Julian calendar as noted above, the same day was 15 Feb. 1716. 

 Calendar First Day of the Year (Country, example) A Day in February A Day in June 
 Gregorian 01 January FRANCE 26 February 1716 21 June 1715 
 Julian 25 March ENGLAND 15 February 1715 10 June 1715 
 Julian 01 January SCOTLAND 15 February 1716 10 June 1715 
Some of the orthodox churches have not yet accepted the Gregorian calendar, which is why Christmas is celebrated thirteen days "too late" in Russia.  After the year 2100, which will be a leap year in the Julian calendar but not in the Gregorian, Christmas will be fourteen days "late" in Russia.
The pope's mathematicians were very clever.  In the last 420 years the Gregorian calendar has changed with regard to the earth's travel around the sun by about three hours.  We will need an adjustment of one day about 3000 years from now.

Look for discrepancies in dates before 1752. This change in the calendar can explain the birth of two children who seem way to close together.  You might find a son born 27 March 1640 and the next child born on 28 February 1640. In this case, the dates were "old style," and there were actually eleven months between the births, instead of negative 27 days!

For more information on entering dates, see Entering Dates in Date Fields.