Mixed-style date
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A mixed-style date is one written in a way designed to minimize confusion about the difference between the Julian and Gregorian calendars.
In 1752, Britain and her colonies switched over from the Julian Calendar to the Gregorian Calendar by skipping eleven days. Furthermore, the system for numbering the year itself was changed: whereas previously it had been customary to iterate the year number on the 25th of March (Lady Day), from 1753 onward the new year began on New Year's Day, the 1st of January.
As a result of this, considerable confusion can result when discussing events which occurred prior to 1753. Should we use the 'old-style' date for the occurrence (as it would have been written by a participant) or the 'new-style' (as we would write it)? As far as the month and day are concerned, there seems no point in recalculating the Gregorian equivalent, especially since some events and holidays are so intimately tied to the date. (Is there any point in saying that Christmas was celebrated on December 14th?) On the other hand, though, as far as the year is concerned, we may very well be interested in how many years ago an event occurred, for which we do need to recalculate the date according to our system.
Based on these considerations, many authors choose to present dates essentially as they would have been written by contemporaries; but if the date falls before the 25th of March, both possible numbers for the year are given, separated by a slash. Thus the birthday of George Washington would be written as February 11, 1731/32 where 1731 is the old-style date and 1732 the new. (A Frenchman on the Gregorian calendar would have put his birth at February 22nd 1732, of course.)
N.B.— While 'old-style' and 'new-style' are widely-used and accepted terms, 'mixed-style' is not. These years are often called double-dated years.