What is a "double date"?
Why do I sometimes see dates written as
January 13, 1718/1719?
There is a very good explanation here: http://www.cree.name/genuki/dates.htm
In short, in various countries at various points in history, the day and month at which a "Year" started was not always the same. In England and Wales, for example, the 'Civil' or 'Legal' year used to start on 25th March, not 1st January (which is when the 'Historical' year started). So a birth recorded in the period between 1st January and 25th March is shown as (e.g.) 3rd March 1733/4 to show that that the date intended was in March of the Civil Year 1733 and the Historical Year 1734, that is, the month before April 1734.
There are other complications to do with calculation of leap years, and changeover from Julian to Gregorian calendars, but I believe I've covered the primary usage of 'double dates'.
A double date comes from the transition from the Julian calendar to the Gregorian calendar. According to the Julian calendar, the first day of the year was March 25 and each year was 365 days and 6 hours long. Not all areas accepted the change to the Gregorian calendar at the same time, however. Because the Julian and Gregorian calendars were long used simultaneously, although in different places, calendar dates in the transition period were often ambiguous, unless it is specified which calendar was being used. For this reason, many people wrote dates falling between January 1 and March 25 with double dates on the original document to clarify. Others used the terms
Old style (OS) meant Julian;
New style (NS) meant Gregorian. The first year in a double date given is the Julian calendar, and the second given is the Gregorian calendar.
Many Catholic countries switched over in 1582, but others waited much longer to switch over. Russia only did so in 1918.
Although this is related to the change from the Julian calendar to the Gregorian calendar the key issue that leads to the need for "double dates" is not that change, but rather the related change from starting the year on 25th March to starting it on 1st January.
In particular, although the Gregorian calendar standardised on 1st January for the start of the year, previous usage had tended to vary, with 25th March being used in some contexts and 1st January in others.
Hence for dates between 1st January and 25th March in years before the introduction of the Gregorian calendar it can be unclear which year is meant, which led to the introduction of the "double date" syntax to make clear that the date is in the first year if you take 1st January as the start of the year, but the second year if you take 25th March to be the start of the year.
In fact, a "double date" is a social event. :-)
The accepted term is a "dual date" to avoid the unfortunate ambiguity. The term "double year" is also in common use but less correct since the year field isn't the only field that might be paired-up. For instance: 12/23 Feb 1750/1751.
In a nutshell, when the Gregorian calendar was adopted, in place of the Julian calendar, and the fact that differnt countries adopted it at different times.