During the period of Bodley’s diplomatic activity, two calendars were in use: the Julian and the Gregorian. The difference in the calendars was 10 days, so the 10 May in the Julian calendar was 20 May in the Gregorian. The Gregorian calendar, known as ‘New Style’ was not taken up by England until 1752/3.
Letters written between the continent and England usually indicate which calendar is being used, by a variety of methods, ‘N.S.’ being the most common, but other methods include ‘stylo vetiano’, ‘stylo antiquo’, and ‘stylo novo’. Occasionally correspondents will note both dates, which are commonly separated by a virgule, e.g. 10/20 May.
For more information, see CR Cheney, A Handbook of Dates for students of British History 2nd Edition, (Cambridge, 2008).