What is the origin of the hymn "Fairest Lord Jesus"?

As I was listening to Christmas music over the holidays, I noticed that the popular Danish Christmas hymn "Dejlig er jorden" ("Beautiful is the world") shares a melody with the hymn "Fairest Lord Jesus", while there are only very vague similarities in the lyrics, such as the motif of the world's natural beauty. Curious about this I checked Wikipedia and learned that both hymns ultimately derive from a German hymn that first appeared in print in 1677. What are the historical details of the progression from the German hymn of 1677 to "Fairest Lord Jesus"?