"The late xxxxs" is sometimes used to mean the later years of a decade, but it is also used to mean the later decades of a century.
"the late 1880s" usually means somewhere around 1887-1889.
"the late 1800s" usually means somewhere around 1870-1899.
However, in the latter example, what if I actually mean 1807-1809? How can I specify that I'm actually talking about the 1807-1809 range?