etymology - Where did the "odd" in "N odd years" come from?

I was reminded this usage by the recent question asking about the origin of "-ish." Odd is often used in a similar way in the stock phrase "odd years" to mean "around" or "about" a certain length of time. For example I might say, "250 odd years" to mean "250 years, give or take." In this usage of odd, I can't really see the connection with its usual given definitions of "strange" or "not divisible by 2." Could I get some history on the evolution of this meaning of "odd"?

Answer

From On the nature of the approximative expression num-odd:

The origin of the suffix ‐odd is, unsurprisingly, the word odd, denoting a surplus or remainder (OED entry for odd, lemma 3a). This use dates back to the 14th–15th century. Based on the OED examples, its earliest uses were with monetary items and with terms denoting weights and measures, but by the 17th century this use had broadened to include other count nouns as well. The structure of such examples is still NUM-odd-N, where odd is an adjective.

From the same page in that paper, the development:

Quite soon after the use described above, odd became used in constructions of the type NUM N1 (and ) odd N2, where N2 is a count noun of lower rank than N1 (OED lemma 3b,c). The meaning of the adjective odd is still one of ‘surplus’, ‘extra’.

(24) a. Than leveth there 38 degrees and odde minutes.
…

Soon N2 can be omitted and elliptically understood, giving a general sense of surplus without exactly specifying the surplus (OED lemma 4a).

(25) a. Distant sixtie miles and odde.
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Since the phrase and odd denotes a surplus of the same kind (though of lower denomination) as N1, it becomes possible to shift the and odd phrase to a position immediately preceding N1, yielding the construction NUM and odd N1 (OED lemma 4b).

(26) …
b. Having ridden post day and night fourscore and odd miles…
…

Finally the and is omitted and odd becomes suffixed to the numeral (OED lemma 4c and paper above). This use dates from the 16th century, based on OED examples.

(27) a. Eightie odd yeares of sorrow haue I seene.
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