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mid-14c., "female spinner of thread," from Middle English spinnen (see spin) + -stere, feminine suffix. Spinning commonly done by unmarried women, hence the word came to denote "an unmarried woman" in legal documents from 1600s to early 1900s, and by 1719 was being used generically for "woman still unmarried and beyond the usual age for it."
Spinster, a terme, or an addition in our Common Law, onely added in Obligations, Euidences, and Writings, vnto maids vnmarried. [John Minsheu, "Ductor in Linguas," 1617]