1520s, "scent with perfumes," back-formation from fumigation. The older verb was simply fume (c. 1400). Meaning "apply smoke or fumes to," especially for cleansing purposes, is from 1781. Related: Fumigated; fumigating; fumigatory.
late 14c., "action of making aromatic smoke as part of a ceremony," from Latin fumigationem (nominative fumigatio) "a smoking," noun of action from past-participle stem of fumigare "to smoke," from fumus "smoke, fume" (from PIE root *dheu- (1) "dust, vapor, smoke") + root of agere "to set in motion, to do, perform" (from PIE root *ag- "to drive, draw out or forth, move"). Sense of "exposure (of someone or something) to aromatic fumes" is c. 1400, originally as a medicinal or therapeutic treatment.
late 14c., "vapor, odorous vapor; exhalation," from Old French fum "smoke, steam, vapor, breath, aroma, scent" (12c.), from Latin fumus "smoke, steam, fume, old flavor" (source also of Italian fumo, Spanish humo), from PIE root *dheu- (1) "dust, vapor, smoke."
In old medicine, an "exhalation" of the body that produces emotions, dreams, sloth, etc; later especially of smokes or vapors that go to the head and affect the senses with a narcotic or stifling quality.
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Harper, D. (n.d.). Etymology of fumigate. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/fumigate