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.
also pet-cock, "a small plug-cock, made to be fastened to a pipe and used for draining water and condensation from steam cylinders, etc.," 1864, from cock (n.2); the signification of the first element is uncertain.
"Finnish steam bath," also the house or room where it is taken, 1881, from Finnish sauna. Originally in a Finnish context; by 1959 in reference to installation in homes and gyms outside Finland.
Middle English reke "smoke, fumes; steam, vapor," from Old English rec (Anglian), riec (West Saxon), "smoke from burning material," probably from a Scandinavian source such as Old Norse reykr, Danish rǿg, Swedish rök "smoke, steam."
These are reconstructed to be from Proto-Germanic *raukiz (source also of Old Frisian rek, Middle Dutch rooc, Old High German rouh, German Rauch, Icelandic reykr "smoke, steam"), from PIE *reug- "to vomit, belch;" also "smoke, cloud."
The sense of "stench" is attested 1650s via the notion of "that which rises" (compare reek (v.)). Century Dictionary (1891) marks the word "Obsolete, archaic, or Scotch." According to OED, "As the word has chiefly survived in northern use the palatalized form reech is comparatively rare." A c. 1250 document refers to the period March-April as Reke-fille "the misty month."