"long narrow passage for admitting air," 1690s, from air (n.1) + shaft (n.2).
Entries linking to air-shaft
air n.1
c. 1300, "invisible gases that surround the earth," from Old French air "atmosphere, breeze, weather" (12c.), from Latin aer "air, lower atmosphere, sky," from Greek aēr (genitive aeros) "mist, haze, clouds," later "atmosphere" (perhaps related to aenai "to blow, breathe"), which is of unknown origin. It is possibly from a PIE *awer- and thus related to aeirein "to raise" and arteria "windpipe, artery" (see aorta) on notion of "lifting, suspended, that which rises," but this has phonetic difficulties.
In Homer mostly "thick air, mist;" later "air" as one of the four elements. Words for "air" in Indo-European languages tend to be associated with wind, brightness, sky. In English, air replaced native lyft, luft (see loft (n.)). In old chemistry, air (with a qualifying adjective) was used of any gas.
To be in the air "in general awareness" is from 1875; up in the air "uncertain, doubtful" is from 1752. To build castles in the air "entertain visionary schemes that have no practical foundation" is from 1590s (in 17c. English had airmonger "one preoccupied with visionary projects"). Broadcasting sense (as in on the air, airplay) is by 1927. To give (someone) the air "dismiss" is from 1900. Air pollution is attested by 1870. Air guitar is by 1983. Air traffic controller is from 1956.
shaft n.2
in mines and earthworks, "long, narrow, vertical or inclined passage sunk into the earth," early 15c., probably from shaft (n.1) on notion of "long and cylindrical," perhaps as a translation of cognate Low German schacht in this sense (Grimm's suggestion, though OED is against it). Or it may represent a separate (unrecorded) development in Old English directly from Proto-Germanic *skaftaz if the original sense is "scrape, dig." The slang sense of shaft (n.1) is punned upon in the song "She Got the Gold Mine, I Got the Shaft," a U.S. hit for Jerry Reed in 1982.