air-gun (n.)

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.

mid-14c., gunne "an engine of war that throws rocks, arrows or other missiles from a tube by the force of explosive powder or other substance," apparently a shortening of woman's name Gunilda, found in Middle English gonnilde "cannon" and in an Anglo-Latin reference to a specific gun from a 1330 munitions inventory of Windsor Castle ("... una magna balista de cornu quae Domina Gunilda ..."). Also compare gonnilde gnoste "spark or flame used to fire a cannon" (early 14c.).

The woman's name is from Old Norse Gunnhildr, a compound of gunnr and hildr, both meaning "war, battle." First element from PIE *gwhen- "to strike, kill" (see bane); for second, see Hilda.

The identification of women with powerful weapons is common historically (such as Big Bertha, Brown Bess, Mons Meg, etc.).

Or perhaps gun is directly from Old Norse gunnr "battle." The word was perhaps influenced by or confirmed by (or possibly from) Old French engon, dialectal variant of engin "engine."

Meaning grew with technology, from cannons to firearms as they developed 15c.; popularly applied to pistols and revolvers from 1744. In modern military use the word is restricted to cannons (which must be mounted), especially long ones used for high velocity and long trajectory. Hence great guns (1884 as an exclamation) distinguished from small guns (such as muskets) from c. 1400. Meaning "thief, rascal" is from 1858. For son of a gun, see son. To jump the gun (1912, American English) is a figurative use from track and field. Guns "a woman's breasts" (especially if prominent) attested by 2006.

[G]un covers firearms from the heaviest naval or siege guns (but in technical use excluding mortars and howitzers) to the soldier's rifle or the sportsman's shotgun, and in current U.S. use even the gangster's revolver. In the other European languages there is no such comprehensive word, but different terms for the small or hand gun of the soldier or sportsman (even these, sometimes differentiated) and the heavy naval guns or artillery pieces .... [Buck, 1949]