1560s, "single number regarded as an undivided whole," alteration of unity on the basis of digit. Popularized in John Dee's English translation of Euclid, to express Greek monas (Dee says unity formerly was used in this sense). Meaning "single thing regarded as a member of a group" is attested from 1640s. Extended sense of "a quantity adopted as a standard of measure" is from 1738. Sense of "group of wards in a hospital" is attested from 1893.
"melody, tune, connected rhythmic succession of distinct musical sounds," 1580s, nativized from Italian aria (see aria), perhaps via French.
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.
1590s, "manner, appearance" (as in an air of mystery); 1650s, "assumed manner, affected appearance" (especially in phrase put on airs, 1781), from French air "look, appearance, mien, bearing, tone" (Old French aire "reality, essence, nature, descent, extraction" (12c.); compare debonair), which is perhaps from Latin ager "place, field, productive land" (from PIE root *agro- "field") on the notion of "place of origin."
But some French sources connect this Old French word with the source of air (n.1), and it also is possible these senses in English developed from or were influenced by air (n.1); compare sense development of atmosphere and Latin spiritus "breath, breeze," also "high spirit, pride," and the extended senses of anima.
also airspace, by 1821 in reference to stove and furnace construction, from air (n.1) + space (n.). From 1852 in reference to the cubic contents of a room (with reference to the persons in it) in sanitary regulations for boarding rooms, hospitals, etc. In firearms, "a vacant space between the powder charge and the projectile" (1847). By 1910 as "portion of the atmosphere controlled by a country above its territory."