alar (adj.)

1540s, "imaginary motionless straight line around which a body (such as the Earth) rotates," from Latin axis "axle, pivot, axis of the earth or sky," from PIE *aks- "axis" (source also of Old English eax, Old High German ahsa "axle;" Greek axon "axis, axle, wagon;" Sanskrit aksah "an axle, axis, beam of a balance;" Lithuanian ašis "axle").

The general sense of "straight line about which parts are arranged" is from 1660s. The figurative sense in world history of "alliance between Germany and Italy" (later extended unetymologically to include Japan) is from 1936. The original reference was to a "Rome-Berlin axis" in central Europe. The word later was used in reference to a London-Washington axis (World War II) and a Moscow-Peking axis (early Cold War).

1610s, "slope of the edge of a cutting tool," also "groove by which a stone is held in its setting," from Old French *besel (13c.; Modern French biseau), cognate with Spanish and Portuguese bisel; of uncertain origin, perhaps literally "a stone with two angles," from Vulgar Latin *bis-alus, from bis- "twice" (from PIE root *dwo- "two") + ala "wing, side" (see alar).

The meaning "oblique face of a gem" is from c. 1840. The verb meaning "grind (a tool) down to an edge" is from 1670s. Compare bevel.