"statement of self-evident truth," late 15c., from Middle French axiome, from Latin axioma, from Greek axioma "authority," literally "that which is thought worthy or fit," from axioun "to think worthy," from axios "worthy, worth, of like value, weighing as much," from PIE adjective *ag-ty-o- "weighty," from root *ag- "to drive, draw out or forth, move."

Axioms in philosophy are not axioms until they are proved upon our pulses. [Keats, letter, May 3, 1818]

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").

General sense of "straight line about which parts are arranged" is from 1660s. Figurative sense in world history of "alliance between Germany and Italy" (later extended unetymologically to include Japan) is from 1936. 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).

"pole or pin upon which a wheel revolves" (properly, the round ends of the axle-tree which are inserted in the hubs or naves of the wheels), 1630s, from Middle English axel-, from some combination of Old English eax and Old Norse öxull "axis," both from Proto-Germanic *akhsulaz (source also of Old English eaxl "shoulder," oxta, ohsta "armpit," which survived as dialectal oxter; also Old Saxon ahsla, Old High German ahsala, German Achsel "shoulder"), from PIE *aks- "axis" (see axis, which is from the Latin cognate of this Germanic word). Found only in compound axle-tree before 14c.