Word Origin & History
O.E. æces (Northumbrian acas), later æx, from P.Gmc. *akusjo (cf. O.S. accus, O.N. ex, O.Fris. axe, Ger. Axt, Goth. aqizi), from PIE *agw(e)si- (cf. Gk. axine, L. ascia). Meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967.
The ax in fig. sense of cutting of anything (expenses, workers, etc.), especially as a cost-saving measure, is from 1922, probably from the notion of the headman's literal axe (itself attested from mid-15c.). To have an axe to grind is from an 1815 essay by U.S. newspaper editor Charles Miner (17801865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. Misattributed to Benjamin Franklin in Weekley, OED print edition, and many other sources.
"The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain." [OED]