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Old English æces (Northumbrian acas) "axe, pickaxe, hatchet," later æx, from Proto-Germanic *akusjo (cf. Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- (cf. Greek axine, Latin ascia).
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]Meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967. The axe in figurative 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. editor and politician Charles Miner (1780-1865) 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, though "better on every ground, of etymology, phonology, & analogy" (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler]
1670s, "to shape or cut with an axe," from axe (n.). Meaning "to remove, severely reduce," usually figurative, recorded by 1922. Related: Axed; axing.
[musical instrument sense fr the resemblance in shape between a saxophone and an ax, and possibly fr the rhyme with sax]
used in the Authorized Version of Deut. 19:5; 20:19; 1 Kings 6:7, as the translation of a Hebrew word which means "chopping." It was used for felling trees (Isa. 10:34) and hewing timber for building. It is the rendering of a different word in Judg. 9:48, 1 Sam. 13:20, 21, Ps. 74:5, which refers to its sharpness. In 2 Kings 6:5 it is the translation of a word used with reference to its being made of iron. In Isa. 44:12 the Revised Version renders by "axe" the Hebrew _maatsad_, which means a "hewing" instrument. In the Authorized Version it is rendered "tongs." It is also used in Jer. 10:3, and rendered "axe." The "battle-axe" (army of Medes and Persians) mentioned in Jer. 51:20 was probably, as noted in the margin of the Revised Version, a "maul" or heavy mace. In Ps. 74:6 the word so rendered means "feller." (See the figurative expression in Matt. 3:10; Luke 3:9.)