Etymology
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mallet (n.)

"small wooden hammer," chiefly used for driving another tool, late 14c., from Old French maillet "mallet, small wooden hammer, door-knocker," diminutive of mail, from Latin malleus "a hammer, mallet," from Proto-Italic *molalo-, *molklo- "hammer," from PIE *molkh-tlo- "crushing instrument," source also of Russian molot, Czech mlat "hammer," from PIE root *mele- "to crush, grind." It is wielded with one hand, while the heavier mall or maul requires both.

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machete (n.)

"heavy knife or cutlass," used as a weapon and tool by the Spanish in the Americas, 1590s (in pseudo-Spanish form macheto), from Spanish machete "a chopping knife," probably a diminutive of macho "sledge hammer," alteration of mazo "club," which is probably [Barnhart] a dialectal variant of maza "mallet," from Vulgar Latin *mattea "war club" (see mace (n.1)). An alternative explanation traces macho to Latin marculus "a small hammer," diminutive of marcus "hammer," from a base parallel to that of Latin malleus (see mallet).

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*mele- 

*melə-, Proto-Indo-European root meaning "to crush, grind," with derivatives referring to ground or crumbling substances and crushing or grinding instruments.

It forms all or part of: amyl; amyloid; blintz; emmer; emolument; immolate; maelstrom; mall; malleable; malleolus; mallet; malleus; maul; meal (n.2) "edible ground grain;" mill (n.1) "building fitted to grind grain;" millet; mola; molar (n.); mold (n.3) "loose earth;" molder; ormolu; pall-mall.

It is the hypothetical source of/evidence for its existence is provided by: Hittite mallanzi "they grind;" Armenian malem "I crush, bruise;" Greek mylos "millstone," myle "mill;" Latin molere "to grind," mola "millstone, mill," milium "millet;" Old English melu "meal, flour;" Albanian miel "meal, flour;" Old Church Slavonic meljo, Lithuanian malu, malti "to grind;" Old Church Slavonic mlatu, Russian molotu "hammer."

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beetle (n.2)
"heavy wooden mallet used to drive wedges, pack earth, etc.," Old English bietl "mallet, hammer," from Proto-Germanic *bautilo-z, from *bautan "to beat," from PIE root *bhau- "to strike."
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maul (n.)

c. 1200, mealle, "heavy wooden hammer or mallet; sledgehammer," from Old French mail "hammer," from Latin malleus "hammer" (from PIE root *mele- "to crush, grind").

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sledge (n.1)
"heavy hammer," Old English slecg "hammer, mallet," from Proto-Germanic *slagjo- (source also of Old Norse sleggja, Middle Swedish sleggia "sledgehammer"), related to slege "beating, blow, stroke" and slean "to strike" (see slay (v.)). Sledgehammer is pleonastic.
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*bhau- 
Proto-Indo-European root meaning "to strike."

It forms all or part of: abut; baste (v.3) "beat with a stick, thrash;" battledore; beat; beetle (n.2) "heavy wooden mallet;" botony; boutonniere; butt (n.1) "thick end;" butt (v.) "strike with the head;" buttocks; button; buttress; confute; halibut; rebut; refute; sackbut; turbot.

It is the hypothetical source of/evidence for its existence is provided by: Latin *futare "to beat" (in compounds); Old English beadu "battle," beatan "to beat," bytl "hammer, mallet."
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mullet (n.2)

"hairstyle short the sides and long in back," 1996, perhaps from mullet-head "stupid, dull person" (1857). Mullet-head also was a name of a type of North American freshwater fish with a large, flat head (1866) with a reputation for stupidity. The term in reference to the haircut seems to have emerged into pop culture with the Beastie Boys song "Mullet Head."

#1 on the side and don't touch the back
#6 on the top and don't cut it wack, Jack
[Beastie Boys, "Mullet Head"]

As a surname, Mullet is attested from late 13c., thought to be a diminutive of Old French mul "mule." Compare also mallet-headed, in reference to the flat tops of chisels meant to be struck with a mallet.

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battledore (n.)
mid-15c., "bat-like implement used in washing clothes," of unknown origin, perhaps from Old Provençal batedor, Spanish batidor "beater, bat," from batir "to beat;" perhaps blended with Middle English betel "hammer, mallet" (see beetle (n.2)). As a type of racket used in a game, from 1590s, from similarity of shape.
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mall (n.)

1737, "shaded walk serving as a promenade," generalized from The Mall, name of a broad, tree-lined promenade in St. James's Park, London (so called from 1670s, earlier Maill, 1640s), which was so called because it formerly was an open alley that was used to play pall-mall.

This was a once-popular game played with a wooden ball in a kind of smooth alley boarded in at each side, in which the ball was struck with a mallet to send it through an iron arch placed at the end of the alley. The game's name is from French pallemaille, from Italian pallamaglio, from palla "ball" (see balloon (n.)) + maglio "mallet" (from Latin malleus "a hammer, mallet," from PIE root *mele- "to crush, grind"). Modern sense of "enclosed shopping gallery" is from 1962 (from 1951 in reference to city streets set aside for pedestrians only). Mall rat "one who frequents a mall" is from 1985 (see rat (n.)).

The short history of malls goes like this: In 1954, Victor Gruen's Northland Center, often credited as the first modern shopping mall (though earlier examples existed), opens in Southfield, Michigan. The suburban location is fitting because the rise of the automobile, helped along by the Federal-Aid Highway Act, led to the widespread creation of large shopping centers away from urban centers. This, among other factors, nearly killed downtowns, and malls reigned supreme for some 40 years. By the 1990s, however, a new urbanism movement revived the urban shopping experience and eroded the dominance of malls. Next, the rise of big box stores and online shopping sounded the death knell for mall culture. [Steven Kurutz, "An Ode to Shopping Malls," New York Times, July 26, 2017]
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