Etymology
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mattock (n.)

"instrument for loosening soil in digging, shaped like a pickaxe but with broad instead of pointed ends," Middle English mattok, from Old English mttoc, formerly said to be probably from Vulgar Latin *matteuca "club," which is related to Latin mateola, a kind of mallet (see mace (n.1)), but this is not certain, and synonymous Russian motyka, Polish motyka, Lithuanian matikas, as well as  Old High German medela "plow," Middle High German metz "knife" suggest rather a PIE *mat- as source of the Latin, Germanic, and Slavic words. OED says similar words in Welsh and Gaelic are from English.

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hack (n.1)
"tool for chopping," early 14c., from hack (v.1); cognates: Danish hakke "mattock," German Hacke "pickax, hatchet, hoe." Meaning "a cut, notch" is from 1570s. Meaning "an act of cutting" is from 1836; figurative sense of "a try, an attempt" is first attested 1898.
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hoe (n.)
"implement for digging, scraping, or loosening earth," mid-14c., from Old French houe (12c.), from Frankish *hauwa, from Proto-Germanic *hawwan (source also of Old High German houwa "hoe, mattock, pick-axe," German Haue), from PIE *kau- "to hew, strike" (see hew).
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*sek- 

Proto-Indo-European root meaning "to cut." It forms all or part of: bisect; dissect; hacksaw; insect; intersect; resect; saw (n.1) "cutting tool;" Saxon; scythe; secant; secateurs; sect; section; sector; sedge; segment; skin; skinflint; skinny; transect.

It is the hypothetical source of/evidence for its existence is provided by: Hittite shakk- "to know, pay attention to;" Latin secare "to cut," sectio "a cutting, cutting off, division;" Old Church Slavonic seko, sešti "to cut," sečivo "ax, hatchet," Russian seč' "to cut to pieces;" Lithuanian įsėkti "to engrave, carve;" Albanian šate "mattock;" Old Saxon segasna, Old English sigðe "scythe;" Old English secg "sword," seax "knife, short sword;" Old Irish doescim "I cut."

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sap (v.1)

1590s, intransitive, "dig a covered trench toward the enemy's position," from French saper, from sappe "spade," from Late Latin sappa "spade, mattock" (source also of Italian zappa, Spanish zapa "spade"), which is of unknown origin. The transitive sense of "undermine (a wall, etc.), render unstable by digging into or eating away the foundations" is from 1650s.

The extended transitive sense (of health, confidence, etc.), "weaken or destroy insidiously," is by 1755 and perhaps has been influenced or reinforced in sense by the verb form of sap (n.1), on the notion of "draining the vital sap from," and later by sap (v.2) "beat with a club or stick."

It also sometimes is used as a noun, "a narrow, covered ditch or trench by which a fortress or besieged place can be approached under fire" (1640s). Sap (n.) in the "tool for digging" sense also occasionally is met in 16th century English. Related: Sapped; sapping.

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