Etymology
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stool (n.)

Old English stol "seat for one person," from Proto-Germanic *stōla- (source also of Old Frisian stol, Old Norse stoll, Old High German stuol, German Stuhl "seat," Gothic stols "high seat, throne"), from PIE *sta-lo-, locative of root *sta- "to stand, make or be firm."(source also of Lithuanian pa-stolas "stand," Old Church Slavonic stolu "stool").

Originally used of thrones (as in cynestol "royal seat, throne"); decline in sense began with adoption of chair (n.) from French, which relegated stool to small seats without arms or backs, then to "privy" (early 15c.) and thence to "bowel movement" (1530s).

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stool-ball (n.)
outdoor game similar to cricket, in 16c. and 17c. generally played by women alone, late 15c., from stool (n.) + ball (n.1). "The 'stool' was the wicket ... perhaps it was originally an ordinary stool" [OED].
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cucking-stool (n.)
The Cucking-stool was a means adopted for the punishment of scolds and incorrigible women by ducking them in the water, after having secured them in a chair or stool, fixed at the end of a long pole, serving as a lever by which they were immersed in some muddy or stinking pond. [Willis's Current Notes, February 1851]

early 13c., from verbal noun from cuck "to void excrement," from Old Norse kuka "feces," from PIE root *kakka- "to defecate." So called because they sometimes resembled the old close stool of the pre-plumbing days, a portable indoor toilet that looked like a chair with a box under the seat. Old folk etymology made the first element a corruption of cotquean. For second element, see stool. Also known as trebucket and castigatory, it was used on fraudulent tradesmen, in addition to disorderly women, either for public exposure to ridicule or for ducking.

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barstool (n.)
also bar-stool, bar stool, 1910, from bar (n.2) + stool.
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footstool (n.)

also foot-stool, "stool, usually small and low, to rest the feet on while sitting," 1520s, from foot (n.) + stool. Earlier was fotsceomel, from Old English fotsceamel; for the second element of which see shambles. Figurative sense of "one who is the abject thrall of another" is from 1530s.

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toadstool (n.)
late 14c., apparently just what it looks like: a fanciful name from Middle English tadde "toad" (see toad) + stole "stool" (see stool). Toads themselves were regarded as highly poisonous, and this word is "popularly restricted to poisonous or inedible fungi, as distinct from edible "mushrooms" [OED]. Compare toad-cheese, a poisonous fungi; toad's meat (1886), a "rustic" term for toadstool.
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stool pigeon (n.)
"police informer," 1859, American English; earlier "one who betrays the unwary (or is used to betray them)," 1821, originally a decoy bird (1812); said to be from decoys being fastened to stools to lure other pigeons. But perhaps related to stall "decoy bird" (c. 1500), especially "a pigeon used to entice a hawk into the net" (see stall (n.2)). Also see pigeon.
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*sta- 

*stā-, Proto-Indo-European root meaning "to stand, set down, make or be firm," with derivatives meaning "place or thing that is standing."

It forms all or part of: Afghanistan; Anastasia; apostasy; apostate; armistice; arrest; assist; astatic; astatine; Baluchistan; bedstead; circumstance; consist; constable; constant; constitute; contrast; cost; desist; destination; destine; destitute; diastase; distance; distant; ecstasy; epistasis; epistemology; establish; estaminet; estate; etagere; existence; extant; Hindustan; histidine; histo-; histogram; histology; histone; hypostasis; insist; instant; instauration; institute; interstice; isostasy; isostatic; Kazakhstan; metastasis; obstacle; obstetric; obstinate; oust; Pakistan; peristyle; persist; post (n.1) "timber set upright;" press (v.2) "force into service;" presto; prostate; prostitute; resist; rest (v.2) "to be left, remain;" restitution; restive; restore; shtetl; solstice; stable (adj.) "secure against falling;" stable (n.) "building for domestic animals;" stage; stalag; stalwart; stamen; -stan; stance; stanchion; stand; standard; stanza; stapes; starboard; stare decisis; stasis; -stat; stat; state (n.1) "circumstances, conditions;" stater; static; station; statistics; stator; statue; stature; status; statute; staunch; (adj.) "strong, substantial;" stay (v.1) "come to a halt, remain in place;" stay (n.2) "strong rope which supports a ship's mast;" stead; steed; steer (n.) "male beef cattle;" steer (v.) "guide the course of a vehicle;" stem (n.) "trunk of a plant;" stern (n.) "hind part of a ship;" stet; stoa; stoic; stool; store; stound; stow; stud (n.1) "nailhead, knob;" stud (n.2) "horse kept for breeding;" stylite; subsist; substance; substitute; substitution; superstition; system; Taurus; understand.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit tisthati "stands;" Avestan histaiti "to stand;" Persian -stan "country," literally "where one stands;" Greek histēmi "put, place, cause to stand; weigh," stasis "a standing still," statos "placed," stylos "pillar;" Latin sistere "stand still, stop, make stand, place, produce in court," status "manner, position, condition, attitude," stare "to stand," statio "station, post;" Lithuanian stojuos "I place myself," statau "I place;" Old Church Slavonic staja "place myself," stanu "position;" Gothic standan, Old English standan "to stand," stede "place;" Old Norse steði "anvil;" Old Irish sessam "the act of standing."

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shambles (n.)
early 15c., "meat or fish market," from schamil "table, stall for vending" (c. 1300), from Old English scamol, scomul "stool, footstool" (also figurative); "bench, table for vending," an early Proto-Germanic borrowing (Old Saxon skamel "stool," Middle Dutch schamel, Old High German scamel, German schemel, Danish skammel "footstool") from Latin scamillus "low stool, a little bench," ultimately a diminutive of scamnum "stool, bench," from PIE root *skabh- "to prop up, support." In English, sense evolved from "place where meat is sold" to "slaughterhouse" (1540s), then figuratively "place of butchery" (1590s), and generally "confusion, mess" (1901, usually in plural).
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foot-rest (n.)

"stool or short bench used to support as person's foot," 1844, from foot (n.) + rest (n.).

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