Etymology
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scheme (n.)
1550s, "figure of speech," from Medieval Latin schema "shape, figure, form, appearance; figure of speech; posture in dancing," from Greek skhema (genitive skhematos) "figure, appearance, the nature of a thing," related to skhein "to get," and ekhein "to have, hold; be in a given state or condition," from PIE root *segh- "to hold."

The sense "program of action" first is attested 1640s. Unfavorable overtones (selfish, devious) began to creep in early 18c. Meaning "complex unity of coordinated component elements" is from 1736. Color scheme is attested from 1884.
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scheme (v.)
"devise a scheme," 1767 (earlier "reduce to a scheme," 1716), from scheme (n.). Related: Schemed; scheming.
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schemer (n.)
1724, "a contriver, plotter," agent noun from scheme (v.).
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schematic (adj.)
"pertaining to schemes," 1701, from Latin stem of scheme (n.) + -ic. Noun meaning "diagram" is first attested 1929. Related: Schematical (1670s).
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sketch (n.)
"rough drawing intended to serve as the basis for a finished picture," 1660s, from Dutch schets or Low German skizze, both apparently 17c. artists' borrowings from Italian schizzo "sketch, drawing," which is commonly said to be from Latin *schedius (OED compares schedia "raft," schedium "an extemporaneous poem"), from or related to Greek skhedios "temporary, extemporaneous, done or made off-hand," related to skhema "form, shape, appearance" (see scheme (n.)). But according to Barnhart Italian schizzo is a special use of schizzo "a splash, squirt," from schizzare "to splash or squirt," of uncertain origin.

Extended sense of "brief account" is from 1660s; meaning "short play or performance, usually comic" is from 1789. Sketch-book recorded from 1820. German Skizze, French esquisse, Spanish esquicio are likewise from Italian schizzo.
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proposal (n.)

"a plan or scheme offered for acceptance," 1650s, from propose + -al (2); specific sense of "offer of marriage" is by 1749.

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machinate (v.)

c. 1600, "to lay plots, intrigue," a back-formation from machination, or else from Latin machinatus, past participle of machinari "to contrive skillfully; to design, scheme, plot." Transitive sense of "to plan, contrive, form (a plot, scheme, etc.)" also is from c. 1600. Related: Machinated; machinating; machinator.

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scrap (n.2)
"fight," 1846, possibly a variant of scrape (n.1) on the notion of "an abrasive encounter." Weekley and OED suggest obsolete colloquial scrap "scheme, villainy, vile intention" (1670s).
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project (n.)

c. 1400, projecte, "a plan, draft, scheme, design," from Medieval Latin proiectum "something thrown forth," noun use of neuter of Latin proiectus, past participle of proicere "stretch out, thrust out, throw forth," from pro- "forward" (see pro-) + combining form of iacere (past participle iactus) "to throw" (from PIE root *ye- "to throw, impel").

Meaning "scheme, proposal, mental plan" is from c. 1600. Meaning "group of government-subsidized low-rent apartment buildings" is recorded from 1935, American English, short for housing project (1932). Related: Projects. Project manager is attested from 1913.

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contrivance (n.)

1620s, "a plan or scheme for attaining some end," from contrive + -ance. Meaning "act of contriving" is from 1640s; sense of "the thing contrived, planned, or invented" is from 1660s. Earlier was contrival (c. 1600).

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