Etymology
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vagary (n.)
1570s, "a wandering, a roaming journey," from Italian vagare or directly from Latin vagari "to wander, stroll about, roam, be unsettled, spread abroad," from vagus "roving, wandering" (see vague). The infinitive appears to have been adopted in English as a noun and conformed to nouns in -ary, "but this can hardly be explained except as an orig. university use" [Century Dictionary]. Current meaning of "eccentric notion or conduct" (1620s) is from notion of mental wandering. Related: Vagaries.
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vagrant (n.)
mid-15c., "person who lacks regular employment, one without fixed abode, a tramp," probably from Anglo-French vageraunt, also wacrant, walcrant, which is said in many sources to be a noun use of the past participle of Old French walcrer "to wander," from Frankish (Germanic) *walken, from the same source as Old Norse valka "wander" and English walk (v.).

Under this theory the word was influenced by Old French vagant, vagaunt "wandering," from Latin vagantem (nominative vagans), past participle of vagari "to wander, stroll about" (see vagary). But on another theory the Anglo-French word ultimately is from Old French vagant, with an unetymological -r-. Middle English also had vagaunt "wandering, without fixed abode" (late 14c.), from Old French vagant.
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machine (n.)

1540s, "structure of any kind," from Middle French machine "device, contrivance," from Latin machina "machine, engine, military machine; device, trick; instrument" (source also of Spanish maquina, Italian macchina), from Greek makhana, Doric variant of Attic mēkhanē "device, tool, machine;" also "contrivance, cunning," traditionally (Watkins) from PIE *magh-ana- "that which enables," from root *magh- "to be able, have power." But Beekes, on formal grounds, objects to the connection to words in Germanic and Slavic. He finds the Greek word isolated and is convinced that it is Pre-Greek.

Main modern sense of "device made of moving parts for applying mechanical power" (1670s) probably grew out of mid-17c. senses of "apparatus, appliance" and "military siege-tower."  It gradually came to be applied to an apparatus that works without the strength or skill of the workman.

From 17c.-19c. also "a vehicle; a stage- or mail-coach; a ship," and, from 1901, "a motor-car." Also in late 19c. slang the word was used for both "penis" and "vagina," one of the few so honored.

The political sense "a strict organization of the working members of a political party to secure a predominating influence for themselves and their associates" is U.S. slang, attested by 1876. Machine age, a time notable for the extensive use of mechanical devices, is attested by 1882, though there is this:

The idea of remodelling society at public meetings is one of the least reasonable which ever entered the mind of an agitator: and the notion that the relations of the sexes can be re-arranged and finally disposed of by preamble and resolution, is one of the latest, as it should have been the last, vagary of a machine age. ["The Literary World," Nov. 1, 1851]

Machine for living(in) "house" translates Le Corbusier's machine à habiter (1923).

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