Etymology
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mitre (n.)

mid-14c., "bishop's tall hat," from Old French mitre and directly from Latin mitra "headband, turban," from Greek mitra "headband, turban," earlier a belt or cloth worn under armor about the waist, perhaps from PIE root *mei- "to bind, attach" (source also of Sanskrit mitra- "friend, friendship," Old Persian Mithra-, god name; Russian mir "world, peace"). The Greek word might be borrowed from Indo-Iranian.

In pre-Christian Latin, in reference to a type of head-dress anciently worn by inhabitants of Lydia, Phrygia, and other parts of Asia Minor, "the wearing of which by men was regarded in Rome as a mark of effeminacy" [OED]. But the word was used in Vulgate to translate Hebrew micnepheth, the sacerdotal head-dress of the ancient Jewish high priests.

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miter (n.1)

in reference to a type of head-dress, an alternative spelling of mitre (see -re).

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mitral (adj.)

c. 1600, "resembling a mitre, of or pertaining to a mitre," from French mitral, from Modern Latin mitralis, from Latin mitra (see mitre). The mitral valve of the heart is so called from 1705, from Modern Latin mitrales valvulae.

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miter (n.2)

in carpentry, "a joint at a 45 degree angle," 1670s, of uncertain origin, perhaps from mitre, via notion of joining of the two peaks of the folded cap. As a verb, to make or join with a miter-joint," from 1731. Related: Mitered. Miter-box is attested from 1670s.

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mir 

1877, "a Russian commune or village," also (with capital M-) the name of a late 20c. space station, Russian, literally "peace, world," also "village, community," from Old Church Slavonic miru "peace," from Proto-Slavic *miru "commune, joy, peace" ("possibly borrowed from Iranian" [Watkins]), from PIE root *mei- (4) "to bind, tie" (see mitre). Old Church Slavonic miru was "used in Christian terminology as a collective 'community of peace' " [Buck], translating Greek kosmos. Hence, "the known world, mankind."

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