Etymology
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leave (v.)

Old English læfan "to allow to remain in the same state or condition; to let remain, allow to survive; to have left (of a deceased person, in reference to heirs, etc.); to bequeath (a heritage)," from Proto-Germanic *laibjanan (source also of Old Frisian leva "to leave," Old Saxon farlebid "left over"), causative of *liban "remain" (source of Old English belifan, German bleiben, Gothic bileiban "to remain"), from PIE root *leip- "to stick, adhere."

The Germanic root seems to have had only the sense "remain, continue" (which was in Old English as well but has since become obsolete), which also is in Greek lipares "persevering, importunate." But this usually is regarded as a development from the primary PIE sense of "adhere, be sticky" (compare Lithuanian lipti, Old Church Slavonic lipet "to adhere," Greek lipos "grease," Sanskrit rip-/lip- "to smear, adhere to."

Originally a strong verb (past participle lifen), it early switched to a weak form. Meaning "go away, take one's departure, depart from; leave behind" (c. 1200) comes from notion of "leave behind" (as in to leave the earth "to die;" to leave the field "retreat"). From c. 1200 as "to stop, cease; give up, relinquish, abstain from having to do with; discontinue, come to an end;" also "to omit, neglect; to abandon, forsake, desert; divorce;" also "allow (someone) to go."

Colloquial use for "let, allow" is by 1840, said by OED to be chiefly American English. Not related to leave (n.). To leave out "omit" is from late 15c. To leave (something) alone is from c. 1400; to leave (something) be is from 1825. To leave (something/nothing) to be desired is from 1780. To leave it at that is from 1902. Leave off is from c. 1400 as "cease, desist" (transitive); early 15c. as "stop, make an end" (intransitive).

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leave (n.)
"permission, liberty granted to do something," Old English leafe "leave, permission, licence," dative and accusative of leaf "permission," from Proto-Germanic *laubo (source also of Old Norse leyfi "permission," and, with prefix, Old Saxon orlof, Old Frisian orlof, German Urlaub "leave of absence"), from PIE root *leubh- "to care, desire, love," the original idea being "approval resulting from pleasure." It is a noun relative of lief "dear" (adj.); and compare belief. In the military sense, it is attested from 1771.
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leave-taking (n.)
late 14c., from the verbal phrase to take leave, originally "obtain or receive formal permission" in any sense (c. 1300); see take (v.) + leave (n.). Sense evolution was through "receive formal permission to depart;" by 16c. it had begun to mean "depart with an expression of farewell," and in some cases came to mean the farewell itself. Give (someone) leave (v.) "allow, permit" is from 12c.
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left 
"remaining," past tense and past participle of leave (v.).
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beleave (v.)
Old English belæfan, "to cause or allow to remain behind, to leave something behind," a general Germanic compound (compare Gothic bilaibjan) from be- + Old English læfan "to leave" (see leave (v.)). Obsolete from 17c. In Middle English sometimes contracted to bleve. For further development, see belive.
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belive (v.)
obsolete verb from Old English belifan "remain," intransitive form of belæfan "cause to remain" (see beleave). A general Germanic compound (cognates: Old Saxon bilibhon, Gothic bileiban, Old High German biliban, German bleiben, Dutch blijven). It was confused in early Middle English with beleave and merged into it, which gave beleave two clashing senses ("to leave," also "to remain") which might be why the compound word, the cognate of important verbs in other Germanic languages, was abandoned in English and only leave (v.) remains.
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*leip- 
Proto-Indo-European root meaning "to stick, adhere; fat."

It forms all or part of: adipose; beleave; delay; leave (v.); lebensraum; life; liparo-; lipo- (1) "fat;" lipoma; liposuction; lively; live (v.); liver (n.1) "secreting organ of the body;" Olaf; relay.

It is the hypothetical source of/evidence for its existence is provided by: Greek liparein "to persist, persevere," aleiphein "anoint with oil," lipos "fat;" Old English lifer "liver," læfan "to allow to remain."
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*leubh- 

Proto-Indo-European root meaning "to care, desire, love."

It forms all or part of: belief; believe; furlough; leave (n.) "permission, liberty granted to do something;" leman; libido; lief; livelong; love; lovely; quodlibet.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit lubhyati "desires," lobhaya- "to make crazy;" Persian ahiftan "to be tangled, be hit down, be in love;" Latin lubet, later libet "pleases," libido, lubido "desire, longing; sensual passion, lust;" Old Church Slavonic l'ubu "dear, beloved," ljubiti, Russian ljubit' "to love;" Lithuanian liaupsė "song of praise;" Old English lufu "feeling of love; romantic sexual attraction," German Liebe "love," Gothic liufs "dear, beloved."

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furlough (n.)

1620s, vorloffe, "leave of absence," especially in military use, "leave or license given by a commanding officer to an officer or a soldier to be absent from service for a certain time," from Dutch verlof, literally "permission," from Middle Dutch ver- "completely, for" + laf, lof "permission," from Proto-Germanic *laubo-, from PIE root *leubh- "to care, desire, love." In English, the elements of it are for- + leave. The -gh spelling predominated from 1770s and represents the "f" that had been pronounced at the end of the word but later disappeared in English.

The spelling furloe occurs in the 18th century, but furlough appears to be the earliest spelling (as in Blount's Gloss., ed. 1674). As the spelling furlough does not follow that of the orig. language, it was prob. intended to be phonetic (from a military point of view), the gh perhaps as f and the accent on the second syllable .... [Century Dictionary]

By 1946 in reference to temporary layoffs of workers (originally of civilian employees in the U.S. military); by 1975 applied to conditional temporary releases of prisoners for the purpose of going to jobs (work-release).

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leaved (adj.)
"having a leaf or leaves," past-participle adjective from verb leave "to put forth leaves," mid-13c., from leaf (n.).
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