"state of not being present," late 14c., from Old French absence "absence" (14c.), from Latin absentia, abstract noun from absentem (nominative absens), present participle of abesse "be away from, be absent," from ab "off, away from" (see ab-) + esse "to be" (from PIE root *es- "to be"). Absence makes the heart grow fonder is a line from the song "Isle of Beauty" by English poet and composer Thomas Haynes Bayly (1797-1839).
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).
"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.
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.
mid-14c., "leave or permission to depart," from Old French conget, congié "permission, leave of absence, dismissal, ceremonial leave-taking" (Modern French congé), from Medieval Latin commeatus "leave, permission to depart," in classical Latin "passage, going to and fro," hence "leave of absence," from commeare, from com "with, together" (see com-) + meare "to go, pass" (from PIE root *mei- (1) "to change, go, move"). Probably lost 17c. and revived 19c. from Modern French as congé. Also as a verb.
1783, "grant leave of absence" (to a soldier), from furlough (n.). Of employees, "lay off or suspend temporarily," by 1940. Related: Furloughed; furloughing.
c. 1600, transitive, "to leave, abandon," either in a good or bad sense; 1640s, in reference to military service or duty, "leave without permission;" from French déserter "cause to leave," literally "undo or sever connection," from Late Latin desertare, frequentative of Latin deserere "to abandon, to leave, forsake, give up, leave in the lurch," from de "undo" (see de-) + serere "join together, put in a row" (from PIE root *ser- (2) "to line up"). Intransitive sense of "quit (a service or post) without permission" is from 1680s. Related: Deserted; deserting.