Etymology
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Words related to behoof

behoove (v.)

Old English behofian "to have need of, have use for," verbal form of the ancient compound word represented by behoof (q.v.). From c. 1200 as "be fit or meet for, be necessary for," now used only in the third person, with it as subject. Related: Behooved; behooving.

Historically, it rimes with move, prove, but being now mainly a literary word, it is generally made to rime with rove, grove, by those who know it only in books. [OED]
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be- 

word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).

The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).

heave (v.)
Old English hebban "to lift, raise; lift up, exalt" (class VI strong verb; past tense hof, past participle hafen), from Proto-Germanic *hafjan (source also of Old Norse hefja, Dutch heffen, German heben, Gothic hafjan "to lift, raise"), from PIE *kap-yo-, from root *kap- "to grasp." The sense evolution would be "to take, take hold of," thence "lift."

Related to have (Old English habban "to hold, possess"). Meaning "to throw" is from 1590s. Nautical meaning "haul or pull" in any direction is from 1620s. Intransitive use from early 14c. as "be raised or forced up;" 1610s as "rise and fall with alternate motion." Sense of "retch, make an effort to vomit" is first attested c. 1600. Related: Heaved; heaving. Nautical heave-ho was a chant in lifting (c. 1300, hevelow).
*kap- 

Proto-Indo-European root meaning "to grasp."

It forms all or part of: accept; anticipate; anticipation; behave; behoof; behoove; cable; cacciatore; caitiff; capable; capacious; capacity; capias; capiche; capstan; caption; captious; captivate; captive; captor; capture; case (n.2) "receptacle;" catch; catchpoll; cater; chase (n.1) "a hunt;" chase (v.) "to run after, hunt;" chasse; chasseur; conceive; cop (v.) "to seize, catch;" copper (n.2) "policeman;" deceive; emancipate; except; forceps; gaffe; haft; have; hawk (n.); heave; heavy; heft; incapacity; inception; incipient; intercept; intussusception; manciple; municipal; occupy; participation; perceive; precept; prince; purchase; receive; recipe; recover; recuperate; sashay; susceptible.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit kapati "two handfuls;" Greek kaptein "to swallow, gulp down," kope "oar, handle;" Latin capax "able to hold much, broad," capistrum "halter," capere "to grasp, lay hold; be large enough for; comprehend;" Lettish kampiu "seize;" Old Irish cacht "servant-girl," literally "captive;" Welsh caeth "captive, slave;" Gothic haban "have, hold;" Old English hæft "handle," habban "to have, hold."