Etymology
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be (v.)

Old English beon, beom, bion "be, exist, come to be, become, happen," from Proto-Germanic *biju- "I am, I will be." This "b-root" is from PIE root *bheue- "to be, exist, grow," and in addition to the words in English it yielded the German present first and second person singular (bin, bist, from Old High German bim "I am," bist "thou art"), the Latin perfective tenses of esse (fui "I was," etc.), Old Church Slavonic byti "be," Greek phu- "become," Old Irish bi'u "I am," Lithuanian būti "to be," Russian byt' "to be," etc.

The modern verb to be in its entirety represents the merger of two once-distinct verbs, the "b-root" represented by be and the am/was verb, which was itself a conglomerate. Roger Lass ("Old English") describes the verb as "a collection of semantically related paradigm fragments," while Weekley calls it "an accidental conglomeration from the different Old English dial[ect]s." It is the most irregular verb in Modern English and the most common.

Collective in all Germanic languages, it has eight different forms in Modern English: BE (infinitive, subjunctive, imperative); AM (present 1st person singular); ARE (present 2nd person singular and all plural); IS (present 3rd person singular); WAS (past 1st and 3rd persons singular); WERE (past 2nd person singular, all plural; subjunctive); BEING (progressive & present participle; gerund); BEEN (perfect participle).

The paradigm in Old English was: eom, beo (present 1st person singular); eart, bist (present 2nd person singular); is, bið (present 3rd person singular);  sind, sindon, beoð (present plural in all persons); wæs (past 1st and 3rd person singular); wære (past 2nd person singular); wæron (past plural in all persons); wære (singular subjunctive preterit); wæren (plural subjunctive preterit).

The "b-root" had no past tense in Old English, but often served as future tense of am/was. In 13c. it took the place of the infinitive, participle and imperative forms of am/was. Later its plural forms (we beth, ye ben, they be) became standard in Middle English and it made inroads into the singular (I be, thou beest, he beth), but forms of are claimed this turf in the 1500s and replaced be in the plural. For the origin and evolution of the am/was branches of this tangle, see am and was.

That but this blow Might be the be all, and the end all. ["Macbeth" I.vii.5]
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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).

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-on 

subatomic particle suffix, from ion.

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on- 

the preposition and adverb on used as a prefix; Old English on-, an-.

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on (prep., adv.)

"in a position above and in contact with; in such a position as to be supported by;" also noting the goal to which some action is or has been directed; "about, concerning, regarding; in a position to cover;" as an adverb, "in or into a position in contact with and supported by the top or upper part of something; in or into place; in place for use or action; into movement or action; in operation," Old English on, unstressed variant of an "in, on, into," from Proto-Germanic *ana "on" (source also of Dutch aan, German an, Gothic ana "on, upon"), from PIE root *an- (1) "on" (source also of Avestan ana "on," Greek ana "on, upon," Latin an-, Old Church Slavonic na, Lithuanian nuo "down from").

Also used in Old English in many places where we now would use in. From 16c.-18c. (and still in northern England dialect) often reduced to o'. Phrase on to "aware" is from 1877.

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ball (n.1)

"round object, compact spherical body," also "a ball used in a game," c. 1200, probably from an unrecorded Old English *beal, *beall (evidenced by the diminutive bealluc "testicle"), or from cognate Old Norse bollr "ball," from Proto-Germanic *balluz (source also of Dutch bal, Flemish bal, Old High German ballo, German Ball), from PIE root *bhel- (2) "to blow, swell."

The meaning "testicle" is from early 14c. (compare ballocks). Ball of the foot is from mid-14c. The meaning "rounded missile used in warfare" is from late 14c.

The meaning "a game played with a ball" is from mid-14c. The baseball sense of "pitch that does not cross the plate within the strike zone" is by 1889, probably short for high ball, low ball, etc.

Ball-point pen is by 1946. Ball of fire when first recorded in 1821 referred to "a glass of brandy;" as "spectacularly successful striver" it is attested from c. 1900. Many phrases are from sports: To have the ball "hold the advantage" is from c. 1400. To be on the ball is from 1912; to keep (one's) eye on the ball in the figurative sense is by 1907, probably ultimately on golf, where it was oft-repeated advice. Figurative use of ball in (someone's) court is by 1956, from tennis.

The head must necessarily be steady, for it is most important that you should keep your eye fixedly on the ball from the moment that the club-head is lifted from the ground until the ball is actually struck. "Keep your eye on the ball," should be your companion text to "Slow back." [Horace G. Hutchinson, "Hints on the Game of Golf," 1886]
Once a meeting is over, someone will be expected to do something. Make sure it is someone else. This is known as keeping the ball in their court. [Shepherd Mead, "How to Get Rich in TV Without Really Trying," 1956]
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ball (n.2)

"dancing party, social assembly for dancing," 1630s, from French, from Old French baller "to dance," from Late Latin ballare "to dance," from Greek ballizein "to dance, jump about," literally "to throw one's body" (ancient Greek dancing being highly athletic), from PIE root *gwele- "to throw, reach." The extended meaning "very enjoyable time" is American English slang from 1945, perhaps 1930s in African-American vernacular.

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ball (v.)

1650s, "make into a ball," from ball (n.1). Intransitive sense of "become like a ball, form a compact cluster" is from 1713; that of "to copulate" is first recorded 1940s in jazz slang, either from the noun sense of "testicle" or "enjoyable time" (from ball (n.2)). Related: Balled; balling.

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be-in (n.)

"a public gathering of hippies" [OED], 1967, from be + in (adv.).

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