ballgame (n.)

"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."

Meaning "testicle" is from early 14c. (compare ballocks). Ball of the foot is from mid-14c. Meaning "rounded missile used in warfare" is from late 14c. A ball as an object in a sports game is recorded from c. 1200; meaning "a game played with a ball" is from mid-14c. 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 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 an oft-repeated item of 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]

c. 1200, from Old English gamen "joy, fun; game, amusement," common Germanic (cognates: Old Frisian game "joy, glee," Old Norse gaman "game, sport; pleasure, amusement," Old Saxon gaman, Old High German gaman "sport, merriment," Danish gamen, Swedish gamman "merriment"), said to be identical with Gothic gaman "participation, communion," from Proto-Germanic *ga- collective prefix + *mann "person," giving a sense of "people together."

The -en was lost perhaps through being mistaken for a suffix. Meaning "contest for success or superiority played according to rules" is first attested c. 1200 (of athletic contests, chess, backgammon). Especially "the sport of hunting, fishing, hawking, or fowling" (c. 1300), thus "wild animals caught for sport" (c. 1300), which is the game in fair game (see under fair (adj.)), also gamey. Meaning "number of points required to win a game" is from 1830. Game plan is 1941, from U.S. football; game show first attested 1961.