eye-shot (n.)

c. 1200, from Old English ege (Mercian), eage (West Saxon) "eye; region around the eye; apperture, hole," from Proto-Germanic *augon (source also of Old Saxon aga, Old Frisian age, Old Norse auga, Swedish öga, Danish øie, Middle Dutch oghe, Dutch oog, Old High German ouga, German Auge, Gothic augo "eye"). Apparently the Germanic form evolved irregularly from PIE root *okw- "to see."

HAMLET: My father — methinks I see my father.

HORATIO: Where, my lord?

HAMLET: In my mind's eye, Horatio.

Until late 14c. the English plural was in -an, hence modern dialectal plural een, ene. Of potatoes from 1670s. Of peacock feathers from late 14c. As a loop used with a hook in fastening (clothes, etc.) from 1590s. The eye of a needle was in Old English. As "the center of revolution" of anything from 1760. Nautical in the wind's eye "in the direction of the wind" is from 1560s.

To see eye to eye is from Isaiah lii.8. Eye contact attested from 1953. To have (or keep) an eye on "keep under supervision" is attested from early 15c. To have eyes for "be interested in or attracted to" is from 1736; make eyes at in the romance sense is from 1837; gleam in (someone's) eye (n.) "barely formed idea" is from 1959. Eye-biter was an old name for "a sort of witch who bewitches with the eyes."

Middle English shot "a missile, arrow, dart" (senses now archaic or obsolete); "a swift movement, a gushing out," from Old English scot, sceot "a shot, a shooting, an act of shooting; that which is discharged in shooting, what is shot forth; darting, rapid motion."

This is from Proto-Germanic *skutan (source also of Old Norse skutr, Old Frisian skete, Middle Dutch scote, German Schuß "a shot"), from PIE root *skeud- "to shoot, chase, throw." The Old English noun is related to sceotan "to shoot." The meaning "discharge of a bow, missile," also is from related Old English gesceot.

The noun was extended to other projectiles (balls, bullets) by mid-15c. Especially "lead in small pellets, a small ball or pellet," a number of which are combined in one charge, which is attested by 1770 (shortened from earlier small shot, 1727).

The general sense of "an attempt to hit with a projectile" is by 1650s. Extended to sports (hockey, basketball, etc.) by 1772, originally in curling. It is attested by early 15c. as "range or distance of a missile in flight," hence "range" in general (c. 1600), as in earshot. 

Another original meaning, "payment" (perhaps literally "money thrown down") is preserved in scot-free; also see scot (n.). The notion of "throwing down" might have led to the meaning "a drink," first attested 1670s; the more precise meaning "small drink of straight liquor" is by 1928.

The sense of "hypodermic injection" is attested from 1904; the figurative phrase shot in the arm "stimulant" is by 1922. The broad meaning "a try, an attempt" is by 1756; the sense of "remark meant to wound" is by 1841. The meaning "an expert in shooting with a firearm" is from 1780; the sense of "a rocket flight" is by 1934. The camera-view sense is by 1958.

To call the shots "control events, make decisions" is American English, 1922, perhaps from sport shooting. Shot in the dark "uninformed guess, random attempt" is by 1885. Big shot "important person" is from 1861. 

By the rude bridge that arched the flood,

   Their flag to April’s breeze unfurled,

Here once the embattled farmers stood

   And fired the shot heard round the world.

[Emerson, from "Concord Hymn"]