bite (v.)youdaoicibaDictYouDict
Old English bitan (class I strong verb; past tense bat, past participle biten), from Proto-Germanic *bitan (cognates: Old Saxon bitan, Old Norse and Old Frisian bita, Middle Dutch biten, Dutch bijten, German beissen, Gothic beitan "to bite"), from PIE root *bheid- "to split, crack" (see fissure).

To bite the bullet is said to be 1700s military slang, from old medical custom of having the patient bite a lead bullet during an operation to divert attention from pain and reduce screaming. Figurative use from 1891; the custom itself attested from 1840s. To bite (one's) tongue "refrain from speaking" is 1590s. To bite the dust "die" is 1750 (Latin had the same image; compare Virgil's procubuit moriens et humum semel ore momordit). To bite off more than one can chew (c. 1880) is U.S. slang, from plug tobacco.
bullet (n.)youdaoicibaDictYouDict
1550s, from Middle French boulette "cannonball, small ball," diminutive of boule "a ball" (13c.), from Latin bulla "round thing, knob" (see bull (n.2)). Earliest version of bite the bullet recorded 1891, probably with a sense of giving someone a soft lead bullet to clench in the teeth during a painful operation.