bourn (n.1)

also bourne, "small stream," especially of the winter torrents of the chalk downs, Old English brunna, burna "brook, stream," from Proto-Germanic *brunnoz "spring, fountain" (source also of Old High German brunno, Old Norse brunnr, Old Frisian burna, German Brunnen "fountain," Gothis brunna "well"), ultimately from PIE root *bhreu- "to boil, bubble, effervesce, burn." The southern England form of northern burn.

"destination," 1520s, from French borne, apparently a variant of bodne "limit, boundary, boundary stone" (see bound (n.1)). Used by Shakespeare in Hamlet's soliloquy (1602) and elsewhere, from which it entered into English poetic speech. He meant it probably in the correct sense of "boundary," but it has been taken to mean "goal" (Wordsworth, Matthew Arnold) or sometimes "realm" (Keats).

The dread of something after death, The vndiscouered Countrey; from whose Borne No Traueller returnes. ["Hamlet" III.i.79]