there are several theories, all similar, and deriving the word from the tendency to say "both the." One is that it is Old English begen (masc.) "both" (from Proto-Germanic *ba, from PIE *bho "both") + -þ extended base. Another traces it to the Proto-Germanic formula represented in Old English by ba þa "both these," from ba (feminine nominative and accusative of begen) + þa, nominative and accusative plural of se "that." A third traces it to Old Norse baðir "both," from *bai thaiz "both the," from Proto-Germanic *thaiz, third person plural pronoun. Compare similar formation in Old Frisian bethe, Dutch beide, Old High German beide, German beide, Gothic bajoþs.