An Etymological Dictionary of the German Language/Annotated/halb
halb, adjective, ‘half,’ from Middle High German halp, Old High German halb (genitive halbes). adjective; compare Old Saxon and Low German half, Dutch half, Anglo-Saxon healf, English half, Old Icelandic hálfr, Gothic halbs, adjective; the common Teutonic adjective for High German halb; there are no undoubted cognates in the non-Teutonic languages (Teutonic halba-, from pre-Teutonic kalbho-). The feminine of the adjective is used in Old Teutonic as a substantive in the sense of ‘side, direction’; Gothic halba, Old Icelandic halfa, Old High German halba, Middle High German halbe, Old Saxon halƀa; hence it might seem as if the adjective had originally some such meaning as ‘lateral, that which lies on one side.’ But in any case the adjective in the sense of ‘half’ was purely a numeral in primitively Teutonic; the Modern High German method of reckoning anderthalb (112), drittehalb (212), viertehalb (312), is common to Teutonic; compare Old Icelandic halfr annarr (112), halfr þriþe (212), halfr fjorþe (312); Anglo-Saxon ôþer healf, þridde healf, feorþe healf; even in Middle English this enumeration exists (it is wanting in English); in High German it has been retained from the earliest period.
halb, halben, preposition, ‘on account of,’ from Middle High German halp, halbe, halben, ‘on account of, by reason of, from, concerning’; properly a case of the Middle High German substantive halbe, feminine, ‘side,’ mentioned under halb (adjective), hence construed with the genitive; Middle High German mîn-halp, dîn-halp, der herren halbe, sëhens halben, ‘on my, thy account, on the gentlemen's account, for the sake of seeing.’ Similarly the Modern High German halber, ‘on account of,’ recorded in the 15th century, is a petrified form of the inflected adjective; so too halben, dative plural, halbe, halp, from Old High German halb, probably an instrumental singular (since Notker halb has been used as a preposition). This usage is also found in the other Teutonic languages; compare Old Icelandic af-halfu, Middle English on-, bi-halfe; Gothic in þizai halbai, ‘in this respect.’