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The chancel was also originally screened off in Roman Catholic churches, but under Protestant influences the barriers were eventually done away there also.
In the ancient Church the altar was understood to have taken the place of the Holy of Holies. Screening the altar follows from Jewish tradition, where the Holy of Holies was covered by a veil (Exodus 27) and could be accessed only by the priests (Hebrews 9:7). See On the Divine Liturgy by written by Germanus, Patriarch of Constantinople in the 8th century.