Charles Buck Theological Dictionary
An officer in the Greek church, whose business it is to visit the provinces allotted to him, in order to inform himself of the lives and manners of the clergy; take cognizance of ecclesiastical causes; the manner of celebrating divine service; the administration of the sacraments, particularly confession; the observance of the canons; monastic discipline; affairs of marriages, divorces, &c. but above all, to take an account of the several revenues which the patriarch receives from several churches, and particularly as to what regards collecting the same. The exarch, after having enriched himself in his post, frequently rises to the patriarchate himself. Exarch is also used in the Eastern church antiquity, for a general or superior over several monasteries, the same that we call archimandrite; being excepted by the patriarch of Constantinople from the jurisdiction of the bishop.
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Buck, Charles. Entry for 'Exarch'. Charles Buck Theological Dictionary. https://www.studylight.org/dictionaries/eng/cbd/e/exarch.html. 1802.