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As The Catholic Encyclopedia article on dancing illustrates, the Church does not condemn all ballroom dancing:
Undoubtedly old national dances in which the performers stand apart, hardly, if at all, holding the partner’s hand, fall under ethical censure scarcely more than any other kind of social intercourse. But, aside from the concomitants—place, late hours, décolleté, escorting, etc.—common to all such entertainments, round dances, although they may possibly be carried on with decorum and modesty, are regarded by moralists as fraught, by their very nature, with the greatest danger to morals. To them perhaps, but unquestionably still more obviously to masked balls, should be applied the warning of the Second Council of Baltimore, against “those fashionable dances, which, as at present carried on, are revolting to every feeling of delicacy and propriety.”
Note that the Church has concerns about ballroom dancing that involves women wearing low-cut dresses (decollete’) and “masked balls,” in which the identities of dancers are not readily apparent. In addition, the Second Plenary Council of Baltimore (1866) warned against dances as “at present carried on” not because ballroom dancing is intrinsically wrong. It is those dances that “are revolting to every feeling of delicacy and propriety,” not ballroom dances necessarily, let alone dancing in general.
Some groups of Protestant Christians categorically ban dancing, not the Catholic Church, though the Church warns against the moral perils of some types of dancing, including because of the immodest clothing of the participants.