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That was the line taken by other Federalists in the House. The six states whose House delegations were dominated by Federalists all voted for Burr. Jefferson carried eight states, with the remaining two states split and therefore not voting. So Jefferson was one short of the nine states needed for election.
The deadlock lasted for four days and 33 ballots, and endangered the peaceful transfer of power. When Federalists began to talk of an "interim" appointed president (who would be a Federalist), Jefferson threatened Adams with "resistance by force and incalculable consequences."
Young Congressman Bayard had tried to pull a handful of Republican Congressmen to swing three states for Burr, but failed. He then decided to abandon Burr, handing Delaware and the presidency to Jefferson. Did he do so to avert a constitutional crisis? In truth, Jefferson agreed to retain two federal port collectors sponsored by Bayard. "You are safe," Bayard wrote to one of them.
The federal union had averted an early crisis, but not everybody was happy. Abigail Adams wrote to her husband: "'What an inconsistency,' said a lady to me today, 'the bells of Christ Church ringing peals of rejoicing for an infidel president!'" We do have a long tradition of finding it easier to be partisan than conciliatory.