What obstacles did Thomas Jefferson have to overcome to become the President?
The electoral system at that time made no distinction between candidates for President and Vice-President, the Vice_Presidency simply going to the runner-up in the Presidential race. The Democratic-Republican electors voted solidly for Jefferson and Burr, producing a tie vote which had to be resolved in the House of Representatives. Burr had always been viewed as the Vice-Presidential candidate, but refused to withdraw, and it took 36 ballots before Jefferson was finally elected. Burr became Vice-President, but unsurprisingly was not renominated in 1804.
