Election Day is upon us with polls showing President Trump and Vice President Harris neck and neck, but Americans need not dread an Electoral College tie. The Constitution, one might say, has an app for that. The new House will choose America’s next president even if both candidates win 269 electoral votes, meaning whoever has longer coattails down-ballot will take on an importance not seen in two centuries.

When the Constitution was drafted, the two-party system hadn’t yet emerged, raising the prospect that no candidate would get a majority in the Electoral College. This first happened in 1800, when each elector cast two votes. The candidate who earned the most would be president, and the runner-up vice president.

Jefferson earned 73 electoral votes with his running mate, Senator Aaron Burr, beating President John Adams’s 65. The first Chief Justice of the United States, John Jay, made sure Adams would get more electors than his running mate, the minister to France, Charles Pinckney. Because Jefferson didn’t have any wrangler, he and Burr tied.

It fell to the House to choose, and Burr decided that he’d rather not settle for second place. He and Jefferson deadlocked for a week and 36 ballots. As a snowstorm raged at Washington, a Federalist leader, Alexander Hamilton, tried to thaw the frozen machinery of government. Although an enemy of Jefferson, a Democratic-Republican, Hamilton declared him “not so dangerous” as Burr.

Hamilton would meet his fate four years later when Burr killed him in a duel. On that snowy evening, with the presidency in the balance, his act of putting the country first broke the ice on the 37th ballot. To ensure that no future vice-presidential candidate could again pull a Burr, the 12th Amendment was ratified in 1804, mandating separate ballots for president and vice president.

In 1824, four major figures vied for the White House. This “War of the Giants” ended the Era of Good Feeling under President Monroe, who had won reelection with all but one electoral vote. The dissenter was a New Hampshire senator, William Plumer. A pleasing myth endures that he’d thought only President Washington should have the honor of a unanimous election. In truth, he judged Monroe incompetent.

President John Quincy Adams, secretary of state, President Jackson, a senator, emerged with 84 and 99 electoral votes respectively. The Treasury secretary, William Crawford, banked 41 and the speaker of the House, Henry Clay, 37. The 12th Amendment required the House to choose between the top three candidates, but Crawford had suffered a stroke. It came down to Adams versus Jackson.

Each state delegation was given one vote in 1824, as would be the case for Trump and Ms. Harris. Although Jackson was a hero in the War of 1812, he was an enemy to “the Great Compromiser,” Clay. “I cannot believe,” Clay said of Jackson, “that killing 2,500 Englishmen at New Orleans qualifies for the various, difficult, and complicated duties of the Chief Magistracy.” He threw his support to Adams, who prevailed, by a single vote, on the first ballot.

In the current House, Republicans control the 26 delegations and Democrats 24. That raises the possibility that neither Trump nor Ms. Harris will get the required majority if the new House is split 25 to 25. Should that deadlock linger through Inauguration Day, January 20, it may be Burr who gets the last laugh.

While the House picks a president, the new Senate will choose a vice president, and he’d be the acting president until the state delegations settle on a permanent chief executive. That raises the possibility that, if the two chambers are in hands of opposite parties, Mr. Walz could end up serving under Trump or Mr. Vance under Ms. Harris.

FiveThirtyEight’s statistician Nate Silver judges the odds of a Trump-Harris tie at just 0.4 percent. Yet if a repeat of Jefferson-Burr happens, Americans can rest assured that the Constitution provides a mechanism for a solution, and presidential coattails will wag the dog. There will be a new president come January, ready to take up the various, difficult, and complicated duties of leading the republic.