In this extraordinary presidential campaign, where every rule in the book has been torn up, they are the forgotten men: running mates in a year when the names of Donald Trump and Hillary Clinton drown out all else.

Yet Tim Kaine and Mike Pence matter. They’re the vice presidential nominees, and if history is any guide, either of them could be summoned at short notice to even higher things.

Vice presidents count for much more than they used to. No longer do they toil unnoticed in the vineyard or, worse, kick their heels doing nothing. No longer does the old saying about the job being worth less than a bucket of warm spit (in the politer version of the expression) apply.

The first truly consequential vice president of modern times was Walter Mondale, under Jimmy Carter. During George W Bush’s presidency, Dick Cheney rose to become the most consequential of them all, wielding unprecedented influence to the point of sometimes seeming to run a parallel administration.

One reason for the growing importance of the post is simply that there’s more to do. The modern presidency is the most crushing job on earth; a vice president can share at least some of the burden, though how much depends on the personal relationship between the two. If Cheney’s star waned in Bush’s second term, Joe Biden has become a steadily more central figure under Barack Obama, in large measure simply because the two men have grown closer.

Donald Trump's most controversial quotes Show all 14 1 /14 Donald Trump's most controversial quotes Donald Trump's most controversial quotes On Isis: "Some of the candidates, they went in and didn’t know the air conditioner didn’t work and sweated like dogs, and they didn’t know the room was too big because they didn’t have anybody there. How are they going to beat ISIS?" Getty Donald Trump's most controversial quotes On immigration: "I will build a great wall — and nobody builds walls better than me, believe me —and I’ll build them very inexpensively. I will build a great, great wall on our southern border, and I will make Mexico pay for that wall. Mark my words." Reuters Donald Trump's most controversial quotes On Free Trade: "Free trade is terrible. Free trade can be wonderful if you have smart people. But we have stupid people." PAUL J. RICHARDS | AFP | Getty Images Donald Trump's most controversial quotes On Mexicans: "When Mexico sends its people, they’re not sending their best. They’re sending people that have lots of problems. They’re bringing drugs. They’re bringing crime. They’re rapists." Getty Donald Trump's most controversial quotes On China: "I just sold an apartment for $15 million to somebody from China. Am I supposed to dislike them?... I love China. The biggest bank in the world is from China. You know where their United States headquarters is located? In this building, in Trump Tower." Getty Images Donald Trump's most controversial quotes On work: "If you're interested in 'balancing' work and pleasure, stop trying to balance them. Instead make your work more pleasurable." AP Donald Trump's most controversial quotes On success: "What separates the winners from the losers is how a person reacts to each new twist of fate." Donald Trump's most controversial quotes On life: "Everything in life is luck." AFP Donald Trump's most controversial quotes On ambition: "You have to think anyway, so why not think big?" Getty Images Donald Trump's most controversial quotes On his opponents: "Bush is totally in favour of Common Core. I don't see how he can possibly get the nomination. He's weak on immigration. He's in favour of Common Core. How the hell can you vote for this guy? You just can't do it." Reuters Donald Trump's most controversial quotes On Obamacare: "You have to be hit by a tractor, literally, a tractor, to use it, because the deductibles are so high. It's virtually useless. And remember the $5 billion web site?... I have so many web sites, I have them all over the place. I hire people, they do a web site. It costs me $3." Getty Images Donald Trump's most controversial quotes On Barack Obama: "Obama is going to be out playing golf. He might be on one of my courses. I would invite him. I have the best courses in the world. I have one right next to the White House." PA Donald Trump's most controversial quotes On himself: "Love him or hate him, Trump is a man who is certain about what he wants and sets out to get it, no holds barred. Women find his power almost as much of a turn-on as his money." Getty Images Donald Trump's most controversial quotes On America: "The American Dream is dead. But if I get elected president I will bring it back bigger and better and stronger than ever before and we will make America great again." GETTY

Where Kaine might fit on this spectrum is clear: it’s not difficult to imagine him developing a relationship with Clinton akin to that between Obama and Biden. But Pence? Sometimes during the campaign Trump has sounded as if he didn’t have a running mate at all. It’s hard to see how that will change should the egocentric billionaire take up residence at Pennsylvania Avenue.

However a vice president’s most vital role resides in that heartbeat of distance that separates him from the presidency itself. Three times in the past 75 years, a sitting vice president has been called upon in extremis, when the supreme office has suddenly become vacant through death or resignation: Harry Truman in 1945, Lyndon B Johnson in 1963, and Gerald Ford in 1974. So out of the loop, incidentally, was Truman that Roosevelt hadn’t even informed him of the Manhattan Project developing a nuclear weapon when FDR died in April 1945.

Who would dare rule out another dramatic succession, given the ever more polarised state of US politics, where scandals real and imagined are permanently in the air and threats of impeachment are bandied around daily – not to mention the intensely hostile feelings that both Hillary Clinton and Donald Trump arouse, in a nation awash with guns?

Americans are aware of all this. George H W Bush was not helped (to put it mildly) by his choice of Dan Quayle as vice president. And nothing did more to destroy John McCain’s reputation for sound judgement than that the 2008 Republican nominee, a man of 72 with a less than perfect health record, should choose Sarah Palin as his running mate and thus potential successor.

Vice presidents are a factor for another reason too. The job guarantees national name recognition and thus an automatic place in any early list of White House hopefuls the next time around. Of the last 12 vice presidents, seven have either become president or run for president.

On all these scores, both Kaine and Pence pass muster. They are no Sarah Palins. Unlike their bosses, they are popular. Indeed, in the darkest hours of Trump’s campaign this autumn, some wished (and a few may even plotted) for him to be forced out and Pence take over.

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Pence is a family values conservative through and through, but was well liked on all sides during his 12 years as an Indiana Congressman in Washington. Ditto Kaine, a moderate Democrat for whom Republicans would struggle to find an unkind word during his stints as mayor of Richmond, Virginia’s capital city, governor of the state and then as one of its two US senators.

Both are amply qualified to take over; indeed, it’s easy to imagine either man as president (in Pence’s case a good deal more easy than his boss).