“It would be easy,” Jared Cohen writes, “to look at the timing of this book and assume it was inspired by all the impeachment talk surrounding Donald Trump.”

It would. When we meet at his home in New York to discuss the book, Accidental Presidents, rumour is rife on the wind. In Washington, the Mueller report is imminent. Later in the day, it is submitted to the attorney general. Two days later the AG says collusion is not proven, obstruction of justice will not stick.

As this is the United States of America in the third year under Trump, this does not put out the fire.

Cohen has written a history of eight vice-presidents who stepped up when their president was removed by fate. It covers the assassinations everyone knows, Lincoln and Kennedy, those some may not, Garfield and McKinley, and what happened when presidents died from natural causes: Harrison, Taylor, Harding, Roosevelt.

With a verve born of deep reading, Cohen also discusses near-misses – illness, accident, shots missed or only wounding – and the instance in 1974 when Richard Nixon was in rude health but resigned in disgrace and Gerald Ford took over. Dick Cheney, a White House staffer then, vice-president under the second Bush, is a key source. Ronald Reagan was shot in 1981 but survived and we are in the longest period in US history without the replacement of a president in office.

Records are made to be broken. And so in his final chapter Cohen considers the 25th amendment, which governs presidential succession or removal and has yet to be tested in full. He finished his book in September 2018. The same month, an anonymous official wrote in the New York Times that cabinet members had discussed Trump’s fitness for the presidency and whether it was time to remove him.

All hell broke loose. Months later, the hoopla has not died down. Cohen insists again he did not write with Trump in mind. But he’s ready to be asked about him.

“Yeah,” he says, “why not? It’s remarkable how timely the book has proven. I was writing a book to rationalise a hobby, and instead I’ve written a book that provides, I think, a very important historical frame of reference for what we’re going through right now.”

Evidence of that hobby is to be found all around us. It starts with a children’s book on the coffee table, by Alice Provensen and called The Buck Stops Here. It’s a rhyming history of the presidents, in which the eight-year-old Cohen gravitated to the more ghoulish ditties. In some cases, it’s more than a bit Edward Gorey:

Cohen holds his own private Rosebud with care, turning the pages slowly. Then he gives a tour of what was spawned by the book: a staggering collection of memorabilia and arcana. There are the campaign buttons that began it all, there are historic front pages – “Dewey defeats Truman” – and there are documents bearing the signature of every president “except Jefferson and Trump”. There are framed locks of hair, Lincoln included, and a portrait in oils by Eisenhower. There is even a box of “Win with Dick” bubblegum Nixon cigars.

“I’ve always been interested in the topic of the presidents who died in office and how history is changed by a heartbeat,” Cohen says. “I just always found the drama of that, the unexpected nature of it, to be this story in our history that isn’t really told.

“We talk about Lincoln’s assassination because of the context in which it happened, and Kennedy’s assassination because there’s people alive who really remember it and watched it on television. But the actual abrupt transfers of power, at some of the most consequential moments in history, are largely forgotten. So I always wanted to kind of put them in one place, and see what the story looks like when you told a history of America anchored around those eight instances.

“So then the question became, ‘When do you find the time to do this when you’re a CEO of a tech company?’”

Which explains both the collection and the Upper West Side apartment it fills. Cohen is chief executive of Jigsaw, formerly Google Ideas, the tech giant’s New York incubator.

The answer to how he came to write Accidental Presidents, an idea long held, is that “when my wife and I were pregnant with our first daughter five years ago”, Cohen found himself needing “a nesting activity”.

“So because I get antsy it was, ‘You know what? I’m finally gonna do this.’ There was something about a giddiness … and memories of my childhood that were evoked from knowing there was a kid on the horizon.”

That kid buzzes happily about. She will soon be the oldest of three. Cohen’s wife, the attorney Rebecca Zubaty, is in attendance too. In the afternoon Cohen will go to his office at Jigsaw – where lurks a life-size wax Teddy Roosevelt, naturally – but for now he drinks coffee from a cup used by Reagan and talks research and development.

He’s only 37 but he has also been an adviser to secretaries of state under George W Bush and Barack Obama (he calls Condoleezza Rice his mentor), a fellow at the Council on Foreign Relations, one of Time’s 100 most influential people, a member of countless advisory boards and a juror for the Tribeca Film Festival. He’s a regular at Davos and he’s written three other books, two on his own and one with Eric Schmidt of Google. The publicity tour for that one included a chat with the pope.

Life at such altitudes demands the ability – and the resources – with which to compartmentalise time, in this case while ranging far from his home archive, sometimes writing for “10 minutes here and there on a plane”.

“I will confess,” he adds, “there were some instances where I used my desire for archival research as an excuse to purchase a signed presidential document or two.”

But he also spent time in presidential libraries, for example in Hyde Park, New York, “sitting down with letters between Eleanor Roosevelt and Harry Truman”. His descriptions of his work take on a spiritual edge. He talks of putting himself in presidents’ shoes, of trying to think like them, of working “almost like an actor”. Reading Eleanor Roosevelt’s letters, he says, was “like getting visibility into the ghost of FDR, and what FDR would’ve said to Truman”.

Of course, Accidental Presidents is really a book about vice-presidents. Discussing Truman, Cohen makes a point relevant to each of his subjects.

“Talk about a lesson that leaders should look to. Truman in his 82 days as vice-president didn’t receive a single intelligence briefing, didn’t meet a single foreign leader, never set foot in the map room where the war was being planned, had no idea what was happening in the war, was basically out socialising. Less than a week before he ascended to the presidency, he described himself to a friend as ‘a political eunuch’.

“And then one day wakes up and he’s president, and what does he inherit? Hitler’s still driving World War II from a bunker. The war in Japan is raging on, they’re in the middle of the Battle of Okinawa, which is the worst in the history of the United States.

“You have a massive bureaucratic fight between the army and the navy that threatens the war effort. You have hundreds of thousands of prisoners of war. You have Stalin basically reneging on every single promise from Yalta and Tehran. You have the postwar order to figure out. You have Congress to navigate.

“I’m forgetting a bunch of it, but that’s a pretty hefty task.”

In a phrase that has something to do with Mark Hanna, an operative who had much to do with the rise of Teddy Roosevelt, vice-presidents are “a heartbeat away from the presidency”. Cohen thinks that means their selection should be taken much more seriously.

Not until the 1960s did the constitution provide answers about the succession and they remain imperfect. Cohen’s book is in many ways a history of ambiguity, of situations demanding answers from politicians not sure of the questions. Presidents are replaced by precedent, which is formed by practice, pragmatism and, most often, the simple exercise of power.

Cohen sketches ideas about reform, perhaps legislation stipulating qualifications for any VP. He questions the practice of picking a running mate simply to balance a ticket. Trump’s veep, Mike Pence, is a former congressman and governor but his selection placated the religious right. So the question follows: should Trump leave office before his time is up, does Cohen think Pence is ready?

“I think Pence is in the loop,” he says, carefully. “I think it’s a different question whether he is the right person to lead this nation.”

And then he dives, with brio, back into beloved history.

“I think there’s an interesting analogy to the Harding and Coolidge administration,” he says. “The likely scenario in which Pence becomes president is, if due to scandal or some sort of indictment or whatever, the president doesn’t complete his term. In which case, Pence takes a page from the Calvin Coolidge book.”

Warren Harding, a president more scandal-ridden than most, is often mentioned next to Trump. On 2 August 1923, in San Francisco, he had a heart attack and died.

Of his successor, Cohen says: “The way he dealt with all of the Harding scandals is he cultivated a whole image of Silent Cal: ‘I’m a man so quiet and so reserved that I couldn’t possibly have had anything to do with this.’ He basically doubled down on his insignificance. I suspect that that would be the best page from history to look to.”

Public cheerleading aside, Pence seems to do something similar. According to Michael Wolff, the vice-president has practiced “extreme self-effacement”, avoiding meetings he or his handlers suspect he shouldn’t be in.

“Yeah,” says Cohen, “and it’s interesting. You see a little bit of the Coolidge strategy, the distancing in the vice-president.”

Pence has had much from which to distance himself: the Russia investigation, House oversight, payments to a porn star and a Playboy model, more. But as our conversation comes back round to Trump, Cohen insists much coverage of the 45th president is flawed, subject to “recency bias”.

“Let’s go through the critiques of President Trump,” he says, “and put them in historic context. And I’m not passing judgment one way or the other, but these are critiques that you hear.

“So people like to say he’s the worst president in history. You can’t be the worst president in history without having a context that is so significant, where the gravity of your decisions means that whether you get it right or wrong can have 100–plus-year consequences. We’re not in one of those moments. Those moments are the civil war, reconstruction, World War I, World War II … these are the seminal moments. We’re just not in one of them.”

Later, with a liberal dose of if not l’esprit d’escalier then l’esprit d’96th Street 1 train, I regret not asking if climate change and its dangers, met by Trump with aggressive disdain, might not qualify for such a list. So I email Cohen, and he replies:

Back in the apartment, Cohen addresses “a second critique” of Trump, which is “that he’s the most erratic and impulsive president in history. That’s certainly true over social media, but there haven’t been that many social media presidents. If you wanna look at erratic, John Tyler is the first accidental president” – succeeding after William Henry Harrison caught pneumonia at his inauguration – “and he gets kicked out of his own party.

“The Democrats won’t take him back because they’re still angry he ran as a Whig in 1840. And so he gets angry and pissed off at everybody, and uses covert tactics to annex Texas and provoke war with Mexico, which was the single action that made civil war inevitable as opposed to likely. That’s impulsive and erratic. In my mind, that matters a lot more.”

Of course, these days the only battles over Tyler are fought on paper, between readers of history or those who write it, nowhere near so fierce as those fought out over Trump. So, in the spirit of choosing of sides, gleeful in the company of another history geek, I ask Cohen to name his favourite accidental president and the most consequential.

His favourite is Teddy Roosevelt, “both extraordinary and irresponsible”, a man whose understanding of the “bully pulpit” Cohen compares to reality TV and all it has given to Trump.

He evidently loves Roosevelt but suggests America got lucky when he ruled in peacetime, that such a hothead “didn’t preside over one of these moments of tremendous consequence”. Which leads us to the accidental president he sees as most consequential.

I suggest Lyndon Johnson, Kennedy’s replacement who pushed civil rights laws through Congress but, as it were, met his Waterloo in Vietnam.

“I would say Andrew Johnson,” Cohen counters, of Lincoln’s successor. “Because the victory in the civil war gave the country an enormous opportunity that was completely squandered, and we’re still paying a price for it today. We still have a number of problems today that I think tie directly back to decisions that Andrew Johnson made.”

Among them, endemic racism and inequality and a geographic and political divide every bit as raw as in the 1860s.

Of course, the 17th president is represented on the wall to our left. Among documents signed by Washington and Grant, close by letters written by John and John Quincy Adams, there hangs an unconditional pardon, signed by Johnson for Thomas Ford on 20 February 1867. In the same frame, there is an object that brings the interview full circle. It is a ticket to the Senate chamber, dated 6 May 1868, admitting the bearer to the “Impeachment of the President”.

In such company it feels in slightly poor taste to finish with a cliche wrongly attributed to Mark Twain. But what the hell. History may not repeat itself, but it does sometimes rhyme.