The Maniac was first tested in the summer of 1951, “with a thermonuclear calculation that ran for 60 days nonstop.” About 6 by 2 by 8 feet and weighing a trim half-ton, it was much smaller than the room-size Eniac. But it inherited some of its predecessor’s reliability issues. Dyson quotes engineers’ exasperated entries from the Princeton machine’s logbook. May 7, 1953: “What’s the use? good night.” June 14, 1953: “Damnit — I can be just as stubborn as this thing.” June 17, 1956: “the hell with it.”

Like the nuclear physicist Edward Teller, von Neumann was an apparently unconflicted proponent of the bomb. At the Institute for Advanced Study, his hawkishness clashed with Einstein’s pacifism, and Einstein opposed building his computer there. Virginia Davis, wife of the logician Martin Davis, remembers writing “Stop the Bomb” in the dust on von Neumann’s car. But von Neumann’s second wife, Klári, recalled him being shaken by what his computer might wreak. One night in 1945, John announced, “What we are creating now is a monster whose influence is going to change history, provided there is any history left.” His biggest worry wasn’t the bomb, however, but, as Dyson writes, “the growing powers of machines.” Klári recalled prescribing “a couple of sleeping pills and a very strong drink.”

“Turing’s Cathedral,” incorporating original research and reporting — Dyson interviewed several people present at the institute during von Neumann’s tenure there, including his own father, the physicist Freeman Dyson — is an expansive narrative wherein every character, place and idea rates a digression. A brief history of Olden Farm, the site of the Princeton computer, begins with the Lenni Lenape Indians and carries on through William Penn, George Washington and so forth. One of Dyson’s running jokes is the supposedly abominable climate: Princeton “in summer has been described as ‘like the inside of a dog’s mouth.’ ” Humidity caused the Maniac’s air-conditioning units to freeze solid with ice.

The book brims with unexpected detail. Maybe the bomb (or the specter of the machines) affected everyone. Gödel believed his food was poisoned and starved himself to death. Turing, persecuted for his homosexuality, actually did die of poisoning, perhaps by biting into a cyanide-laced apple. Less well known is the tragic end of Klári von Neumann, a depressive Jewish socialite who became one of the world’s first machine-language programmers and enacted the grandest suicide of the lot, downing cocktails before walking into the Pacific surf in a black dress with fur cuffs. Dyson’s well-made sentences are worthy of these operatic contradictions. One example: “ ‘God does not play dice with the Universe,’ Albert Einstein advised physicist Max Born (Olivia Newton-­John’s grandfather) in 1936.”

Unlike many historians, Dyson has no need to reach for contemporary relevance. He quotes Julian Bigelow, the Maniac’s chief engineer, in a passage that could serve as the book’s précis: “What von Neumann contributed” was “this unshakable confidence that said: ‘Go ahead, nothing else matters, get it running at this speed and this capability, and the rest of it is just a lot of nonsense.’ . . . People ordinarily of modest aspirations, we all worked so hard and selflessly because we believed — we knew — it was happening here and at a few other places right then, and we were lucky to be in on it. . . . A tidal wave of computational power was about to break and inundate everything in science and much elsewhere, and things would never be the same.”