The marriage may have been happy and passionate, but it was cursed by the tragedies of infant mortality, financial despair and Marx’s infidelities. One of them was nonsexual: it was with Marx’s intellectual partner, the wealthy, irrepressibly promiscuous bon viveur, Friedrich Engels, who paid Marx a salary from the profits of his capitalist factories. Here is one of Gabriel’s typical descriptions: “His clear blue eyes sparkled at the prospect of adventure — whether it be revolutionary or, perhaps even better, sexual.” There is a priceless moment reminiscent of the recent Dominique Strauss-Kahn episode, when a fellow leftist, Moses Hess, accuses the womanizer Engels of raping his mistress, Sibylle: “If, by the by, the jackass should persist in his preposterous lie about rape, I can provide him with enough . . . details to send him reeling,” Engels said. “Her rage with me is unrequited love.”

Between Marx’s lovers and his work, Jenny’s life was never easy: “While she pleaded with his family for assistance,” Gabriel writes, “he was having sex with Lenchen on Dean Street.” (At exactly the same time? How does she know where?) Lenchen was the family’s companion and housekeeper, Helene Demuth, with whom Marx fathered a child. Or as the author explains unnecessarily: “It isn’t known whether this was the first or the last time the two had intercourse.” Why this either-or? Surely it may have been the second or the 20th time — and, at the risk of challenging Gabriel’s eerily omniscient sexual-Marxist research, things worth doing once are often worth doing again. Either way, Lenchen gave birth to a son, Freddy. Engels pretended to be the father of the boy, who became one of the secrets of Marx’s biography: Stalin himself ordered it buried in the archives.

Gabriel’s story becomes heartbreaking with the deaths of four of the Marx children: when Franzisca, age 1, died, they lacked the money to buy a coffin. “Our three living children lay down by us, and we all wept for the little angel whose livid, lifeless body was in the next room,” Jenny wrote. When a son, Musch, died, Marx shouted, “You cannot give me back my boy,” and he told Engels, “I’ve already had my share of bad luck, but only now do I know what real unhappiness is. I feel ­broken down.” His only consolation? “The hope that there is still something sensible for us to do together in the world.”

When Jenny died in 1881 and Karl in 1883, their surviving children, Tussy and Laura, and the men in their lives, became the leaders of the movement, especially after Engels left them a significant portion of his $4.8 million estate. But it’s hard not to feel that somehow Karl’s obsessive mission destroyed those who came after: both daughters committed suicide, Tussy in 1897, driven to it by a callous partner; Laura in 1911, in a death pact with her ­husband.

There is a key moment in 1910 when a Russian couple bicycle over to visit Laura when she is living in France. They are Lenin and his wife, Krupskaya, who mused, “Here I am with Marx’s daughter!” Yet the only one of the Marxes still alive to see Lenin and his Marxist Bolsheviks seize Russia was Karl’s secret illegitimate son, Freddy.