Being one of the greatest political thinkers of the 19th century did not always pay very well.

When Karl Marx welcomed the arrival of a new daughter, Eleanor, in 1855 his family was languishing in cramped lodgings at 28 Dean Street, London. Happily, this did not signal a miserable childhood for Eleanor. On the contrary, she thrived in a bookish, loving home and enjoyed, in her father, a wonderful storyteller and a dedicated private tutor.

Karl was sure to impart his adoration of Shakespeare but he also had a sense of fun. When Eleanor turned six she received the complete volumes of Peter Simple by Captain Marryat: who knew that the architect of international socialism had a soft spot for madcap seafaring adventures set in the Napoleonic Wars? If her precocity is anything to go by, Eleanor did very well out of this upbringing. While still a young girl she adopted Abraham Lincoln and Garibaldi as her heroes, she showed a keen interest in Irish affairs, and the stage was set for the daughter to follow in the footsteps of the famous father.

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According to Rachel Holmes, she did so with astonishing courage and success. The adult Eleanor is described here as "one of British history's great heroes". Her life is positioned as "one of the most significant and interesting events in the evolution of social democracy in Victorian Britain". Her contribution to political thought bears comparison with that of Mary Wollstonecraft and she left "a colossal, although unacknowledged, legacy for future generations". These are bold claims but Holmes justifies every one of them.

Eleanor Marx was never short of hobbies and intellectual sidelines. She toyed with the idea of being an actress, she would emerge as one of the leading champions of Ibsen's plays, and she found time to produce the first English translation of Flaubert's Madame Bovary. Politics became her greatest passion, however. Her activism began early: she tutored leaders of workers' movements in everything from basic literacy and numeracy to economic theory, and helped organise relief for impoverished exiles from the 1871 Paris Commune. Later, she made telling contributions to the blossoming of the British Trade Union movement, the expansion of internationalist socialism, and the arrival of the Independent Labour Party. By the late 1880s she was one of the most famous Socialists in the world.

Her work was not always easy. Holmes does a splendid job of recounting the Left's internecine squabbles - socialists versus anarchists, determined revolutionaries against those who saw sense in pursuing electoral success - and Eleanor was also, of course, a woman. Nineteenth-century Socialism was not always as forward-thinking as one might expect when it came to gender: at the First International in 1864 it was still possible for French delegates to announce that "to men belong labour and the study of human problems; to women child care and adornment of the worker's home".

Such sentiments would both infuriate and inspire Eleanor. For her, any socialism worthy of the name had to strike at all kinds of oppression and involve both sexes on an equal footing.

This, along with promoting internationalism over parochialism, was the lodestone of Eleanor's career and she emerges from these pages as one of the founding-mothers of modern feminism: a concept, as Holmes puts it, that began in the 1870s not the 1970s, "contrary to current popular misconceptions". Her greatest achievement was to apply sophisticated economic theorising to this worthy cause.

Eleanor Marx did not always have a happy life. The man she fell for, Edward Aveling, was a frequently unscrupulous individual and it is hard to determine what Eleanor saw in him: George Bernard Shaw described him as "short, with the face and eyes of a lizard, and no physical charm except a voice like a euphonium". His character (the wellspring of infidelities and financial shenanigans) was no more appealing. There were also Marx family secrets and scandals with which to cope and things ended tragically for Eleanor. The coroner's report, produced when Eleanor died at just 43, declared "suicide by swallowing prussic acid at the time labouring under mental derangement", though there have always been those who wonder quite what happened in the spring of 1898.

It was a short but extraordinary life. It cannot always have been cosy in Karl Marx's shadow. Some people detested Eleanor on sight and even among the comrades there were problems. She could work inordinately hard to protect her interpretation of her father's legacy or assemble his writings and letters, but she also had to witness his ideas being twisted by those who claimed him as their guiding light. As Eleanor once put it in a moment of frustration, "Heaven save Karl Marx from his friends." What would she have made, one wonders, of the 20th-century despoilment of Marx's philosophy?

Please read this book. It takes you to a time when hastily written pamphlets had an outside chance of changing the world: a time when the Reading Room of the British Museum was one of the hubs of the intellectual universe. Eleanor sat there, just like her father, and her story is intoxicating.

It is hard not to like someone who asked "who is the fiend who invented house-keeping? I hope his invention may plague him in another world."