The book that made Stephen Hawking a publishing phenomenon was called "From the Big Bang to Black Holes: A Short History of Time" until his editor at Bantam, Peter Guzzardi, turned it around and changed "Short" to "Brief". It was a moment of inspiration that surely helped Hawking amass record sales. The formula is much copied: there are now Brief Histories of all sorts. There is even a Brief History of Thyme.

Hawking's memoir, My Brief History, is a skip across the surface of the Cambridge cosmologist's life, from his quirky upbringing in London and St Albans to his latest work on the beginning of time and the evolution of the universe. The details are sketched, but the brevity makes for a bold picture. Hawking's intellectual activity soars as his illness takes hold and eventually puts an intolerable burden on his marriages.

Books about Hawking are apt to point out that he was born exactly 300 years after the death of Galileo, as if the fact might help us comprehend the origins of his genius. Hawking repeats the line but only for his amusement. He estimates that 200,000 other babies were born on 8 January 1942. "I don't know whether any of them was later interested in astronomy."

Hawking's family lived in a tall Victorian house his parents had bought cheap during the war, when everyone thought London would be bombed flat. His mother worked as a tax inspector and then as a secretary, which was how she met his father. An Oxford-trained doctor, he specialised in tropical diseases, and kept infected mosquitoes at the National Institute for Medical Research in Mill Hill. The son of a poor Yorkshire farmer, he was cautious with money. Instead of installing heating when the family moved to St Albans, he wore several jumpers and a dressing gown on top of his normal clothes.

The young Hawking had a passion for model trains, and it was through toys and the complicated games he invented – one war game was played on a board of 4,000 squares – that he satisfied his urge to understand how complex systems worked. In later life this urge was met by his work in cosmology. "If you understand how the universe operates," he writes, "you control it, in a way."

Hawking did not excel at school in St Albans. He was never more than halfway up the class, and his teachers were dismayed at his untidy coursework. When he was 12 years old a friend of his bet another friend a bag of sweets that he would never amount to much. Hawking is (rather needlessly) self-deprecating here, offering: "I don't know if this bet was ever settled, and if so, which way it was decided."

From St Albans, Hawking went to Oxford University, where the prevailing attitude was anti-work. The students affected an air of boredom – nothing was worth making an effort for. Hawking calculates he put in 1,000 hours over three years, an average of only an hour a day.

It was in his last year at Oxford that the illness that came to shape Hawking's life began to make itself known. He became clumsy. After falling down some stairs, he went to see the doctor, who told him to lay off the beer.

After scraping a first-class degree, Hawking left Oxford to pursue a PhD at Cambridge. His clumsiness worsened and soon after his 21st birthday, he went to hospital for tests. The doctors took muscle samples and injected his spine with a radio-opaque fluid and watched it go up and down as they tilted his bed. They didn't say what he had (motor neuron disease), only that it wasn't multiple sclerosis. But Hawking was left in no doubt that his condition would worsen and there was nothing the doctors could do.

He was in shock. He started listening to Wagner. He lost all motivation to finish his PhD. But the gloom did not last. While Hawking was in hospital, a boy he vaguely knew in a bed opposite died of leukaemia. "Whenever I feel inclined to be sorry for myself, I remember that boy," he writes.

To Hawking's surprise he started to enjoy life more. He got engaged to a girl called Jane Wilde. If he was to marry, he needed a job, and to get a job, he needed to finish his PhD. He started to work properly for the first time in his life.

What followed was a remarkable trajectory. He helped demolish the steady-state theory of the universe, which claimed that the expanding cosmos was forever being filled with fresh matter. He found that black holes were not so black, and instead could emit radiation. He developed the concept of "imaginary time", and argued it made no sense to talk of a time before the universe began: to ask what happened before the beginning of the universe is as meaningless as asking what is south of the South Pole. This is not a book to learn Hawking's science from, but it is an introduction to the cosmos he unveiled.

In the slender chapter on marriage, the brevity of My Brief History is more brutal. Hawking describes Jane asking the local church organist to move in, with a view to marrying him after Hawking died. "I would have objected, but I too was expecting an early death and felt I needed someone to support the children after I was gone," he writes. "In the end, I could stand it no more." Both that marriage, and the second to his nurse, Elaine Mason (a "passionate and tempestuous" relationship) failed under the appalling pressure of his illness.

The trouble with being the world's most famous scientist is that when you come to write your memoir, much of it has been said before. My Brief History reads like a farewell letter from a man who, faced with the prospect of an early death, made so much of his life.