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NOSTALGIA, said the satirist (and mathematician) Tom Lehrer, is not what is used to be. Neither is mathematics.

Jan Gullberg, a Swedish surgeon, has offered up a huge dollop of nostalgia in his 1100-page volume, Mathematics, From the Birth of Numbers. For any British scientist educated in the 1950s or 1960s, the book will surely bring back many childhood memories.

The first memory will be of those heavy volumes, found in even the most non-bookish household, that promised to tell you everything about the human body or the world of science, or even claimed to offer up 鈥渆verything about everything鈥�. In part encyclopedias, these books showed that their authors had laboured long and hard to develop themes and narrative lines鈥攊n short, readability. As a result, no matter where you delved, you would find yourself being drawn into the thread. They were a boon for any harassed mother faced with an inquisitive teenager kept home sick in bed, unable to go to school.

The second memory will be of what, in the Britain of the two decades just after the Second World War, was called A and S-level mathematics. And this is almost exactly the point at which Gullberg鈥檚 volume ends. After a long introductory section on numbers and number systems, the bulk of the material covered either focuses on, or is centred around, calculus. Looking back to my own school days, this is almost exactly the view I had of mathematics at the time.

Thirty years later, having spent the entire intervening period as a professional mathematician, I realise just how narrow and, as a result, uncharacteristic that view was, even for the mathematics of the time. But back then it excited me and was enough to persuade me that mathematics was the life for me.

Mathematics is one of the great creations of the human mind, a magnificent part of our culture, and it bothers me to see it maligned鈥攁s it so often is these days鈥攁lmost always as a result of ignorance. I am also bothered when the true nature of the subject is misrepresented.

Gullberg is quite clearly a mathophile, a nonmathematician who loves mathematics, so there is no way he is going to malign the discipline. But he does misrepresent it鈥攏ot intentionally, but in exactly the same way I would have had I sat down and tried to write a book about mathematics when I was fresh from my GCE A-level and S-level studies. He gives us a lot of breadth in essentially one part of mathematics, but little by way of depth. Large parts of the discipline are not even mentioned.

So, as a 鈥済eneral treatment of the field of mathematics鈥�, which seems to be how the book is being marketed, I think it fails. But when I put the marketing to one side, what remains is a fascinating, though curious, addition to the mathematical literature. Gullberg has developed an interest in鈥攅ven a love of鈥攎athematics, that has inspired him to spend 10 years of his life learning about the subject and putting what he has learnt into this book.

And that interest shines through. Not on every page, since large sections of the book read like lecture notes that have been cleaned up and filled out for publication. Rather, it is when you look at the work as a whole, at the care and attention that has gone into its preparation, that you cannot but admire the author鈥檚 dedication and commitment.

Yet what Gullberg has given us is not the insider鈥檚 view of mathematics. How could he? He is not a mathematician. Instead it is very much the impression of a keen amateur, from the outside. And while those of us 鈥渋n the business鈥� may find fault with the result, about 10 000 readers in the US have already decided that what he has to say is sufficiently interesting to buy the book.

This begs the really fascinating question of why so many people are buying it. Certainly not because it describes any new mathematics. Indeed, the view of mathematics it gives is for the most part frozen in the 1950s. Even the occasional mention of more recent developments, such as the very brief treatment of fractals, has a dated look to it.

The reason for the book鈥檚 success, I would suggest, is bound up with the notion of nostalgia that I began with. Mathematics has changed so much during the past 200 years that much of what you see in even the best popular expositions of mathematics is alien territory for most people. It does not resemble the mathematics that they learnt as teenagers. (The same can be said for physics and, indeed, any of the sciences.)

To see what I mean, take a look at any of the more recent popular books on mathematics: William Dunham鈥檚 The Mathematical Universe, Ian Stewart鈥檚 two books Nature鈥檚 NumbersandThe Problems of Mathematics, John Casti鈥檚 Five Golden Rules, or my own Mathematics: The Science of Patterns. Those books try to convey some of the incredible variety of modern mathematics. But in order to do so in a way accessible to the general reader, they all have to leave out most of the detail. In that respect, they may well leave some readers unsatisfied in a way that Gullberg鈥檚 book does not.

For many readers now in their late forties and upward, who have not studied mathematics since adolescence, Gullberg鈥檚 book provides a reassuringly familiar look at a field that in their youth they found fascinating. They, I suspect, are the ones buying the book. And since no mathematician would dream of writing such a book, Gullberg has a clear field.

So I am left in two minds. On the one hand, I am bothered that Gullberg鈥檚 book presents such a narrow, dated view of mathematics. On the other hand, Gullberg鈥檚 book does provide a well-crafted encapsulation of a view of mathematics that is not uncommon among mathematically literate nonmathematicians of a certain age.

If I had to vote on my dilemma, I would come down on Gullberg鈥檚 side, and for this reason. Let鈥檚 suppose I am right, and that middle-aged people are buying the book because it offers them the promise of being able to really 鈥済et back into鈥� some serious mathematics that looks familiar to them. Mathematics, From the Birth of Numbers then becomes probably the only book with genuine mathematical content they possess. Then, the next time a sick son or daughter鈥攐r maybe a grandson or granddaughter鈥攈as to miss school and stay home in bed because of illness, Gullberg鈥檚 book is brought out to while away the hours. By the end of that week in bed, maybe鈥攋ust maybe鈥攁 spark of interest will have been kindled in a young mind, and a future mathematician will have been born.

The fact is, if I had been such a sick teenager, Gullberg鈥檚 book would have had exactly that effect on me. In the final analysis, that alone is reason enough for me to say thank you to Gullberg for writing it.