This is a review of David Acheson’s new book, which we were kindly sent a copy of to read.

In *The Spirit of Mathematics: Algebra And All That*, David has pulled together a collection of what he refers to as ‘elegant mathematics using only simple materials’ – neat, short algebraic proofs and definitions, models of physical systems and mathematical tricks and curiosities.

He includes all the classics, from proof by induction to Fibonacci numbers to hitting a snooker ball, and each is presented with enthusiasm, alongside stories of mathematicians – and fearlessly including all the equations and derivations (if every equation really did halve your readership, as Stephen Hawking believed, this would be a very brave book to publish). But the maths is well-explained and very approachable, and it’s refreshing to see it featured so prominently outside of a textbook.

The book is also filled with helpful diagrams and illustrations, as well as humorous asides, cartoons and pictures of many mathematicians (sadly, only one female mathematician is featured, and she’s included only for her joke about how hard she’s found it to get a proof…) – but the book is well-produced and clearly laid out, with well-defined, short chapters each with a clearly defined topic.

The result is a compendium of intriguing ideas which would fascinate and compel a keen mathematician wanting to learn more, and provide hours of intrigue and jumping-off points for further investigation. Most topics are only covered briefly, so a deeper understanding would need research elsewhere, but for an enthusiastic reader this would happen naturally. Each discovery is motivated by a real-world example, or an interesting puzzle or curiosity, and all the key topics from algebra are touched on in one way or another.

However, this book wouldn’t suit an inexperienced mathematician – given which steps in the calculations are described as ‘simple’, a reasonable level of maths is assumed, and I’d imagine a strong GCSE or A-level student, particularly one already keen to learn more, would get much more out of it than a younger student. It’d also suit an adult wishing to refresh their mathematical knowledge from school and pick up some new ideas. But despite the blurb on the back claiming ‘for those who dread the subject, this book may be an eye-opener’, I suspect that such a reader might struggle in places.

Overall, this is a well-presented celebration of the best parts of mathematics, and showcases just how powerful maths can be.