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Vector - Robyn Arianrhod ****

This is a remarkable book for the right audience (more on that in a moment), but one that's hard to classify. It's part history of science/maths, part popular maths and even has a smidgen of textbook about it, as it has more full-on mathematical content that a typical title for the general public usually has.

What Robyn Arianrhod does in painstaking detail is to record the development of the concept of vectors, vector calculus and their big cousin tensors. These are mathematical tools that would become crucial for physics, not to mention more recently, for example, in the more exotic aspects of computing.

Let's get the audience thing out of the way. Early on in the book we get a sentence beginning ‘You likely first learned integral calculus by…’ The assumption is very much that the reader already knows the basics of maths at least to A-level (level to start an undergraduate degree in a 'hard' science or maths) and has no problem with practical use of calculus. Although the historical aspects do bring in, for example, the concept of vectors through early and basic forms, we are soon soaring into mathematics that will be outside of the experience of many readers.

For me, when I used the operators of vector calculus in physics at university it seemed as if they had sprung from nowhere - I would have loved a book like this to put them into context. And I think someone like me at the time - a physics undergraduate with an interest in the history as well as the physics itself - would be an ideal audience. As Charles Seife notes on the back of the book, there is plenty of history of science/popular science on the development of physics from Newton to Einstein, but little that really fills in how the necessary mathematics was developed in parallel. While it's true the early days of calculus have been widely covered, there has been far less on the aspects that Arianrhod gives us, making this a classic find for the right readership.

There are some niggles that take it down from a potential five stars. It's very slow paced, sometimes getting a little bogged down in the detail of the historical development of the mathematical tools. (When I read Arianrhod's Seduced by Logic, I had a very similar problem with too much detail.) And the print is poor - it's both small and light weight, making it almost impossible, for example, to distinguish bold and normal fonts on the page, something that is used as a convention in this branch of mathematics.

Clearly not for everyone. But if you fit into the book's target audience, it's a must-have for your shelves.

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