Book review: Imagine an arrow on a computer screen – in PowerPoint, say. You can drag it around the screen and it’s the same arrow. You can make different arrows by stretching it or rotating it to point in a different direction. Each of these arrows can be described by two numbers: horizontal and vertical distance, or angle and length, or in many other equivalent ways.
In Vector, Robyn Arianrhod describes the history of this concept. The equivalence of the description by geometry and the description by sets of numbers is basic to today’s mathematics, even at elementary levels, but it took a surprisingly long time to emerge.
Even though this equivalence is now regarded as basic, the question of how much to emphasise arrows, and operations on arrows versus rows and columns of numbers, is still a matter of discussion and disagreement in mathematics education. Computations are done on the grids of numbers, but intuition is often easier with the arrows.
Arianrhod does a good job of defamiliarising the equivalence of numbers and geometry and guiding the reader through the often controversial history. There are familiar names, such as Newton and Hamilton, and less familiar ones such as Grassman and (Émilie) du Châtelet. Arianrhod talks about how gender, religion and social class excluded some researchers from the formative discussions in elite universities and learned societies.
The second key topic of the book is about putting arrows together. Suppose one arrow describes where you walked and another describes the slope of the hill you were walking over. You can combine the two arrows to work out how much higher you were at the end than at the start. This number, your elevation gain, is a real thing. It doesn’t depend on how you choose to represent the two arrows, so any representation must give the same answer. If you change the units of distance from metres to feet, the numbers for the distance walked will get bigger, and the numbers for the elevation gain per unit must get smaller in an exactly compensating way.
This idea of going from multiple arrows to single numbers in an invariant way is the basis of tensor algebra and tensor analysis. Vector describes how it is fundamental to much of mechanics and especially to the theory of relativity. Tensor analysis is also important to machine learning, but there the book is more superficial.
Compared with applied mathematics as I learnt it, there is more of an emphasis on the three-dimensional setting in Vector. In particular, there is a lot of history of quaternions, a generalisation of complex numbers that provide a compact and convenient representation of motions and rotations in three-dimensional space. Three-dimensional space is obviously important since it’s where we live, but there is a tension between mathematical approaches that work in any dimension and those that are specifically limited to three dimensions. I didn’t encounter quaternions at all as a student, but their history and the debates between coordinate-based and quaternion enthusiasts were new to me and fascinating.
I enjoyed Vector, but it’s not clear who else is in the target audience. From the topics covered it fits nicely as a book on the history of mathematical ideas for a general audience. However, more mathematics background is assumed than many potential readers will have.
For example, comfort with complex numbers and calculus is taken for granted. A paragraph in chapter two begins, “You likely first learned integral calculus by approximating the area under a curve …” and in chapter four, Arianrhod writes of scalar and vector products of two vectors, “I’ve given the modern component forms of these in the endnote, although you’re likely familiar with them from maths classes.” Someone interested in the 18th- and 19th-century history, and who was willing to skip over unfamiliar equations, would find the first half of the book fairly accessible, but I think the sections on tensors and relativity would be harder going.
Vector by Robyn Arianrhod (UNSW Press, $49.99) is out now.
