Vector is a brave attempt to present a readable, integrated contextual and technical history of mathematics – a task that few dare on this scale. The book is about 330 pages long, excluding the extensive end matter, and ranges geographically from Asia to North America and chronologically from Mesopotamia to the present, but concentrates on nineteenth- and twentieth-century Europe.

Arianrhod sets out her two main themes – efficient symbolic representation of information, and the best ways of calculating with it – in a prologue that starts with the Mesopotamians, whizzes through the Greeks, looks ahead to the present and concludes with the Mad Hatter’s Tea Party (which ridiculed quaternion ideas). The themes, and the chronological dotting, about continue in the first two introductory chapters, ‘The liberation of algebra’ (as symbols replace words and diagrams) and ‘The arrival of calculus’.

The story proper begins in Chapter 3 and its coherence improves. Beginning with Isaac Newton and Thomas Harriot, this chapter traces how concepts we now recognize as vectors, such as force, and results relating to them, like the divergence theorem, intersected in the mid-nineteenth century with representations of complex numbers and symbolic algebra, enabling William Rowan Hamilton to formulate quaternions in 1843 (Chapter 4). Quaternions resulted from his quest for a mathematical object to represent three-dimensional rotations in space, much as complex numbers were viewed as doing for two-dimensional rotations. Hamilton showed that multiplying two quaternions, each comprising ‘scalar’ and ‘vector’ parts, could represent the composite effect of two successive rotations. Accounts follow of Hermann Grassmann’s independent development of vectors, Peter Guthrie Tait’s adoption of quaternions, and James Clerk Maxwell’s use of their vector part in his 1873 Treatise on Electricity and Magnetism, before discussion of the ‘vector wars’ in which mathematicians and physicists such as William Kingdon Clifford, Oliver Heaviside and Josiah Willard Gibbs established vectors, rather than quaternions, as the more usable representation.

For Arianrhod, the importance of either a quaternion or a vector is notational, that it is a single mathematical object expressed as a single symbol – typically today a letter in bold (for example v to represent velocity) – rather than a collection of components. This ultimately enabled abstraction from the three-dimensional physical world, and the development of multidimensional vectors and of tensors describing relationships between sets of vectors.

The development of tensor calculus occupies the second half of the book. The story of vector analysis has been told many times, most thoroughly in Michael J. Crowe’s A History of Vector Analysis (1967), but that of tensor analysis is less well known. Arianrhod hooks the reader in Chapter 10, ‘Curving spaces and invariant distances’, discussing how vectors, which served Einstein well for special relativity, were inadequate for general relativity, and Einstein’s consequent search for mathematics better suited to his ideas. He, or rather his friend Marcel Grossman, found it in Gregorio Ricci and Tullio Levi-Civita’s 1900 textbook Méthodes de calcul différentiel absolu et leurs applications, which laid out tensor theory, itself a development of Carl Friedrich Gauss and Bernhard Riemann’s work on intrinsic curvature and Augustin-Louis Cauchy’s on stress in continuous media. Subsequent chapters cover post-1915 uptake of general relativity and tensor-based theories of gravity and cosmology; Emmy Noethe’s route to Noethe’s theorem, which revealed the relationship between conservation laws and the symmetries of a physical system; and multifarious uses of tensors in current science and technology, from relativistic quantum mechanics to artificial intelligence.

Less successful than the technical history outlined is Arianrhod’s attempt to provide social context for mathematical developments, which is occasionally natural and seamless, but more often patchy and apparently random in what to contextualize. The text is enlivened by engaging biographical vignettes of the main players and, in connection with William Thomson and Maxwell, comments about the Industrial Revolution and rapidly changing communications in the nineteenth century, but such themes are not sustained.

Arianrhod is aware of recent trends to surface lesser-known names and non-European contributions. In the case of women, this is unsurprising from the author of Seduced by Logic (2012), a dual biography of Emilie du Chatelet and Mary Somerville. But apart from Emmy Noethe, and possibly Sophie Germain and Sofia Kovalevskaya, who play bit parts, the introduction of female, and Asian, names seems contrived and fails to convince that these people really contributed to the story.

In other respects, Arianrhod accepts without question traditional historiographic attitudes, such as a steady development from concrete to abstract mathematics through symbolism, that abstraction is necessary for generality, and that reasoning must be through a verbalizable series of (algebraic) steps. The usefulness of such attitudes is challenged today by histories and philosophies of mathematical practice such as Karine Chemla (ed.), The History of Mathematical Proof in Ancient Traditions (2012); Niccolò Guicciardini (ed.) Anachronisms in the History of Mathematics (2021); or Fenner Tanswell, Mathematical Rigour and Informal Proof (2024).

Vector does not present original research but is based on extensive, if sparsely referenced, scholarly literature. The audience at which it aims is unclear. Its racy style, with the most difficult material relegated to ample explanatory endnotes, suggest a trade book, as might the attempt to explain the technicalities from a basis of school-level mathematics. But although the prologue states that ‘you might remember from school that a vector can encode’ both size and direction (p. x), by page 85 we read, ‘by now, though, if you’ve taken a linear algebra course’, and Arianrhod does not shy away from equations – lots of them. There seems insufficient interest in the contextual history to hold the attention of readers not already familiar with the mathematics. For mathematicians and physicists, though, strengths of the book may be the exploration of the entanglement of mathematics and physics, and a synthesis of mathematical developments not over-burdened with historiographic considerations. Vector demonstrates clearly the difficulty and pitfalls inherent in answering recent calls, such as Karen Parshall’s (2025, https://www.newton.ac.uk/seminar/44492/) for integration of technical and social history of mathematics on anything but a microhistorical scale.