Robyn Arianrhod, <i>Vector: A Surprising Story of Space, Time, and Mathematical Transformation</i> Chicago: University of Chicago Press, 2024. Pp. 376. ISBN 978-0-226-82110-8. $28.00 (cloth).

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 threedimensional 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.