In 1679, Leibnitz expressed the need for a ‘geometry of situation’ that would be independent of the geometry of magnitude. It seems that he was really looking for a means of transforming geometric figures as we now do by means of linear algebra. He made a start on this daunting task, and his abortive efforts are clearly explained in the early pages of this extremely interesting and innovative book by Michael J. Crowe.

First published in 1967 by Notre Dame Press, this is a reprint of the Dover edition of 1994, and it remains the most substantial work available on the history of vector analysis. In keeping with the subtitle, Michael Crowe takes the reader on a 200 year trip that begins with Leibniz’s loosely formed ideas of 1679 to the early 20^{th} century, which saw the emergence, and acceptance, of the modern system of vector analysis

Among the earliest recognisable vectorial systems are the barycentric coordinates of Ferdinand Möbius and the calculus of equipollences devised by Giusto Bellavitas. More well-known is Hamilton’s work on quaternions, which began as a three-dimensional analogy of complex numbers. A whole chapter is devoted to Hamilton’s life and work, and the book contains many other biographical vignettes. Perhaps the longest and most detailed of these is devoted to Hermann Grassmann, who responded more specifically to Leibniz’s quest for a mathematical system that would represent spatial ideas independent of the ‘algebra of magnitude’.

Although quaternions were mainly regarded as being of relevance to electrical theory, Grassmann’s *n*-dimensional system was so broad and so general that mathematicians found it difficult to see that the idea of a vector space was contained within it. On the other hand, his earlier work (on the theory of tidal flow) provided the first clear idea of scalar and vector products.

Oliver Heaviside’s modified use of quaternions in electrical theory in the late 19^{th} century enabled him to arrive at a version of vector analysis as it is known today. In the process, he stripped down the quaternion system to the bare bones of its vectorial component, whilst simultaneously taunting Phillip Guthrie Tait about his devotion to quaternions. The following quotation appeared in Heaviside’s book on electromagnetic theory.

“Quaternion” was, I think, defined by an American schoolgirl to be “an ancient religious ceremony”. This, however, was a complete mistake. The ancients — unlike Prof. Tait — knew not, and did not worship Quaternions.

It is also shown in this book that Willard Gibbs independently formulated a version of vector analysis that was similar to Heaviside’s. Principally motivated by an interest in Maxwell’s electrical theory, Gibbs forged his ideas from quaternion (as opposed to Grassmannian) elements. But this led to him being lampooned by Hamilton’s fervent disciple Phillip Tait, who believed that undiluted quaternions could answer most of the needs of contemporary physics. Tait’s indignations is reflected in the following quotation.

Even Prof. Willard Gibbs must be ranked as one of the retarders of Quaternion progress, in virtue of his pamphlet on Vector Analysis; a sort of hermaphrodite monster, compounded of the notations of Hamilton and Grassmann.

In short, this history of vector analysis is highly fascinating — not only for its mathematical content, but also for the manner in which Michael Crowe describes the controversies surrounding its development. The only quibble regarding his mathematical judgement concerns his view (expressed in 1994) that quaternions offer little value in terms of application. Nonetheless, this boo will be enjoyed by anyone interested in the history of science or mathematics.

Peter Ruane’s working life was mainly concerned with the training of mathematics teachers.