A History of Vector Analysis: The Evolution of the Idea of a Vectorial System

Description

On October 16, 1843, Sir William Rowan Hamilton discovered quaternions and, on the very same day, presented his breakthrough to the Royal Irish Academy. Meanwhile, in a less dramatic style, a German high school teacher, Hermann Grassmann, was developing another vectorial system involving hypercomplex numbers comparable to quaternions. The creations of these two mathematicians led to other vectorial systems, most notably the system of vector analysis formulated by Josiah Willard Gibbs and Oliver Heaviside and now almost universally employed in mathematics, physics and engineering. Yet the Gibbs-Heaviside system won acceptance only after decades of debate and controversy in the latter half of the nineteenth century concerning which of the competing systems offered the greatest advantages for mathematical pedagogy and practice.
This volume, the first large-scale study of the development of vectorial systems, traces he rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers created by Hamilton and Grassmann to the final acceptance around 1910 of the modern system of vector analysis. Professor Michael J. Crowe (University of Notre Dame) discusses each major vectorial system as well as the motivations that led to their creation, development, and acceptance or rejection.
The vectorial approach revolutionized mathematical methods and teaching in algebra, geometry, and physical science. As Professor Crowe explains, in these areas traditional Cartesian methods were replaced by vectorial approaches. He also presents the history of ideas of vector addition, subtraction, multiplication, division (in those systems where it occurs) and differentiation. His book also contains refreshing portraits of the personalities involved in the competition among the various systems.
Teachers, students, and practitioners of mathematics, physics, and engineering as well as anyone interested in the history of scientific ideas will find this volume to be well written, solidly argued, and excellently documented. Reviewers have described it a s "a fascinating volume," "an engaging and penetrating historical study" and "an outstanding book (that) will doubtless long remain the standard work on the subject." In 1992 it won an award for excellence from the Jean Scott Foundation of France.

**

discovered here

U. Notre Dame prof. (now emeritus)

Since historical publications on modern developments are rare, this book is welcome. It does orient the reader to the men, dates and publications which played a role in the rise of vector analysis. Starting from the geometric representation of complex numbers by Argand, Wessel and Gauss, it presents the work of Hamilton and Grassmann, the selection and adoption by Maxwell, Gibbs, and Heaviside of the basic vector concepts imbedded in Hamilton's quaternions and Grassmann's extensive quantities or hypercomplex numbers, notably the vector itself, scalar and vector product of two vectors, and gradient, divergence and curl of functions. This is followed by some account of the controversy between the advocates of quaternions and the advocates of vectors.
The book deals only with the basic vector concepts and does nothing with Gibbs' later creation of the linear vector function, dyadics, the transformation theorems of Stokes and Gauss now commonly taught in vector analysis, or the concept of a vector as an invariant under transformation of coordinates. Biographical material, quotations and references to sources are ample. The subtitle is a better description of the contents than the title proper.

Reviewed by M. Kline