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

Michael J. Crowe
Publisher: 
Dover Publications
Publication Date: 
1994
Number of Pages: 
288
Format: 
Paperback
Price: 
19.95
ISBN: 
9780486679105
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
P. N. Ruane
, on
04/14/2012
]

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 20th 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 19th 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.

 

Chapter One THE EARLIEST TRADITIONS
  I. Introduction
  II. The Concept of the Parallelogram of Velocities and Forces
  III. Leibniz' Concept of a Geometry of Situation
  IV. The Concept of the Geometrical Representation of Complex Numbers
  V. Summary and Conclusion
    Notes
  Chapter Two SIR WILLIAM ROWAN HAMILTON AND QUATERNIONS
  I. Introduction: Hamiltonian Historiography
  II. Hamilton's Life and Fame
  III. Hamilton and Complex Numbers
  IV. Hamilton's Discovery of Quaternions
  V. Quaternions until Hamilton's Death (1865)
  VI. Summary and Conclusion
    Notes
  "Chapter Three OTHER EARLY VECTORIAL SYSTEMS, ESPECIALLY GRASSMANN'S THEORY OF EXTENSION"
  I. Introduction
  II. August Ferdinand Möbius and His Barycentric Calculus
  III. Giusto Bellavitis and His Calculus of Equipollences
  IV. Hermann Grassmann and His Calculus of Extension: Introduction
  V. Grassmann's Theorie der Ebbe und Flut
  VI. Grassmann's Ausdehnungslehre of 1844
  VII. The Period from 1844 to 1862
  VIII. "Grassmann's Ausdehnungslehre of 1862 and the Gradual, Limited Acceptance of His Work"
  IX. Matthew O'Brien
    Notes
  Chapter Four TRADITIONS IN VECTORIAL ANALYSIS FROM THE MIDDLE PERIOD OF ITS HISTORY
  I. Introduction
  II. Interest in Vectorial Analysis in Various Countries from 1841 to 1900
  III. Peter Guthrie Tait: Advocate and Developer of Quaternions
  IV. Benjamin Peirce: Advocate of Quaternions in America
  V. James Clerk Maxwell: Critic of Quaternions
  VI. William Kingdom Clifford: Transition Figure Notes
  Chapter Five GIBBS AND HEAVISIDE AND THE DEVELOPMENT OF THE MODERN SYSTEM OF VECTOR ANALYSIS
  I. Introduction
  II. Josiah Willard Gibbs
  III. Gibbs' Early Work in Vector Analysis
  IV. Gibbs' Elements of Vector Analysis
  V. Gibbs' Other Work Pertaining to Vector Analysis
  VI. Oliver Heaviside
  VII. Heaviside's Electrical Papers
  VIII. Heaviside's Electromagnetic Theory
  IX. The Reception Given to Heaviside's Writings
    Conclusion
    Notes
  Chapter Six A STRUGGLE FOR EXISTENCE IN THE 1890'S
  I. Introduction
  II. "The "Struggle for Existence"
  III. Conclusions
    Notes
  CHAPTER SEVEN THE EMERGENCE OF THE MODERN SYSTEM OF VECTOR ANALYSIS: 1894-1910
  I. Introduction
  II. Twelve Major Publications in Vector Analysis from 1894 to 1910
  III. Summary and Conclusion
    Notes
  Chapter Eight SUMMARY AND CONCLUSIONS
    Notes
  Index

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