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A History of Vector Analysis: The Evolution of the Idea of a Vectorial Systemby Michael J. Crowe
Synopses & Reviews
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.
Prize-winning study traces the rise of the vector concept from the discovery of complex numbers through the systems of hypercomplex numbers to the final acceptance around 1910 of the modern system of vector analysis.
The first large-scale study of the development of vectorial systems, awarded a special prize for excellence in 1992 from Frances prestigious Jean Scott Foundation. Traces the 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. Concentrates on vector addition and subtraction, the forms of vector multiplication, vector division (in those systems where it occurs), and the specification of vector types.
Table of Contents
Chapter One THE EARLIEST TRADITIONS
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
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
"Chapter Three OTHER EARLY VECTORIAL SYSTEMS, ESPECIALLY GRASSMANN'S THEORY OF EXTENSION"
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
Chapter Four TRADITIONS IN VECTORIAL ANALYSIS FROM THE MIDDLE PERIOD OF ITS HISTORY
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
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
Chapter Six A STRUGGLE FOR EXISTENCE IN THE 1890'S
II. "The "Struggle for Existence"
CHAPTER SEVEN THE EMERGENCE OF THE MODERN SYSTEM OF VECTOR ANALYSIS: 1894-1910
II. Twelve Major Publications in Vector Analysis from 1894 to 1910
III. Summary and Conclusion
Chapter Eight SUMMARY AND CONCLUSIONS
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