A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Dover Books on Mathematics) [Paperback]

Michael J. Crowe (Author)

Book Description

Series: Dover Books on Mathematics | Publication Date: November 2, 2011

This prize-winning study 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, and more.

Product Details

• Paperback: 270 pages
• Publisher: Dover Publications (November 2, 2011)
• Language: English
• ISBN-10: 0486679101
• ISBN-13: 978-0486679105
• Product Dimensions: 8.5 x 5.5 x 0.6 inches
• Shipping Weight: 11.2 ounces (View shipping rates and policies)
• Average Customer Review: 4.8 out of 5 stars  See all reviews (8 customer reviews)
• Amazon Best Sellers Rank: #721,419 in Books (See Top 100 in Books)

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29 of 29 people found the following review helpful:

4.0 out of 5 stars Thoughtful, Detailed History of Vector Analysis, November 22, 1999

By 

Michael Wischmeyer (Houston, Texas) - See all my reviews
(VINE VOICE)    (REAL NAME)   

This review is from: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Dover Books on Mathematics) (Paperback)

How were the concepts of vector analysis developed? How did modern vector notation become widely accepted? Who were the key players and why did quaternions fail to gain acceptance? This book is extensively documented, scholarly in its approach, sometimes a bit slow, but overall it is a fascinating look at these specific questions as well as the fundamental issue of what factors promote or delay acceptance of revolutionary ideas in science and mathematics.

I did not become immediately engaged with Crowe's style and even set the book aside after reading the prefaces and first chapter. A few months later I returned to chapter two (in part due to a previous reviewer's high rating). And what a surprise - I suddenly found myself intrigued with Crowe's discussion of Sir William Hamilton's single minded focus on quaternions, the perseverance and genius of Hermann Grassmann, the critical roles played by Peter Tait and James Maxwell, and the pragmatic way in which Josiah Gibbs and Oliver Heaviside independently extracted key vectorial concepts from Hamiliton-Tait's quaternion analysis.

Crowe's book was originally published in 1967 by University of Notre Dame, Dover reprinted it in 1985, Crowe recieved the Jean Scott Prize by the Maison des Sciences de l'Homme (Paris)in 1992, and Dover reprinted it again in 1992. Dover should be commended for making such reprints readily available at affordable prices.

The discussion of Hamilton's quaternions does not require familiarity with quaternions, but some prior acquaintance might be helpful. I encountered quaternions in another Dover reprint: Matrices and Transformations by Pettofrezzo. Section 2-3 introduces quaternion notation, simple manipulations, and shows that addition and multiplication of quaternions is isomorphic with two particular sets of matrices.

Has quaternion analysis survived? See Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace, and Virtual Reality by Jack Kuipers. The reviews by readers are all five stars.

7 of 7 people found the following review helpful:

5.0 out of 5 stars Interesting summary of the history of an important idea, October 29, 2004

By 

Bruce R. Gilson (Wheaton, MD United States) - See all my reviews
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This review is from: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Dover Books on Mathematics) (Paperback)

Although several others made important contributions, this book is primarily a study about the ideas of four people: Hamilton, Grassmann, Gibbs, and Heaviside. Hamilton's creation of the algebra of quaternions, while an important mathematical innovation, was thought of in many minds as primarily a physical tool, to be used in many of the applications that today are done by vectorial methods (and, in fact, the terms "scalar" and "vector" were invented by Hamilton, but with slightly different meanings than their present ones). Grassmann developed a quite different system, much closer to our present vector algebra, but unrecognized because of his obscurity and his books' unreadability. The true founders of modern vector analysis were the American physical chemist Josiah Willard Gibbs and the British physicist Oliver Heaviside, working independently of each other. What is interesting is that both Gibbs and Heaviside arrived at identical systems by modifying Hamilton's quaternion algebra to make it more accurately reflect the needs of physical scientists. While both Gibbs and Heaviside started with Hamilton's methods, the system they both arrived at was closer to Grassmann's in structure. And all this is clearly put forth in Crowe's book.

One other thing that the book makes clear is that J. Willard Gibbs, far more humbly than most scientists involved in priority disputes, clearly recognized that Grassmann had anticipated his ideas, although Grassmann's books had not come to Gibbs' attention until Gibbs had completely worked out his own system. And Gibbs, though he had based his ideas on Hamilton's, also recognized that Grassmann had the superior approach. (Though this may have NOT been a sign of humility, because in this regard Gibbs ended up using Grassmann's ideas to justify his own.)

Crowe's book is very readable, makes all these points quite clearly, and is highly recommended if you are interested in the subject.

4 of 4 people found the following review helpful:

4.0 out of 5 stars A reception-of-ideas history of vectorial systems, February 6, 2006

By 

Viktor Blasjo - See all my reviews
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This review is from: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (Dover Books on Mathematics) (Paperback)

The story of vectorial systems is the story of a search for an algebra of space. In chapter 1 we see that the need for such a theory was recognised already by Leibniz. We also study the rise of the geometry of complex numbers. Since complex numbers are an extremely successful fusion of plane geometry and algebra, one is tempted to look for a three-dimensional number system to do the same for space. Hamilton did so (chapter 2), and although he had to settle for four-dimensional quaternions, their "vectorial part" may still serve the purpose of an algebra of space quite well. Grassmann achieved much the same things when working to form a sort of general algebra of multidimensional magnitudes (chapter 3). In fact, Grassmann didn't even know about the geometry of complex numbers, and had to be told about it by Gauss. As is perfectly sensible, the ideas of Hamilton and Grassmann were poorly received. Both were inclined to an annoying "metaphysical style of expression" (Hamilton's phrase; p.36), and neither of them solved a single outstanding mathematical problem. One instead needs to be "astounded" by things like "the simplicity of the calculations resulting from this method" (Grassmann; p. 56). Basically this is what happened once vectorial ideas were freed from the smothering love of their creators (chapter 4); for instance we have Maxwell claiming that vector methods are useful "especially in electrodynamics" where things "can be expressed far more simply by a few expressions of Hamilton's, than by the ordinary equations" (p. 135). By now all the main ideas of the modern theory is in place, so the rest of the story is less interesting. A new generation began to detach vector ideas from quaternions (chapter 5), which led to a heated debate with quaternionists (chapter 6), but of course the reformists succeeded and the modern formulation of the theory was well established by the turn of the century (chapter 7).

On the whole, this book is little more than a compilation of historical information. Crowe barely treats the mathematics at all, and certainly not to the extent that would be necessary to understand "the evolution of the idea of a vectorial system".