Customer Reviews

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

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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.

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.

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".

The vector story is very smart, very passionate, and very, very, very good.

To most science and engineering graduates, nowadays, the algebra and calculus of vectors, i would imagine, strikes us all as a tradition that might have been handed down from the ancient Greeks. But such a sense of historical omnipresence stands in sharp contrast to the actual story of their historically brief existence and the extraordinarily dramatic events that have led to their widespread adoption.

This book is that story.

For another perspective on vectors, I also recommend: Geometrical Vectors (Chicago Lectures in Physics)

This book is a model of science history. Crowe manages to give one a clear view of the trends and moods of the time (1840-1900), the personalities of the various figures involved (Esp. Hamilton, Grassman and Gibbs), without sacrificing his most significant asset: the facts.

Lessons we learn early become so much a part of our thinking that we cannot imagine the world otherwise. So it is with arithmetic and yes, vector analysis too. Yet logic forces us to admit that there once was a time without arithmetic or vector analysis. Crowe's carefully written and well researched book makes clear that that time for vector analysis was not so long ago. Prior to vector analysis, the description of commonplace processes such as the movement of an object was done by Cartesian analysis. So, if one wanted to describe the motion of a particle, separate equations were written for the particle's motion in each of the three spatial dimensions. Thus Newton's deceptively simple force law, F=ma becomes three equations, one for the x-axis, a second for the y-axis and a third for the z-axis. With vector analysis, these three equations collapse into one, (vector F) = (mass)x(vector a), and, it is this simplicity of expression, which is much more evident when bold fonts are available to the reviewer, that makes vectors so appealing.

Crowe opens by taking us back to a time before vector analysis and provides a view of its origins in the parallelogram of forces, the geometry of complex numbers, and the early ruminations of Leibniz on space analysis. After an enlightening discussion of complex number geometry, he eases us into the invention of quaternions by Sir W. R. Hamilton and Hamilton's intellectual struggles preceding the invention of these hypercomplex numbers. For us who are not mathematicians by trade, this struggle is instructive, for, not only do we learn something about the creative process and the history of mathematical thought, we also, if we care to retain it, learn some math. Furthermore, we are reminded of Hamilton's genius as a theoretical physicist. I, for one, knew nothing of his work in geometrical optics and his exciting prediction, later experimentally verified, of conical diffraction.

From quaternions Crowe leads us into a mind opening discussion of various efforts to describe geometrical objects in three dimensions. Here we discover not only some of the storied names of nineteenth century mathematics but also, if we are alert and receptive, some, perhaps, wholly unexpected mathematical ideas, for example the multiplication of points to obtain a line. Rightly so, it is the work of Hermann Grassmann, that commands the most attention here. Regrettably, Grassmann's product was so involved, so monumental even, that Crowe left this reader wishing for more, a wish that could only be satisfied by other books.

Then we meet Gibbs and Heaviside, those two splendid scientists, who, standing on the shoulders of those two giants Hamilton and Grassmann, created the practical system of vector analysis that we all learn today in beginning calculus, or if not there, then in beginning mechanics or electricity and magnetism. Here we see the clash of the physicists' pragmatism with the mathematicians desire for logical unity, a clash that reverberates for years. Eventually, in the realm of the physical sciences, Gibbs and Heaviside carry the day but not without controversy and struggle. It is during this period in the late nineteenth century that we discover once again that intellectuals are just people subject to the same passions as anyone else. In fact, with the play "Copenhagen" in mind which provides a marvelous fictional portrayal of scientists in conflict, I suggest that the struggles over vector analysis have all the elements of good fiction within them.

In this book, Crowe traces in detail, the history of the thought processes that led to our modern system of vector analysis. Though the reading may be dry, there is great excitement lurking just below the surface and between the lines. If you have ever wondered how we got where we are vector wise, you can find out here. Crowe includes biographical sketches that humanize the leading figures in this story. And, if you enjoy footnotes, Crowe is masterful with some of his. This book is the product of prodigious intellectual effort and self discipline and will greatly reward the patient reader.

The book I ordered was in better than stated condition, was sent in a timely and seamless manner. I was delighted to find a copy of this book in hardcover and in such splendid condition. Hats off.

This is a great book. It's refreshing to read a work of history by a historian, rather than the usual assortment of stories that you hear mathematicians themselves tell. The problem with that is that history books can be difficult, but this is very light, and also (as you can already tell from the title) structured like an unfolding drama. Anyone with knowledge of vector calculus and related subjects (!) could take great pleasure in reading here about the emergence and early history of the subject. Not only Grassmann, Heaviside, Gibbs, but several other lesser known figures are woven into the tale. It's a shame it's out of print. Now I feel like I lucked out, finding my copy for sale in a local bookstore.