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

by Michael J. Crowe

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 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. 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.

Genres: Mathematics, History, Science, History Of Science, Nonfiction, Historical

270 pages, Paperback

First published January 1, 1967

Reviews

[Manny · Rating 4 out of 5]

[Before reading]

I'll bet dollars to dimes that Pynchon used this as background when researching Against the Day. Check out the Wikipedia article if you're skeptical.

Unable to resist... just ordered.
_____________________

[After reading]

Crowe's book did not disappoint, and I recommend it both to Pynchonheads and to people interested in finding out how modern mathematics got to be the way it is. If you fall into both categories, you are insane if you don't go and order a copy now.

Regarding Against the Day, I am now certain that Pynchon used it as a major source when creating his book. In fact, I can't help wondering if the whole novel wasn't inspired by this rather fine quote from Benjamin Peirce's 1855 book Analytical Mechanics:

But the strongest use of the symbol i is to be found in its magical power of doubling the actual universe, and placing by its side an ideal universe, its exact counterpart, with which it can be compared and contrasted, and, by means of curiously connecting fibres, form with it an organic whole, from which modern analysis has developed her surpassing geometry.

I'm curious to hear from people who know more about Pynchon and can say whether this has been discussed earlier.

If you just want to find out about the history of mathematics, Crowe absolutely delivers there too. I was first exposed to vector spaces and the scalar and vector products in my early teens, and have long regarded them as something so obvious that they scarcely required an explanation. I knew about quaternions, and was vaguely aware that they historically antedate modern vector analysis, but it wasn't until I read Pynchon that I started becoming interested in discovering what actually happened. What was the link between these two frameworks? Where did the other systems of vector analysis come from, in particular the one pioneered by Grassmann? Who the hell was Grassmann, anyway? I now know the answers to all these questions. The thing that surprised me most: the scalar and vector products turn out to be parts of the original quaternion product, which have been cut up and rebranded. It really is a startling lesson in the art of how science gets done.

I have trouble understanding how this excellent book can only have fourteen ratings on Goodreads. Don't pay any attention to me; but if the great Pynchon was prepared to spend several years and over a thousand pages writing an extended trailer for it, you'd expect just a little more buzz.

[Arthur Pesah · Rating 5 out of 5]

Wonderful trip into the universe of 19th century mathematicians, written in a very clear prose and coherent narrative!

You learn how Hamilton, Grassman, Tait, Gibbs, Heaviside, and many others have defined the terms and concepts that are now fundamental tools in physics and mathematics: vectors, linear functions, curl, divergence, scalar and cross products, etc. You observe the evolution in the philosophy of mathematics that occurred during this century (from intuition to abstraction) and the long process that it takes to accept revolutionary ideas. You witness one of the hottest debates in the history of mathematics (quaternions vs vectors) and how it has shaped modern physics and mathematics.

It's interesting to see how many arguments against the use of vector analysis in the 19th century ("what can you prove with vectors that can't be proven with coordinates?", "what is the point of shorter notations if they are harder to intepret?", "vectors are too hard to learn and unnecessary") are still used today in the exact same form against new levels of abstraction, such as category theory. It shows how hard it can be to judge a new system before it has shown undeniable applications.

I highly recommand this book to anyone interested in the history and philosophy of modern physics and mathematics, and the process of scientific creation.

[Shashvat Shukla · Rating 5 out of 5]

Well structured book. Very reflective on how history of maths should be done.

The book captures the nuanced genealogy of the concept of vectors, giving credit widely where it is due. It captures the spirit of the times in many excerpts of published materials and correspondence - and also captures the spirit by showing many lines of development that happened in parallel.

The concept of a vector was a hard one for mathematicians to accept. The need to describe Electricity and Magnetism made it absolutely critical and thus more widely accepted. The controversial and doubtful development before that came from various systems, but most notably those of Hamilton and Grassman. I have known about quaternions but never realised their significance in the descent of vectors.

The organisation of this book reflects the precision of mathematicians, we need more of these!

[Jenny Squibb · Rating 5 out of 5]

Excelent history of the evolution of vector method

I learned a great deal on the origin of the current Gibbs-Heaviside vector system used in engineering and mathematics. I do wish that more was said on Clifford, but the level of coverage seems to be appropriate for the level of influence Clifford had.

[Blaine · Rating 4 out of 5]

This is written for a limited audience, but every single member of that audience should pick it up. If you know what a “vector” is, then you will grasp the bulk of it. If you have done vector calculus and worked with complex numbers, then you have all the background necessary. Direct, well organized, informative, and engaging. What more do you want from a book like this?

[Saikiran · Rating 5 out of 5]

great and helped to find origins of vector and scalars

great sir its so illustrative in away so nicely explained concepts of vectors ,scalars and tensors also thanks sir great

[Des Bladet · Rating 5 out of 5]

The main goal of this book is to describe the historical processes by which the Gibbs-Heaviside vector calculus became hegemonic in physics, beating out the rival systems of Grassman's exterior algebra (whose impenetrable texts won few converts) and Hamilton's octonians (which Gibbs and Heaviside basically cannibalised for parts, as Crowe makes clear).

Crowe's view of what counts as "modern" is pretty quirky though: he describes someone's treatment of linear vector functions by dyadics as "modern" and frankly I've never heard of that. (It seems to be an obsolete version of the tensor product, but trust me: this is not how it is taught now, or even back in the 1980s when I was an undergraduate.)

He discusses Clifford as a transitional figure but doesn't discuss Clifford algebra/geometric algebra in any detail (possibly not at all; I can't tell!). And Emil Cartan doesn't even rate an index entry!

In Crowe's account, the cross product is triumphantly Gibbs-Heavisidely a vector in 3-space, bi-vectors are banished and 3 space is really the only space that counts anyway and mathematicians versions of geometry are a niche interest of no particular relevance to his tale.

(It is, indeed, notable that both Gibbs and Heaviside were physicists, not mathematicians, and there decisive clout came from their successes in physics.)

Crowe's story ends in 1910 and takes it for more granted that nothing much has changed since. In fact there is, even in physics, a heterodox tradition (that of Flanders, Burke, and their followers) in which Cartan's differential forms - which build on Grassman's algebra and extend it with a notion of differentials that turns out to be exactly what you need to do integrations on manifolds, that has in the last century largely failed to have any impact on the undergraduate physics curriculum, although it is understood to be indispensible for serious geometrical work in physics.

[liv · Rating 3 out of 5]

⚠️NERD ALERT⚠️

the mathematician drama was really funny