A history of vector analysis : the evolution of the idea of a vectorial system
He wrote two exhaustive books, Lectures on Quaternions and Elements of Quaternions , detailing not just the algebra of quaternions but also how they could be used in geometry. At one point, Hamilton wrote, "I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions was for the close of the seventeenth. At about the same time that Hamilton discovered quaternions, Hermann Grassmann was composing The Calculus of Extension , now well known by its German title, Ausdehnungslehre.
In , Grassmann began development of "a new geometric calculus " as part of his study of the theory of tides, and he subsequently used these tools to simplify portions of two classical works, the Analytical Mechanics of Joseph Louis Lagrange and the Celestial Mechanics of Pierre Simon Laplace In his Ausdehnungslehre , first, Grassmann expanded the conception of vectors from the familiar two or three dimensions to an arbitrary number, n , of dimensions; this greatly extended the ideas of space. Second, and even more generally, Grassmann anticipated a good deal of modern matrix and linear algebra and vector and tensor analysis.
Unfortunately, the Ausdehnungslehre had two strikes against it. First, it was highly abstract, lacking in explanatory examples and written in an obscure style with an overly complicated notation. Second, Grassmann was a secondary school teacher without a major scientific reputation compared to Hamilton. Even though his work was largely ignored, Grassmann promoted its message in the s and s with applications to electrodynamics and to the geometry of curves and surfaces, but without much general success.
In , Grassmann published a second and much revised edition of his Ausdehnungslehre, but it too was obscurely written and too abstract for the mathematicians of the time, and it met essentially the same fate as his first edition.
In the later years of his life, Grassmann turned away from mathematics and launched a second and very successful research career in phonetics and comparative linguistics. Finally, in the late s and s, the Ausdehnungslehre slowly began to be understood and appreciated, and Grassmann began receiving some favorable recognition for his visionary mathematics.
A third edition of the Ausdehnungslehre was published in , the year after Grassmann died. During the middle of the nineteenth century, Benjamin Peirce was far and away the most prominent mathematician in the United States, and he referred to Hamilton as, "the monumental author of quaternions. Rather, Peirce expanded on what he called "this wonderful algebra of space" in composing his Linear Associative Algebra , a work of totally abstract algebra.
James Clerk Maxwell was a discerning and critical proponent of quaternions. Maxwell and Tait were Scottish and had studied together in Edinburgh and at Cambridge University, and they shared interests in mathematical physics. In what he called "the mathematical classification of physical quantities," Maxwell divided the variables of physics into two categories, scalars and vectors.
"A History of Vector Analysis" by Michael J. Crowe - Book Reviews
Then, in terms of this stratification, he pointed out that using quaternions made transparent the mathematical analogies in physics that had been discovered by Lord Kelvin Sir William Thomson, between the flow of heat and the distribution of electrostatic forces. However, in his papers, and especially in his very influential Treatise on Electricity and Magnetism , Maxwell emphasized the importance of what he described as "quaternion ideas … or the doctrine of Vectors" as a "mathematical method … a method of thinking.
Essentially, Maxwell was suggesting a purely vectorial analysis. In his Elements of Dynamic , Clifford broke down the product of two quaternions into two very different vector products, which he called the scalar product now known as the dot product and the vector product today we call it the cross product. For vector analysis, he asserted "[M]y conviction [is] that its principles will exert a vast influence upon the future of mathematical science.
The development of the algebra of vectors and of vector analysis as we know it today was first revealed in sets of remarkable notes made by J. Willard Gibbs for his students at Yale University.
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Gibbs was a native of New Haven, Connecticut his father had also been a professor at Yale , and his main scientific accomplishments were in physics, namely thermodynamics. He concluded that vectors would provide a more efficient tool for his work in physics. 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.
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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 that quaternions offer little value in terms of application. Skip to main content. Search form Search. Login Join Give Shops. Halmos - Lester R. Ford Awards Merten M.
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