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  • Print publication year: 2009
  • Online publication date: October 2011

Afterword

from Algebra and Number Theory

Summary

In chapter 10 of his A History of Algebra [26], B. L. van der Waerden repeats much of what he wrote about Hamilton's discovery of quaternions. Interestingly, there he mentions Caspar Wessel as one of the originators of the geometric interpretation of complex numbers, while in the current article he ignores Wessel. But he also goes on to discuss Cayley's own use of quaternions to describe rotations in three-space, meanwhile pointing out the earlier results of Rodrigues. In addition, he deals with some applications of quaternions to the question of representing integers as sums of four squares. He concludes by discussing Hermann Hankel's 1867 book that presents many of Grassmann's results, but in a form that was easier to understand.

Simon Altmann writes in his article that we know “next to nothing” about Olinde Rodrigues, but in the next fifteen years he remedied this situation, publishing the results in his recent biography, Mathematics and Social Utopias in France: Olinde Rodrigues and His Times [1]. Similarly, Karen Parshall went on to do further research on the life and work of Sylvester. Her results appear in her edition of Sylvester's letters [18] as well as in her magnificent biography of the English mathematician [19].

Israel Kleiner has expanded his paper on group theory and some of his other work on the history of algebra into a new book, A History of Abstract Algebra [15]. Leo Corry's Modern Algebra and the Rise of Mathematical Structures [5] is another recent work that concentrates specifically on the development of abstraction in the nineteenth and twentieth centuries, but claims that true abstraction did not come into being until the work of Emmy Noether in the 1920s.