In chapter 10 of his A History of Algebra , 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 . 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  as well as in her magnificent biography of the English mathematician .
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 . Leo Corry's Modern Algebra and the Rise of Mathematical Structures  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.
One of the most important aspects of geometry in the nineteenth century was the development of non-Euclidean geometry, and this chapter begins with two brief studies of aspects of its development. In the first, George Bruce Halsted reviews volume VII of Gauss's Werke and concludes from a study of many of Gauss's letters first published in that volume that Gauss's ideas on the subject had no influence on the independent discoveries of János Bolyai and Nikolai Lobachevsky, or on the earlier publication by Ferdinand Karl Schweikart (1780–1859). In the second article, Florence P. Lewis gives us a whirlwind tour through the history of the parallel postulate, from Proclus to Bolyai and Lobachevsky. She then proposes how this history, and its effect in how mathematicians understood the nature of a postulate, could affect the teaching of geometry in schools. In particular, she emphasizes that one reason for the study of geometry is its role in “training the mind.”
Another major aspect of nineteenth-century geometry was the development of projective geometry. That development began in the seventeenth-century work of Girard Desargues and Blaise Pascal, but, as Julian Lowell Coolidge notes, it then “dragged along” for about a century. In his article, Coolidge takes up the story in the nineteenth century, summarizing the work of such mathematicians as Jean-Victor Poncelet, Michal Chasles, Jacob Steiner, and Johann Karl Christian von Staudt. But then Coolidge remarks that by the end of the nineteenth century, the field of synthetic projective geometry was “pretty much worked out.”
G. B. Halsted names several mathematicians who participated in the Second International Congress, most of whom are not household names today. These include, first, the representatives of the U. S., Charlotte Angas Scott (1858–1931), from Bryn Mawr College  of Japan, Rikitaro Fujisawa (1861–1933); and of Spain, Zoel Garcia de Galdeano y Yanguas (1846–1924). The last of these was the academic advisor of Julio Rey Pastor, who later became the central figure in the development of mathematics in Argentina in the twentieth century. Then there were Halsted's “interesting personalities”. One of these was Samuel Dickstein (1851–1939), a Jew from Russian Poland who at the time of the Congress was the principal of a science-oriented secondary school in Warsaw that had introduced Hebrew into its curriculum and who later was one of the first professors of mathematics at the University of Warsaw. There were also Karl Gutzmer (1860–1924), a German mathematician who worked on differential equations; Emile Lemoine (1840–1912), a French civil engineer who did some work in geometry; Alessandro Padoa (1868–1937), an Italian logician who lectured on a new system of definitions for Euclidean geometry at the Congress; and Dmitrii Sintsov (1867–1946), who created a school of geometry at Kharkov University in Russia. One wonders what Halsted's criteria for “interesting personalities” were. (Biographies of most of the mathematicians Halsted mentions are available online at Wikipedia or at the St. Andrews MacTutor website.)
For more details on the work of Cauchy, the best book is Judith Grabiner's own work, The Origins of Cauchy's Rigorous Calculus . A slightly more recent work on nineteenth-century analysis in general is Umberto Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass .
It is interesting to compare Cauchy's work with the contemporary work of Bolzano. Bolzano's important paper on the intermediate value theorem is available in , while  is the complete collection of his mathematical works. For more discussion of the work of Bolzano, see I. Grattan-Guinness . Grattan-Guinness claims that Cauchy took the central ideas of his definitions of continuity and convergence from Bolzano. But H. Freudenthal and H. Sinaceur disagree , . In particular, the latter article claims that Bolzano and Cauchy represented two different mathematical traditions.
It is also interesting that Cauchy's definition of convergence had been essentially given by José Anastácio da Cunha (1744–1787), a Portuguese scholar, in a comprehensive textbook written in Portuguese in 1782. Unfortunately, even though the book was translated into French in 1811, it apparently was little noticed. For more on da Cunha, see the articles A. J. Franco de Oliveira , João Filipe Queiró , and A. P. Youschkevitch .
Near the beginning of his article on functions, Kleiner notes that there was a long “prehistory” of functions, but does not discuss this at all.
This final chapter contains three survey articles on mathematics, dating from 1900, 1951, and 2000, as well as a brief and subjective account of the Second International Congress of Mathematicians, held in Paris in August 1900.
George Bruce Halsted was one of the American delegates to the Congress and wrote a report for the Monthly shortly after he returned. The major part of the paper deals with his reactions to Hilbert's famous address on the problems of mathematics, an address that set the agenda for twentieth-century work in mathematics. But the other talk that particularly interested Halsted was one on Japanese mathematics, by Rikitaro Fujisawa (1861–1933). Fujisawa's conclusions as to the Japanese independent discovery of both zero and the square root of – 1 are not accepted today.
In a report written for the beginning of the twentieth century, G. A. Miller discusses some “new fields” of mathematics, fields that seemed to him to be particularly fertile. Among the important areas currently under active investigation, Miller picked the arithmetization of analysis, the development of set theory, and the study of groups as particularly worthy of further attention. He also noted that practical applications of mathematics were important; in particular, he was impressed with the discovery of a linkage that would construct a straight line.
A half-century later, Hermann Weyl discussed the mathematics of the first half of the twentieth century.
For over one hundred years, the Mathematical Association of America has been publishing high-quality articles on the history of mathematics, some written by distinguished historians and mathematicians such as J. L. Coolidge, B. L. van der Waerden, Hermann Weyl and G. H. Hardy. Many well-known historians of the present day also contribute to the MAA's journals, such as Ivor Grattan-Guinness, Judith Grabiner, Israel Kleiner and Karen Parshall.
Some years ago, we decided that it would be useful to reprint a selection of these papers and to set them in the context of modern historical research, so that current mathematicians can continue to enjoy them and so that newer articles can be easily compared with older ones. The result was our MAA volume Sherlock Holmes in Babylon, which took the story from earliest times up to the time of Euler in the eighteenth century. The current volume is a sequel to our earlier one, and continues with topics from the nineteenth and twentieth centuries. We hope that you will enjoy this second collection.
A careful reading of some of the older papers shows that althoughmodern research has introduced some new information or has fostered some new interpretations, in large measure they are neither dated or obsolete. Nevertheless, we have sometimes decided to include two or more papers on a single topic, written years apart, to show the progress in the history of mathematics.
We wish to thank Don Albers, Director of Publications at the MAA, and Gerald Alexanderson, former chair of the publications committee of the MAA, for their support for the history of mathematics at the MAA in general, and for this project in particular.
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