In 1905,
Alfred Bucherer acknowledged the influence of the encyclopedia on
vector notation in the second edition of his book: :When I wrote the first edition of this small work, the discussions and deliberations concerning a uniform symbolism for
vector analysis were still in flux. Since that time through the adoption of a suitable method of designation by those working on the
Encyklopädie an important system of symbolism has been put forward. In 1916,
George Abram Miller noted: :One of the great advantages of this large encyclopedia is that it tends to avoid duplication by establishing a higher minimum of general mathematical knowledge. ... The vastness of the new [mathematical] literature, combined with the fact that some of the new developments appeared first in somewhat obscure places, has often made it difficult for an author to determine whether his results were new. While some of this difficulty remains, yet the large encyclopedia, in which related important results are carefully associated, tends to reduce the difficulty materially. in 1922, with the publication of a volume including articles by
Charles Osgood on
analytic functions and
functions of several complex variables, and other topics, a notice was published by the
Bulletin of the American Mathematical Society. In his review of the
Encyclopedic Dictionary of Mathematics,
Jean Dieudonné raised the specter of Klein's encyclopedia while denigrating its orientation to
applied mathematics and historical documentation: :A tremendous gain of space has been achieved by eliminating much of the discursiveness of the old
Encyklopädie; the great majority of its historical information (which would have been a mere duplication); a large amount of results of secondary importance which needlessly cluttered many articles; and finally, all the parts devoted to astronomy, geodesy, mechanics, and physics which had no significant mathematical content. It has thus been possible to compress into about one-tenth of the bulk of the
Encyklopädie a more valuable amount of information on a science which certainly at present is ten times more extensive than it was in 1900. Librarian Barbara Kirsch Schaefer wrote: :Despite its age it remains a valuable source of reference, for its period of publication spans one of the most fruitful periods of mathematical research. Noted for its comprehensive treatment and well-documented scholarly articles, it is aimed at the specialist. In 1982. a history of aeronautics noted the following: :As organizer and editor of the monumental
Encyclopedia of Mathematical Sciences Including Their Applications, [Klein] compiled a collection of definitive studies that became the standard reference in
mathematical physics. Early in the thirty-year enterprise Klein solicited the esteemed
Sebastian Finsterwalder, professor of mathematics at the Munich polytechnic (and incidentally, one of Prandtl's teachers), to write an essay on
aerodynamics. This review article is significant in the history of aerodynamics because of its comprehensive scope and because it was submitted in August 1902. The date is more than a year before the Wrights achieved their powered flights at Kitty Hawk, North Carolina, and two years before Prandtl introduced his theory of the
boundary layer. It is therefore kind of a prenatal record of the science we now call aerodynamics. More to the point, however, it was then a rare compendious account of the state of the art of aerodynamics, a first reference to be found in much subsequent research in the field. Klein's encyclopedia as a whole, moreover, provided the model for the later publication of
Aerodynamic Theory, the six-volume encyclopedia of the science of flight that
William F. Durand edited in the mid-1930s...
Ivor Grattan-Guinness observed in 2009: : Many of the articles were the first of their kind on their topic, and several are still the last or the best. Some of them have excellent information on the deeper historical background. This is especially true of articles on applied mathematics, including
engineering, which was stressed in its title. He also wrote, "The mathematicians at Berlin, the other main mathematical pole in Germany and a citadel for
pure mathematics, were not invited to collaborate on the EMW and are reputed to have sneered at it." In 2013. Umberto Bottazzini and
Jeremy Gray published
Hidden Harmony in which they examined the history of
complex analysis. In the final chapter concerned with
textbooks, they used Klein's and Molk's encyclopedia projects to contrast the approaches in Germany (
Weierstrass and
Riemann) and France (
Cauchy). In 1900 an element of an
algebra over a field (usually
\mathbb{R} or
\mathbb{C}) was known as a
hypercomplex number, exemplified by
quaternions (\mathbb{H}) which contributed the
dot product and
cross product useful in
analytic geometry, and the
del operator in analysis. Explorative articles on hypercomplex numbers, mentioned by Bottazzini and Gray, written by
Eduard Study (1898) and
Elie Cartan (1908), served as advertisements to twentieth century algebraists, and they soon retired the term
hypercomplex by displaying the structure of algebras. ==French edition==