Editing Hermann Grassmann (section)

==Response==
In the 1840s, mathematicians were generally unprepared to understand Grassmann's ideas.<ref name="Prasolov">{{cite book |last1=Prasolov |first1=Viktor V. |title=Problems and Theorems in Linear Algebra |date=1994 |publisher=[[American Mathematical Society]] |location=Providence, RI |isbn=0-8218-0236-4 |translator1-first=Dimitry A. |translator1-last=Leites}}</ref> In the 1860s and 1870s various mathematicians came to ideas similar to that of Grassmann's, but Grassmann himself was not interested in mathematics anymore.{{r|Prasolov|p=46}}

[[Adhémar Jean Claude Barré de Saint-Venant]] developed a vector calculus similar to that of Grassmann, which he published in 1845. He then entered into a dispute with Grassmann about which of the two had thought of the ideas first. Grassmann had published his results in 1844, but Saint-Venant claimed that he had first developed these ideas in 1832.

One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was [[Hermann Hankel]], whose 1867 ''Theorie der complexen Zahlensysteme''.<ref>{{cite encyclopedia |last=Crowe |first=Michael J. |encyclopedia=Dictionary of Scientific Biography |isbn=0-684-10114-9 |publisher=Charles Scribner's Sons |entry=Hankel, Hermann |chapter-url=https://www.encyclopedia.com/science/dictionaries-thesauruses-pictures-and-press-releases/hankel-hermann}}</ref>
{{blockquote|[…], he developed […] some of Hermann Grassmann's algebras and W.R. Hamilton's [[quaternion]]s. Hankel was the first to recognise the significance of Grassmann's long-neglected writings and was strongly influenced by them.}}

In 1872 [[Victor Schlegel]] published the first part of his ''System der Raumlehre'', which used Grassmann's approach to derive ancient and modern results in [[plane geometry]]. [[Felix Klein]] wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann. Schlegel followed in 1875 with a second part of his ''System'' according to Grassmann, this time developing higher-dimensional geometry. Meanwhile, Klein was advancing his [[Erlangen program]], which also expanded the scope of geometry.<ref name="Rowe">{{cite journal |last1=Rowe |first1=David E. |author1-link=David E. Rowe |title=Debating Grassmann's Mathematics: Schlegel Versus Klein |journal=The Mathematical Intelligencer |year=2010 |volume=32 |issue=1 |pages=41–48 |doi=10.1007/s00283-009-9094-2 |publisher=Springer }}</ref>

Comprehension of Grassmann awaited the concept of [[vector space]]s, which then could express the [[multilinear algebra]] of his extension theory.  To establish the priority of Grassmann over Hamilton, [[Josiah Willard Gibbs]] urged Grassmann's heirs to have the 1840 essay on tides published.<ref>[[Lynde Wheeler]] (1951), ''Josiah Willard Gibbs:  The History of a Great Mind'', 1998 reprint, Woodbridge, CT:  Ox Bow, pp. 113-116.</ref>  [[A. N. Whitehead]]'s first monograph, the ''Universal Algebra'' (1898), included the first systematic exposition in English of the theory of extension and the [[exterior algebra]]. With the rise of [[differential geometry]] the exterior algebra was applied to [[differential form]]s.

In 1995 Lloyd C. Kannenberg published an English translation of The Ausdehnungslehre and Other works. For an introduction to the role of Grassmann's work in contemporary [[mathematical physics]] see ''[[The Road to Reality]]'' by [[Roger Penrose]].<ref>{{cite book |last1=Penrose |first1=Roger |author1-link=Roger Penrose |title=The Road to Reality: A Complete Guide to the Laws of the Universe |date=February 2005 |publisher=Alfred A. Knopf |location=New York |isbn=0-679-45443-8 |chapter=2. An Ancient Theorem and a Modern Question, 11. Hypercomplex numbers}}</ref>