One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme
- ... developed some of Hermann Grassmann's algebras and Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ...
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 geometry. Meanwhile Klein was advancing his Erlangen Program which also expanded the scope of geometry.
Comprehension of Grassmann awaited the concept of vector spaces which then could express the multilinear algebra of his extension theory. 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 forms.
For an introduction to the role of Grassmann's work in contemporary mathematical physics see The Road to Reality by Roger Penrose.
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.
Read more about this topic: Hermann Grassmann
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