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These are $n$-tuples of reals, added componentwise and multiplied via the “exterior product.” They were introduced by Grassmann in 1844 as part of a brilliant attempt to construct a vector algebra in $n$-dimensional space. Grassmann’s style was far from simple and his approach was ahead of its time

History of Abstract Algebra, Israel Kleiner

To my understanding, differential forms is the reason why Grassman algebras are so famous. But they only appeared in 1899, so how did Grassman's interest in Grassman algebras arise right in 1844?

I have thought that maybe it is related to some differential geometry matters. Googling tells me that Gauss put forth "Disquisitiones generales circa superficies curvas" in 1827. So, maybe Grassman was influenced by this? I am not sure.

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    $\begingroup$ Keep in mind that Grassmann didn't know that the differential forms would be the main use of the algebras when he started. $\endgroup$
    – Mark Olson
    May 14 at 12:28
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    $\begingroup$ Where is this opening quote from? Cite your sources $\endgroup$
    – Mauricio
    May 14 at 14:21
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    $\begingroup$ Grassman's motivation was purely philosophical, he wanted to develop an algebra that describes linear geometry, including its multi-dimensional analogs, see Fearnley-Sander. After a geometric interpretation made complex numbers acceptable this was a popular preoccupation. Hamilton discovered quaternions while trying in vain to put an algebra on triplets. Various other hypercomplex numbers were explored. Grassman just went more abstract and more general than others, which is why he was so hard to understand. $\endgroup$
    – Conifold
    May 16 at 2:53
  • $\begingroup$ @Conifold When I read "n-tuples" of reals, I was reminded of Riemann's 1854 (?) text which mentions "Notion of an n-ply extended magnitude". I would have placed Riemann's text before Grassman, but apparently it is the other way around. Any link between the two? (emis.de/classics/Riemann/WKCGeom.pdf) $\endgroup$
    – Frank
    May 16 at 13:50
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    $\begingroup$ @Frank Riemann showed little interest in algebraization of multidimensional geometry, even in quaternions, and Grassman's 1844 treatise was impenetrable and almost unknown. His ideas started to percolate, very slowly, only after he rewrote it in 1862. Here is Struik's poetic contrast:"One mind, essentially dialectical in nature, hews down barriers and discovers new relationships, and the other, operating more formally, builds up new symbolisms to control the field again through rigid, even frozen, methods." $\endgroup$
    – Conifold
    May 16 at 15:45

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