What is the origin of the term "exterior" in "exterior calculus"? How does this term relate to "interior products" and "inner products", if it does at all?



Etymology of "exterior" in "exterior calculus"
I think it's well known to have been introduced by **Grassmann**. He explains the word choice in *Die lineale Ausdehnungslehre* [(1844, pp. x-xi)](https://archive.org/stream/dielinealeausde00grasgoog#page/n15):



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> I have shown how one can understand as product of two segments the parallelogram (...); this product is distinguished in that factors can only be interchanged with a sign change, while at the same time the product of two parallel segments vanishes. This concept stood in contrast to another (...) Namely when I projected one segment perpendicularly onto another, then the arithmetic product of this projection with the segment projected upon represented likewise a product of the two, insofar as the distributive relation with addition also held. But that product was of a whole other kind than the first, as its factors were interchanged without sign change, and the product of two perpendicular segments vanished. ***I named the former product exterior, the latter interior,*** reflecting that the former was nonzero only when involving independent directions, the latter only when involving a shared i.e. partly common one.
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and again in the second edition [(1862, p. 51)](https://archive.org/stream/dieausdehnugsle00grasgoog#page/n69):



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> I chose the name ***exterior multiplication*** to emphasize that the product is only nonzero when one factor lies entirely outside the span of the other. Exterior multiplication stands in contrast with the interior one (chap. 4).
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