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...
In Grassmann's Aus1862 he describes general structures to form products of two or more vectors. In modern language, Chapter 2 begins by saying that if you express two vectors in some basis
\begin{aligned}
u
& = \sum_i c_i e_i \\
v
& = \sum_i d_i e_i \\
\end{aligned}
and you have some vector product operation $\circ$ you want to define, then it suffices to describe the result of the operation $\circ$ on the basis vectors $e_i$, since
...
Later in his discussion of symmetric tensors, Grassmann refers to ``ein beliebiges Produkt'' $P_{a_1, a_2, ..., a_n}$ (``an arbitrary product''~\cite[p. 196, \S 353]{Grassmann2000}), which Gibbs in his Multiple Algebra (collected works, Vol 2, p 109) picked up on as being of significance and bequeathed the name ``indeterminate product'', being the most general product from which all the other products Grassmann discusses can be derived.
Grassmann's other contribution to this topic is the concept of ,,offne Produkt`` or ``open product''~\footnote{Somewhat confusingly, Grassmann's 1862 book defines open products, or ``product[s] with n
{interchangable \{interchangable\} openings''~\cite[\S\S 353, p. 196]{Grassmann2000}, in a way which we would recognize today as symmetric tensors of rank $n$.}, which he writes in Ausdehnungslehre 1844, Sec 172, p 267 (English pp 271-2) with the notation $[A() . B]$, acting on a vector $P$ by
\[
[A() . B] P = AP . B,
...
or in modern notation,
\[
(b a^T) p =
(a\cdotp) (a\cdot p) b,
\]
transcribing vectors into lower case letters in line with Householder's convention. In other words, Grassmann's open product is what we would call today the outer product, where ``open'' refers to the presence of the empty parenthesis denoting an ``opening''. Gibbs recognized the open product as a matrix in Multiple Algebra, Collected Works vol 2, p 94, but it is specifically a rank 1 matrix. (\textbf{TODO} Possibly the notion of rank did not exist then...?)
