Hermann Grassmann (1809-1877) became a Sanskrit scholar, but in his spare time turned to mathematical research because of his father's interests. In 1824, his father, Günther published a book for elementary instruction wi thi passage: "[T]he rectangle itself is the true geometrical product, and the construction of it is really geometrical multiplication.... A rectangle is the geometric [Cartesian] product of its base and height, and this product behaves in the same way as the arithmetic product." Thus the idea of Book II of Euclid'sElements was finally rewritten in arithmetic terminology by the father, to be developed by the son.

Says Hestenes: "Hermann Grassmann completed the algebraic formulation of basic ideas in Greek ideas begun by Descartes. The Greek theory of ratio and proportion is now incorporated in the properties of scalar and vector multiplication. The Greek idea of projection is incorporated in the inner product. And the Greek geometrical product is expressed by outer multiplication. ....Only in ... Grassman's outer product is it possible to understand that the careful Greek distinction between number and magnitude as real geometric signigicance ... correspond[ing] roughly to the distinction between a scalar and a vector....Only in the work of Grassmann are the notions of direction, dimension, orientation and scalar magnitude finally disentangled ...impossible without the earlier vague distinction of the Greeks and perhaps without its reformulation in quasi-aruthmetic terms by his father....Grassmann was the first to define multiplication simply by specifying a set of algebraic rules."

In the almost unreadable Ausdehninglehre (Calculus of Extension), Grassmann built "a calculus of extensive magnitudes", a kind of vector algebraization of geometry, beginning with vector sum and difference. It's been said that Grassmann created many different kinds of vector products, but the notable ones are interior product (inner product), exterior product (related to outer product) and, late in life, a multiproduct, as sum of these two product to recapture Hamilton's quaternions. Today, in differential forms, interior product and exterior product reside in DIFFERENT SYSTEMS, bridged by an recondite transformation -- in stark contrast to their complementary relation in Clifford Algebra, and my own derivation of outer product from inner product and derivation of multiproduct from both.

The difference between a Clifford Algebra and what is today "a Grassmann Algebra" is articulated by a multiproduct added to its reversion:

Hestenes says: "It is fair to say, for example, that Grassmann laid the foundations for linear algebra .... Yet the standard linear algebra grew up without his contribution. The frequent duplication of Grassmann's discoveries is not a mark of limited originality but rather a sign that Grassmann was keenly attuned to a powerful thematic force driving mathematical development, namely, the subtle interplay between geometry and algebra. What sets Grassmann ahead of other creative mathematicians is his systemic vision of a universal geometric calculus. This marks him as one of the great conceptual synthesizers of all time."

The American combinatorist, Gian-Carlo Rota has also commented upon another misunderstanding of Grassmann's work: "With the rise of functional analysis, another dogma was making headway namely, the distinction between a vector space V and its dual V*, and the pairing of the two viewed as a bilinear form. ....Grassmann's idea was to develop a calculus for the join and meet of linear varieties in projective space, a calculus that is actually realized by the progressive [outer] and the regressive [inner] products....[T]he dual space V* of a vector space V [is] a hyperplane ... living in V, and ... identification [of V*] with a linear functional is a step backwards in clarity." Grassmann's exterior product was also discovered by x. Saint-Venant in 1845; by ?. O'Brien in 1846; and by Augustin Cauchy in 1853.