On the Exterior Calculus of Invariant Theory

The full extent of the tragedy of Hermann Grassmann, which has been unfolding since his death, by a succession of misadventures and misunderstandings of his work unique in the history of modern mathematics, is yet to be fully realized. It began with a deliberate ambiguity on the part of the author of the “Ausdehnungslehre,” whereby one and the same notation was chosen to designate two different algebraic operations, originally known as the progressive and the regressive product. This choice was not an oversight on Grassmann’s part, as we shall see, but was rather dictated by the author’s deep understanding of the philosophical reach of his discovery, an understanding which lay well beyond the powers of the