Geometric Algebra

In 1844 when Hamilton published his invention of quaternions, the German mathematician and schoolteacher, Hermann Gunther Grassmann [1809–1877], published the first edition of his geometric calculus Lineale Ausdehnungslehre, which offered an original algebra for describing geometric operations. The word ‘Ausdehnungslehre’ translates as ‘theory of extension’ and the principal algebraic product of the theory was the exterior product. The notion of extension is something that Euclid was aware of, in that the product of two lengths create an area, and the product of a length and an area create a volume. Grassmann discovered an algebra where his exterior product of vectors created areas, volumes and higher-dimensional objects.