text: Mathematician One of the many examinations for which Grassmann sat required that he submit an essay on the theory of the tides. In 1840, he did so, taking the basic theory from Laplace's Traité de mécanique céleste and from Lagrange's Mécanique analytique, but expositing this theory making use of the vector methods he had been mulling over since 1832. This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what is now called linear algebra and the notion of a vector space. He went on to develop those methods in his Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (A1) and its later revision Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (A2).
  In 1844, Grassmann published his masterpiece (A1) commonly referred to as the Ausdehnungslehre, which translates as "theory of extension" or "theory of extensive magnitudes". Since A1 proposed a new foundation for all of mathematics, the work began with quite general definitions of a philosophical nature. Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded.
  Fearnley-Sander describes Grassmann's foundation of linear algebra  as follows:
  Following an idea of Grassmann's father, A1 also defined the exterior product, also called "combinatorial product" (in German: kombinatorisches Produkt or äußeres Produkt “outer product”), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule epep = 0 by the rule epep = 1. (For quaternions, we have the rule i2 = j2 = k2 = −1.) For more details, see Exterior algebra.