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 Discoveries of less value, which in part covered those of Grassmann and Hamilton, were made by Saint-Venant (1797–1886), who described the multiplication of vectors, and the addition of vectors and oriented areas; by Cauchy, whose "clefs algébriques" were units subject to combinatorial multiplication, and were applied by the author to the theory of elimination in the same way as had been done earlier by Grassmann; by Justus Bellavitis (1803–1880), who published in 1835 and 1837 in the Annali delle Scienze his calculus of æquipollences. Bellavitis, for many years professor at Padua, was a self-taught mathematician of much power, who in his thirty-eighth year laid down a city office in his native place, Bassano, that he might give his time to science.[65]

The first impression of Grassmann's ideas is marked in the writings of Hermann Hankel (1839–1873), who published in 1867 his Vorlesungen über die Complexen Zahlen. Hankel, then docent in Leipzig, had been in correspondence with Grassmann. The "alternate numbers" of Hankel are subject to his law of combinatorial multiplication. In considering the foundations of algebra Hankel affirms the principle of the permanence of formal laws previously enunciated incompletely by Peacock. Hankel was a close student of mathematical history, and left behind an unfinished work thereon. Before his death he was professor at Tübingen. His Complexe Zahlen was at first little read, and we must turn to Victor Schlegel of Hagen as the successful interpreter of Grassmann. Schlegel was at one time a young colleague of Grassmann at the Marienstifts-Gymnasium in Stettin. Encouraged by Clebsch, Schlegel wrote a System der Raumlehre which explained the essential conceptions and operations of the Ausdehnungslehre.

Multiple algebra was powerfully advanced by Peirce, whose theory is not geometrical, as are those of Hamilton and