The speculative science of the Germans must be regarded as dating from Leibnitz. The name of Puffendorf, indeed, is generally placed first among those writers who have contributed to tbe formation of an indigenous German philosophy; but he cultivated only one branch?jurisprudence. Leibnitz was in every sense universal. Our limited space obliges us to condense to the utmost as plain an account a...

1. Historical Developments About 150 years ago, in 1844, the German high school teacher Hermann Grassmann published an ambitious work entitled The Linear Extension Theory, A New Branch of Mathematics. For Grassmann this was indeed The Branch of mathematics, which in his own words “far surpasses” all others. His subsequent work Geometric Algebra won the prize of 45 gold ducats set out by the Pri...

By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with usual Leibnitz rule as well as undeformed gauge transformations with a twisted Leibnitz rule. Explicit structures of the gauge generators in either case are computed. It is shown that, in the former case, the relation mapping the generator with t...