Geometric Algebra – Leibnitz’ Dream

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 Princely Jablonowski Society for the recreation and further establishment of the geometric calculus invented by G.W. Leibniz. Grassmann went on to prove the usefulness of his extension theory by applying it to the theory of tides and other phenomena in physics. Grassmann’s influence was far reaching. The English mathematician W.K. Clifford published in 1878 his Applications of Grassmann’s extensive algebra, describing “geometric algebra”. Clifford had been a student of James Maxwell. Clifford’s desire to understand the mathematical basis of Maxwell’s equations partly motivated his research in geometric algebra. He started by clarifying the relation of Grassmann’s method to (Hamilton’s) quaternions. Clifford “profoundly admired” Grassmann’s Ausdehnungslehre, with the “conviction that its principles will exercise a vast influence upon the future of mathematical science.” Now this algebra is often simply referred to as “Clifford algebra.” And the Italian G. Peano published in 1888 his Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann. Four years later, in 1892 Felix Klein himself successfully began to push for a complete posthumous republication of Grassmann’s works by the Royal Saxonian Society of Sciences. But, due to the early death of Clifford, J.W. Gibbs’ and O. Heaviside’s vector analysis dominated most of the 20 century, and not Clifford geometric algebra. Yet today, at the beginning of the 21 century, some people believe, that based on Grassmann’s and Clifford’s work soon more or less all of mathematics may be formulated as a single unified universal geometric calculus, with concrete geometrical foundations. The algebraic “grammar” such a geometric calculus uses is Clifford geometric algebra.