This new edition of a classic text fashions on Dirac algebra into a unified, coordinate-free mathematical language for both classical and quantum physics that articulates smoothly with standard vector algebra and provides a powerful alternative to matrix and tensor algebras. The standard set of Maxwell's equations, for example, are reduced to a single equation with a single differential operator that is identical to the one used in the Dirac equation. Moreover, reformulation of the Dirac equation reveals hidden geometric structure and new insight into the role of complex numbers in quantum mechanics. That is incorporated into a novel gauge theory approach to quantum mechanics and General Relativity.