Tovacian Geometric Dimensional Analysis I (Path A):Aikman's Dimensional Grid in the Basis of Resistance, Length, Time and Vorticity, with an Audit of Maxwell's Original Equations

J. Aikman's Geometrical Dimensional Analysis (GDA) places every physical quantity at a point of a four-dimensional grid of exponents. Tovacian GDA (TGDA) writes the four coordinates as the components s + x i + y j + z k of a quaternion in A. L. Baker's notation. Path A is the reading in which quaternion addition is physical multiplication. We recover, from the coordinates themselves, the units of the four axes: the real axis is the ohm, i the metre, j the second and k the square root of a pascal. The change of basis from SI has determinant 1. In this basis charge is √(action/resistance), which is the author's Mattie's law written as a statement about units, and the real coordinate counts reversions: 0 for mechanical quantities, ±½ (Baker's mean reversion) for electromagnetic ones and 1 for resistance. An audit of Maxwell's twenty original equations finds them homogeneous once three transcription errors are corrected. Path A is a change of basis; it organizes units and cannot fix dimensionless numbers.