ABSTRACT
Euclidean geometry is a metrical geometry. Length and angle are concepts fundamental to it, and, as we shall see, in Pythagoras' proposition they are related in the simplest feasible fashion. If, therefore, we can give good reason for having space Euclidean, we have thereby made it metrical. We have invoked some notion of nearness as regards space in ascribing to it the topological property of continuity. Since space has more than one dimension, we shall need the unit to be given in more than one standard direction. Merely in establishing a metric mesh, there will be a strong temptation to set up a linear space, with a number of unit vectors, spanning the space, as basis, and the rule of scalar multiplication as formalizing an iterated measuring process. When we come further to establish some rule for measuring distances not in any cardinal direction, the pressure to Pythagoreanism will be felt.
