We develop an algebraic and algorithmic theory of principality for the recently introduced framework of intersection type calculi with dimensional bound. The theory enables inference of principal type information under dimensional bound, it provides an algebraic and algorithmic theory of approximation of classical principal types in terms of computable bases of abstract vector spaces (more precisely, semimodules), and it shows a systematic connection of dimensional calculi to the theory of approximants. Finite, computable bases are shown to span standard principal typings of a given term for sufficiently high dimension, thereby providing an approximation to standard principality by type inference, and capturing it precisely for sufficiently large dimensional parameter. Subsidiary results include decidability of principal inhabitation for intersection types (given a type does there exist a normal form for which the type is principal?). Remarkably, combining bounded type inference with principal inhabitation allows us to compute approximate normal forms of arbitrary terms without using beta-reduction.