The transcendental rank of the formulas of an $\aleph_1$-categorical theory

In this paper we investigate the transcendental rank introduced by Morley. We introduce the concept of a fibering and prove a series of lemmas about fibering, on the basis of which we prove the main theorems: on an estimate for the rank of a formula (Theorem 3); on the summation of ranks (Theorem 2); and a reproof of Baldwin's theorem on the finiteness of the rank of $\aleph_1$-categorical theories (Theorem 1).