We apply Heine’s method—the key idea Heine used in 1846 to derive his famous transformation formula for 2ϕ1 series—to multiple basic series over the root system of type A. In the classical case, this leads to a bibasic extension of Heine’s formula, which was implicit in a paper of Andrews which he wrote in 1966. As special cases, we recover extensions of many of Ramanujan’s 2ϕ1 transformations. In addition, we extend previous work of the author regarding a bibasic extension of Andrews’ q-Lauricella function, and show how to obtain very general transformation formulas of this type. The results obtained include transformations of an n-fold sum into an m-fold sum.