  The main goal of this paper is to develop a new embedding method which we use
to show that some finite metric spaces admit low-distortion embeddings into all
non-superreflexive spaces. This method is based on the theory of
equal-signs-additive sequences developed by Brunel and Sucheston (1975-1976).
We also show that some of the low-distortion embeddability results obtained
using this method cannot be obtained using the method based on the
factorization between the summing basis and the unit vector basis of $\ell_1$,
which was used by Bourgain (1986) and Johnson and Schechtman (2009).
