  The solution term by term to the scattering of all consistent string theories
is given. The moduli space of M-theory is derived and connects the various
string theories. The solutions contain both the perturbative and
non-perturbative sectors of the string. Modular forms found by differential
equations on subspaces of the M-theory moduli space and transfinite algebras
play an essential role in deriving the coefficients. Various results and
identities in algebra are found from the explicit solution. Archetypes and
models are presented in accord with phenomenology and cosmology.
