The purpose of this paper is to state and/or prove multiple hypergeometric transformations by using known quadratic hypergeometric transformations, hypergeometric summation formulas and using formulas for Pochhammer symbols and Gamma functions. The new theorems are proved in detail and connections between several conjectures are proved. All possible cases of parameters for elimination regarding quadratic transformations are summarized in tables. We also find corresponding reduction and summation formulas for Horn functions. As a by-prodoct, q-analogues of some of the formulas are proved. 2020 Mathematics Subject Classification Primary 33C65; Secondary 33C20,33C05,33D15