The number and type of measurements needed to ascertain the amplitudes for pseudoscalar meson photoproduction are analyzed in this paper. It is found that 8 carefully selected measurements can determine the four transversity amplitudes without discrete ambiguities. That number of measurements is one less than previously believed. We approach this problem in two distinct ways: (1) solving for the amplitude magnitudes and phases directly; and (2) using a bilinear helicity product formulation to map an algebra of measurements over to the well-known algebra of the $4\times 4$ Gamma matrices. It is shown that the latter method leads to an alternate proof that 8 carefully chosen experiments suffice for determining the transversity amplitudes completely. In addition, Fierz transformations of the Gamma matrices are used to develop useful linear and nonlinear relationships between the spin observables. These relationships not only help in finding complete sets of experiments, but also yield important constraints between the 16 observables for this reaction.