We present a comprehensive phenomenological examination of the Veneziano ansatz for pion-nucleon and kaon-nucleon processes. Using invariant amplitudes constructed as sums of beta-function terms, we attempt to fit simultaneously all the relevant high- and low-energy scattering data as well as the elastic widths of baryon resonances. We discuss a useful technique for ensuring that the theoretical amplitudes will possess the observed spin-parity structure of the physical spectrum of baryon states.