We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\ensuremath{\varphi}}}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension
Dc, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\ensuremath{\varphi}}}^{4}$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above
Dc. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N\ensuremath{\le}1024 strongly support our theoretically obtained scaling relations.