Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as t=(T-Tc)Tc and the ordering field, h, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, gt=t+bth2+ctt2+ and gh=h(1+cht+). The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, L, of the form (1+aLt+bLt2+), etc., where the correction amplitudes aL,bL are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term bth2 also generates nonanalytic contributions such as an additive, energylike term, varying as | t |1- in the zero-field susceptibility, and factors like (1+cL| h |2-1) on the critical isotherm, t=0. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in Tc and various amplitudes although "universal" ratios remain constant.