The microscopic mechanics discovered by Nos\'e, of which Gauss's isokinetic mechanics is a special case, makes it possible to simulate macroscopic irreversible nonequilibrium flows with purely reversible equations of motion. The Gauss-Nos\'e and Nos\'e-Hoover equations of motion explicitly include time-reversible momentum and energy reservoirs. Computer simulations of nonequilibrium steady-state systems described by Gauss-Nos\'e mechanics invariably evolve in such a way as to increase entropy. The corresponding phase-space distribution functions, which include reservoir degrees of freedom, collapse onto stable strange attractors. Hypothetical time-reversed motions, which would violate the second law of thermodynamics, cannot be observed for two reasons: First, such reversed motions would occupy zero volume in the phase space; second, they would be dynamically unstable. Thus, Nos\'e's reversible mechanics is fully consistent with irreversible thermodynamics, in the way forecast by Prigogine. That is, the consistency follows from the formulation of new microscopic equations of motion.