A perturbation theory for acoustic ray propagation is presented which is based on the analogy between Hamilton’s Principle of Least Action (independent variable: time) and Fermat’s Principle of Least Time (independent variable: space coordinate). In a vertical ocean section with x positive to the right and z positive upward, the Lagrangian variables z and ż (a dot denotes differentiation with respect to x; ż=dz/dx) are first replaced by the canonical variables (z, p) of the corresponding Hamiltonian formulation. For the zeroth-order case, where the sound c(x,z) reduces to the form c0(z), a canonical transformation, z, p → ω0, J0, can be made. The new variables are the analogs of angles-action variables in dynamics, and are constants. The generator for this transformation is just the Eikonal function for zeroth order and, hence, is recognized as the travel time between a source and receiver of sound. Finally, when the x-dependent perturbations are turned on, ω0 and J0 cease to be constants and a second, infinitesimal, canonical transformation, ω0, J0 → ω, J, is made. For range-dependent sound channels, ω=ω(x) and J=J(x). The canonical equations for these variables are exhibited, and how they may be solved by standard perturbation theory techniques is shown. Finally, some examples of the theory as applied to perturbations which vary slowly in x or, alternatively, which vary rapidly in x compared with the basic variation in the range-independent channel are given. The various perturbations to travel time are included in these calculations.