A geometic formulation of the seismic travel time problem is given based upon the use of slowness as an independent variable. Many of the difficulties in the conventional treatment (e.g., singular kernels) are thereby avoided. Furthermore, it is shown that the inverse problem possesses an inherently linear formulation. In this formalism we are able to provide extremal solutions giving upper and lower depth bounds using linear programing. This approach has been compared with two well-known nonlinear extermal inversions. We find our technique to be easier to implement and find that it often generates superior results.