The recently developed methods of analyzing stress and strain distributions in the plane plastic flow of an ideally plastic material are applied to the problem of machining. It is shown that the idealized stress-strain relationship involves justifiable assumptions for this application. Analytical expressions are obtained for orthogonal machining which give the machining force, the chip thickness, and the chip deformation in terms of the tool geometry, the relevant coefficients of friction, and the appropriate yield stress of the work. With increasing friction at the tool face or decreasing rake angle the development of a built-up nose arises naturally as a consequence of the analysis. The theory includes this phenomenon. The results of this theory are compared with published experimental results, and with other theoretical analyses. Satisfactory agreement with experiment is obtained.