This paper describes an application and extension of the work of H. J. Maehly [1] on the rational approximation of arc tan x, and of E. G. Kogbetliantz [2], who developed Maehly's procedure so as to be applicable to the computer programming of elementary transcendental functions. It is to be shown here that certain modifications, such as the introduction of terms which are easily computed on specific computers, lead to considerable iprovements. In particular, the application of the modified method to several elementary functions will be described and corresponding final results will be given. Some of these approximations have been used with great success to develop subroutines for the IBM 704 and 709 computers. Our experience indicates that the method of Maehly and Kogbetliantz, as modified below, is superior to other current numerical procedures.