This thesis examines the viability of the Farrer theory of how The Gospel According to Luke was written in light of the influential criticisms of that theory by F.G. Downing. Downing argues that on Farrer's theory, Luke has departed from known compositional procedures of Graeco-Roman authors in deserting the common witness of his sources, in picking out Matthew's additions to Mark from Matthew's gospel to use in his own work, and in removing the Markan parallels from the Matthean additions that he has picked out. This thesis will argue to the contrary that in following one of his sources at a time rather than trying to follow both simultaneously, and in using material from his second (Matthean) source to supplement his main (Markan) source, Farrer's Luke appears to be following accepted ancient compositional methods, and that he has no demonstrable tendency to remove Markan parallels from his use of Matthew.