Numerical methods are used to investigate the steady two-dimensional motion of a viscous incompressible fluid past a flat plate of finite breadth at zero incidence to a uniform stream. Before application of numerical techniques, the governing partial differential equations for the stream function and vorticity are reduced to ordinary differential equations by an adaptation of methods normally used to solve Oseen's linearized equations. The complete range of the Reynolds number R is considered, from indefinitely small to indefinitely large. All the results are intended to represent solutions of the full Navier-Stokes equations of motion, although in practice approximations are inevitable. These are mainly brought about by the necessity of limiting the size of the calculations. At the lower end of the Reynolds-number range, the calculated frictional drag coefficient agrees well with the results of Tomotika & Aoi (1953) based on Oseen's equations. At intermediate and higher Reynolds numbers there is good agreement with the experimental results of Janour (1951) and with the improvement of the Blasius solution given by Kuo (1953). Finally a limiting solution is obtained as R → ∞. This shows that the drag coefficient is proportional to R−½, in accordance with boundary-layer theory. The actual calculated value of the coefficient is about 4% higher than the Blasius value. Although the present results tend generally to confirm the trend of the recently published results at R = 0·1, 1 and 10 of Janssen (1957), there are substantial discrepancies in the detailed results in a number of instances. In particular, the drag values obtained at R = 1 and 10 are some 20% higher than Janssen's although there is reasonable agreement at R = 0·1. It seems possible that Janssen's analogue is a little crude at the higher Reynolds numbers.