The calculation of hill slope in the form of downhill gradient and aspect for a point in a digital elevation model (DEM), is a popular procedure in the hydrological, environmental and remote sensing. The most commonly used slope calculation algorithms employed on DEM topography data make use of a three by three search window, or kernel, centred on the grid point (grid cell) in question in order to calculate the gradient and aspect at that point. A comparison of eight frequently used slope calculation algorithms for digital elevation matrices has been carried out using both synthetic and real data as test surfaces. Morrison's surface III, a trigonometrically defined surface, was used as the synthetic test surface. This was differentiated analytically to give true gradient and aspect values against which to compare the results of the tested algorithms. The results of the best-performing slope algorithm on Morrison's surface were then used as the reference against which to compare the other tested algorithms on a real DEM. For both of the test surfaces residual gradient and aspect grids were calculated by subtracting the gradient and aspect grids produced by the algorithms on test from the true/reference gradient and aspect grids. The resulting residual gradient and aspect grids were used to calculate root-mean-square (RMS) residual error estimates that were used to rank the slope algorithms from "best" (lowest value of RMS residual error) to "worst" (largest value of RMS residual error). For Morrison's test surface, Fleming and Hoffer's method gave the "best" results for both gradient and aspect. Horn's method (used in ArcInfo GRID) also performed well for both gradient and aspect estimation. However, the popular maximum downward gradient method (MDG) performed poorly, coming last in the rankings. A similar pattern was seen in the gradient and aspect rankings derived using the Rhum DEM, with Horn's method performing well and the MDG method poorly.