Euler's method for solving initial value problems is an excellent vehicle for observing the relationship between discretization error and rounding error in numerical computation. Reductions in stepsize, in order to decrease discretization error, necessarily increase the number of steps and so introduce additional rounding error. The problem is common and can be quite troublesome. We examine here a simple device, well known to those versed in the fixed point computations employed many years ago, that can help delay the onset of this problem.