We construct a new finite difference method for computing {\it reference} numerical solutions to the one--parameter family of Blasius' problems arising from incompressible laminar flow past a thin flat plate with mass transfer by both suction and blowing. We show that, by studying several representative problems in the family, the method generates nodal approximations, at a finite number of nodes, to the solution and its derivatives, the piecewise linear interpolants of which provide global pointwise accurate approximations to the solution and its derivatives on the semi--infinite domain $[0, \infty)$. Using an experimental error estimate technique we determine orders of convergence and error constants of the reference numerical solutions and their discrete derivatives. Algebraic formulae for realistic pointwise error bounds, in terms of the number of mesh subintervals used in the discrete problem, determine the number of mesh points required to achieve a given preassigned guaranteed accuracy in the reference numerical solutions of Blasius' problem. Such reference numerical solutions to Blasius' problem can be used to construct $Re$--uniformly accurate approximations to the components $u_P(x,y)$, $ v_P(x,y)$ of the solution of Prandtl's problem and to their first order partial derivatives.