The Schläfli function$f_{n}(x)$allows to compute volume of a regular$(n-1)$-dimensional spherical simplex of the dihedral angle$2\alpha =$arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function$f_{n}(x)$for$n\geq 4$in the frequently used in applications interval$x\in [n-1,n+1]$. The computation is based on the Chebyshev approximation of the function$q_{n}(x)$, which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.