The authors discuss the problem of numerical evaluation of Cauchy principal value integrals. Their approach is novel because they consider using integrand values at complex points taken from a circular arc outside of the real integration interval. Previously published methods have mostly used points from within the integration interval. The formulas described by the authors are of the Gauss-Kronrod type, and were developed in an earlier paper by the same authors. The authors compare numerical results obtained using their new method with results from two methods that use integrand evaluation points taken from within the integration interval. The methods are applied to several test integrands. The new method is significantly more accurate when the singularity is not near one of the endpoints of the integration interval, but is less accurate when the singularity is near one of the endpoints. The new method is also compared with one other method that uses complex integrand evaluation points. The new method is more accurate for the one test problem for which results are given.