  We present a new computational scheme, GRIP (Geometric Random Inner
Products), for testing the quality of random number generators. The GRIP
formalism utilizes geometric probability techniques to calculate the average
scalar products of random vectors generated in geometric objects, such as
circles and spheres. We show that these average scalar products define a family
of geometric constants which can be used to evaluate the quality of random
number generators. We explicitly apply the GRIP tests to several random number
generators frequently used in Monte Carlo simulations, and demonstrate a new
statistical property for good random number generators.
