A lattice point is any point in the plane having integer Cartesian coordinates. If C is a circle in the plane, the lattice points of C are those lattice points on the boundary or in the interior of C. If C is a circle of radius square root of r, and if C is centered at 0,0, Ar denotes the number of lattice points of C and Er denotes the difference between Ar and one-half the circumference of C. Numerical information is considered for the functions Ar, Er, and Erthe 4th root of r. A method is devised for computing Ar on a digital computer for all values of r which are perfect squares in the closed interval 1,4 time 10 to the 10th power. The method is then utilized in a computer program, and Ar is evaluated. Knowing Ar, approximate evaluations of Er and Erthe 4th root of r are readily obtained. The results of all computations are given in tabulated form.