Three different estimates were compared with complete nuclear counts made over 200 fields (20 sets of 10) selected from histologic sections of 18 breast carcinomas. All counting used a 10 x 10 square grid eyepiece graticule at x 400. Two area sampling techniques agreed well with overall counts (59,210 cells counted in all 200 fields; estimates were 58,500 and 59,805 cells), but an estimate based on linear intersection of nuclei yielded a lower estimate (23,382 cells). The most efficient estimate was 10 times the number of carcinoma cell nuclei in a random sample of 10 small graticule squares; summation over 10 fields yielded a mean ratio of estimated to counted total of 98.2% with limits of agreement of 89.9% to 107% for individual-case 10-field estimates. To improve efficiency further a simplified version of this estimate was devised and validated on carcinoma cell nuclei in 200 fields (20 sets of 10) selected from 20 invasive cervical carcinomas. An invariant sampling set was randomized to one of four rotations (0 degree, 90 degrees, 180 degrees, or 270 degrees). A total of 102,799 cells was counted in all 200 fields; the estimate was 101,280 cells. Summation over 10 fields yielded a mean ratio of estimated to counted total of 98.3% with limits of agreement of 89.8% to 108%. This simplified procedure facilitated estimation of cell numbers without detriment to agreement with denumerated counts and could be implemented as a custom-made counting graticule.