Previous studies of {\em Drosophila melanogaster} syncytial embryo nuclear division patterns have failed to detect any regularity in the orientation of the mitotic spindles within or between nuclear cycles (Minden et al., 1989; Warn and Magrath, 1982). Here, we describe a computational analysis of time-lapse microscopy sequences of syncytial nuclear cycles 10-13, when the nuclei form an evenly spaced monolayer at the surface of the embryo, that provides quantitative evidence that the direction of syncytial nuclear mitosis is not random, but is clearly influenced by the relative positions of neighboring nuclei. An approximate law governing mitotic direction is introduced that is based on a scheme that compromises among ``votes'' made by neighboring nuclei.