This paper is concerned with the arithmetical consequences of adopting the theory of minimal rank as a means of determining communalities in factor analysis. The complexities and inconsistencies inherent in the algebraic formulation are illustrated by an analysis of Burt's 11 × 11 table of correlations between emotional traits. It is found that, owing to the appearance of Heywood cases, the only solution that meets Thurstone's requirements entails more factors and higher communalities than the minimal rank would imply, and that this solution has eventually to be rejected on psychological grounds. The final conclusion is that, even were a satisfactory method found for determining the communalities implied by minimal rank, the theory would still be unacceptable because it neglects the statistical considerations imposed by the fact that every empirical matrix is obtained by sampling