Caveat on the use of the Delesse principle of areal analysis for estimating component volume densities

Abstract

SummaryIn stereological work, component volume densities are estimated routinely by invoking the Delesse principle of areal analysis. This is true whether the morpho‐metric methods are based on the measurement of transectional areas, linear intercepts or point densities. Following our earlier consideration of the special case of a spherical component in a spherical containing volume, we here present a warning on the use of this fundamental principle. This report demonstrates that estimates of volume densities may be derived by application of two main formulations which are based on the areal analysis of ‘cubical’ and ‘spherical’ mathematical models. It is emphasized that these formulations of the Delesse relationship are not mathematically equivalent in all practical instances; failure to appreciate the correct practical circumstances in which each should be applied may lead therefore to the introduction of a statistical bias into the volume density data which are obtained. The conditions under which the formulations should be applied are stressed and clarified and some biological examples are provided. Whilst the problems are presented in a form which is aimed primarily at the biologist, they are also pertinent to the earth and materials scientists and to morphometrists in general.