SUMMARY Data consisting of the intersections of a planar or linear probe with a field of spheres, with a diameter distribution G(x), is often used to estimate linear functionals or their ratios. It is shown that distribution-free estimators may be poor and that their distribution, even in large samples, depends on knowledge of G for small x that may be unobtainable. The parametric approach is arduous and not robust against errors in the lower tail. It seems that this experimental method should be avoided when there is a practicable alternative. Suppose that a population of particles, geometrically similar with a size distribution function G(x) are randomly dispersed through space. Roughly, their centres will be placed by a Poisson process. The space may be probed in some way; we will consider only planar and linear probes. In the former the data are the intersections of the particles and some area of the probing plane. In the latter the data are a set of chords on some interval of the probing line. From such data, the object is to estimate functionals of the form