Cavalieri published his work in 1635, which today is known simply as the Cavalieri principle. Cavalieri's principle provided a practical alternative to the Archimedes' principle of volume estimation by water displacement.

Cavalieri's method showed that the volume of a population of objects could be estimated from the profile areas of the objects on cut sections. In practice, the Cavalieri approach requires an initial random cut through the reference space of interest, with subsequent cuts at consistent intervals, i.e., systematic-uniform- random sampling.

The Cavalieri principle allows for volume estimation based on the integration of the areas of a defined reference space.

Provided the sections through the reference space are systematic-random, i.e., all sections through the reference space have an equal probability of being sampled, the Cavalieri method gives an unbiased total volume estimate from reference areas on sections. By repeating the estimate on several individuals from the population of interest, one obtains an estimate of the mean total volume for the reference space of interest.

Point Counting for Area Estimation in Cavalieri Volume Estimates

Using the Cavalieri principle, the mean total volume of an arbitrary-shaped reference space can be estimated without bias from the areas on systematic-uniform-random sections. The geometric probe in this case is a 2-D plane; the estimator is the Cavalieri principle. Several methods exist for estimating the areas on each of the cut surfaces. One of the most efficient methods is point-counting.