Cavalieri was interested in mathematics and worked on a problem he called indivisibles. The purpose of his work was to compute areas and volumes. He worked at a time before calculus had been invented. He was not the only one worked on this idea. It had been tossed around in one form or another for quite a while. In fact, Galileo had his own version of this idea. Robertval also worked on this idea.

The general idea is that an area can be decomposed into line segments. The lines are 1 dimension less than the quantity being computed. A volume can be decomposed into planar sections. Cavalieri’s theorem states that if 2 objects have the same height and the length of the line segments (areas of the planar sections) is the same for all positions from top to bottom, then the 2 areas (volumes) are the same.

The method here is a comparison. One object is compared to another. Some measure of the 2 objects is determined to be the same.

In 1902 some geologists examining crystals did something else yet claimed that the Cavalieri theorem applied. That is the first time that the name Cavalieri was associated with a volume estimation technique. Were this an actual application of the Cavalieri theorem then 2 objects would have been compared or an object would have been constructed that had the same volume as the crystals. Neither was done. Instead volumes were determined using a model-based methodology. This mistake was pointed out a long time ago.

Despite that initial error the name Cavalieri has stuck. Today we have the following statement posted all over the internet:

“the volume of non-classically shaped objects could be estimated in an accurate manner from the sum of areas on the cut surfaces of the objects”

This is wrong in several ways.

1. Cavalieri’s work dealt with the solutions to problems of shapes bounded by mathematical equations. He needed to be able to determine the areas and he used what was referred to as quadrature. This is like integration in calculus.

2. Cavalieri did not estimate anything. His theorem and his work required the use of all sections.

3. Cavalieri’s work was not accurate. It was exact.

4. Cavalieri did not work with cut surface except to double check his work. One of the methods of integration was to cut shapes out of heavy paper stock and to weight them. This provides an independent check of the mathematics.

What is done today in stereology has nothing to do with Cavalieri. Cavalieri never estimated. Cavalieri never used point counting. Point counting is not developed until the 1920s.

This mistake or rather misnomer in stereology has led to people making reference to Cavalieri’s books. For a while it appeared to be fashionable to add the reference to Cavalieri. That fad appears to be over.

May 4, 2012 at 4:22 pm |

Steriologist, I’m afraid you can’t do anything about a misnomer that has become a name after such a long time. Somehow, the name is not too badly chosen for the general way to estimate volume from section areas, as Cavalieri’s theorem and the “Cavalieri principle” build on the same mathematical background. But I hope it is still possible to fight effectively against using “Cavalieri principle” as a synonym for point counting volume estimation …