Intermediate Level

Areas

The application illustrates Cavalieri’s two-dimensional principle:

If two plane regions are contained between a pair of parallel lines and any line parallel to those, that intersect the region, does it in cut sections of the same length, then the two regions have equal areas.

How did Bonaventura Cavalieri (1598-1647) come to this conclusion?

Imagine two equal reams of paper placed on a table, one "tidy" and another "messed up" as shown in the picture.
We have no doubt that the volume of the two reams is the same, although the format is different.
We also note that the area of the front flat surfaces of the two reams (marked in the figure) have different shapes, but necessarily have equal areas.

By dividing the two reams at the same height we obtain as sections two equal leaves, one in each ream, so they have the same area and the same dimensions. This experience illustrates Cavalieri's Principle in two and three dimensions.
How did Cavalieri arrive to its formulation?

In the work "Geometria indivisibilibus continuorum nova quadam ratione promota", Cavalieri developed the idea of indivisible quantities.

According to Cavalieri, a line is an infinite set of points, a surface is an infinite set of parallel straight segments, and a solid is an infinite set of parallel plane regions. Thus, a flat region can be thought of as being formed by the juxtaposition of segments, the "indivisibles" of the surface, and a solid as the juxtaposition of flat regions, the "indivisibles" of the volume.

To calculate areas and volumes Cavalieri opted for the Greek method of geometric reasoning. In the case of the areas, he compared one surface with another one that is formed by the same lines and, in the case of solids, he compared one solid with another one that is formed by flat surfaces with the same area.

For his work on indivisibles, the Italian mathematician Bonaventura Cavalieri (1598-1647) is considered one of the precursors of integral calculus.

References

• Pinto A., A teoria dos indivisíveis: Uma contribuição do padre Bonaventura Cavalieri, Tese de Mestrado em História da Ciência, Pontifícia Universidade Católica de São Paulo, 2008
• Struik D.J., História Concisa das Matemáticas, Gradiva, 1989