Cavalieri principle example

It amazes me that the the Cavalieri principle is so poorly understood. Maybe a good way to understand the principle is to use it.

For starters let’s post what the Cavalieri principle states:

If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal (Kern and Bland 1948, p. 26).

The Cavalieri principle also applies to planar areas.

Two objects of equal altitude means that the objects have the same height. The heights are indicated by parallel planes or lines in the 2-dimensional case. Cavalieri uses this idea of bounding planes or lines throughout his tome. The
lines or planes between the bounding planes/lines and parallel to them intersect the objects of interest. The area or length of the intersections is checked to see if the areas or lengths are the same. If they are for all intersecting planes/lines, then the volumes/areas are the same.

Notice that the Cavalieri principle does not compute volumes or areas. This principle only shows that volumes or areas are the same without computing the values. Cavalieri used his method to relate the volume of one unknown object to one or more objects for which he could determine the volume. Thus, an unknown could be related to a known.

Let’s consider the “napkin ring” problem. A ball is converted into a napkin ring by drilling out the center of the ball. The drill passes through the center of the ball and leaves a ring as shown above. The height of all of the rings is the same. A larger drill is used each time the ball is larger so that the height is a xed h.

What is fairly amazing is that the volumes of all of the napkin rings is the same for all rings of height h.
The height h is fixed, but the radius of the ball changes. Let’s call the ball’s radius r. Then the radius of the drill needed to leave a ring of height h is computed as follows:

The divide by 2 is used because the height refers to the total height of the ring and only the half above the center of the ball should be used in the computation.

For simplicity, let’s use the original center of the sphere as the origin. This is the (0,0,0) of our coordinate system. The ring has heights from -h to h. A plane that intersects the napkin ring forms an annulus. The outer radius of the annulus is computed below, and the inner radius was computed above.

The area of the annulus is the outer disc area minus the inner disc area.

The area of the annulus is independent of the radius. It is only dependent on the height of the napkin ring and the position of the intercepting plane. The Cavalieri principle allows us to state that the volumes of all napkin rings of the
same height have the same volume. The Cavalieri principle does not tell us what that volume is.

We can determine the volume of napkin rings by imagining a napkin ring which has a hole of zero radius through it. That napkin ring has the same volume as all of the other napkin rings of the same height. The height of the ball must be h.

Thus all napkin rings of height h have a volume of:

The volume of the napkin ring can be computed directly by subtracting the volume removed from the ball from the volume of the ball. The removed material is a cylinder in the middle plus two end caps.

The same result was obtained. By using the Cavalieri principle the problem was solved by comparing areas of intercepts. The same result was determined using formulas. The Cavalieri principle required two objects to compare. Formulas did not require more than one object.

The Cavalieri principle is not used in stereology since there is only 1 object. There is no comparison to an object of known volume. Instead, formulas are used to estimate the areas of profi les and also estimate the volume.

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