Cavalieri's Principle

Amy Damron & Renea Mielec

Bonaventura Francesco Cavalieri was born in 1598 in Milan. By 1629, Cavalieri developed a method of indivisibles which became a factor in the development of the integral calculus. Cavalieri developed his theory of indivisibles from Archimedes' method of exhaustion and incorporated Kepler's theory of infinitesionally small geometric quantities. The theory of indivisibles was presented in 1635 in Cavalieri's Geometria indivisibilis continuorum nova . It is from this theory that Cavalieri found the area and volume of various geometric figures. Cavalieri's book was widely attacked. So in reply, Cavalieri improved his exposition by publishing Exercitationes geometrica sex . Some other contributions made by Cavalieri are the tables of logarithms of trigonometric functions for astronomers. He wrote on conic sections, trigonometry, optics, astronomy, and astrology. Cavalieri also developed a general rule for the focal length of lenses and described a reflecting telescope.

Cavalieri's Principle is a method for computing the volume of solid regions, much like the present-day integration (except Cavalieri's Principle is not as systematic as integration).

Cavalieri's Principle for Area: If all the lines parallel to some fixed line that meet the plane regions M and M do so in line segments having equal lengths, whose endpoints are the boundary points of the two regions, then Area M = Area M .

> pl1 :=plot({1,3,7},x=0..10,color=blue):

> pl3 :=plot([[0,0],[0,7]],color=magenta):
pl4 :=plot(9-x^2,x=0..3,color=blue):

> initial:=plots[display]([pl1,pl3,pl4]):

>

> pl5 :=plot([4+t+sqrt(9-t),t,t=0..9],color=blue):
pl6 :=plot(x-4,x=4..13,color=red):
pl7 :=plot([[0,0],[0,7]],color=magenta):
pl8 :=plot({1,3,7},x=0..20,color=blue):

>

> final:=plots[display]([pl5, pl6, pl7, pl8]):

>

> plots[display]([initial,final],insequence=true);

Cavalieri's Principle for Volume: If all the planes parallel to some fixed plane that meet the solid regions T and T do so in plane sections having equal areas, whose boundaries lie in the boundaries of the two regions, then Vol T = Vol T .

> restart;

>

> fram:=t->plots[display]([
plots[polygonplot3d]([[0,0,0],[1,0,0],[0,1,0]],color=magenta),
plots[polygonplot3d]( {[[1,0,0],[0,t,3]],[[0,0,0],[0,t,3]],[[0,1,0],[0,t,3]]}),plots[polygonplot3d]([[0,t/2,3/2],[1/2,t/2,3/2],[0,(1+t)/2,3/2]],color=red)]);

>

>

> printlevel:=1;

> plots[display]([seq(fram(3*i/20),i=0..20)],insequence=true,axes=boxed,orientation=[-18,69]);

> plot(fram(1));

Plotting error, empty plot

>

>

A great example of these concepts in everyday life is a set of records or CDs. If you have two equal stacks of records or CDs and take one of the stacks and tilt it slightly to the left or right, the two stacks will still have the same cross-sectional area and volume. Another example is a stack of scrap paper that is twisted or contorted.

Except that if the paper is twisted so that the individual sheets are no longer planar, then Cavalieri's principle no longer applies, but rather some extension of his principle

Cavalieri justified his principle through the following:

Imagine two solids S and S as having been cut into horizontal slices by the same family of parallel planes. If you look at each slice, you will notice that each resembles a right cylinder with the base equal to the cross-section of the solid by the plane and height equal to the thickness of the slice. Since the volume of such a cylinder is equal to the area of the base times the height and since the two cross-sections are assumed to have the same area, we can say that the two slices have approximately the same volume. Adding up the volumes of the slices of S and S , we say that the solids have approximately the same volume. Cavalieri went on and said that you could infinitely slice each slice in which each volume of the new slices of S and S , could be added together so the two solids have the same volume.

>

>

The Volume of a Sphere

Cavalieri developed a method of calculating the volume of a sphere. The first step to finding the volume of a sphere is to circumbscribe the sphere with a cylinder. The following is a diagram of the sphere circumscribed in the cylinder.

> with(plottools):
c := cylinder([1,1,-2.3],3.3,6.6,style=line):
plots[display](c,scaling=constrained):

> d := sphere([1,1,1], 3.3):
plots[display](c,d, scaling=constrained, style=patch, axes=none);

We can easily calculate the areas of the cross sections. The following diagram demonstrates a vertical cross section of our sphere and its circumscribed cylinder. As you can see the cylinder appears as a rectangle and the sphere appears as the circle. We will label the height of the cross section of L to be, h.

The outer radius is R, and the inner radius is defined as r=square root[(R^2)-(h^2)]. The inner radius was found by the Pythagorean theorem. The cross-sectional area of, L, at height, h, is:

>

Area of h = (Pi*(R^2))-(Pi*(r^2))

= [Pi*(R^2-r^2)]

= Pi*[R^2-(R^2-r^2)]

= Pi*r^2.

Now consider the figure L' which is shown as two cones tip to tip. The following is a diagram of the vertical cross section of the figure with a height of, h, and a radius of, R.

>

> icecream := plottools[cone]([0,0,0],0.7,1,color=blue), plottools[cone]([0,0,0],0.7,-1,color=red):
plots[display]([icecream], scaling=constrained, style=patch);

You should format the inline mathematics in your projects. Just select it with your mouse, then choose the "Format" menu at the top of Maple, then choose "convert to", then choose "math expression".

The cross-sectional area at height, h, is

Area of

Therefore, vL = vL'.

But, vL' = {(2)*[(1/3)*(Pi)*(R^2)]*(R)}

= {(2/3)*(Pi)*(R^3)}

Therefore, vL = {(2/3)*(Pi)*(R^3)}

Hence, vS = vC - vL

= {2*(PI)*(R^3)}-{(2/3)*Pi*(R^3)}

= {(4/3)*Pi*(R^3)}.

The final product of the substitution and algebra generates the formula of a sphere.

It is also possible to show the volumes of prisms, cylinders, pyramids, and cones by Cavalieri's Principle. Cavalieri's Principle is thought to be an early pioneer of calculus.

We are all familiar with integration. The Slice Method is another form of Cavalieri's Principle. It states:

Let S be a solid and Px be a family of parallel planes such that:

1. S lies between Pa and Pb.

2. The area of the section of S by Px is an integrable function A(x).

Then the volume of S is equal to:

> int(A(x),x=a..b);

>

Since we are going to be high school math teachers, we wanted to find exercises about Cavalieri's principle which could be worked by a high school student. We were happily surprised to find many rich sources. The following are a couple projects incorporating Cavalieri's principle into a high school math class. The projects make the topic into an interesting and fun exercise.

Cavalieri's principle has even been used as the basis of a portfolio project. The setup of the project is to make a cone, sphere, and cylinder out of modeling clay. Let the diameter of the sphere, the circumference of the base of the cone, and the diameter of the cylinder to be the same length. Then take fishing line or dental floss and cut each object into six slices of equal thickness. Combine the bottom slice of the cone with the bottom slice of the sphere to make a new disk of equal thickness. Then compare the new disk with the bottom slice of they cylinder. From this analysis the students discover the concept of Cavalieri's principle. We also found a tidbit on using twisted pad of scratch paper to illustrate Cavalieri's principle. We may not know how to calculate the volume of the twisted three-dimensional figure but since page-for-page the solids have equal cross-sectional areas the volume of the twisted solid equals that of the cube.

Works Cited

Aichele, Douglas B., et al. Geometry: Explorations and Applications . McDougal Littell, Inc. 1998.

"Bonaventura Francesco Cavalieri." http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cavalieri.html December 1996.

Kay, David C. College Geometry A Discovery Approach . Harper Collins College Publishers. 1994.

Marsden, Jerrold and Alan Weinstein. Calculus . The Benjamin/Cummings Publishing Company, Inc. 1980.

Moise, Edwin E. Elementary Geometry: From An Advanced Standpoint . Addison-Wesley Publishing Company, Inc. 1963.

Smith, David Eugene. A Source Book in Mathematics . volume 2. Dover Publications, Inc. New York. 1959.