The Method of Mechanical Theorems

The method of mechanical theorems is one of Archimedes most proud achievements. It is called the method of mechanical theorems because Archimedes uses his extensive knowledge of levers to balance geometric figures against one another in order to compare and contrast their areas or volumes. It is important to note that Archimedes did not consider this to be a rigorous proof. He used this method to investigate attributes of objects then he would use his findings to aid him in a proof by the method of exhaustion or some other rigorous and popular method of the time. Archimedes begins his investigation in to the volume of a sphere with with Figure 1.

Using Figure 1, Archimedes uses the method of mechanical theorems to show.

1. Any sphere is, in respect to volume, four times greater than that of the cone with base equal to a great circle of the sphere and height that is equal to the radius of the sphere.

2. A cylinder that has a base that is equal to the great circle of the sphere and a height equal to the diameter of the sphere is 1.5 times more in volume than the sphere.

In Figure 1, ●ABCD is a great circle cross section of a sphere. AC, BD are diameters of the circle and AC⊥BD
▲AEF is a cross section of a cone from the apex at A to the base EF
▰LGEF is a cross section of a cylinder.
CH is the length of a lever with A as a center, thus CA=HA.
MN || BD. Let MN be a two dimensional representation of a plane that cuts through the sphere, cylinder, and cone.

MS=CA and QS=AS as ▲QSA∼▲ECA
MS⋅QS=CA⋅AS
CA⋅AS= AO²   because ▲COA and ▲ASO are similar right triangles.

This is when Archimedes applies what is known as the Law of the Lever. He places the center of the circle cross section from the cylinder at point A. He also places the centers of the circles of the sphere and the cone at point H.

The law of the lever states that when two sides of a lever are at equilibrium, the product of the weight, and the distance of the weights from the fulcrum are equivalent. In other words in the figure below A⋅B=C⋅D

Archimedes recognized that the equation HA(Cylinder Circle)=AS(Sphere Circle + Cone Circle) is in the same form, with HA and AS as distances of the objects from the fulcrum point A.

Archimedes then employs a method similar to that of Cavalieri's Principle, which states: If the cross sections of two objects have equivalent area for all corresponding cross sections, then the two objects have the same volume.

With Cavalieri's Principle as a justification, Archimedes places the entire cone and sphere center of mass at point H and the center of mass of the Cylinder at K. Now we find that HA(Circle Volume+Cone Volume)=AK(Cylinder Volume).
Since HA=2AK, then
     
The cylinder=3(cone) so
     

Since the formulas for the volumes of cylinders and cones are already known, we now can find the formula for the volume of a sphere using either of the results above. First we will derive the formula using the volume of the cone and the results that relate cone volumes and sphere volumes from above.

Now we get the same result with the formula for the volume of a cylinder and the results from above that relate the volume of a cylinder to the volume of a sphere.


There is also a Geogebra Applet that allows you to further explore the derivation of the formula for the volume of a sphere and the Cavalieri Principle.