Descartes' Lost Theorem

3. Euler's Theorems

Leonhard Euler
In Euler's papers he states two theorems, describing them as equally important, and emphasizing that they are completely equivalent.

In every solid bounded by planar faces the number of solid angles together with the number of faces is greater by two than the number of edges.

In every solid bounded by planar faces the sum of all the plane angles, which make up the corners of the solid, is equal to four times as many right angles as there are solid angles, minus eight.

The first theorem is the one best remembered today; his second theorem is an exact rediscovery of Descartes' theorem from some 130 years before. Using Euler's symbols of S for the number of solid angles, A for the number of edges and H for the number of faces, the first theorem becomes

S + H - 2 = A or S + H = A + 2

while the second becomes, using pi/2 for a right angle,

(Sum of all plane angles) = (pi/2)(4 S - 8).

As Euler explains, the link between these two theorems is the fact from plane geometry that in a polygon of n sides, the sum of the angles is (n-2)pi.

Using this fact the sum of all the plane angles, which can be rewritten as the sum over all faces of the sum of the angles in that face, becomes the sum over all faces of pi times the number of sides minus 2. Since each edge of the solid appears exactly twice as the side of a face (and we pick up a -2 for each face), we find

(Sum of all plane angles) = pi(2 A - 2 H).

Setting this equal to (pi/2)(4 S - 8) yields the first theorem immediately, in the form A - H = S - 2. This argument runs backwards just as well, and establishes the complete mathematical equivalence of the two theorems.

Euler was extremely and justifiably proud of this work, but mistaken when he states that ``it is surprising that no one before now has thought of these basic principles of solid geometry.''

For further thought. Use the regular polyhedra as examples of these two theorems.

© copyright 1999, American Mathematical Society.