Descartes's Lost Theorem

Euler's theorems

Leonhard Euler 1707-1783

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 solidangles together with the number of faces is greater by two thanthe number of edges. In every solid bounded by planar faces the sum of all theplane angles, which make up the corners of the solid, is equalto 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'stheorem from some one hundred thirty 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, thefirst theorem becomes

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

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

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

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

Using this fact, one finds that the sum of all the plane angles, which can berewritten as the sum over all faces of the sum of the angles inthat face, becomes the sum over all faces of times thenumber of sides minus 2. Since each edge of the solid appearsexactly twice as the side of a face (and we pick up a -2for each face), we find

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

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

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

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

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