Descartes' Lost Theorem

5. Descartes' curvatura

Euler's Theorem is equivalent to Descartes' Lost Theorem, but the methods by which they arrived at their results must have been quite different. In fact Descartes' Theorem appears in his work as a consequence of an observation for which there is no parallel in Euler.

Descartes defines the exterior solid angle at a vertex of a polyhedron as ``that quantity by which the sum of all the plane angles which make up the solid angle is less than four plane right angles,'' i.e. 2 pi minus the sum of the face angles at that vertex, and he states:

Just as in a polygon the sum of the exterior angles is equal to four right angles, so in a polyhedron the sum of the exterior solid angles is equal to 8 solid right angles.

Descartes does not have a completely coherent definition of what a solid right angle should be, and this may be one reason why he never published his Treatise, but computationally he interpreted his statement as meaning:

(Sum of exterior solid angles) = 4 pi,

and in this form it is clearly equivalent to his Lost Theorem.

Descartes expands on his definition of exterior angle: ``By exterior angle I mean the curvature and slope of the planes with respect to each other ...'' . In his use of curvatura he foreshadows by two centuries Gauss' definition of the intrinsic curvature of a surface.

© copyright 1999, American Mathematical Society.