**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 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 *,

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.