That's very interesting! I agree Goldberg's proof is not very persuasive. I hope Auburn university will fix their electronic dissertation library.

There's a 1985 paper by Robert Dawson, _Monostatic simplexes_ (The American Mathematical Monthly, Vol. 92, No. 8 (Oct., 1985), pp. 541-546) which opens with a more convincing proof, which it attributes to John H. Conway:

> Obviously, a simplex cannot tip about an edge unless the dihedral angle at that edge is obtuse. As the altitude, and hence the height of the barycenter, is inversely proportional to the area of the base for any given tetrahedron, a tetrahedron can only tip from a smaller face to a larger one.

Suppose some tetrahedron to be monostatic, and let A and B be the largest and second-largest faces respectively. Either the tetrahedron rolls from another face, C, onto B and thence onto A, or else it rolls from B to A and also from C to A. In either case, one of the two largest faces has two obtuse dihedral angles, and one of them is on an edge shared with the other of the two largest faces.

The projection of the remaining face, D, onto the face with two obtuse dihedral angles must be as large as the sum of the projections of the other three faces. But this makes the area of D larger than that of the face we are projecting onto, contradicting our assumption that A and B are the two largest faces