Here are some examples illustrate what I meant:

Bonnet-Myers:Bonnet in 1855 proved n=2 case, Myers in 1941 extended to any dimension using the same idea.

Bishop-Gromov Volume comparison: Bishop knew the result inside cut locus in 1963, Gromov made a simple observation that beyond cut locus it is still true.

Ok, you got what I mean.

  • 5
    $\begingroup$ This seems unhelpfully broad -- most important theorems get extended by others, and it is very subjective as to what counts as a simple observation (especially in retrospect). Can you be more specific about what you want to know? $\endgroup$ Jan 2, 2010 at 4:27
  • $\begingroup$ Observations that the first one proved the theorem could think of by a more thorough thought or whatever. In some sense, it is unlucky they passed those thoughts. $\endgroup$
    – lemega
    Jan 2, 2010 at 4:37

1 Answer 1


Perhaps Euler's polyhedral formula:

V(vertices) + F(faces) - E(edges) = 2

provides an example of what you mean?

Euler did not give a proper proof but shortly thereafter this result inspired huge advances that had dramatic effects on the evolution of geometry, topology, convexity, and what today is called graph theory.

One measure of how rich this topic is can be seen from the many types and styles of proofs that can be found for this result collected below by David Eppstein:



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