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In many areas of mathematics, there are problems that admit a natural formulation in any dimension. It often happens that such a problem is easier to solve in dimension $n>k$ as compared to dimension $k$. Sometimes, the problem is solved in all but finitely many dimensions.

The examples are, e. g., Poincaré conjecture, or Milnor's solution to the "Can one hear the shape of a drum" problem.

So, the question is, what is the earliest known example of this situation?

One that I can think of is Maxwell's derivation of velocity distribution in a gas, which wouldn't work in dimension 1. But since it's not so clear what the associated mathematical problem is, let's not take it as a cut-off for possible suggestions.

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Desargues theorem. Suppose you have two triangles $(A,B,C), (A',B',C')$ in the plane such that the lines $AA'$, $BB',\; CC'$ intersect at one point. Then the three points of intersection $AB\cap A'B'$, $BC\cap B'C'$ and $CA\cap C'A'$ lie on one line. And conversely.

Proof. Lift the vertices $A$ and $A'$ from the plane to the three space. So that triangles do not lie in the same plane now. The statement becomes almost evident. Then obtain the plane theorem by passing to the limit.

In some sense, there is no other proof: if you want to axiomatise projective geometry in the plane, you have to take this as an axiom.

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  • $\begingroup$ a nice answer, but here the 3D result does readily imply the 2D one (and, I believe, wasn't published earlier, am I right?), so it is not quite what I meant. $\endgroup$
    – Kostya_I
    Apr 29, 2016 at 12:43
  • $\begingroup$ "but here the 3D result does readily imply the 2D one" Yes, but only for some projective planes, namely the ones where you can perform this lifting construction to the $3$-space. In fact, Desargues theorem is valid in all projective spaces of dimension $\geq 3$, but there are some non-desarguesian projective planes. So the case of dimension $2$ is really the tricky part here. Is not this precisely an example of what you are looking for? $\endgroup$
    – Francesco Polizzi
    Apr 29, 2016 at 13:50
  • $\begingroup$ If you put the question as "whether the Desargues theorem is valid in any projective space as defined axiomatically", then, of course, I agree. But, I believe, this view has little to do with how Desargues seen his result - he just proved the theorem for "the plane" that is obviously embeddable to "space". $\endgroup$
    – Kostya_I
    Apr 29, 2016 at 16:24

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