Where does the notation $S^n$ for the $n$-dimensional sphere originate? Just as in the title. The notation is somewhat surprising in its use of a superscript rather than a subscript, but I gather there is a history of using $M^n$ to refer to an $n$-dimensional manifold. Still, it is perhaps a bit awkward in leading us to say things like "$S^1 \times S^1$ is not isomorphic to $S^2$". The notation could perhaps be understood in the context of a smash product, but this is likely anachronistic.

Anyway, enough speculation. The question is simple: where does the use of $S^n$ to denote the $n$-dimensional sphere originate?
We know that on one hand mathematical notation was developing evolutionarily, sometimes even ad hoc. On the other hand, mathematics is huge, requires a lot of notation, so much notation that it is hard to keep it short. Thus, at least in practice, consistency is often not attained across different times and domains.
We know that on one hand mathematical notation was developing evolutionarily, sometimes even ad hoc. On the other hand, mathematics is huge, requires a lot of notation, so much notation that it is hard to keep it short. Thus, at least in practice, consistency is often not attained across different times and domains.