What is $\infty^6$? The title of this question may make you want to close it immediately, but bear with me a moment.  In several older mathematics papers (early 20th century) I have seen statements such as

The motions of three-dimensional space are $\infty^6$.

I am curious what this means.  From context I guess that "being $\infty^6$" means something roughly like what we would nowadays call "being a 6-dimensional manifold".  But did it have a precise meaning?  Who defined it, where and when?  When did it fall out of use?
 A: Yes, indeed, this notation was used to state the dimension of the manifold. The idea of dimenson is very intuitive but it took long time and a lot of labor to formalize. Before the modern definitions of "dimension", "manifold" and "homeomorphism" were spread, people, especially geometers, expressed this fact by saying that something depends on $n$ parameters, or is an "n"-parametric family, and wrote $\infty^n$. This notation is out of date now, but they still speak of $n$-parametric families. When I was an undergraduate student (early 1970s) $\infty^n$ was still used in the lectures on projective geometry, for example. I was puzzled because I knew that Cantor proved that all these things have the same cardinaity, until I read about Brouwer's theory of dimension and "domain preservation property". 
(About a century before, Cantor and Poincare were similarly puzzled until Brouwer clarified the thing:-) I think Cantor was the first who tried to prove that manifolds of different dimensions are not homeomorphic, but he failed.
(Before Cantor it seemed evident to everyone that plane contains more points than the line, I do not know whether anyone cared to prove this, but those who did try, could not).
