The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the line, which in turn can be shown to be connected by proving the transitivity of the the natural action of the braid group on n-tuples of transpositions in $S_n$ with product 1, which generate $S_n$: in this action, a generator $\sigma_i$ of the braid group acts as

$$(g_1, ... g_n) \to (g_1, ... g_{i+1}, g_i^{g_{i+1}}, g_{i+2}, ..., g_n)$$

This argument is often referred to as "a theorem of Clebsch (1872 or 1873), Luroth (1871), and Hurwitz (1891)." Does anyone know the history of this argument more precisely, and in particular which parts are due to Luroth, which to Clebsch, and which to Hurwitz?