Can’t get there from here? Curious logic in the famous paper by Einstein and de Sitter

The Einstein–de Sitter cosmological model, a spatially flat Friedmann–Robertson–Walker cosmological model containing ordinary matter ("dust") but no radiation or cosmological constant, has been studied extensively, both because of its mathematical simplicity and, partially related to that, because it was often used as a fiducial model. It is thus interesting that Einstein and de Sitter performed their main calculation, that of the density based on the Hubble constant, in a very roundabout and confusing way. After providing some historical background, I describe their calculation and explain why their method is technically correct even though it is misleading.