did the Eddington limit [@Tarter73]. Under a few assumptions, the central mass is $M
\propto V^2 R$, where $V$ is the Doppler width of the line and $R$ is the size of the broad-line region (BLR). It is the latter quantity that is difficult to determine. An early attempt to estimate $R$ by [@Dibai80] was based on the assumption of constant emissivity per unit volume, but led to an incorrect dependence on luminosity as in this case, luminosity is proportional to volume, so $R \propto L^{1/3}$. [@Wandel85] inferred the BLR size from the  luminosity. Other attempts were based on photoionization physics [see @Ferland85; @Osterbrock85]. [@Davidson72] found that the relative strength of emission lines in ionized gas could be characterized by an ionization parameter $$U= \frac{Q({\rm H})}{4\pi R^2 c n_{\rm H}},
\label{eq:Udef}$$ where $Q({\rm H})$ is the rate at which H-ionizing photons are emitted by the central source and $n_{\rm H}$ is the particle density of the gas. The ionization parameter $U$ is proportional to the ratio of ionization rate to recombination rate in the BLR clouds. The similarity of emission-line flux ratios in AGN spectra over orders of magnitude in luminosity suggested that $U$ is constant, and the presence of 