  Reconstructions of total solar irradiance (TSI) rely mainly on linear
relations between TSI variation and indices of facular area. When these are
extrapolated to the prolonged 15th - 17th century Sp\"orer and Maunder solar
activity minima, the estimated solar dimming is insufficient to explain the
mid- millennial climate cooling of the Little Ice Age. We draw attention here
to evidence that the relation departs from linearity at the lowest activity
levels. Imaging photometry and radiometry indicate an increased TSI
contribution per unit area from small network faculae by a factor of 2-4
compared to larger faculae in and around active regions. Even partial removal
of this more TSI - effective network at prolonged minima could enable
climatically significant solar dimming, yet be consistent with the weakened but
persistent 11- yr cycle observed in Be 10 during the Maunder Minimum. The
mechanism we suggest would not alter previous findings that increased solar
radiative forcing is insufficient to account for 20th century global warming.
