I just moved to an old house. The toilet has a black line that i haven't been able to clean. I used all I could find in strong liquids. Curiously, I couldn't find thouse handy stones (I think they are called pomez) that my parents used to use in the past,
Pumice stones are not a good idea. Pumice is an abrasive and will scratch porcelain, leaving a surface that is going to be more prone to staining and more difficult to clean. The black is likely to be manganese which is frequently found along with iron.
Try a good cleaner that has as much oxalic acid in it as possible (the oxalic acid will convert iron as rust and manganese to a water soluble salt.) "Bar Keeper's Friend" is a scouring powder found in some grocery stores and good hardware stores that might help. Since it is also slightly abrasive don't use it to scratch the porcelain, rather put a paste on and let it sit.
If properly used a pumice stone is both quite safe and very effective......presoak or keep the stone damp, always use on a damp surface, use gentle even pressure, as the stone dissolves use the paste to rub out the stain. If the bowl is particularly bad use a acid cleaner first (softens the hardwater deposits and sanitizes the bowl) however do not let the acid dry on the surface nor soak excessively as it can damage the surface......then use a pumice to finish up the residue.... For routine cleaning a few quick passes with a pumice will keep rings at bay. Soggy
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