Supercat The hospital has a history of the treatment. This doesn't sound quite right to me. You could say "The hospital has a history of treating the disease".
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SupercatThe hospital has a history of the treatment.This doesn't sound quite right to me. You could say "The hospital has a history of treating the disease". Actually, "a history of" is often used for negative things rather than positive things, though it is not actually wrong to use it of positive things too.
SupercatWell, I meant that 'the hospital has done/experienced' anyway, regardless that it has succeeded in it or not. Either way, is 'a history of~' okay?"The hospital has a history of treating the disease", assuming no further qualification, would normally be understood to refer to successful treatment (or as successful as could be expected).