The phrase "hadst on good authority" means to have received information from a reliable source. There are a number of synonyms for this phrase, including "according to reliable sources", "as reported by trustworthy individuals", "from a credible source", and "on the authority of reliable sources". Other similar phrases include "as attested to by reliable witnesses", "with the assurance of trustworthy sources", and "on the word of reliable informants". Regardless of the phrasing used, the key implication is that the information being shared is of a high degree of reliability and can be trusted to be accurate.