The books, papers, letters and transactions pertaining to the office of treasurer, are at all times during office hours open to the inspection of a committee of the legislature, or either branch thereof, to examine and settle all accounts, or to take copies of the same, and to count all moneys; and when the successor of any such treasurer is appointed and qualified, the state controller must examine and settle all the accounts of such treasurer, remaining unsettled, and give to him a certified statement, showing the balance of moneys, securities and effects for which he is accountable, and which have been delivered to his successor, and report the same to the legislature.