After the assessor’s book shall be corrected and adjusted according to law, but not later than September twentieth of each year, or in the case of any city not within a county or counties with a charter form of government, not later than October first, the county commission shall ascertain the sum necessary to be raised for county purposes, and fix the rate of taxes on the several subjects of taxation so as to raise the required sum, and the same shall be entered in proper columns in the tax book.