The board shall direct that at the time and in the manner required by law for the levying of taxes for county purposes, such board of supervisors shall levy, in addition to such other tax as may be levied by such board of supervisors, at the rate or rates so fixed and determined by the board, a tax upon the real property and improvements thereon, but not on personal property within the district, and it is made the duty of the officer or body having authority to levy taxes within each county to levy the tax so required.