The board of county commissioners within ten days from the receipt of the notice, if in session, and if not in session, within five days after the first day of the next regular or special session, shall designate the width of the draw to be made in such bridge, and the length of span necessary to permit the free flow of water: PROVIDED, That if any persons deem themselves aggrieved by the determination of the matter by the board, they may appeal to the superior court which may hear and determine the matter upon such further notice and on such testimony as it shall direct to be produced.