REGULAR APPROACHES. One meaning of "approaches" is "entrenchments, etc., by which the besiegers draw closer to the besieged" (Oxford Universal Dictionary). When one reads that the attacker "undertook regular approaches," it means that he declined to attempt capturing a place by immediate assault (which often is less costly in the end) and elected the time-consuming and laborious process of formal siege operations. The basic technique was to dig a first "parallel" just outside the defender's artillery range; to run forward a zigzag trench, or sap; and then to dig a second parallel. This process is continued, with successive parallels enabling the besieger to move forward his work parties and artillery until the enemy surrenders or until a final assault can be made against his weakened fortifications.
Onions, C. T., ed. Oxford Universal Dictionary on Historical Principles. 3rd ed. Oxford: Oxford University Press, 1955.