Suppose a hexagon (Fig. 79), fortified according to Vauban's first method, it is evident that all the parts of the circumference of one thousand, and even one thousand eight hundred yards, are commanded by the curtains, the faces of the bastions and the demi-lunes. If the fortress stands in a level country, the enemy cannot occupy any point in that circumference without being exposed to its fire. To raise his first parallel and first batteries, he would have to begin his works at the limit of the range of the rampart guns; and, as we saw in Fig. 70, he must erect these batteries sufficiently near the place to enable their fire to tell upon the defences--_i.e._, at eight or nine hundred yards. At this distance the curtains could be swept, the faces and flanks raked, and the parapets thrown down. As the projectiles reached either point blank, or under an angle of about 10° when the ball rebounded, the besieged could protect himself against it for a very considerable time, and keep his own artillery intact. [Illustration: FIG. 79.] But as the range of siege pieces in the present day extends to eight or nine thousand yards, the conditions both for the besieged and the besieger are very different. Hence (Fig. 80) the enemy raises his batteries on two or three fronts of the circumference, taking advantage of the rise in the ground, a wall, or a wood to protect his works, and when all is ready, he unmasks these batteries, and covers a segment of the fortress with a quantity of explosive projectiles, which, reaching it at an angle of 25° to 30° burst, no matter where--_dans le tas_, to use a common expression--the distance not allowing an aim at flanks or faces in particular--the besieger being in fact unable to distinguish them. Supposing the besieged able to maintain his artillery and reply, exposed as he is to the deluge of iron on his face and flanks, he has to aim at eccentric points which may vary, to whose position he has no clue but the smoke of the guns, and at an enemy, who, taking advantage of an indefinite amount of space to make his arrangements and shelter his men, is completely free. But to maintain his artillery and preserve his men and munitions, the besieged has only a space relatively limited to move in; he is soon encumbered with _débris_ of all kinds, every movement is difficult for him, and he has not even room to repair damages. He tires himself out to no great purpose. If the attack has maintained its fire at a long range for several days, it has introduced such confusion into a great part of the defences, that in two or three nights afterwards the first parallel may be commenced at about one thousand yards, it may be well armed and protected by batteries _en retraite_ and trench-shelters, so as to discourage sorties and allow of an advance to crown the covered ways. In what state are the works of the besieged by this time? The flanks of the bastions are as much damaged as their faces, the