In all mountainous frontiers, or sides of states bordering on large rivers, or chains of lakes, there will necessarily be but few points by which an invader can penetrate into the interior of the country. Let us suppose that, for a frontier of moderate extent, there are _five_ passes, or avenues through which the enemy may approach the interior. To effectually defend these approaches against the invading army will require, for each, an army of ten thousand men. Not being able to decide positively on the plans of the enemy, all these communications must be defended at the same time. This requires a defending army of fifty thousand men. Let us now suppose each of these passes to be fortified in such a way, that one thousand men will be able to hold the enemy in check, and force him to resort to the operations of a siege; or, at least, to retard his advance till an active army can be organized in the interior, and prepared to meet him in the field. We here see that five thousand men, by means of fortifications, can accomplish the same defensive object as fifty thousand men without these artificial means of security. But let us enter a little more into the details of frontier defences, and examine the character of the several systems which have been successively proposed or adopted. Frontiers are divided into four distinct classes, according as the state may be open on one or more sides, or bounded by mountains, large rivers and lakes, or by the sea. An open frontier is the most difficult of defence; and while there exists a perfect uniformity among military men upon the vast importance of fortifying such a frontier, there is an equal diversity of opinion respecting the best manner of arranging these works. We shall here mention three general systems of arranging forts for the defence of an open country, each of which has been advocated at different times, and afterwards received various modifications and additions. These three systems comprise the main features of all others worthy of much consideration. They are:-- 1st. The system of continuous lines, proposed by Montalembert. 2d. A system of three lines of detached works, strongly recommended by D'Arcon and others. 3d. A system proposed by Vauban, and advocated by Rogniat, consisting of lines of very strong works, placed at considerable distances from each other and covering large _intrenched camps_.