I have further warned the reader that, in spite of the way in which the intricacy of organization inseparable from great masses and the manifold disposition of a modern army will mask the general nature of such an operation, that operation cannot be understood unless its simplest lines are clear. I have further insisted that in practice those lines remain only in the idea of the scheme of the whole, and are not to be discovered save in the loosest way from the actual positions of men upon the map. We have seen that this "open strategic square" involved essentially two conceptions--the fixed "operative corner" and the swinging "manoeuvring masses." The manoeuvring masses, at this moment when the great German blow fell upon the Sambre and the Meuse, and when Namur went down immediately before it, were (_a_) upon the frontiers of Alsace and Lorraine, (_b_) in the centre of the country, (_c_) near the capital and to the west of it, and even, some of them, upon the sea. The operative corner was this group of armies before Namur on the Sambre and Meuse, the 4th French Army under Langle, the 5th French Army under Lanrezac, the British contingent under French. We know from what has been written above in this book that it is the whole business of an operative corner to "take on" superior numbers, and to hold them as well as possible, even though compelled to retreat, until the manoeuvring masses can swing and come up in aid, and so pin the enemy. We further know from what has gone before that the whole crux of this manoeuvre lies in the power of the operative corner to stand the shock. It was the business of the French in this operative corner before Namur and of their British Allies there to await and, if possible, to withstand by a careful choice of position the first shock of enemies who would certainly be numerically superior. It was the whole business of the German commanders to make the shock overwhelming, in order that the operative corner should be pounded to pieces, or should be surrounded and annihilated before the manoeuvring masses could swing up in aid. Should this destruction of the operative corner take place before the manoeuvring masses behind it could swing, the campaign in the West was lost to the Allies, and the Germans pouring in between the still separated corners of the square were the masters for good. It behoves us, therefore, if we desire to understand the campaign, to grasp how this operative corner stood, upon what defences it relied, in what force it was, what numbers it thought were coming against it, and what numbers were, as a fact, coming against it. To get all this clear, it is best to begin with a diagram.