You would have seen something like Sketch 31 where the bodies enclosed under the title A were the operative corner; various garrisons and armies in the field, enclosed under the title B, were the manoeuvring mass. But it is only by putting the matter quite clearly in the abstract diagrammatic form that its principle can be grasped. With this digression I will return and conclude with the main points of debate in the use of the open strategic square. We have seen that the operative corner is in this scheme deliberately imperilled at the outset. The following is a sketch map of the actual position, and it will be seen that the topographical features of this countryside are fairly represented by Sketch 39; while this other sketch shows how these troops that were about to take the shock stood to the general mass of the armies. But to return to the diagram (which I repeat and amplify as Sketch 41), let us see how the Allied force in the operative corner before Namur stood with relation to this angle of natural obstacles, the two rivers Sambre and Meuse, and the fortified zone round the point where they met. [Illustration: Sketch 40.] The situation of that force was as follows:-- [Illustration: Sketch 41.] Along and behind βγ stretched the 5th Army of the French, prolonged on its left by the British contingent. I have marked the first in the diagram with the figure 5, the second with the letters Br, and the latter portion I have also shaded. At right angles to the French 5th Army stretched the French 4th Army, which I have marked with the figure 4. It depended upon the obstacle of the Meuse βδ for its defence, just as the French 5th Army depended upon the Sambre, γβ. It must, of course, be understood that when one says these forces "lay along" the aforesaid lines, one does not mean that they merely lay behind them. One means that they held the bridges and prepared to dispute the crossing of them. Now, the French plan was as follows. They said to themselves: "There will come against us an enemy acting along the arrows VWXYZ, and this enemy will certainly be in superior force to our own. He will perhaps be as much as fifty per cent. stronger than we are. But he will suffer under these disadvantages:--