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.

Suppose two lines perpendicular one to the other, and therefore forming a right angle, AB and BC.  Suppose at their junction, B, a considerable zone or segment, SSS, of a circle, as shaded in the following diagram.  Supposing the line AB to be protected along the outer half of it, AK, by no natural obstacle—­the state of affairs which I have represented by a dotted line {~GREEK SMALL LETTER ALPHA~}{~GREEK SMALL LETTER GAMMA~}; but suppose the second half of it, KB, should be protected by a natural obstacle, though not a very formidable one—­such as I have represented by the continuous line {~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER BETA~}.  Supposing the perpendicular line BC to be protected by a really formidable natural obstacle {~GREEK SMALL LETTER BETA~}{~GREEK SMALL LETTER DELTA~}, and supposing the shaded segment of the circle at B to represent a fortified zone (1) accessible to any one within the angle KBC, as from the arrow M; (2) inaccessible (until it was captured or forced) to any one coming from outside the angle, as from the arrows NNN; (3) containing within itself, protected by its ring of fortifications, passages, PP, for traversing the two natural obstacles, {~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER BETA~} and {~GREEK SMALL LETTER BETA~}{~GREEK SMALL LETTER DELTA~}, which meet at the point {~GREEK SMALL LETTER BETA~}.

[Illustration:  Sketch 38.]

There you have the elements of the position in which the advance corner of the great French square was situated just before it took the shock of the main German armies.  The two lines AB and BC are the French and British armies lying behind the Sambre, {~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER BETA~}, and the Middle Meuse, {~GREEK SMALL LETTER BETA~}{~GREEK SMALL LETTER DELTA~}, respectively; but the line of the Sambre ceases to protect eastward along the dotted line {~GREEK SMALL LETTER ALPHA~}{~GREEK SMALL LETTER GAMMA~} beyond the point up to which the river forms a natural obstacle, while from K to B the line is protected by the river Sambre itself.  The more formidable obstacle, {~GREEK SMALL LETTER BETA~}{~GREEK SMALL LETTER DELTA~}, represents the great trench or ravine of the Meuse which stretches south from Namur.  The town of Namur itself is at B, the junction of the two rivers; and the fortified zone, SSS, is the ring of forts lying far out all round Namur; while the passages, PP, over the obstacles contained within that fortified zone, and accessible to the people inside the angle from M, but not to the people outside the angle from NNN, are the bridges across both the Sambre and the Meuse at Namur.