A, thus tackled, has become what may be called “the operative corner of the square.”  It is his task “to retreat and hold the enemy” while B, C, and D, “the masses of manoeuvre,” swing up.  But under that simple phrase “operative corner” is hidden all the awful business of a fighting retreat:  it means leaving your wounded behind you, marching night and day, with your men under the impression of defeat; leaving your disabled guns behind you, keeping up liaison between all your hurrying, retreating units, with a vast force pressing forward to your destruction.  A’s entire force is deliberately imperilled in order to achieve the success of the plan as a whole, and upon A’s tenacity, as will be seen in what follows, the success of that plan entirely depends.

[Illustration:  Sketch 24.]

Well, while A is thus retreating—­say, from his old position at A_1 on the foregoing diagram to such a position as A_2, with Black swarming up to crush him—­the other corners of the square, B, C, and D, receive the order to “swing”—­that is, to go forward inclining to the left or the right according to the command given.

Mark clearly that, until the order is given, the general commanding Black cannot possibly tell whether the “swing” will be directed to the left or to the right.  Either B will close up against A, C spread out farther to the left, and D come in between A and C (which is a “swing” to the left) as in Sketch 25, or C will close up against A, B will spread well out to the right, and D come up between A and B, as in Sketch 26 (which is a “swing” to the right).

[Illustration:  Sketch 25.]

Until the “swing” actually begins, Black, the enemy, cannot possibly tell whether it is his left-hand units (1 to 8) or his right-hand units (9 to 16) which will be affected.  One of the two ends of his line will have to meet White’s concentrated effort; the other will be left out in the cold.  Black cannot make dispositions on the one hypothesis or on the other.  Whichever he chose, White would, of course, swing the other way and disconcert him.

Black, therefore, has to keep his line even until he knows which way White is going to swing.

[Illustration:  Sketch 26.]

Let us suppose that White swings to the left.

Mark what follows.  The distances which White’s units have got to go are comparatively small.  B will be up at A’s side, and so will D in a short time after the swing is over, and when the swing is completed, the position is after this fashion.  Black’s numbers, 1 to 9 inclusive, find themselves tackled by all Black’s twelve.  There is a superiority of number against Black on his right, White’s left, and the remaining part of Black’s line (10 to 16 inclusive), is out in the cold.

If it were a tactical problem, and all this were taking place in a small field, Black’s left wing, 10-16, would, of course, come up at once and redress the balance.  But being a strategical problem, and involving very large numbers and very great distances, Black’s left wing, 10-16, can do nothing of the kind.  For Black’s left wing, 10-16, cannot possibly get up in time.  Long before it has arrived on the scene, White’s 12 will have broken Black’s 9 along Black’s right wing.