10

MANOEUVRING TROOPS.

if it continue to turn, it will approach towards AB, on thecontrary side, till at length, after making a complete circle,it will return to AB.

It is evident that the right line, AX, has made a demi-revolution when it reaches AM from AB. In fact, if thepart of the circle, BAME, were to be folded by the lineMB over the underneath part, the one would cover theother precisely, and would coalesce with it.

In manoeuvring troops, after having placed them in aright line, facing one way, it is very often necessary tomake them turn to the other side. They are commandedto face about, which is effected by each man turning onhis heels, A, fig. 3, pi. 1. To render the movement easy, onefoot, B, is placed behind the other, fig. 4, and at the sametime the man turns on his heels. Each of the feet performshalf a circle, fig. 5. The foot which was placed behindthen comes in front, and is brought back to a line with theother, fig. 6. If the soldier were to face about once more,he would find himself fronting the original direction, andhe would have made a complete circle.

Let us consider the angles formed by the right line AC,with the right line DAB, fig. 7, pi. 1. There are here twoangles ; BAC , a small one, and CAD, a large one: theirsum is always equal to half a revolution of AC, or fromAB to AD.

The angle BAC , therefore, is required in order that theangle DAC may form a complete half revolution, andDAC is required in the same manner for BAC to form ahalf revolution. For this reason, BAC is called the sup-plement, or the supplementary angle to DAC , just as DAC is the supplement or the supplementary angle to BAC .

Let us suppose that the angle BAC , is enlarged by theline AC separating from AB; we shall then see that thesupplementary angle is diminished. As the smaller angle,BAC , is always increasing, and the larger angle, DAC ,diminishing, there will be a time when they will be equal,as in fig. 8. Each of these equal angles is what is called aright angle. Thus we see, that a right angle is made by a