ions of lines and positions by the use of diagrams.

(Fig. 1.) The army at A covers the whole of the ground in rear of the
line DC perpendicular to the line AB, the position of the enemy being at
B.

(Fig. 2.) AJ being equal to BJ, A will still cover every thing in rear
of DC.

(Fig. 3.) If the army A is obliged to cover the point _a_, the army B
will cover all the space without the circle whose radius is _a_ B; and of
course A continues to cover the point _a_ so long as it remains within
this circle _a_ B.

_A line of operations_ embraces that portion of the theatre of war which
an army or _corps d'armee_ passes over in attaining its object; _the
front of operations_ is the front formed by the army as it advances on
this line.

When an army acts as a single mass, without forming independent corps,
the line it follows is denominated a _simple line of operations_.

If two or more corps act in an isolated manner, but against the same
opposing force, they are said to follow _double_ or _multiple lines_.

The lines by which Moreau and Jourdan entered Germany in 1796, were
double lines; but Napoleon's advance by Bamberg and Gera, in 1806,
although moving in seven distinct _corps d'armee,_ formed but a single
line of operations.

_Interior lines of operations_ are those followed by an army which
operates between the enemy's lines in such a way as to be able to
concentrate his forces on one of these lines before the other can be
brought to its assistance. For example, Napoleon's line of operations
in 1814, between the Marne and the Seine, where he manoeuvred with so
much skill and success against the immensely superior forces of the
allies.

_Exterior lines_ present the opposite results; they are those which an
army will form in moving on the extremities of the opposing masses. For
example, the lines of the Marne and the Seine, followed by the army of
Silesia and the grand Austro-Russian army, in the campaign of 1814.
Burgoyne's line of operations, in 1777, was double and exterior.

_Concentric lines_ are such as start from distant points, and are
directed towards the same object, either in the rear or in advance of
their base.

If a mass leaves a single point and separates into several distinct
corps, taking divergent directions, it is said to pursue _eccentric
lines_.

Lines are said to be _deep_, when the end to be attained is very distant
from the base.

The lines followed by a secondary or auxiliary forc