Simple lines of operations are those of an army acting from a frontier when it is not subdivided into large independent bodies.
Double lines of operations are those of two independent armies proceeding from the same frontier, or those of two nearly equal armies which are commanded by the same general but are widely separated in distance and for long intervals of time.1
Interior lines of operations are those adopted by one or two armies to oppose several hostile bodies, and having such a direction that the general can concentrate the masses and maneuver with his whole force in a shorter period of time than it would require for the enemy to oppose to them a greater force.2 Exterior lines lead to the opposite result, and are those formed by an army which operates at the same time on both flanks of the enemy, or against several of his masses.
Concentric lines of operations are those which depart from widely-separated points and meet at the same point, either in advance of or behind the base.
Divergent lines are those by which an army would leave a given point to move upon several distinct points. These lines, of course, necessitate a subdivision of the army.
There are also deep lines, which are simply long lines.
The term maneuver-lines I apply to momentary strategic lines, often adopted for a single temporary maneuver, and which are by no means to be confounded with the real lines of operations.
Secondary lines are those of two armies acting so as to afford each other mutual support,—as, in 1796, the army of the Sambre and Meuse was secondary to the army of the Rhine, and, in 1812, the army of Bagration was secondary to that of Barclay.