[Footnote 12: Some German writers have said that I confound central positions with the line of operations,--in which assertion they are mistaken. An army may occupy a central position in the presence of two masses of the enemy, and not have interior lines of operations: these are two very different things. Others have thought that I would have done better to use the term _radii of operations_ to express the idea of double lines. The reasoning in this case is plausible if we conceive the theater of operations to be a circle; but, as every radius is, after all, a line, it is simply a dispute about words.] [Footnote 13: This assertion has been disputed. I think it is correct; for Mélas, confined between the Bormida, the Tanaro, and the Po, was unable to recruit for his army, barely able to maintain a communication by couriers with his base, and he certainly would have been obliged to cut his way out or to surrender in case he had not been reinforced.] [Footnote 14: The capture of Paris by the allies decided the fate of Napoleon; but he had no army, and was attacked by all Europe, and the French people had, in addition, separated their cause from his. If he had possessed fifty thousand more old soldiers, he would have shown that the capital was at his head-quarters.] [Footnote 15: The inferiority of an army does not depend exclusively upon the number of soldiers: their military qualities, their _morale_, and the ability of their commander are also very important elements.] [Footnote 16: When the fractions of an army are separated from the main body by only a few marches, and particularly when they are not intended to act separately throughout the campaign, these are central strategic positions, and not lines of operations.] [Footnote 17: In the movements immediately preceding the battle of Leipsic, Napoleon, strictly speaking, had but a single line of operations, and his armies were simply in central strategic positions; but the principle is the same, and hence the example is illustrative of lines of operations.] [Footnote 18: I am well aware that it is not always possible to avoid a combat without running greater risks than would result from a check; but Macdonald might have fought Blücher to advantage if he had better understood Napoleon's instructions.] [Footnote 19: It will not be thought strange that I sometimes approve of concentric, and at other times divergent, maneuvers, when we reflect that among the finest operations of Napoleon there are some in which he employed these two systems alternately within twenty-four hours; for example, in the movements about Ratisbon in 1809.] ARTICLE XXII. Strategic Lines.