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territories." And by equal territories it is not necessary to under- stand equal areas, but rather equivalent areas; for otherwise the number of the subjects would not have been equal. In reality, in order to make the partition, they established a balance of social, and especially of economic, forces. These forces are the result of a combination of territory and of population. It was in accord- ance with this balance that the delimitation of the frontiers was traced.
Again, at the death of Charles Mattel, in 741, the Prankish heritage was re-established in its unity, and was even increased by the addition of the duchies of Thuringia and Alemannia ; and Bavaria and Frisia were rendered tributary. The Mussulmans had been completely driven out of Gaul. On the northwest the empire extended as far as the mouth of the Weser ; on the south, to the Pyrenees and the Mediterranean; on the west, except for Bretagne, it touched the ocean; on the east it skirted the Saale, the Erzgebirge, and the Bohmerwald, and included the upper basin of the Danube, with the secondary basins of its southern tributaries. In 768 the empire was again divided. The first Carolingian kings scarcely took account of what we call nation- alities, nor even of the great provincial divisions of Prankish Gaul. Austrasia and Aquitaine, for example, were divided into two zones of almost equal extent, with artificial frontiers. Far from being separations, they were destined to be reunited, from a strategical and political point of view, in such a way that a com- munity of action was naturally imposed. Such was the spirit of the act by which Pepin himself determined the division between his two sons. Karlman having died in 771, the unity of the inheritance was reconstituted in favor of Charlemagne. For a time three great empires coexisted, and the evolution of each of them shows that no society is arrested, either in its extension or in its decline, by physical limits. Neither in an exclusively ethnic sense, nor in one exclusively geographical, are there natural fron- tiers, any more than there are natural laws. There are no laws but social laws, nor any frontiers but social frontiers.
At the death of Mahomet, in 632, the political and religious unity of the Arabian peninsula, shaken for a moment, had been
