To summarize: Carthage launched itself into an offensive war with Hannibal, but with an already advanced Empire reaching its end and without a military disaster. It was about to confront another empire of lesser prestige at that time, but with youth and in its first period of great strategic victories. The Italian campaign by Hannibal begins with his triumphs in the battles of Mount Tesina, Trebbia, Trasimene and Cannae, returning to Carthage where he is defeated in Zama by Scipio. Studying the enemy from the point of view of the mathematical-historical concept, is then, a new factor of the decision that should be analyzed and that completely influences the results in war.� The Macedonian Empire, constituted in the year 750 B.C., ends its first period of great strategic victories in the year 506, with the kingdom of Amintas I, after the great depression (506-413). This Empire is defeated by the Persian Empire. In the second period of great strategic victories (413-323), the Athenians are defeated and the Ionian colonies go on to become part of the Macedonian power. In the year 338 the dominion over Greece is complete.� Under the leadership of Alexander the Great (334-325) in the second period, the Macedonian Empire reaches its maximum point. In the year 221, Phillip the III, confronts the Roman Empire. The latter formed in 270, had its first victory in the first Punic War. As a summary: Phillip the III with his Empire in its stage of disintegration, against the Roman Empire in its first period, which destroys the Macedonian fleet. The war begins again in the year 211 and is prolonged until 205 B.C., when Phillip made peace in Rome. In the space of time 200-197, Phillip is conquered in Cynoscephalae. The third Roman-Macedonian War ends the Macedonian Empire, which is totally absorbed by Rome in 146. DECISIVE WORLD BATTLES.� A. In the mathematical-historical concept, the Marathon result, one of the fifteen decisive world battles, was due to the historical phase that the Greeks were developing, the era of great demographic fractionation in which nations are not conquered, the Persians met the mathematical-historical process in the phase known as military disaster, after having gone through its first period.� B. That of Arbella (331) which, along with the Battles of Granicus (334), Issus (333) and other strategic victories, constituted the second period of great victories of the Macedonian Empire of Alexander the Great.� This battle of Arbella is part of the victory of Macedonia against Persia, an Empire going through a phase of decadence and disintegration.� That is, an Empire undergoing expansion, Macedonia, fulfilling or developing the second aggressive process, against an Empire going through its last phase of disintegration:� Persia.