In the Americas, the Columbian period, and up to the present time, the Unification phase is not produced after the era of Great Fractionation within the evolution of a culture, nor an empire that substitutes another one that is disintegrated within one of the cycles, but it is an empire that is formed with elements coming from other empires, in a full expansive or decadent phase, over a country in the state of primary life. Dr. Deulofeu says: �.... the constitution of a nucleus of unification in these conditions stands back from those that, up to the present time, I have been able to precisely state, that is, those already included within the cultures that unfailingly and with an impressive precision obey the mathematical-historical process. In any case, from the study of the territories that today make up the different nations of the American Continent, from the moment that the Conquistadors arrived and up to the present time, it can be deduced that the constitution of these nuclei over countries in a state of primary life, is the same followed by nuclei within a culture when it reaches the end of its era of Great Demographic Fractionation. Thus, in fact, with the conquistadors the colonizers arrive and they begin to take hold of the country, taking the lands that are pleasing to them and cultivating them. At the beginning these families are totally independent, having no laws except for those they impose on themselves. The aspect is then the patriarchal regime, with the father as the absolute head of the family. As families increase, territories are occupied, and an organization is soon imposed, regulations between families and the fatherland. In the centers where merchants and businessmen meet population nuclei begin to constitute themselves and cities begin to be formed; each time a more precise organization is imposed, and governors and some states will follow. � Finally an assembly constituted by representatives of each one of the states is established. It is exactly the same evolution that we have seen in all known civilizations starting from the feudal phase, serfs and slaves also existing also in them�. In addition to the historical cycle, there are artistic, philosophical, social phases, etc. �Man is basically what education makes of him�, said the great philosophers. Then, cycles, their internal articulations and changes are normally peaceful. Dr. Deulofeu indicates that the difference between the beginning and the development of the United States, the Latin American Countries and Brazil, is due to the fact that when they became independent from their fatherlands, the phase in Great Britain was that of expansion while the Spanish and Portuguese Empires were declining.� In the mathematics of history, there takes place the beginning of the relative Independence phase that is characterized by an effervescence, political and social instability so that afterwards a period of progress begins.