Two points are to be observed concerning the right ordering of rulers in a state or nation. One is that all should take some share in the government: for this form  of constitution ensures peace among the people, commends itself to all, and is most enduring, as stated in Polit. ii, 6. The other point is to be observed in respect of the kinds of government, or the different ways in which the constitutions are established. For whereas these differ in kind, as the Philosopher states (Polit. iii, 5), nevertheless the first place is held by the ‘kingdom,’ where the power of government is vested in one; and ‘aristocracy,’ which signifies government by the best, where the power of government is vested in a few. Accordingly, the best form of government is in a state or kingdom, where one is given the power to preside over all; while under him are others having governing powers: and yet a government of this kind is shared by all, both because all are eligible to govern, and because the rulers are chosen by all. For this is the best form of polity, being partly kingdom, since there is one at the head of all; partly aristocracy, in so far as a number of persons are set in authority; partly democracy, i.e. government by the people, in so far as the rulers can be chosen from the people, and the people have the right to choose their rulers.

>>>>>>>>>>>>>>>>>>– St Thomas Aquinas, Summa Theologiae IaIIae, 105, 1.
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To most of our contemporaries the absurdity of hereditary monarchy or aristocracy seems self evident. Why would we select our rulers or even our figureheads on the basis not of talent or virtue but of birth? Few people (still, thank God) would be willing to suggest that such attributes may be guaranteed by selective breeding and a quick glance through the genealogies of the great European dynasties suffices to show that the effort would be vain. Monarchs of great virtue seem forever to be succeeded by decadents and rulers of great ability by incompetents. In fact the conjunction of one or both of virtue and ability in two successive rulers is so unusual that it tends to alter dramatically the course of history. It seems obvious that we should seek the most talented and most virtuous rulers. Indeed, as appointing people to appoint them would itself require a quest for the most talented and virtuous (establishing an infinite regress), there seems no alternative to doing so by universal franchise.
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And yet hereditary monarchy, at least of the limited kind, seems to work and to produce greater stability and prosperity than the system reason would appear to uphold (and expounded by St Thomas). There are a number of negative reasons why this might be. Power may tend to corrupt but it seems the pursuit of power corrupts a great deal faster and more comprehensively. A highly talented and utterly corrupt ruler extremely adept at the simulation of virtue may actually end up being the result of the most rational constitution and such a ruler, insofar as he does not desire to rule for the common good but for his private good and insofar as he still requires the consent of the multitude to rule, will seek to corrupt the multitude into hedonists the easier to bribe them. The ability of such a population to select for talent and virtue will thus be subverted and a downward spiral of corruption and mediocrity will ensue. Hereditary government is a lottery but no more or less than a hereditary citizenry and, so long as the principle is preserved inviolate, it eliminates the corrupting factor of the pursuit of power. Yet it is easy to see that, as in seventeenth and eighteenth century France, power does tend to corrupt and the mores and manners of the aristocracy and ruling dynasty may become so corrupt as to equal the negative effects of ambition and destroy the ruling elite in precisely the same way. The English House of Commons as it was until the seventeenth century would seem to counteract this tendency. By providing a popular veto on new legislation and the raising of subsidy it pushes the elite to operate within certain bounds of competence and decency.
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Furthermore, popular forms of government too often give rise to the idea that legitimate rule as such comes from the will of the people. This is a highly pernicious error leading to tyranny and racialism, for it becomes impossible to determine the frontiers of the people without an unjustified and dangerous hyper-realism about ‘the nation’ and the only way of avoiding permanent revolution becomes the claim that the will of the people has been mystically transferred to their representative(s). These representatives accordingly acquire unlimited power over everything and everyone because, by the logic of necessarily popular sovereignty, each and every citizen must have agreed in advance to whatever their rulers do.
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But these are all highly negative arguments against popular government and hereditary forms have their drawbacks too. Not only do they prescind  from virtue and ability but they are also oppressive to the spirit of those born without fortune as well as breeding contempt and pride of blood in its beneficiaries. Even ignoring these factors it would seem that the arguments presented thus far still entail that a popular selection of rulers is the rational constitution and only the ravages of sin could deflect us into a hereditary model. Are there no positive arguments for heredity? Aristotle maintains that all the early states were hereditary monarchies because they arose from the original sovereignty of the family which would seem to argue that hereditary monarchy is quite natural (if a discarded stage of human development). Furthermore, though He at first condemned the people’s desire for a king other than Himself, God Himself then seemed to endorse hereditary government not only by His grudging appointment of Saul but by His oath to David and before that to Abraham and by His use of an hereditary covenant all down to the birth of the Messiah Himself. Could one not argue therefore that the hereditary stage of human political development has passed with the enthronement of the universal hereditary monarch Jesus Christ and it behoves us henceforth to live in a republican polity within the context of His kingship (that is, subject to the correction and supervision of the ecclesiastical hierarchy He established and guarantees)?
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But what of the tendency to corruption which, we have argued heredity works against, and the error of popular sovereignty and the need to give due recognition to the family as the fundamental unit of society? As far as the tendency towards corruption goes it would seem we are in the presence of incommensurable goods. One might seek to avoid it through the institution of a hereditary ruler limited by an elective legislative aristocracy or one might prefer the advantages of a ruler chosen freely from and by the whole population and rely exclusively upon a life tenure to prevent corruption and impeachment procedures or the Papal plenitude of power to identify and dethrone a tyrant. Might not the guarantee of infallibility (combined with the prohibition upon the direct exercise of temporal power by the clergy) replace the safeguards we hitherto provided by heredity? As to the error of popular sovereignty, this can flourish even in the context of a limited hereditary monarchy and can work to destroy an unlimited one. It must be faced head on in any system and refuted by open argument. The question of the family requires a more subtle consideration. One might restrict the franchise to the pater familias or his widow. One could extend the franchise to all the baptised of any age and empower the father or widowed mother of those who had not reached the age of majority to vote on their behalf. Furthermore, as the Roman Republic demonstrates, the avoidance of a hereditary aristocracy need not rule out a hereditary nobility. Boethius grudgingly concedes that nobility at least inspires the descendants to emulate the virtues and achievements of their ancestors. We might add that it would provide an incentive for the new man to work for posterity and give permanent recognition to the constitutive social role of the family across generations as well as in the present.
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I would go so far as to suggest that the form of republicanism outlined above fits with the political ideals of Scripture, St Augustine, St Severinus Boethius, St Benedict, St Dominic and St Thomas Aquinas as well as commending itself to reason and the example of history. Not only that, but it more or less represents the form of government that theoretically obtained in the Roman Empire, at least on paper,  at the moment of the Incarnation.