Bodin was deeply concerned with the question of harmony and order in a very disordered time. For Bodin, the common good depended on order, and order in society could only exist through a well-established and properly functioning monarchy.
In Bodin's view the end of law is to secure order in the Commonweale. He even goes so far as to say that it is 'better to have an evil Commonweale than none at all'.
The state should be built with relation to the concord of numbers. The three types of progression–arithmetic, geometric, and harmonic–he called the three daughters of Themis, representing order, justice, and peace. The middle term included the other two. The arithmetic progression was more suited to a democratic state, since it denoted equality. Plato, in building an aristocratic state, preferred that it should be governed according to the geometric system. But the harmonic ratio, developed from the other two, portrayed the relationship of overlord and vassal and was therefore suited to a monarchy. It represented peace, and this was the highest objective of all empires. Here Bodin entered upon a discussion of musical intervals, probably drawn from Boethius or Macrobius, which sought to show a parallel between the well-tempered state and concord in music.
The conclusion is that a state can best avoid danger from within or from without if it is built on harmonic principles, which for Bodin meant a monarchy administrated in the interests of all.
"As for the fact that Plato wished his state to be governed according to geometric ratio, Aristotle decided subtly and cleverly that this concerned rewards only. Arithmetic ratio he related to honoring pledges and to penalties. How rightly, I will not discuss; but about the harmonic ratio neither said anything. Yet I think this ratio, as the most beautiful of all, pertains to the form of the best empire. First because it is developed from arithmetic and geometric ratios alone, yet is unlike each. The harmonic ratio cannot pertain to penalties or rewards, or to pledges, since in pledges an arithmetic equality inheres, in penalties and rewards an equable geometrical similarity. In the harmonic alone inheres the relationship of the superior and the inferior."
Of the three kinds of justice, Distributive, Commutative, and Harmonical: and what proportion they have unto an estate Royal, Oligarchic, and Popular.
>Let us then say in continuing of our purpose, that it is not enough to maintain, that a Monarchy is the best estate of a Commonweal, & which in it has the least inconvenience; except we also (as we said) add thereunto, a Monarchy Royal. Neither yet suffices it to say, that the Royal Monarchy is most excellent, if we should not also show that unto the absolute perfection thereof it ought to be fast knit together by an Oligarchic and Popular kind of government: which are proper unto the estates Oligarchic, and Popular. In which doing, the estate of the Monarchy shall be simple, and yet the government so compound and mixt, without any confusion at all of the three kinds of Estates, or Commonweales. For we have before shewed, that there is a great difference betwixt the mingling, or rather confounding of the three estates of Commonweales in one (a thing altogether impossible) and the making of a government of a Monarchy, to be Oligarchic and Popular. For as amongst Monarchies, the Royal Monarchy so governed (as I have said) is the most commendable: even so amongst kingdoms, that which holds most, or comes nearest unto this Harmonical Justice, is of others the most perfect. Justice therefore I say to be The right division of rewards and punishments, and of that which of right unto every man belongs. For that by these, as by most certain guides, wee must enter into this most religious and stately temple of Justice. But this equal division which we seek for, can in no wise be accomplished, or performed, but by a moderate mixture, and confusion of equality, and similitude together, which is the true proportion Harmonicall, and whereof no man hath as yet spoken.
>Plato having presupposed the best form of a Commonweale, to be that which was composed of a Tyrannicall and Popular estate: in framing the same, is contrary unto himself, hauing established a Commonweale not only Popular, but altogether a∣so Popularly governed; giving unto the whole assembly of his citizens, the power to make, and to abrogate laws, to place and displace all manner of officers, to determine of peace and warre, to judge of the goods, the life, and honour, of every particular man in sovereignty: which is indeed the true Popular estate, and Popularly also governed. And albeit that he had so (as we say) formed his Commonweale, yet neuerthelesse hee said, That the Commonweale could never be happy, if it were not by Geometrical proportion governed; saying that God (whom every wise lawmaker ought to imitate) in the government of the world always useth Geometrical proportion.
>Now certain it is, that Distributive, or Geometrical Justice, is most contrary unto the Popular estate and government by Plato set down: the people still seeking after nothing more, than for equalitie in all things; a thing proper unto Commutative, or Arithmetical Justice. Which was the cause for which Xenophon (Plato his companion, and both of them jealous one of another's glory) being of opinion, That Commonweales ought to be framed, and the laws administered according unto Arithmetical proportion and equality, brings in Cyrus yet a boy, corrected and chastised, for that he being chosen king, had changed but the servants garments, appointing better apparel unto them of the better sort, and meaner unto them of the meaner sort: as having therein regard unto decency, and the proportion Geometrical After which chastisement, Cyrus is by his master taught, to give unto every man that which unto him belongs, and to remember that he was a Persian borne, and was therefore to use the Persian laws and customs, which gave unto every man that which was unto him proper: and not the manners and fashions of the Medes, who thought it meet, that to be unto every man given, which was decent and convenient for him. Which writings of Xenophon, Plato having read, and knowing right well that it was himself, and not Cyrus, which had been corrected; forthwith reproved the Cyropaedia, without naming of any partie. This diversity of opinions, betwixt Xenophon and Plato (famous among the Greeks) was the cause of two great factions, the one of the Nobility and richer sort, who held for Geometrical Justice, and the Oligarchical estate; the other of the baser and poorer sort, who maintained Commutative or Arithmetical Justice, and therefore wished to have had all estates and Commonweales Popular. Now of these two factions arise a third, which was of opinion, That in euery Commonweale Arithmetical Justice was to be kept in just equality, when question was of the goods of any one in particular, or for the recompensing of offences and forfeitures: but if question were of common rewards to be bestowed out of the common treasure, or for the division of countries conquered, or for the inflicting of common punishments, that then Distributive, or Geometrical Justice, was to be observed and kept, having regard unto the good or evil deserts, and the qualities or calling of every man: insomuch that these men used two proportions, and yet for all that diversely, sometime the one and some∣time the other: as Aristotle said it ought to be done, but yet not naming either Plato or Xenophon, who yet had both first touched this string.