"Within the infant rind of this small flower
Poison hath residence and medicine power:
For this, being smelt, with that part cheers each part;
Being tasted, slays all senses with the heart.
Two such opposed kings encamp them still
In man as well as herbs, grace and rude will;"
Havelock Vetinari went to the Assassins' School. He knows about poison. He knows which ones work fast and which slow, which can be delayed in their action and how, which can mimic natural diseases of the body, and which are mild or even beneficial by themselves but become deadly in combination with other equally innocuous drugs. Of course he knows all the antidotes as well. The poets will write of him in after years, 'Vetinari, he died old.'
Statecraft he learned from actual practice, bar a few pointers from people like his aunt and Lady Margolotta of Uberwald. Of course he came to it with his theories of government already formed: remembering history, Vetinari is determined not to repeat it.
But being a synthesist, Vetinari uses his assassin's knowledge of how the human body may be killed, to direct his statesman's determination to keep the body politic in health.
The prime poison to a state can be summarized in three words: rule of one. Where one man holds power over many men, whether as king, dictator, elected president, platoon sergeant or, most certainly, patrician, the black rot of despotism will eventually erupt. Vetinari would have wondered why Lord Acton needed to state his famous dictum: surely everyone knows that two and two make four? The anomaly is that no government is more stable or efficient than the rule of one man, nor potentially more beneficial to the state. One must simply maintain its inherent strengths while putting a halt to its tendency to produce tyranny.
Vetinari does it by the application of other physics which, noxious in themselves, become paradoxically benign when interacting with an existing poison. Vimes, for instance. Vimes is one of Vetinari's chief antidotes to the Patrician, and the one he's most proud of.
For police are another plague sore on the body politic. What man can resist the temptation inherent in the power to imprison other men? Venality is the best one can hope for; violence and the nasty small boy's pleasure in causing pain is much more likely. It's that much worse because the victims are small and voiceless and lack the resources and hired assassins of the rich and influential.
Hence Vimes. A sentimental man, which is dangerous just by itself. When a sentimentalist's beliefs are (always, inevitably) betrayed, he goes from thinking all men are basically good to knowing all men are verminous scum in need of putting down. But Vimes won't ever think that because he has Vetinari to think it for him, and he refuses to be Vetinari. It's in his blood: his ancestor executed a tyrant and some atavistic instinct in Vimes is itching to do the same. Thus the arm of government in most constant contact with Ankh-Morpork's unlovely citizenry remains humane, while the Patrician is permitted the occasional sigh over that citizenry's pig-headed stupidity without taking the matter further.
Equally Vimes is a moral man. He believes in the Law above everything. That sort of thing usually leads to well-filled prisons and well-loaded gibbets, when it doesn't lead to a dangerous confusion of identity expressible by the syllogism
The Law is absoluteVetinari doesn't believe any Law is absolute but knows it's necessary, when dealing with Vimes, to act as if he does. He makes the Patrician subject to the Law: if the head of state is a mere individual before that sublime abstract, the Law, the head of police can be no less. Automatic checkmate to both of them.
I represent the Law
Therefore I am absolute.
And a good thing too. Because Vetinari recognizes instinctively that Vimes is what Vetinari is not, a natural-born killer. Vetinari will kill, and has killed, when circumstances make it necessary. Vimes will kill, and quite possibly has killed, because the violence inside him wants to kill. Vimes' romanticism, Vimes' dogged clinging to the superiority of the Law, are the means he uses to keep that yellow-eyed killer caged inside him, but Vetinari thinks it useful to have the counter-balance of the Patrician as well. As long as Vimes detests the Patrician and all he stands for and all he potentially can be, Vimes will remain a just man. And as long as Vimes is a just man, Vetinari has a cold-eyed conscience looking at his every deed and examining it for any trace of expediency and dictatorial tendency. If worst comes to worst, the executioner's sword is still in Vimes' hand to cut the poisoned part from the city's body.