In this sentence:
With George, his perfect manservant, and Miss Lemon, his perfect secretary, order and method ruled supreme in his life.
Why is ruled followed by supreme instead of by supremely?
Use of adjectives rather than adverbs in such constructions is common. The adjective modifies not so much the verb as the verb’s subject. Such adjectival predication is by no means confined to such more or less copulative verbs as be and seem, but works with more active verbs as well. Thus New Hampshire’s license-plate motto, which is controversial only for its belligerence, not its grammar:
Live free or die.
Or for a more august exemplar, try the opening of Lamentations of Jeremiah, in the 1611 King James Version:
How doth the city sit solitary, that was full of people!
Participles can work in this way too. Many a candidate for office has been said to “run unopposed.”
Many verbs take a Predicative Complement. This is a phrase that fills a special slot set up by the verb, one that portays either the Subject or the Object. Here are some examples:
In the sentences above, the word treasurer portrays the Subject, she. The phrase ecstatic describes the Subject elephants. It was the Object me that was furious in the third example. And the word warm describes the Subject she in the last.
Adverbs and adverb phrases can almost never, if ever, be Predicative Complements. Predicative Complements are nearly always noun phrases, adjective phrases or preposition phrases.
Adverbs and adverb phrases as Adjuncts
Adverbs like warmly, supremely or ecstatically often modify verb phrases. This means that they give us extra information about the manner in which the subject did what they did. These are usually Adjuncts in the clause structure, in other words they have no special relationship with the verb and are entirely extra in terms of both the meaning and the syntax.
Predicative Adjuncts and the Original Poster's example
The word supreme in the Original Poster's example describes the Subject. We might consider it a Predicative Complement of the verb rule. However, it could feasibly be an Adjunct in the clause structure too. We also often use Predicative Adjuncts to describe the Subjects of sentences. This type of Adjunct does not semantically modify the verb phrase, but instead gives us more information about the Subject. They are different from Predicative Complements because they have no special relationship with the verb. In fact they are supplements. They are are similar to an extra phrase that we might put in brackets. In addition, we can move them to either the beginning or the end of the sentence. We can't do this with Predicative Complements without the sentence sounding very marked:
The word supreme is either a Predicative Adjunct or a Predicative Complement, it's a little difficult to tell which. The following sentences seem to mean the same thing:
He ruled over his Subjects supreme.
He ruled supreme over his subjects.
However, the word supreme does seem to have a special relationship with the verb RULE. In addition, if supreme was really a Predicative Adjunct and not a Predicative Complement, we might expect it to be surrounded by commas:
This however seems to have a slightly different flavour from the first example.
Whatever the function of supreme is, we can be certain that we need the adjective supreme and not the adverb supremely; the phrase is describing the Subject of the sentence, it is not modifying the verb phrase. It doesn't directly talk about the manner of the ruling.
Take the sentence, Order and method ruled supreme in his life.
We would expect, Order and method ruled supremely in his life.
Now look at the sentence, Order and method were supreme in his life.
That looks correct.
I suggest that the original sentence is shorthand for, Order and method ruled[, and were] supreme in his life.
Adjective instead of adverb. Elvis used to sing: "Love me tender, love me sweet, ... Love me tender, love me true", in one of his most famous songs, when he meant tenderly and truly. He did not teach English, but he was pretty successful around the world. I think that following his "mistakes" is not so bad.
I suspect this has to do with the popular title Supreme Ruler, assigned to someone with absolute authority over a wide domain. In this case, to rule supreme doesn't directly translate to "rule in a supreme manner", but to "rule under the title of Supreme", and merely implies supremeness by that title.