Why does "An Advanced English Syntax" say the infinitive in these sentences is adverbial?

Question

In sentences like the following the Infinitive is probably Adverbial, and therefore the italicized part will be put in the Adjunct column.

You seem to be ill.
He is known to be reliable.
It was felt to be very unkind.

(p. 22, An Advanced English Syntax, by C. T. Onions, 1904)

They use the word ‘probably’ to describe those as adverbial (no further information). The book has no grounds why they function as adverbial. Albeit it is a traditional authority (1904), I have a yen for the authors’ intentions. I’m not sure whether today English grammarians can say that they, the infinitive phrases, function as adverbial complements. I was told once that infinitive complements after 'seem', 'feel' are adjectival, modifying the subject (He seems to be happy, he to be happy). My conjecture about why the infinitives are described as adverbial is that they think Infinitives after copular verbs modify the verb, not the subject, to complete the predicate. i.e., What modifies the verb is adverbial (what modifies the noun is adjectival). What do you think of that?

Answer 1

I like to think of this in terms of questions answered by the specific bit of sentence, rather than in terms of inherent potentialities of a specific word ("ill" vs "happy"). Consider:

--"How are you?" --"Ill!" --"How is he?" --"Reliable." --"How was the president's address?" --"Unkind."

--"How do I look?" --"You seem to be ill." --"How is he known?" --"He is known to be reliable." --"How was the president's address received?" --"It was felt to be very unkind."

In the first set the reply offers a modifier for a noun. In the second set a modifier for a verb.

Hope this helps.

Answer 2

In these sentences "seem", "known" and "felt" are verbs. In the examples given, these verbs are described by the adverbial phrases given. In the first example we could say instead "You seem ill", with "ill" being an adverb describing the verb "seem." Instead in that example we use the phrase "to be ill" as instead, so it is an adverbial phrase.