This use of the infinitive reminds us of a substantive clause in apposition. In fact, both the adjective infinitive and the appositive are substitutes for clauses, the former taking the place of a restrictive adjective clause, the latter of either an unrestrictive adjective clause or a substantive clause.
In the sentence, “This is a day on which to travel,” we have an infinitive phrase made from an adjective clause and still retaining the relative pronoun. The relative has an antecedent but does not join to it as usual a dependent proposition.
Note.—When we say that an infinitive is used as an adjective or a noun, we do not mean for an instant that it takes the place of any adjective or noun in the language. So far from this being the case, the infinitive is employed simply because there are no nouns or adjectives to say what the infinitive says for us.
II. The infinitive may be related to a verb in various ways.
1. As subject; as, “To pass this bridge was the severest trial.”—Irving. If we ask, what was the severest trial? the answer is the whole phrase, to pass this bridge; but the base of this phrase is the infinitive to pass, hence it is the essential subject.
A common way of using the infinitive as subject is that arrangement by means of the anticipative subject it, which enables us to put the real subject after the verb; as, “It is a most earnest thing to be alive in this world.”—Carlyle.
Note.—Some grammarians prefer to call it a neuter personal pronoun, subject of the verb, and the infinitive its appositive, or explanatory modifier.
2. As complement.
(a) Object of a transitive verb. Not all transitive verbs can take infinitives for objects. Most of those that do are verbs denoting some activity of the feelings or the will; as, “He resolved to go”; “They hoped to surprise this post.” The infinitives here answer the questions, resolved what? hoped what? They are equivalent to substantive clauses,—He resolved that he would go, they hoped that they might surprise this post; but when the action denoted by the infinitive is to be performed by the same person as that denoted by the subject of the predicate verb, the infinitive is chosen instead of the clause. It is simpler than the clause and has become by frequent use equally clear. When the two actors are different, two subjects must be expressed, hence the clause is employed; for example, They resolved that I should go; They hoped that their allies would surprise this post.
Note.—The contraction of a noun clause to an infinitive phrase is very evident in such a sentence as this,—“He did not know what to say.” The word what leads us to expect a subject and predicate to follow, but we have merely an infinitive. The second subject if expressed would be he, the same as that of the principal verb.