Verbal noun after a passive construction - dangling or not?

Question

Is this sentence correct (in a mathematical text):

"This formula is established by observing that A+B=C" ?

My feeling is that everything is OK because there IS identity of subjects: the reader who observes one formula can establish the other one. In other words, the passive voice makes "this formula" a "fake" subject.

Dangling doesn't come into this use: it's a gerund, not a participle, and it acts as object of the preposition by. "This formula is established by Fred" is the same construction. — StoneyB on hiatus, Jul 09 '15 at 22:44
@StoneyB Wouldn't you say that the whole clause, rather than the ---ing form, is the complement of the preposition? — Araucaria - Him, Jul 10 '15 at 10:10
@Araucaria Yes, indeed. "It is the head of the clause which is the object &c" — StoneyB on hiatus, Jul 10 '15 at 10:13

score 1 · Answer 1 · answered Jul 09 '15 at 21:12

There's nothing "fake" about the subject "This formula." It's a perfectly valid subject of the passive verb "is established." In the transposition from the active voice, the subject is ordinarily moved to the object of the preposition "by." Thus your active-voice version might be

The reader established this formula by observing that A+B=C.

In the passive, this becomes

This formula is established (by the reader) by observing that A+B=C.

When the active-voice subject is unknown or unimportant, it is often elided, as indicated by the parenthesis. This leaves the second "by" introducing an adverbial prepositional phrase of manner to modify the verb. Perhaps you feel a small bit of unease at that "by" echoing the missing one.

A dangling modifier would be one that can only modify the missing active-voice subject:

Having observed that A+B=C, the formula is established.

Verbal noun after a passive construction - dangling or not?

1 Answers1