Dangling Phrases / Dangling Participles

Rule: A participial phrase (or other similar phrase) at the beginning of a sentence refers to the grammatical subject.

Walking slowly down the road, he saw a woman accompanied by two children.

The participle "walking" refers to the subject of the sentence ("he"), not to the woman.  If you wish to make "walking" refer to the woman, you must recast the sentence:

He saw a woman, accompanied by two children, walking slowly down the road.

Nouns in apposition, adjectives, and adjective phrases come under the same rule if they begin the sentence.  The sentences in the lefthand column below are wrong; they should be rewritten as shown in the righthand column.

On arrival in Chicago, his friends met him at the station. When he arrived (or: On his arrival) in Chicago, his friends met him at the station
A soldier of proven valor, they entrusted him with the defense of the city. A soldier of proven valor, he was entrusted with the defense of the city.
Young and inexperienced, the task seemed easy to me. Young and inexperienced, I thought the task easy.
Without a friend to counsel him, the temptation proved irresistible. Without a friend to counsel him, he found the temptation irresistible.
At the age of ten, my father bought me a puppy. When I was ten, my father bought me a puppy.

Sentences that violate the rule are often ludicrous:

Being in a dilapidated condition, I was able to buy the house very cheap.

Wondering irresolutely what to do next, the clock struck twelve.

Born at the age of forty-three, the baby was a great comfort to Mrs. Wooster.

Tail wagging merrily, Bertie took the dog for a walk.

On returning home, Maxine's phone rang.

Adapted from William Strunk, Jr. and E. B. White, The Elements of Style (New York: Macmillan, 1959), pp. 8-9, with additions from Patricia T. O'Conner, Woe Is I (New York: Putnam's, 1996), pp. 159-66.