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Of American-English origin, the expression dangling participle designates:

a participle or participial phrase, often found at the beginning of a sentence, that appears from its position to modify an element of the sentence other than the one it was intended to modify.—definition: Collins Dictionary.

A dangling participle occurs, for example, in the first sentence of the following passage from Jalna, by the Canadian author Mazo de la Roche (1879-1961)—as published in The Atlantic Monthly: A Magazine of Literature, Science, Art, and Politics (Boston and New York: The Atlantic Monthly Company) of July 1927 [page 123, column 2]:

Sitting in a tearoom, the first lines of a new poem began to form in his mind. Pushing his plate of cinnamon toast to one side, he jotted them down on the back of an envelope. A quiver of nervous excitement ran through him. He believed they were good.

The earliest occurrences of the expression dangling participle that I have found are as follows, in chronological order:

1-: From The Correction of Bad English, as a Requirement for Admission to Harvard College, a paper read before the Massachusetts Association of Classical and High School Teachers on Friday 4^th April 1890, by the U.S. educator LeBaron Russell Briggs (1855-1934), published in The Academy: A Journal of Secondary Education (Boston, Massachusetts, USA) of September 1890 [page 305]:

What shall I print as “Specimens of Bad English?” It is idle to say, “Confine your choice to what every educated man knows to be bad.” The obscure is prohibited; the illiterate insults both boy and teacher,—not to mention the University: but outside of the obscure and the illiterate there is nothing that every educated man knows to be bad.
What is more vulgar than you was?—yet some teachers defend it; more illegitimate than it don’t?—yet many teachers use it; more slipshod than I don’t know as?—yet most teachers never notice it; more inexact than dangling participles?—yet good authors employ them; more offensive to a trained eye or ear than to thoroughly appreciate, or to cordially thank?—yet of such phrases professors (even professors of English) are guilty again and again.

2-: From a review of English Prose: Its Elements, History, and Usage (London: Smith, Elder, & Co., 1890), by the British scholar John Earle (1824-1903)—unsigned review published in The Academy: A Journal of Secondary Education (Boston, Massachusetts, USA) of April 1891 [page 175]:

Under the head Lucidity come naturally to be treated the participles and the pronouns. Of the feebly connected, or “dangling,” participle as a cause of obscurity the author has less to say than we could wish [cf. note 1]. He does well to lay down no hard and fast rule. So long as the sentence remains clear, he seems not to take offence at constructions that would shock the follower of conventional rules. Not to go beyond the text-books, one would imagine that concurrent good usage is strict in its management of the placing and relating of participles. Much exploring in the good prose of the earlier centuries is the only cure for too much deference to rules. To know what English idiom is and to trust this idiom without misgiving should be the aim of the English writer. The English sentence pays no homage to Greek and Latin grammar.
The following sentence from a juvenile “composition” gave no offence to a class of young persons, who found it altogether lucid and euphonious:—“Looking over the fence, a lovely landscape lay before us.” Modern writing, not juvenile, abounds in such constructions. Only graduates, fresh from their rhetorics, have the wit to be annoyed by them. Were the English participle inflected, as it once was, so that in the above sentence, e.g., looking might by its ending be put into relation with us, all would be well. But the English reader puts the looking into the right relation just as well without the ending. It is futile to insist that the participle must connect itself with the substantive coming nearest to it or having any special function in the sentence. Had the sentence above quoted been,—Looking over the fence, the children presented to us a beautiful appearance,—there might well be ambiguity.

3-: From The Foundations of Rhetoric (New York: Harper & Brothers Publishers, 1892), by the U.S. journalist and rhetorician Adams Sherman Hill (1833-1910) [Part II. Sentences. Book I. Sentences Good and Bad. Chapter IV. Force. Section II. Force as Affected by Choice of Words; page 235]:

Dangling Participles.—Weak writers often misuse participial phrases.
I. Turning down the shawl, she disclosed a baby’s face.
II. She turned the shawl down revealing a baby’s face.
In this sentence as originally written, “revealing a baby’s face” hangs loose in the sentence. “Revealing” is—to borrow an apt expression—a “dangling participle.”
I. The hero is a Scottish youth who has come to France to seek his fortune.
On this land Elizabeth founded a town which she at first called Calumet, an Indian name, and afterwards Taunton.
II. The hero is a Scottish youth, having come to France to seek his fortune.
On this land Elizabeth founded a town, calling it at first by the Indian name Calumet, and changing that name later to Taunton.

4-: From a letter to the Editor of The Brooklyn Daily Eagle, in which Dr. Caskie Harrison (1848-1902), Head of the Brooklyn Latin School, criticised the “reprehensible” “deviations from correct English” found in the writings of the U.S. journalist and author Edwin Lawrence Godkin (1831-1902), the then Editor of The Evening Post (New York City, New York, USA)—letter published in The Brooklyn Daily Eagle (Brooklyn, New York, USA) of Saturday 8^th February 1896 [page 6, column 5]:

Mr. Godkin has (6) peculiar constructions. On page 308 we read, but cannot understand, “Finally the boarder must remember that though the cottager, like woman, when he is bad is very bad, when good is delightful;” and on page 312 we puzzle over the force of the preposition in “we suspect that nine-tenths of the ministers would confess that rest meant to them getting away from their parishioners, and not in getting away from the sermons.” Mr. Godkin is fond of (7) “dangling” participles, and he would be hard put to it to defend the grammar of “uniting” in the closing passage of the “Role of the Universities in Politics,” which is too long to quote. [cf. note 2]

Notes:

1 The British scholar John Earle (1824-1903) did no use the expression dangling participle, but the noun slipshod, in English Prose: Its Elements, History, and Usage (London: Smith, Elder, & Co., 1890) [Chapter V. The Leading Characteristics of Prose Diction. § 2. Of Lucidity. Participle [page 186]:

We come now to the Participle, which is a verbal Adjective, or adjectival Verb, adapting itself with great agility now to verbal and now to adjectival functions. This pliability of the Participle is freely made use of in English prose, and the matter is deserving of attention, because it is to the abuse of this pliant and ever subservient instrument, more than to any other assignable cause, that we may trace that loose and negligent manner of performance which is called Slipshod. […] It concerns Lucidity to secure that the relation of the participle with its subject should be manifest and, if possible, uninterrupted […]. There ought to be no uncertainty even for a moment as to the subject of a Participle, or else it may be a pitfall to the reader.

2 This is the closing passage of Rôle of the Universities in Politics, as published in Reflections and Comments 1865–1895 (New York: Charles Scribner’s Sons, 1895) [page 163], by the U.S. journalist and author Edwin Lawrence Godkin (1831-1902)—“the struggle” designates the American Civil War (1861-65):

The view of the nature of the struggle which is sure to gain ground all over the country as the years roll on is that it was a fierce and passionate but inevitable attempt to settle at any cost a controversy which could be settled in no other way; and that all who shared in it, victors or vanquished, helped to save the country and establish its government on sure and lasting foundations. This feeling cannot grow without bringing forcibly to mind the fact that the country was saved through the war that virtue might increase, that freedom might spread and endure, and that knowledge might rule, and not that politicians might have a treasury to plunder and marble halls to exchange their vituperation in; thus uniting the best elements of Northern and Southern society by the bonds of honest indignation as well as of noble hopes.