A direct allusion is one in which one literary work makes direct reference to another. Sometimes it can be in the form of naming a specific character or event, often as a sort of shorthand. For example, in T. S. Eliot's "The Love Song of J. Alfred Prufrock", the narrator says:
No! I am not Prince Hamlet, nor was meant to be;
Am an attendant lord, one that will do.
To swell a progress, start a scene or two ...
Here, Eliot's narrator is saying that he is not an important hero, but just one of those minor characters (what would be "extras" in film) who fills out a crowd scene. The allusion to Hamlet is a quick way of explaining something about Eliot's narrator by creating a parallel with the minor attendants we've encountered in Shakespearean plays.
A veiled allusion is different in that is doesn't explicitly name the person or work it is referring to, but evokes its indirectly.
Veiled allusions are often used in satire or in references to real, living people to avoid legal problems of slander, such as in the "roman a clef", a novel based on the lives of real people.
One highly entertaining series of veiled allusions can be found in the works of Byron and Lady Caroline Lamb. Although Lady Caroline was married, she and Bryon had a very public and scandalous affair, which Byron ended. Lady Caroline wrote published entire novel about the affair, Glenarvon, and then Bryon made a series of scathing veiled allusions to her in his poetry.
An allusion is a literary device common in poetry, prose, movies, plays, and even in everyday language. We use allusions whenever we reference something we expect everyone to understand, so that we don't have to explain things fully. Allusions are a kind of cultural shorthand. Allusions are typically brief, and rely on the reader to understand them.
For example, if I call my neighbor a "Scrooge," then I am alluding to Dickens' character in A Christmas Carol. Without having to go into a lengthy discussion I can convey in one word exactly what I think of my neighbor.
A veiled allusion is something a little bit different. To "veil" something means to cover it. A veiled allusion is an allusion that the writer is trying to cover in such a way that not everyone will understand the allusion... perhaps only one person, or just a few people will understand what is really meant. Think of it as kind of an inside joke. Veiled allusions are more subtle than direct allusions, and usually involve a little bit of "reading between the lines."
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