In the second stanza, Arnold alludes to a passage in the ancient Greek play Antigone by Sophocles, in which Sophocles says the gods can decide to ruin people from one generation to the next, like a swelling tide driven by winds. What does the addition of the allusion to Antigone give to this poem?

The addition of the allusion shows that Arnold does not feel alone in his melancholy, as he recognizes that other authors had the same feeling.

We can arrive at this answer because:

He does this to show that Sophocles also felt melancholy and that's why his work portrayed sad moments that accompanied an entire family.

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In the poem "Dover Beach" by Matthew Arnold, the poet complains about the fact that he lives in a sad world. He believes that the time period he was born in is one that has forgotten all about values and faith, and is instead lost and melancholy.

However, he also thinks of how other writers at other time periods might have thought the same. He compares himself to Sophocles, and reflects on the sad ideas that he expressed in his plays. Arnold makes a reference to one of this, Antigone, and talks about how the gods in this play also behaved in sad, cruel ways, such as ruining several generations of a family.

Arnold believes that such a sad plot shows that Sophocles also had a gloomy view of his own world. Therefore, the addition of this allusion to Antigone helps strengthen the connection that the poet feels with Sophocles.

they fear for their lives

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