Explain the irony in the following quote from The Taming of the Shrew and what type of irony it is. "You will be schoolmaster, And undertake the teaching of the maid: That's your device." (Tranio-I.i.196-197

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This quote is directed to Lucentio who has disguised himself as “Cambio,” a teacher of Latin, in order to woo Bianca.  The irony is that he is now in the best position to woo Bianca as he will be in close proximity to her daily as her teacher, and that he does not having "teaching" on his mind at all.  For that matter, neither does Bianca  She is not the sweet, demur, innocent, and well-behaved girl everyone else thinks she is.

The type of irony here is dramatic irony, where Bianca's father and Biance do not know who Cambio really is or what his true mission for applying for the position of Bianca's tutor is, but the audience, Lucentio, and his servant, Tranio, all know the truth. 

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This line, coming as it does in Act I scene 1, helps develop the exposition of the plot. Tranio and Lucentio have just witnessed the exasperated Baptista's final effort to marry off both of his daughters. He has decreed that Bianca, his youngest daughter who is by fair the sweetest and most gentle, can only marry once the older sister, Katharina, who is known as "the shrew" for her argumentative nature and her lack of patience with suitors, herself is married.

Lucentio is struck at once with fervent and passionate love for Bianca, and thus finds himself in a very difficult situation - how can he hope to woo Bianca under such restrictions? Thus, talking with his manservant, Tranio, they devise a plan that will enable him to court Bianca in secret - Lucentio will impersonate a schoolmaster. I think there is some verbal irony in what Tranio says:

You will be a schoolmaster,

And undertake the teaching of the maid;

That's your device.

Although overtly Lucentio will play this role, he will be seeking to "teach" Bianca something apart from her lessons he will be supposedly teaching.

Thus the quote you have selected contains verbal irony and a possible innuendo concerning how Lucentio intends to court Bianca.

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