The term ‘birthday’ is thought to have originated during Shakespeare’s lifetime in the 1570s, and although the concept of birthdays being celebrated as we do today was not common in Shakespeare’s era, “birth-day” is a word found in three of his plays:
In Julius Caesar, a few moments before Cassius and Brutus engage in battle with Antony and Octavius Caesar, Cassius, feeling his imminent death upon him, says, ‘this is my birthday; as this very day was Cassius born.’ There is no more about it, just that wistful statement, that comes out of the blue.
“This is my birth-day; as this very day
Was Cassius born.”
Julius Caesar: Act 5, Scene 1
In Antony and Cleopatra, though, we see Cleopatra grasping the opportunity for an unexpected celebration on her birthday. Antony has just had Caesar’s messenger whipped. He’s in a foul mood but somehow gets a new wind and calls for the servants to fill the bowls for some late-night drinking. Cleopatra who has been feeling lonely and rejected is delighted with this change in mood. She says: ‘It is my birthday. I had thought to have held it poor,’ meaning that she had not expected anything for her birthday. But now she has something – not her birthday, though – to celebrate: Antony’s change of mood. ‘Since my lord is Antony again,’ she says, I will be Cleopatra.’
“It is my birth-day:
I had thought to have held it poor: but, since my lord
Is Antony again, I will be Cleopatra.”
Antony and Cleopatra: Act 3, Scene 13
In Pericles, Pericles comes across 3 fishermen he has a conversation with, who tell him about the upcoming birthday and celebrations of the Kings daughter:
“Marry, sir, half a day’s journey: and I’ll tell
you, he hath a fair daughter, and to-morrow is her
birth-day; and there are princes and knights come
from all parts of the world to just and tourney for her love.”
Pericles: Act 2, Scene 1
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https://www.youtube.com/watch?v=l7aOMDV6v-o