Book 4- Some thoughts 2

The logical mind of Hermione Granger. A scientific temperament that works on a deduction based on observation and knowledge, and requiring proof. She cannot thus abide by Trelawney's star-gazing techniques.

In HP 1, it was the poisons puzzle. "This isn't magic, this is logic."

In HP 2, it was the solving of the mystery of the attacker's identity. "that faraway look in her eyes"

In HP 3, it was the decoding of the mystery of Lupin's condition, based on observation and the study of werewolves set by a malicious Snape.

In HP 4, it was her getting to the bottom of the bugging Rita Skeeter mystery. On an aside, this is another example of an interesting use of words by Rowling. A Skeeter is a mosquito. Rita Skeeter is as irritating as a buzzing mosquito, and the animagus form used by her is the beetle. And Hermione gets to the bottom of the mystery on the use of the word 'bug' by Harry in the context of an electronic eavesdropping device, which obviously does not work in Hogwarts.

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Harry's scar, and the role played by it in the connection with Voldemort's mind, becomes a big thing in this book. We had a brief taste of it in HP 1 itself, when it would burn whenever Voldemort/Quirell were near. It plays a significant role in the subsequent books, and shapes much of the narrative. We shall look at it again later.

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The Hermione-Ron relationship. A time when feelings are still not crystallized or expressed. There is the classic 'taking for granted' scenario. And Hermione standing up for herself- 

"Just because it's taken you three years to notice, Ron, doesn't mean that no one else has spotted I'm a girl."

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Three more things that are significant. 

Voldemort takes in a bit of Harry's blood to cut across the protection his mother's sacrifice has provided. This has another, unexpected (by Voldemort) consequence, which would be significant in the last chapters of the last book.

The Death Eaters. The masks are off. Or, in the Death Eaters' case, the masks are on. We, and Harry, face Voldemort and the possibility of mortality.

Priori Incantatum and the phenomenon of the twin cores. These will play a larger role later.