This chapter opens with a letter from Mina Murray to Lucy Westenra, dated May 9.
Just as a reminder—May 9 puts us back in time a bit. May 9 is the day after Jonathan Harker cut himself while shaving and discovered that Dracula doesn't reflect in mirrors.
Mina writes to Lucy apologizing for not having written sooner, but says it's hard to find the time with her job as an assistant schoolteacher.
She says they'll be together soon by the sea for vacation.
Mina tells Lucy that she's been practicing writing in shorthand and at typing on a typewriter so that she'll be able to help Jonathan (Harker) in his work after they're married.
She's keeping a journal in shorthand every day to practice.
She tells Lucy that she's gotten a short letter from Jonathan saying that he'll be returning in about a week.
She asks Lucy to write her back with news—especially about a certain good-looking someone.
Lucy writes back promptly, describing all her social engagements and parties.
Apparently Lucy is of a higher social class than Mina: Mina has to work for a living, while Lucy gets to be a social butterfly.
Lucy says that the good-looking person Mina asked about must be Mr. Arthur Holmwood, who is an upper-class, rich, good-looking guy.
She says they've also been introduced to a friend of Holmwood's named Dr. John Seward, who is also good-looking.
Dr. Seward visits them a lot now, Lucy says.
Lucy confides to Mina that she's in love with Arthur Holmwood and says that she can't wait for Mina to join her on vacation at the seaside so that they can gossip and chat in person.
Lucy writes to Mina again—this time it's dated May 24. For reference, that's a few days before Harker tried to send letters to Mina and Peter Hawkins in secret with the Szgany.
Lucy tells Mina that even though she's almost twenty and has never been proposed to previously, she's actually had three proposals of marriage in one day!
She accepted one of them, but had to refuse the first two.
The first was Dr. John Seward (a.k.a. Jack Seward to his friends). He proposed and was refused— he was disappointed, but took it pretty calmly and promised to be her friend for life if ever she needed anything.
The second was an American from Texas, Quincey P. Morris. Quincey Morris is also a friend of Dr. Seward and Arthur Holmwood.
He asked her to marry him, and she refused as she did with Jack Seward. She admitted to him that she loves someone else, and Quincey Morris promised to be her friend for life.
Lucy is so emotional as she writes about it to Mina that she can't bring herself to describe the third proposal—Arthur Holmwood's—in any detail.
Suffice to say that she's engaged to Arthur and is happy about it.
Dr. Seward's Diary
The chapter switches to Dr. Seward's diary, which, we're told, was kept in a "phonograph," or an early recording device—very high-tech when the novel was written (1897).
May 25 (the day after Lucy rejects him): Seward says that he tried to distract himself from disappointment by working hard.
He runs a mental hospital (a.k.a. a "lunatic asylum"), so he goes down to check on his patients.
One of them, R.M. Renfield, is particularly interesting—and potentially dangerous.
A Letter and Telegram Between Quincey Morris to Arthur Holmwood
The chapter switches to a letter from Quincey Morris to Arthur Holmwood, dated May 25
Morris reminds Arthur of all the good old times they've had together on hunting trips in America, and invites him out for a drink with Jack Seward.
It'll be like old times, and he and Jack can congratulate Arthur on his engagement to Lucy.
Even though he and Jack Seward are both disappointed, they're generous enough to be happy for Arthur.
The chapter closes with a telegram from Arthur Holmwood to Quincey Morris, dated May 26, accepting the invitation.
Historical context note: Telegrams were messages sent by Morse code before long-distance phone service was invented or widely available (and, obviously, long before the internet or cell phones). Telegrams were much faster than handwritten letters, but writers were more limited as to length, since they paid by the word.