The narrator attends one of the Time Traveller's Thursday-night dinner parties and hears the discussion about time travel. He's not entirely convinced by the Time Traveller's demonstration of a model time machine.
On Friday the narrator runs into the Medical Man at the Linnaean Society in London. The Medical Man tells him how he once saw someone else do a similar trick.
The narrator attends the Time Traveller's dinner party again the following Thursday and listens to the Time Traveller's story.
He shares a cab home with the Editor, who clearly thinks the story is made-up.
No one else believes the Time Traveller, but the narrator sort of does.
The narrator comes back the next day to talk to the Time Traveller.
The Time Traveller is not in the lab, so the narrator examines the Time Machine.
He talks to the Time Traveller, then he catches a glimpse of him vanishing.