Final answer:
The narrator and Cathy were waiting for their turn to view the kinetoscope, a notable invention in motion picture technology at the time.
Step-by-step explanation:
The narrator and Cathy were waiting for their turn to view the kinetoscope, which was mentioned as 'the crowning scientific achievement of the nineteenth century' in the passage provided. The kinetoscope was an early motion picture exhibition device, which is why they were surrounded by performers and other entertainments while they waited.
This event was likely an exciting and novel experience, as it represented one of the newest technological advancements in entertainment during that period.The question does not provide specific information about the story or context, so it is difficult to determine what the narrator and Cathy were waiting their turn for. Without more details, it is impossible to provide a concrete answer.