Dr. Mathias examines Dr. Russell when she returns, and says her blood pressure is "80 over 120" and declares it normal. He has the numbers reversed: 120 over 80 is normal.
As Helena looks at her thermograph, she is looking at it backwards as her name can be read from our (the camera's) position.
The third Eagle is shown with the docking tube connected to the starboard side, but it launches with the tube on the port side.
Koenig had Alan cut the Eagle's engines, but the switch used to do so was clearly on his side of the cabin so it would have been quicker and easier for both of them if he did it himself.
Koenig tells Paul to try to make contact using sound, light, and radio wave patterns. However, as sound cannot travel through the vacuum of space, it was pointless to suggest it.
The title of the episode "Rings Around the Moon" refers to Saturn, a planet with rings. When Dr. Helena Russell has to play a game to save her life, it is reminiscent of the children's game "Ring Around the Rosie."
Koenig: Why do doctors make the worst patients?
Dr. Russell: Probably because we know too much.
Dr. Mathias: I just want to satisfy my curiosity.
Dr. Russell: What is it?
Dr. Mathias: You have a lovely optical nerve, Dr. Russell.
Koenig: Why so pensive, Victor?
Bergman: Oh, I was just thinking about that planet, Triton. All that knowledge, and yet... Perhaps knowledge isn't the answer after all.
Koenig: Then what is?