Revealing mistake: A Roman guard empties half a magazine of bullets from his submachine-gun into Flavius, but there are no bullet holes in his body, nor blood on the floor from the wounds inflicted by the bullets.
Visible crew/equipment: Set lights are reflected in Spock's tricorder screen before he closes the cover.
Continuity mistake: Kirk is talking to Scotty on his monitor, after they have finished talking Kirk goes to switch off the monitor by pressing the button just below it. If you watch very closely you will notice Mr Scotts image fades from the screen before Kirk even touches the button. (00:07:05)
Revealing mistake: Isis the cat was disinclined to hold still for special effects shots. When Gary Seven beams off the Enterprise with the cat in his arms, the position of her head jumps several inches as they're dematerializing. (00:10:10)
Continuity mistake: Gary Seven is talking to his computer, we then switch to Kirk and Co and after a few moments with them we cut back to Gary Seven. Somehow in that time, while talking to his computer he changed his clothes. (00:13:40)
Continuity mistake: After Gary turns on the typewriter and tells Roberta to give her report but not with her fingers, she responds, "Well, how do you expect me to type, with my nose?" But the typewriter types, "How do you expect me to type, with my nose?" omitting the "Well." It didn't edit her question, because we see from the rest of the scene that it types absolutely every word she says until Gary turns it off. (00:17:45)
Continuity mistake: When Isis meows and enters the inner office, the door is open several inches and the lights inside the room are on. When Gary Seven follows her in a few moments later, the door has nearly closed itself and the lights inside are off. (00:20:25)
Plot hole: Scotty is receiving images of the launch, supposedly from an orbiting satellite. But many of the camera views are obviously from ground level, and couldn't possibly have been taken from orbit. It's not a rebroadcast from a ground camera, either. The implication is clearly that Scotty is is picking up a live satellite feed. (00:34:30)
Revealing mistake: After Scotty finally locates Seven on the gantry while he's tampering with the rocket, it cuts to a quick shot of Kirk and Spock, then in the following shot of Roberta when she turns the vault's combination locks, the stunt wire is visible behind her as it moves up to turn the door's wheel handle. (00:35:25)
Continuity mistake: Isis is an all-black cat, but in a few shots, her "stunt double" with white paws can be spotted. The cat lying on the floor of the transporter room when Gary Seven beams aboard is a white-pawed double.
Continuity mistake: As Gary Seven beams in, Scotty's hair changes between shots.
Plot hole: Spock claims that history is unchanged at the end of the episode according to the library tapes - how would he know? If history changed, the tapes would change too. And unlike in "City on the Edge of Forever", there's no Guardian around to keep people from being influenced by an altered timeline.
Continuity mistake: Gary Seven pushes a pen holder over to open the doors. The pen wasn't returned to the upright position, but was up when Kirk came in.
Continuity mistake: When the computer opens and begins operation in Gary Seven's office, it's white. In the close up, it goes from white to purple when asking for identification. Afterwards, the color continues to change.
Continuity mistake: As Gary Seven starts to talk to the computer, it has a light gray exterior. When it first talks, the trim turns purple, then it's back to light gray.
Character mistake: Mr Seven calls the security guard "Sergeant" but he is a corporal.
Continuity mistake: The second time McCoy is shown in the Sickbay, he is first shown sitting and writing. A moment later, he's standing bent over the terminal.
Continuity mistake: To open the secret doors, Seven tilts one of the pen holders down to the top of the desk. After he's gone, Ms. Lincoln comes in looking for him, and the pen is upright without it having been pulled back up, unless it was supposed to reset itself.
Other mistake: Early on, there is a shot of the control room. There is a read out that is counting down from "5 04 15", meaning that the launch would be in just over five hours, not less than one minute.
Continuity mistake: The shots of the rocket Scotty cycles through are of different times. There is one shot of the rocket at the launch site, followed by one of the transport taking the rocket to the site.
Answer: Kirk was getting his physical and Dr. McCoy probably turned off communications, because if he hadn't, Kirk would have left and headed straight for the bridge, leaving McCoy irritated.