Plot hole: Trelane says he studied Earth images that travelled to him at light speed, and earlier they establish Gothos is 900 years from Earth. But Trelane references Napoleon and Hamilton, who weren't around until 1800 or so. That would put this episode in 2700, but the original Trek episodes are set in the 2200's.
Continuity mistake: When Kirk and Sulu enter the records room, they pick the lock. Later when they beam the officer back down, he enters the room without unlocking the door. The room should be locked since they beamed him down in the "past" erasing their having been on Earth and in the records room.
Deliberate mistake: Special effects of the ship in space were very expensive in the 60s and couldn't be wasted. So when a larger model was built with slightly different nacelles, shots of both versions became common, even within the same episode. This is why the Enterprise sometimes had red needle-tipped nacelles and sometimes lighted "spinning" ones, and in aft views she had either round white spheres or perforated vents at the nacelles' ends.
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Plot hole: "The Cage": Pike says he doesn't want women on his bridge, Number One being the one and only exception, and he keeps ordering Colt off it. His apology to Number One indicates that A)he doesn't think of her as female, and B)she's the only exception to his rule. So how does he overlook the very female crew-woman seated at the science station?
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different.
Other mistake: Spock and McCoy have a brief argument over whether Charlie could have survived on his own all those years. Because they're talking at the same time, Spock saying "Doctor" as McCoy says "Mr. Spock," the closed captioner apparently misunderstood McCoy's line. The caption has him calling Spock "Dr. Spock."
Continuity mistake: Kirk and McCoy are in the Exeter's engineering section when Kirk uses the intercom to ask if anyone is aboard. Several shots of the deserted ship follow as his voice echoes off the walls, including one of engineering - with Kirk and McCoy suddenly, mysteriously absent.
Revealing mistake: "The Cage": When the laser cannon starts firing at the Talosian rockface, a styrofoam block falls and rolls away on the left side of the screen. After a brief shot of the crew, we see the cannon still firing, and the exact same block falls and rolls off screen on the left again.
Revealing mistake: While the interior sets for the shuttle craft provided adequate room for the actors to stand, the mock-up built for exterior shots was undersized, and too short to stand up in by several inches. So when entering or exiting the shuttle, the actors always had to duck.
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