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The Enterprise's mission is to study in detail key events from Earth's history in 1968 to better understand how Earth civilizations made it through crucial events. The ship has traveled back in time by using the gravitational sling-shot effect that was first used in season 1's "Tomorrow is Yesterday" to get the Enterprise back to its correct time.
The ship unintentionally intercepts a powerful transporter beam that originated light-years away and whose destination was the Earth. A man with a cat materializes on the Enterprise's transporter pad. Kirk questions him, but he'll only give his name as Gary Seven. He quickly deduces that the Enterprise is from the future, but insists that he is not. He beseeches Kirk to let him continue on to the Earth, or history will be altered. Kirk is torn, but decides to imprison Seven. This doesn't last long, though, because once in the brig, Seven uses some kind of device to neutralize the forcefield and put the guard in a happy daze. (Don't the Enterprise's security guards ever search their prisoners?) Then he beams himself and the cat down to a contemporary office suite, with period decorating that hides computers and other equipment.
The man's name is Gary Seven, and he is a human that has been living with aliens. The alien civilization apparently visited the Earth generations ago and took some humans for breeding stock. The descendants of these humans have been trained to work as operatives in 1960s society. The aliens are using the human operatives to try to keep humans from destroying themselves with their rapid advances in technology. Gary Seven's specific mission is to prevent the launch of an orbital missile satellite by the US, which is being launched as a response to the USSR's similar satellite emplacement.
Seven was sent to the Earth at this time because contact has been lost with the operatives previously running the mission; they were killed in a car accident. The new secretary they hired, Roberta Lincoln, arrives and Seven mistakes her for one of the other operatives. He weirds her out a bit with unusual technology and requests until he realizes she's not another operative. She has freaked out, but he employs a little patriotism and gets her to stick around for her secretarial duties.
Meanwhile, Kirk has decided that he can't let someone from off-world just beam to the Earth - what if they do something to change history? Clearly no on on the Earth had transporter technology at this time. He and Spock don period clothes and use signals from Scotty on the transporter to triangulate Seven's location in his office.
They burst into Seven's office, but Roberta makes a scene for long enough that Seven is able to escape via his closet transporter. Roberta calls the police, and they arrive just in time to be swept up by the transporter beaming Kirk and Spock back to the Enterprise. Kirk quickly has them beamed back down.
Seven had beamed himself to McKinley Rocket Base, where the launch will happen shortly. He sneaks onto the launch gantry and up to the level of the electronics that he plans to sabotage. He and the cat get to work.
Spock has learned that the most important even that will happen very soon is the launch, so Kirk reasons that that's what Seven is going to interfere with. He and Spock beam down to the rocket base to try to find Seven, but they are quickly taken into custody by security guards. Meanwhile, on the Enterprise, Scotty is tapping into the images from spy satellites to try to locate Seven on the rocket base. He finds Seven and begins beaming him up. However, Roberta has been playing with the controls she found in Seven's office and inadvertently diverts the transporter beam to the transporter in the office, so Seven goes there.
While Kirk and Spock watch helplessly, the rocket launches. From his office, Seven somehow is able to signal commands to the electronics on the rocket. When Roberta realizes he's messing with a US military rocket launch, she has second thoughts about his intentions, and whacks him on the head with a book and steals his do-it-all device and holds him at "gunpoint".
At the rocket base, the controllers are becoming panicked, because the rocket is going off-course for an impact in the eastern hemisphere, and they can't get it to destruct. In the confusion, Kirk and Spock manage to call Scotty and have him beam them directly to Seven's office. They walk in on Roberta holding up Seven, but Roberta has even more doubts about them, and Seven takes advantage of her confusion to grab his device from her.
Kirk and Spock are still in control of the situation, however, and Kirk orders Spock to use Seven's computer to detonate the rocket. Spock is unable to get through the computer's security in time, so Kirk is forced to allow Seven to do it. Seven detonates the rocket a little more than 100 miles above the Earth's surface.
To wrap up the episode, Kirk and Spock return to Seven's office (after having been to the Enterprise) to let Seven know that events occurred just as the were "supposed" to. Roberta has agreed to keep working for Seven, with full knowledge of what he is and who the Starfleet men are, and they allude to her and Seven's future adventures together.
This episode is very uneven. Knowing that it was supposed to lead to a spin-off series involving Gary Seven and Roberta Lincoln makes the episode make more sense, but as a pure TOS episode it's mediocre.
The idea that Starfleet would intentionally send a starship back in time for research strikes me as unbelievable because of its foolhardiness. Any number of random incidents could have altered Earth history, even if no person had ever visited the surface. I can't believe that this chance would be taken. Even if it was, surely the mission would be to observe a specific event, not just the general flow of events. There was no mention of a specific event, and Kirk wasn't even aware of the rocket launch until Spock found out about it later.
I appreciate that the Enterprise went back to 1968 specifically because that's when the episode was originally aired. I found it odd, however, that the key event in the episode (the rocket launch) was an event that would happen in the near future when the episode was aired. Obviously that event did not happen, which dated the episode almost immediately. I suppose that at that time, no one was thinking about the idea of re-runs or long-term appreciation of the series, but it still strikes me as weird. It might have been more interesting to have based the episode on a key event from the recent past and "explain" what actually happened during that event.
The background of Gary Seven (a human raised on an alien world) was intriguing, and the concept of human operatives trying to guide events on Earth with the help of aliens was also very interesting. It's too bad that this idea didn't get the chance to go anywhere. On the other hand, Roberta Lincoln's ditziness was annoying; one would hope that she would have improved if there had been a series.
The main drag on this episode was the fact that our characters were reacting to events in the episode instead of motivating them. In fact, Kirk and Spock spent a lot of time completely neutralized by security in the rocket base while we watched Seven do whatever he wanted. This is not really why I watch TOS.