Continuity mistake: When Reed instructs Archer how to disarm the second arming mechanism of the mine, he tells him to turn the first component clockwise, but then Archer is shown turning it counter-clockwise. Then Reed tells him to turn the rest of the components counter-clockwise, but afterwards he must reverse the last component to keep the mine from rearming, which means turning it clockwise. But when Archer reverses the last component, he turns it counter-clockwise.
Continuity mistake: In the scene in T'Pol's quarters, after she and Tolaris have mind melded and she asked him to leave, he walks out, and the door starts to close before the shot changes to T'Pol. But when the shot changes, the light from the door on T'Pol shows the door still opening, and then closing.
Plot hole: In 'Oasis,' several times in the episode, specifically when they took Trip captive, the holograms appeared out of nowhere brandishing weapons. The holograms can walk through walls, which we saw, but the guns were real. We saw them hit the deck when Liana deactivated the holograms. So how did they appear with the guns?
Other mistake: Two Borg are discovered North of the Arctic Circle. One is covered in ice, the other encased in ice. Carbon dating shows the wreckage to be over 100 years old (to tie it in with Star Trek:First Contact). The Borg are inactive and unable to repair themselves until they are "defrosted". The temperature above the Arctic Circle doesn't usually drop below -100°F. The Borg have been shown to work very well in the vacuum of space which is 3-4K (roughly -454°F). If they can survive that, even being frozen in a solid block of ice would not slow them down. The nanites have to be in their "blood" and operate at the cellular level. That cannot be damaged by re-entry and the crash. [This is one of those entries that keeps being corrected and recorrected - I'll leave it up to people to make up their own minds.]
Revealing mistake: While Archer and the crew discuss the capabilities of the Xindi vortex technology and whether or not to go to Azati Prime, Archer asks Trip to see the data that Trip has collected from the Degra's ship. The camera pans to the display screen, on which a Mac mouse pointer is visible. It is positioned on the line going downward from the red giant.
Continuity mistake: At 38:20, T'Pol (as acting captain) says that they have resumed their course to Azati Prime; however, from 39-42 minutes, Captain Archer and Trip discuss the "current" status - that they are still orbiting the planet and then the Captain orders Trip to tell Travis to set a course for Azati Prime at maximum warp.
Factual error: When Capt. Archer joins Travis in a weightless environment he refers to in an earlier episode as the ship's "sweet spot", Travis discusses the bad news he received from The Horizon, his parents' cargo ship, with Capt. Archer, and you see tears rolling down (and off) Travis's face. In a weightless environment the tears would not react in this way.
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Continuity mistake: The captain's log date of Jan. 8, 2154, doesn't align with the timeline of previous episodes. Harbinger, two episodes earlier, was said to take place Dec. 27. The episode in between that and Hatchery, Doctor's Orders, wasn't given an on screen date, but T'Pol compared the anomaly in that episode to the one in Harbinger, which she said was encountered a few weeks earlier. Even if that was just two weeks earlier, that would put Doctor's Orders at Jan. 10, and that episode spanned at least four days, so the earliest Hatchery could take place would be Jan. 15.