It's the beginning of the end for the fourth Doctor, as I watch the first two episodes of his final story. There's definitely a theme of endings running through these two parts, as the Doctor (somewhat more somber than usual) reflects on the unstoppable force of entropy. But we get some firsts, as well: it's the first time we hear the Cloister Bell (that toll of doom that persists into the 21st century version), and the first time we hear the TARDIS's (broken) ability to disguise itself as being the result of the chameleon circuit. And it's this circuit that arguably sets these events in motion, as the Doctor resolves to fix it by measuring a real police box and taking the dimensions to a place called Logopolis, where the locals there can help the Doctor fix the problem.
But despite this feeling of somberness and decay, it doesn't really seem to affect the Doctor himself until halfway through part two, and before that he seems more or less his usual self. He does seem awfully worried about the gravity bubble that results from materializing around the police box that's actually a TARDIS (the Master's TARDIS, in fact; it's not made 100% clear, but it's not exactly a big secret either), though, and while we've seen something of this sort before in The Time Monster (where the Master's TARDIS is inside the Doctor's and the Doctor's is inside the Master's), it hasn't been as fully explored as here, with some memorable imagery of TARDISes inside TARDISes. (And note that they've pulled the old seasons 14-17 TARDIS prop back into service for this: you can tell by the windows and the lack of a door handle, among other things.)
There's also the stuff about the Master being back ("So he did escape from Traken"), and, as About Time points out, the Watcher figure is likely meant to be a bluff, to make us think that that's in fact the Master (even though the last shot of The Keeper of Traken spoils this). But we're back to shrinking people as a method of killing. And while all this is going on, an Australian air hostess named Tegan Jovanka, of all things114, accidentally wanders into the TARDIS and spends the better part of two episodes trying to find her way back out again.
There are still some nice moments -- I like the part where the Doctor and Adric brace themselves against the TARDIS doors, expecting the Thames to rush in and flush the Master out, only to find they've landed on a barge instead of in the river, and the weary way in which the detective knocks on the police box's door, fully expecting the Doctor and Adric to be inside, is understated genius.
But once the Doctor meets up with the Watcher on the bridge, there is a real sense of inevitability about the proceedings ("I've just dipped into the future. We must be prepared for the worst," the Doctor tells Adric), as the Doctor heads to Logopolis. Of course, even with that there's still time for a terrible pun (after Tegan asks about her aunt, whom the Doctor has seen the shrunken corpse of, the Doctor replies that he's seen "a little of her"), but still, on to Logopolis, where mathematics is the all-important language of the universe. But the Master is wreaking havoc on the planet (having hitched a ride in the Doctor's TARDIS), and Nyssa, from the last story, is there too for some reason. And then the cliffhanger (which seems to show the Doctor planning on leaving Adric and Tegan behind on Logopolis) instead focuses on a problem with the Logopolitans' calculations: the TARDIS is rapidly shrinking...
114 Tegan is a Welsh name, while Jovanka is the name of a Slavic nymph. Allegedly, producer John Nathan-Turner was trying to decide between the two when script editor Christopher H. Bidmead saw the two names and mistakenly thought it was a first and last name.