The book was Full Circle by Andrew Smith, who also wrote the original TV story. Any Doctor Who aficionado worthy of the name will recognise this as the first story of the classic E-Space trilogy and the one in which Adric (the companion that most fans evidently love to hate, although I always quite liked him) was introduced.
The quote appears on the first page of Chapter 1 (which isn’t the start of the book as this one has a prologue) and reads:
The [Doctor’s] face was at once immensely cheerful and yet tinged with the sadness of one who has known too many people for too short a time.
I’m nowhere near 750 years old (the Doctor’s approximate age at the time of this story), however much I may sometimes feel like it, and I’ve obviously not met anything like as many people as he had. However, I’ve been living in or near a university town for the best part of the last 20 years (and, in case you know me and think I’ve miscounted, I’m referring to two separate universities), and these are notable for the transitory nature of large chunks of the population. Therefore, whether or not it’s reflected in my face, I can certainly relate to the sadness of knowing many (though perhaps not too many) people for all too short a time.
It doesn’t help that my track record for keeping in touch with people when they (or I, though mostly I’m the one staying put) leave is generally pretty poor. Of course, staying in touch is a two-way business so it would be unfair to apportion all or even most of the blame in one direction or the other. Suffice it to say that my contact with some people I’ve known (and in some cases known very well and got on with excellently) is limited while for others it is non-existent.
Long ago, I came to the conclusion that (at least in the cases where you get on well with each other, which for me seems to be most of the time) it’s better to be able to enjoy the pleasure of someone’s company for a short while than never to have met them at all.
And if you are someone I used to know and have dropped out of touch with, please (a) accept my apologies, especially if you made attempts to stay in touch which weren’t reciprocated, and (b) feel free to drop me a line. [And if you’re one of my former English teachers, please accept my further apologies for starting two consecutive sentences with the word “and” :-)]