THE DOCTOR: To the rational mind, nothing is inexplicable; only unexplained.
LEELA: So explain to me how this “TARDIS” is larger on the inside than the out?
THE DOCTOR: Hmm? All right, I’ll show you. It’s because insides and outsides are not in the same dimension. [holds up two black cubes of different sizes] Which box is larger?
LEELA: [points to the larger of the two] That one.
THE DOCTOR: Alright. [places the larger cube on the console, walks over next to Leela, holds up the smaller one] Now, which is larger?
LEELA: [scoffs, points at the cube on the console] That one!
THE DOCTOR: But it looks smaller.
LEELA: Well, that’s because it’s further away!
THE DOCTOR: Exactly! If you could keep that exactly that distance away, and have it here, the large one would fit inside the small one.
LEELA: That’s silly.
THE DOCTOR: That’s trans-dimensional engineering: a key Time Lord discovery!
Doctor Who (1963–1989) British science fiction television series, original run (BBC)
14×05 “The Robots of Death,” Part 1 (1977-01-29) [w. Chris Boucher]
(Source)
Quotations about:
dimensions
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JO: I don’t believe it! It’s bigger inside than out!
THE DOCTOR: Yes. That’s because the TARDIS is dimensionally transcendental.
JO: What does that mean?
THE DOCTOR: It means that it’s bigger inside than out.
Doctor Who (1963–1989) British science fiction television series, original run (BBC)
08×04 “Colony in Space,” Part 1 (1971-04-10) [w. Malcolm Hulke]
(Source)
(Source (Video); dialog confirmed).
The difference between the interior and exterior of the TARDIS was mentioned in the very first episode, 01x01 "An Unearthly Child" (1963), by Susan:I thought you'd both understand when you saw the different dimensions inside from those outside.
The first use of the "bigger [on the] inside" line was in 10x01 "The Three Doctors" (1972), when Sergeant Benton first goes inside, and the Third Doctor is actually the one to comment (video):Well, Sergeant, aren't you going to say it that it's bigger on the inside than it is on the outside? Everybody else does.
(Which implies that others have said it before, we've just not seen it.)
Conversely, in (2005 revival) 10xCS "The Snowmen" (2012), Clara changes the perspective (video):CLARA: But it's. Look at it, it's --
THE DOCTOR: Go on, say it. Most people do.
(Clara makes a circuit of the exterior of the TARDIS, then steps back in.)
CLARA: It's smaller on the outside.
THE DOCTOR: Okay, that is a first.
A compilation of the theme can be found here.