How big the next iPad will be
Assume the next iPad will have a retina display. It's got to*.
From this assumption, we know that the pixel pitch will be ~325 dots per inch. Approximately the same as the iPhone 4 and new iPod Touch.
Also, doubling the screen resolution on each dimension is a really good thing for third-party apps. Everything just works out of the gate, and you have naught to do but make 2X images to make your app look sweet.
This means that the screen will be 2048x1536.
Using the DPI we got from assumption #1, we get screen dimensions of 6.3" x 4.72".
Using good old Pythagoras, we get a diagonal screen measure of 7.8"
The problem of manufacturing a display with 3,145,728 pixels at low enough cost to use in a $500 device, we leave as an exercise to the reader. Or to Apple and their screen vendors.
Update: I listened to John Gruber & Dan Benjamin on their podcast. They actually covered this same idea quite a bit. Gruber points out that there are a number of technical challenges (I agree). The real killer thing though against the speculation above is what Dan pointed out: all the touch targets are going to be about 20% smaller. Ouch.
So, assuming you could overcome the technical issues of 2048x1536, you'd need to stay within 10% or so of 9.7" screen. That means the DPI would have to drop to 290 dpi on an 8.73" display. Nitpickers would complain that it's not "retina" anymore. But it would look pretty good.
Update 2: Given Jobs' recent speech on the Apple quarterly results conference call, it seems very unlikely that Apple will reduce the physical screen dimensions at all. So, assuming that they want to double the DPI on the new iPad, and assuming they can overcome any manufacturing challenges, we get 264 pixels per inch. The current iPad is 132 pixels per inch. I don't know exactly how 264 dpi looks compared to 325, but it's a start.
* I say "it's got to," and here is my detailed reasoning: the iPhone 4 display is so beautiful, so goddamned gorgeous, it actually makes the iPad screen seem really lame by comparison. As soon as I got an iPhone 4, I started spending way less time on the iPad and that time went entirely to the iPhone 4. And it wasn't because of infatuation with the newest gadget. Even now, when the new has worn off both devices, I'll use the iPhone 4 for reading Kindle books or news apps instead of the iPad. It's only when I want more interactive space, like answering email, that I go to the iPad. I believe that Apple (and especially Jobs) will not tolerate such a disparity in beauty between their flagship devices.