How many would you say didn’t click on that one?
The difference engine is actually digital. (MONIAC, using water levels, is a better example of a mechanical analog computer).
The difference engine worked with specific decimal numbers. It was basically an adding machine at heart, but its output was very specific. It allowed you to find the value of a polynomial equation, based on a just a few constants. For example, I might have f(x) = 4x^3 + 2x^2 + 5x - 3; the machine will produce the values of f(x) for as many values of x that I want, saving the time of calculating that by hand.
So no prebuilt CPU? Perhaps some random ICs, breadboards, resistors, etc? Perhaps they’d be able to test a homebuilt CPU using solar power or chemical energy derived from fruit/plants/whatever. Some examples here:
http://web.cecs.pdx.edu/~harry/Relay/index.html
http://members.iinet.net.au/~daveb/simplex/simplex.html
Or do you mean mechanical?
I’d heard of pneumatic computers, using air pressure to build digital computers. I came across Lego Pneumatics. You’d need to build an air tank that could be pressurized manually using bellows, but then the rest is just assembling Legos.
Q: What could be a better approach than playing with Lego?
A: Nothing!
I win the thread. 
= Whoosh =
Any current general computer is theoretically equivalent to a turing machine (the only real limit is the amount of storage). Turing machines are extremely simple compared to modern computers, and a skilled mechanical engineer could build one out of wood, or lego, or whatever.
The upshot is that you can build - given enough material of any sort, and some way of powering it (water wheels, humans turning a crank, wind mills etc) - a computer that equals the current electronic ones. Except that it would be a lot slower (as in millions if not billions of times slower). If you could find some way to use natural light or something like that, it could possibly be faster, but that would require techniques that are probably not achievable - if at all - without using electricity.
ETA: Turing machines are digital. There’s no need for electricity for a computer to be digital, though there are interesting analogue computers that use pneumatics or similar techniques (there are also electrical analogue computers).
The main advantage of digital computers is that in digital computers it’s easier to write a program as data, and vice versa read some data and interpret it as instructions - meaning you can write down a program and feed it to a computer without modifying the computer itself (re-routing a bunch of wires or tubes).
Ok, now I am getting stupider, but why exactly would we want to have that equation solved by an extremely expensive Babbage machine? Why not by hand? What pressing need was there in 1850 for solving function equations in such number?
I know I am pushing into the ignorance zone here, but what specific uses did any of these older computers have, such as the MONIAC? The wikipedia page talks more about how they were built rather than what they did. So it’s 1951 and you have a vacuum tube jobber in your lab, what are you doing with it, other than playing analog Pong? Windows 0.001? Vacuum tube-based shareware? Rusty & Eddie’s BBS (Beta)?
Sadly I have forgot my high school algebra . . .
I’m sure I’m going to be ambushed by a flock of angry Dopers but…
… how can a computer not use electricity? And how can anything that’s DIGITAL not use electricity?
Isn’t digital information essentially electricity? Or what? Wait? What is digital information? I mean, what form does it take?
Head explodes.
MONIAC was an interactive and visualised way of modelling a specific kind of system - so it could be used to experiment with various scenarios at a high level, without being acquainted with the underlying maths.
Early digital computers were often used (indeed specifically designed) to be of use in real-world number-crunching situations - often in a military context, such as codebreaking or missile trajectory calculation. They were labour-saving devices - intended to perform the same kind of calculations as humans with pencils, but faster, more intensively and more reliably.
Digital information is data expressed or stored in one of a finite set of possible states - usually binary in this context. It may be electrical (i.e. a pulse or charge equals a 1 and a gap or lack of charge equals a zero), but digital information can also be expressed or stored mechanically - so a hole (or not) in a paper punch tape is still digital information, but not electrical.
A digital mechanical computer could operate by clockwork, water, wind, hand-cranked power, and could process digital information by mechanical components interacting with mechanically-stored digital data, without any electricity at all.
(Unless of course you bring in the fact that matter itself is all based on electromagnetic charges etc)
ETA: and of course electrically-transmitted information need not be digital - conventional TV and radio broadcasts are analogue, for example - because the data is a stream of continuously-variable values or waves, not compartmentalised chunks of information that can only be one choice out of a fixed set of values.
I remember back at Uni we built the equivalent of an SR flip-flop out of pneumatic components. With a propagation delay of about 3 minutes, I doubt it would have any practical applications though.
Other examples of digital information stored without electricity:
-fingers (digits) raised or lowered to represent ones or zeroes. Thumbs up/Thumbs down to indicate (dis)approval. This is mechanical storage of digital information.
-pits/flats on a CD/DVD/Blu-Ray disc. This is optical storage of digital information.
-magnetic polarization states on a hard drive/tape drive. This is magnetic storage of digital information.
-lanterns in a church steeple. This is, uh, oil/combustion storage of digital information. Two digital bits, where first bit indicates whether British are coming, second indicates mode of travel:
00 = British are not coming.
01 = British are coming, mode of travel is by land.
10 = meaningless state. first and second bits were not uniquely identifiable, so the presence of one lantern could only mean they were coming by land, i.e. a 01.
11 = British are coming, mode of travel is by sea.
-letters typed or written on a page. This is ink/paper storage of digital information. Whereas computers work on a binary (base 2) system, written text might be thought of as base 26 (or a higher number if you consider capital letters, punctuation and spaces). When you see a letter on a page, you aren’t “measuring” it; that would be analog. Instead you are identifying it as either this character, or that character, same as a computer examines a bit and identifies it as a either a one or a zero; that’s digital information. Attemping to copy a drawing by hand is an analog feat; transcribing a book is a digital one.
-Morse code. This is base-3: a bit can be a dash, a dot, or silence, with each letter of the alphabet coded as a sequence of bits. This is auditory transmission of digital information.
I never thought of Morse code as base 3, but you are correct.
Sinking ships? Babbage was sponsered by the Royal Navy, astronomical tables were used for navigation (they didn’t has GPS yet) these were calculated by people, who made mistakes. Either in the calculation or the transcription. The difference engine 1) wouldn’t make any mistakes 2) included a printer, so no human in the loop.
Here’s a series of cartoons about a water-based computer. There are a dozen or so entries, just keep clicking on the “next entry” link (it looks like a date) at the bottom of the page.
My favorites:
Basically, it was a calculator. Calculators are useful, right? It was a calculator that could evaluate polynomials (specified in a particular way). Many functions of mathematical/physical/engineering/practical importance, even if not themselves polynomials, happen to be well approximated by polynomials. So it was useful to to be able to automatically calculate their values. Granted, the Difference Engine was big and clunky and not available to use on demand for your average person, but it could be used to print tables of values for various functions which others could then use as needed (indeed, previously, such tables were often painstakingly calculated by hand and then published). In an era with powerful pocket calculators, this may all seem rather quaint, but think of it as one of the first precursors to automatic calculators, and its use becomes more apparent (note that it was rather far off from general purpose computation; the later plans for the Analytical Engine were more along that line).
If you were on Discworld you can use ants.
Actually, you only need one - all of 2 value boolean logic can be implemented with either a NAND gate or a NOR gate. Practically speaking, though, you need some sort of storage also. Then you are good to go.
BTW, the Computer Museum in Mountain View had a working model of the Difference Engine on display (it was on loan) and I saw a demo. It is awesome - and not something you are going to be able to make very easily out of Radio Shack parts.
To expand on the earlier answers: mathematical tables, especially astronomical ones. Baggage’s unoriginal point was that the calculation of any mathematical table could ultimately be reduced to the grinding though of some higher-order polynomial of such a form. The process was what we’d now describe as an algorithmic approximation and was what he’d significantly more originally realised was hence potentially mechanical. As a competing standard, tables of the period had indeed to be labouriously calculated by hand by teams of low-skilled, underpaid computers, who were just mindlessly churning the numbers out anyway. And it’s worth realising that publishing astronomical tables are the 19th century equivalent of Big Science: using such tables to calculate longitude is handy for a dominant naval power and these sorts of issues are by far the variety of science best funded by the British government of the day.
Babbage (and others) had highlighted how error-prone this human-based system was. Hence his proposal to replace it all by his Difference Engine. Which, unfinished, wound up being funded to about the cost of a battleship
Whether Babbage’s Difference Engines would actually have been worthwhile in a Victorian context is still a matter of debate amongst the specialists. That it could have been finished, given better project management on his part, has been decisively settled by the Science Museum rebuild project(s).
Yet what is rarely realised is that, while Babbage failed to complete his versions, simpler difference engines were subsequent commercially manfactured and sold by the Scheutzs. People did then did try to apply them to much the same problems that he’d hoped to use them for. They really don’t seem to have been any great success, though why this was is still a matter of debate.
Hand calculation of mathematical tables continued until roughly WWII, with a major 1930’s expansion in the US being courtesy of the WPA absorbing mathematically-illiterate personel. If anything, mathematical tables were calculated by hand until computing power rendered them pretty much redundent. (When was the last time you looked up a log table?) I recommend David Grier’s history When Computers Were Human (Princeton, 2005).
I’ve no proof, but my understanding is that the various MONIACs never served much purpose beyond being exceptionally cute teaching examples to economics students. If there’s evidence to the contrary, I’d be interested.