Well, shoot. I guess I'd better tackle this one while I've got two hours left in the weekend, and I've been putting it off for long enough: gear ratios. (Dun dun dun.) I'll admit that I'm treating this topic a bit like I treat the calculus I've forgotten: it'll probably be fine once I've read through it once or twice, but I don't want to admit how much I don't know about it. And don't worry...I hear you. "Gear ratios?" you're saying. "Why don't you just write about taxes or spreadsheets or spark plugs or something maybe minimally more exciting than gear ratios?" My response is: I KNOW. And I have a counter-response, which is this:

http://www.lisaboyer.com/Claytonsite/swoopypage1.htm

Which is to say, I WANT one, and also to say that wooden clock plans are free and available online, and they're not gonna make themselves. So: on to mathy things.

Very generally, you want to have a firm grasp of gear ratios when:

• you have one thing spinning very quickly and you want to slow it down
• you have something spinning very slowly and you want to speed it up
• you own a bike and live in a very hilly area
• you own a manual-transmission car and live in a very hilly area
• you want to transmit rotational motion from point A to point B, which are fixed

Picture two gears, of equal size. the first one is connected to a motor (the driver gear) and the second one (the driven gear) meshes with the first. When the first one rotates - let's say at 1 revolution per second - the second one follows suit exactly. That's the trivial case - the ratio is 1:1, because both gears are rotating at the same speed. Here's a non-trivial case: 

Licensed under a Creative Commons license from here:
http://howto.nicubunu.ro/gears

Let's call the gear at the lower left the driver gear. It has 12 teeth. The next gear in the train has 25 teeth, and is correspondingly bigger. Ignoring the third gear for the moment, what's going on with these two? Say the small gear still spins at a constant rate of 1 revolution/second (that would be omega, for those of you with a physics textbook). When all 12 teeth have gone around once, the bigger gear has advanced by 12 teeth as well - only about half of a full revolution. It takes the bigger gear longer to make a full revolution, so it has a slower rotational velocity. How much slower? A gear ratio is defined as Omega_driven:Omega_driver or N_teethdriven:N_teethdriver. (Omega and N - the number of teeth - are proportional, so which equation you use doesn't matter.) So for this example, 

N_driven:N_driver = 25/12 = 2.08

The small gear spins 2.08 times faster than the big one. Put another way, the small gear has to make 2.08 revolutions to get the big gear to turn once.

Let's isolate the second and third gears, now. We know that the second gear has 25 teeth. The third has 18. Now the equation gets us:

N_driven:N_driver = 18/25 = 0.72

So the third gear spins one revolution for every 0.72 turns of the middle one. New question, then: how many turns of the driver gear does it take for the third gear to make one revolution? Wikipedia says that you can just multiply the individual gear ratios in the train to get the overall ratio for the entire train, provided all the gears contact each other. (I'm not sure I understand mathematically why that's so, but I'll run with it for now.) That leaves us with:

 2.08*0.72 = 1.4976

The driver gear makes 1.4976 revolutions for every one turn of the third gear. And now you should be getting suspicious, because the first gear has 12 teeth and the third gear has 18, and 18/12 is awfully close to 1.4978. It turns out that, provided the gears contact each other, you can simply take the ratio of the first and last gears to get the total gear ratio between them. (The discrepancy between the answer calculated above and the 1.5 you'd get from dividing 18/12 is because I rounded 2.083333333...into something easier to type.)

So...err....not that bad after all, at least not yet. Some questions I can think of (namely, "If my escapement gear has 12 teeth, how many leaves should my fourth wheel pinion have, and how many teeth should my fourth gear have, and how many leaves should my third wheel pinion have (etc., etc.), so that the center wheel makes one revolution every hour?") might be considerably more complicated. I'm sure you have exactly the same questions.

Are you done with gear ratios for now? I'm done.