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To add to DoctorNo's excellent w/u above, I should point out that overdrive was an option on many manual transmission cars of the past. In my own vehicle's case, there was an electric overdrive option available from the factory (and I wish that mine had it!) It does pretty much exactly what is described above - it trades torque for engine speed, just as if it was a higher gear. The difference (in my car's case, at least) is that the overdrive is a separate unit from the transmission. It is activated using a dashboard control. As a separate component, it can be in effect over multiple gears (from 2 to 4, in my case).

The best example for this I've seen is the front derailleur on a ten-speed bicycle. While the rear derailleur is changing between its five gear ratios, the front has two (or more) to switch among that affect the ratio of all of the available rear gears. So too with the overdrive.

One reason it was done this way was that five-speed transmissions were a bit too complex to fit into older, smaller cars; also, the overdrive was envisioned as a highway cruising mode, allowing you to switch gears to adjust speed on the highway (engine braking, for example) without switching out of overdrive.

My favorite thing about the overdrive I wish I had is its name - the Laycock de Normanville J-type. Heehee.