I know that a vernier scale can be used to accurately read a linear scale, such as in vernier calipers. I'm wondering if there is a way the methods behind a vernier scale could be adapted for usage with a non-linear scale, such as a logarithmic scale. The reason I ask this is because I am designing a slide rule (actually a slide rule bracelet, it's pretty cool really) and I'm wondering if it's possible to read the results of multiplication and division to more significant figures without, of course, increasing the size of the slide rule. This doesn't seem possible to me, but I'm hoping somebody else might have some insights about logarithmic scales that I don't. The problem seems to be that since the scale is linear, there would need to be a unique vernier scale for every graduation on the main scale, perhaps even a vernier scale for every possible combination of matched graduations on the two logarithmic scales...
I am also trying to do something similar, however, it seems to me that you need an interestingly shaped scale, when I did it the hard way, I found that for 1-2, the scale ended at the original 5 line, and I've been trying to go even deeper into it, not sure where this will take me, I'll be experimenting with a slanted slide rule next, see where it goes. Good luck!