Isn't the "L" scale on a slide rule linear? See the photo at the top of the Wikipedia page [1] for example. (I may be confused here, since you know slide rules better than me.)
Sure, but to do addition you need two linear scales, one on the fixed part and one on the slider. The L scale is for computing logarithms, not for doing additions.
If the L scale is on the slider, and you’re not too much concerned accuracy (the procedure does 5 ‘align scales’ operations before the final ‘read the result’) one linear scale is sufficient, if you use the hairline on the cursor as a kind of memory:
”Example: calculate 0.23 + 0.45
- "Reset" the rule so that all the scales are lined up.
- Move the cursor to 0.23 on the L scale.
- Move the leftmost 0 on the L scale to the hairline.
- Move the cursor to 0.45 on the L scale.
- Reset the rule again so that all the scales are lined up.
- The cursor should now be at 0.68 on the L scale, which is the sum of 0.23 + 0.45.”
Using a sledge hammer, break the slide rule into N pieces, taking care to insure that N is greater than the sum you wish to compute. Now to compute the sum of A and B, count out A pieces into a paper bag, then count out B pieces into the same paper bag. Now empty the bag onto a clean workspace and count the total number of pieces to produce the result.
[1] https://en.wikipedia.org/wiki/Slide_rule