L scales on spiral models
A lot of calculators with Spiral scales use a 10 turn spiral to accommodate an L (logarithm mantissa) scale on an outer circle.
The advantage of an L scale in these models is that it allows for convolution and evolution (raising to powers), on a calculator that is otherwise largely occupied by the spiral scale. In a model that uses circular scales, you can crowd a lot of power scales such as a square, cube, square roots, etc.
I had a bit of difficulty understanding why a 10 turn spiral with an L scale was so key. Here's why inventors liked it so much. If you imagine a normal log scale and an L scale are laid out in a line, then a 10 turn spiral would be the equivalent of taking the scales and chopping them up as shown above. Note that each turn of the spiral cuts the L scale exactly on a boundary of 1/10 of the full L scale.
This means each spiral turn can be represented by one of those "10th" parts of the L scale. Because they are identical, all 10 of them are represented by one scale on the circumference. It goes from '0' to '10' (or '100, '1000', etc. depending on how many subdivisions you can fit. So, what happens is each turn of the spiral uses the same "L" scale.
10 turn spiral
Turn 5 is shown in red represents the segment beginning at 0.5 on the L scale
Note that the cursor arms in such models are inscribed with turn counts so you know which part of the L scale you are on, and that set's the first digit.
You reconstruct the mantissa by prefixing with 0.n (the turn count) and add the digits at the L scale. The right cursor is approximately at 335 (spiral scale), and the L scale logarithm of 3.35 is shown to be .525 on the L scale.
This method will also work with 100 turns, or 1000 turns if you need a calculator the size of a dining table.
Spirals with different number of turns
There are two other numbers that might be used for turns: 2 (and powers of 10 x 2), and 5 (and powers of 10 x5). These will work because 2 and 5 are divisors of 10, and that works for 10 x and 100x those numbers. I find 20 and 50 occasionally in spiral slide rule patents.
Both lead to some complexities in the design that translate to more difficulty in operation.
5 Turn Spiral
Here is the same linear version as before, changed for a 5 turn model.
In this case, 1/10th of the L scale only covers 1/2 a turn. In fact, the L scale must be a 2 cycle version:
In this case, the logarithm for a number is still read in the same way (put the cursor on a number, read digits from the L scale), but there is a new problem: the first digit is not the turn.
The logarithm of 3.76 is about 0.575. The 0.075 is directly read on the outer circle.
However, the first digit is not the turn count "3". In order to find the first digit, you would need to add an additional index to the cursor, that maps the first digit to the appropriate numbers.
A result is a more 'busy' cursor - possibly a more difficult to use calculator.
20 turn spiral
A 20 turn spiral is somewhere in between the 10 and 5 in complexity. There are 2 turns in each 1/10th of the linear scale. However, this means you can draw one L circle on the circumference just as with the 10 turn models.
Two labels would be needed for each marked point (e.g., 00 and 50, 01 and 51, etc.). Turns 1, 3, 5...(all odds) use the 00...01 marks, and Turn 2, 4, 6...(evens) use the 50...51 marks.
As with the 5 turn model, the cursor needs a scale to hold the first digit or digits. My next project is a 20 turn model (J Michaelson), and I'll update the site with scales for it presently.