This first example is a 3 digit computation, ignoring the vernier for now. The way that the inventor assumed you'd use this calculator is that you'd be holding onto the (rigid) vernier casing in one hand, and spinning the disk and managing the index with the other hand. I have modified his method a bit, to fit the simplified constructions. You don't HAVE to do record keeping on the dials, if you don't wish to.
ZERO the disk ( put the vernier curor over the 1000); also line up the index with origin on disk.
CONNECT the index (i.e, tape it to the disk)
Rotate the vernier cursor counter-clockwise until 950 is under the vernier; Record 1 in the characteristic dial (95 is 9.5 x 10; the power of ten is 1),
Read 9.5 on the cycle scale; record 9 on the first mantissa dial .5 on the second mantissa.
LOCK the index (i.e., move the tape so it holds the vernier and index together)
ZERO the disk . The index should be pointing at about 835 on the outer spiral, and the Vernier cursor is on 1000.
CONNECT the index
Rotate Vernier cursor counter-clockwise until 750 is under the vernier; add 1 in the characteristic dial, add 8.5 on the mantissa dials (9.5 + 8.5 = 18, so 7 is recorded on the first mantissa, zero is on the second)
LOCK the index.
Rotate the index back to 1000.
Under the vernier cursor read 712 (on coil 7) and a bit (the vernier points to .5). The answer including the characteristic (2) would be 712 plus what's on the vernier.
NOTES: With a standard gunter circle, you can set and read a value from either cursor.
With the Log Spiral design, you have but one (vernier) cursor, and the index is strictly a memory store, accumulating a distance. The CONNECT index, Rotate disk, LOCK pattern served to accumulate the angular distance of the two numbers. When you put the index back on the 1000 at the last step you can read the result on the vernier cursor.
on step 7, I stated 'add 1 to the characteristic'. The patent algorithm handles this (you rotate the disk and add 1 when you pass 1000 in one direction, subtracting 1 when you pass 1000 in the opposite direction). In this case, I'm simply doing the power of 10 math in my head. 7.5 x 9.5 is about 70 = 7 x 10, so the characteristic (power of 10) is 1.